create utils trust region
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[thresh_delta]
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type: double precision
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doc: Threshold to stop the optimization if the radius of the trust region delta < thresh_delta
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interface: ezfio,provider,ocaml
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default: 1.e-10
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[thresh_rho]
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type: double precision
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doc: Threshold for the step acceptance in the trust region algorithm, if (rho .geq. thresh_rho) the step is accepted, else the step is cancelled and a smaller step is tried until (rho .geq. thresh_rho)
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interface: ezfio,provider,ocaml
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default: 0.1
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[thresh_eig]
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type: double precision
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doc: Threshold to consider when an eigenvalue is 0 in the trust region algorithm
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interface: ezfio,provider,ocaml
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default: 1.e-12
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[thresh_model]
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type: double precision
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doc: If if ABS(criterion - criterion_model) < thresh_model, the program exit the trust region algorithm
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interface: ezfio,provider,ocaml
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default: 1.e-12
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[absolute_eig]
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type: logical
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doc: If True, the algorithm replace the eigenvalues of the hessian by their absolute value to compute the step (in the trust region)
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interface: ezfio,provider,ocaml
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default: false
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[thresh_wtg]
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type: double precision
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doc: Threshold in the trust region algorithm to considere when the dot product of the eigenvector W by the gradient v_grad is equal to 0. Must be smaller than thresh_eig by several order of magnitude to avoid numerical problem. If the research of the optimal lambda cannot reach the condition (||x|| .eq. delta) because (||x|| .lt. delta), the reason might be that thresh_wtg is too big or/and thresh_eig is too small
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interface: ezfio,provider,ocaml
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default: 1.e-6
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[thresh_wtg2]
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type: double precision
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doc: Threshold in the trust region algorithm to considere when the dot product of the eigenvector W by the gradient v_grad is 0 in the case of avoid_saddle .eq. true. There is no particular reason to put a different value that thresh_wtg, but it can be useful one day
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interface: ezfio,provider,ocaml
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default: 1.e-6
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[avoid_saddle]
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type: logical
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doc: Test to avoid saddle point, active if true
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interface: ezfio,provider,ocaml
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default: false
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[version_avoid_saddle]
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type: integer
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doc: cf. trust region, not stable
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interface: ezfio,provider,ocaml
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default: 3
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[thresh_rho_2]
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type: double precision
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doc: Threshold for the step acceptance for the research of lambda in the trust region algorithm, if (rho_2 .geq. thresh_rho_2) the step is accepted, else the step is rejected
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interface: ezfio,provider,ocaml
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default: 0.1
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[thresh_cc]
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type: double precision
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doc: Threshold to stop the research of the optimal lambda in the trust region algorithm when (dabs(1d0-||x||^2/delta^2) < thresh_cc)
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interface: ezfio,provider,ocaml
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default: 1.e-6
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[thresh_model_2]
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type: double precision
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doc: if (ABS(criterion - criterion_model) < thresh_model_2), i.e., the difference between the actual criterion and the predicted next criterion, during the research of the optimal lambda in the trust region algorithm it prints a warning
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interface: ezfio,provider,ocaml
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default: 1.e-12
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[version_lambda_search]
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type: integer
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doc: Research of the optimal lambda in the trust region algorithm to constrain the norm of the step by solving: 1 -> ||x||^2 - delta^2 .eq. 0, 2 -> 1/||x||^2 - 1/delta^2 .eq. 0
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interface: ezfio,provider,ocaml
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default: 2
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[nb_it_max_lambda]
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type: integer
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doc: Maximal number of iterations for the research of the optimal lambda in the trust region algorithm
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interface: ezfio,provider,ocaml
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default: 100
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[nb_it_max_pre_search]
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type: integer
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doc: Maximal number of iterations for the pre-research of the optimal lambda in the trust region algorithm
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interface: ezfio,provider,ocaml
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default: 40
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hartree_fock
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# Utils trust region
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The documentation can be found in the org files.
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# Org files
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The org files are stored in the directory org in order to avoid overwriting on user changes.
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The org files can be modified, to export the change to the source code, run
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```
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./TANGLE_org_mode.sh
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mv *.irp.f ../.
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```
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! Algorithm for the trust region
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! step_in_trust_region:
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! Computes the step in the trust region (delta)
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! (automatically sets at the iteration 0 and which evolves during the
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! process in function of the evolution of rho). The step is computing by
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! constraining its norm with a lagrange multiplier.
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! Since the calculation of the step is based on the Newton method, an
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! estimation of the gain in energy is given using the Taylors series
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! truncated at the second order (criterion_model).
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! If (DABS(criterion-criterion_model) < 1d-12) then
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! must_exit = .True.
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! else
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! must_exit = .False.
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! This estimation of the gain in energy is used by
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! is_step_cancel_trust_region to say if the step is accepted or cancelled.
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! If the step must be cancelled, the calculation restart from the same
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! hessian and gradient and recomputes the step but in a smaller trust
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! region and so on until the step is accepted. If the step is accepted
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! the hessian and the gradient are recomputed to produce a new step.
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! Example:
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! !### Initialization ###
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! delta = 0d0
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! nb_iter = 0 ! Must start at 0 !!!
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! rho = 0.5d0
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! not_converged = .True.
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!
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! ! ### TODO ###
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! ! Compute the criterion before the loop
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! call #your_criterion(prev_criterion)
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!
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! do while (not_converged)
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! ! ### TODO ##
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! ! Call your gradient
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! ! Call you hessian
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! call #your_gradient(v_grad) (1D array)
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! call #your_hessian(H) (2D array)
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!
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! ! ### TODO ###
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! ! Diagonalization of the hessian
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! call diagonalization_hessian(n,H,e_val,w)
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!
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! cancel_step = .True. ! To enter in the loop just after
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! ! Loop to Reduce the trust radius until the criterion decreases and rho >= thresh_rho
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! do while (cancel_step)
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!
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! ! Hessian,gradient,Criterion -> x
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! call trust_region_step_w_expected_e(tmp_n,W,e_val,v_grad,prev_criterion,rho,nb_iter,delta,criterion_model,tmp_x,must_exit)
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!
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! if (must_exit) then
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! ! ### Message ###
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! ! if step_in_trust_region sets must_exit on true for numerical reasons
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! print*,'algo_trust1 sends the message : Exit'
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! !### exit ###
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! endif
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!
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! !### TODO ###
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! ! Compute x -> m_x
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! ! Compute m_x -> R
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! ! Apply R and keep the previous MOs...
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! ! Update/touch
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! ! Compute the new criterion/energy -> criterion
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!
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! call #your_routine_1D_to_2D_antisymmetric_array(x,m_x)
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! call #your_routine_2D_antisymmetric_array_to_rotation_matrix(m_x,R)
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! call #your_routine_to_apply_the_rotation_matrix(R,prev_mos)
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!
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! TOUCH #your_variables
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!
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! call #your_criterion(criterion)
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!
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! ! Criterion -> step accepted or rejected
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! call trust_region_is_step_cancelled(nb_iter,prev_criterion, criterion, criterion_model,rho,cancel_step)
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!
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! ! ### TODO ###
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! !if (cancel_step) then
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! ! Cancel the previous step (mo_coef = prev_mos if you keep them...)
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! !endif
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! #if (cancel_step) then
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! #mo_coef = prev_mos
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! #endif
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!
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! enddo
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!
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! !call save_mos() !### depend of the time for 1 iteration
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!
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! ! To exit the external loop if must_exit = .True.
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! if (must_exit) then
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! !### exit ###
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! endif
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!
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! ! Step accepted, nb iteration + 1
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! nb_iter = nb_iter + 1
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!
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! ! ### TODO ###
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! !if (###Conditions###) then
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! ! no_converged = .False.
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! !endif
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! #if (#your_conditions) then
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! # not_converged = .False.
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! #endif
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!
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! enddo
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! Variables:
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! Input:
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! | n | integer | m*(m-1)/2 |
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! | m | integer | number of mo in the mo_class |
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! | H(n,n) | double precision | Hessian |
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! | v_grad(n) | double precision | Gradient |
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! | W(n,n) | double precision | Eigenvectors of the hessian |
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! | e_val(n) | double precision | Eigenvalues of the hessian |
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! | criterion | double precision | Actual criterion |
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! | prev_criterion | double precision | Value of the criterion before the first iteration/after the previous iteration |
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! | rho | double precision | Given by is_step_cancel_trus_region |
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! | | | Agreement between the real function and the Taylor series (2nd order) |
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! | nb_iter | integer | Actual number of iterations |
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! Input/output:
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! | delta | double precision | Radius of the trust region |
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! Output:
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! | criterion_model | double precision | Predicted criterion after the rotation |
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! | x(n) | double precision | Step |
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! | must_exit | logical | If the program must exit the loop |
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subroutine trust_region_step_w_expected_e(n,n2,H,W,e_val,v_grad,prev_criterion,rho,nb_iter,delta,criterion_model,x,must_exit)
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include 'pi.h'
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!BEGIN_DOC
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! Compute the step and the expected criterion/energy after the step
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!END_DOC
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implicit none
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! in
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integer, intent(in) :: n,n2, nb_iter
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double precision, intent(in) :: H(n,n2), W(n,n2), v_grad(n)
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double precision, intent(in) :: rho, prev_criterion
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! inout
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double precision, intent(inout) :: delta, e_val(n)
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! out
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double precision, intent(out) :: criterion_model, x(n)
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logical, intent(out) :: must_exit
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! internal
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integer :: info
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must_exit = .False.
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call trust_region_step(n,n2,nb_iter,v_grad,rho,e_val,W,x,delta)
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call trust_region_expected_e(n,n2,v_grad,H,x,prev_criterion,criterion_model)
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! exit if DABS(prev_criterion - criterion_model) < 1d-12
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if (DABS(prev_criterion - criterion_model) < thresh_model) then
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print*,''
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print*,'###############################################################################'
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print*,'DABS(prev_criterion - criterion_model) <', thresh_model, 'stop the trust region'
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print*,'###############################################################################'
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print*,''
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must_exit = .True.
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endif
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if (delta < thresh_delta) then
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print*,''
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print*,'##############################################'
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print*,'Delta <', thresh_delta, 'stop the trust region'
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print*,'##############################################'
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print*,''
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must_exit = .True.
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endif
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! Add after the call to this subroutine, a statement:
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! "if (must_exit) then
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! exit
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! endif"
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! in order to exit the optimization loop
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end subroutine
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! Variables:
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! Input:
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! | nb_iter | integer | actual number of iterations |
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! | prev_criterion | double precision | criterion before the application of the step x |
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! | criterion | double precision | criterion after the application of the step x |
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! | criterion_model | double precision | predicted criterion after the application of x |
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! Output:
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! | rho | double precision | Agreement between the predicted criterion and the real new criterion |
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! | cancel_step | logical | If the step must be cancelled |
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subroutine trust_region_is_step_cancelled(nb_iter,prev_criterion, criterion, criterion_model,rho,cancel_step)
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include 'pi.h'
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!BEGIN_DOC
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! Compute if the step should be cancelled
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!END_DOC
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implicit none
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! in
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double precision, intent(in) :: prev_criterion, criterion, criterion_model
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! inout
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integer, intent(inout) :: nb_iter
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! out
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logical, intent(out) :: cancel_step
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double precision, intent(out) :: rho
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! Computes rho
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call trust_region_rho(prev_criterion,criterion,criterion_model,rho)
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if (nb_iter == 0) then
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nb_iter = 1 ! in order to enable the change of delta if the first iteration is cancelled
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endif
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! If rho < thresh_rho -> give something in output to cancel the step
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if (rho >= thresh_rho) then !0.1d0) then
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! The step is accepted
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cancel_step = .False.
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else
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! The step is rejected
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cancel_step = .True.
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print*, '***********************'
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print*, 'Step cancel : rho <', thresh_rho
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print*, '***********************'
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endif
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end subroutine
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@ -0,0 +1,85 @@
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! Apply MO rotation
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! Subroutine to apply the rotation matrix to the coefficients of the
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! MOs.
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! New MOs = Old MOs . Rotation matrix
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! *Compute the new MOs with the previous MOs and a rotation matrix*
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! Provided:
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! | mo_num | integer | number of MOs |
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! | ao_num | integer | number of AOs |
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! | mo_coef(ao_num,mo_num) | double precision | coefficients of the MOs |
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! Intent in:
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! | R(mo_num,mo_num) | double precision | rotation matrix |
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! Intent out:
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! | prev_mos(ao_num,mo_num) | double precision | MOs before the rotation |
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! Internal:
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! | new_mos(ao_num,mo_num) | double precision | MOs after the rotation |
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! | i,j | integer | indexes |
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subroutine apply_mo_rotation(R,prev_mos)
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include 'pi.h'
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!BEGIN_DOC
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! Compute the new MOs knowing the rotation matrix
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!END_DOC
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implicit none
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! Variables
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! in
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double precision, intent(in) :: R(mo_num,mo_num)
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! out
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double precision, intent(out) :: prev_mos(ao_num,mo_num)
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! internal
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double precision, allocatable :: new_mos(:,:)
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integer :: i,j
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double precision :: t1,t2,t3
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print*,''
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print*,'---apply_mo_rotation---'
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call wall_time(t1)
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! Allocation
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allocate(new_mos(ao_num,mo_num))
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! Calculation
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! Product of old MOs (mo_coef) by Rotation matrix (R)
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call dgemm('N','N',ao_num,mo_num,mo_num,1d0,mo_coef,size(mo_coef,1),R,size(R,1),0d0,new_mos,size(new_mos,1))
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prev_mos = mo_coef
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mo_coef = new_mos
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if (debug) then
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print*,'New mo_coef : '
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do i = 1, mo_num
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write(*,'(100(F10.5))') mo_coef(i,:)
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enddo
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endif
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! Save the new MOs and change the label
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mo_label = 'MCSCF'
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!call save_mos
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call ezfio_set_determinants_mo_label(mo_label)
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!print*,'Done, MOs saved'
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! Deallocation, end
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deallocate(new_mos)
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call wall_time(t2)
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t3 = t2 - t1
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print*,'Time in apply mo rotation:', t3
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print*,'---End apply_mo_rotation---'
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end subroutine
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@ -0,0 +1,61 @@
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! Matrix to vector index
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! *Compute the index i of a vector element from the indexes p,q of a
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! matrix element*
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! Lower diagonal matrix (p,q), p > q -> vector (i)
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! If a matrix is antisymmetric it can be reshaped as a vector. And the
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! vector can be reshaped as an antisymmetric matrix
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! \begin{align*}
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! \begin{pmatrix}
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! 0 & -1 & -2 & -4 \\
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! 1 & 0 & -3 & -5 \\
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! 2 & 3 & 0 & -6 \\
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! 4 & 5 & 6 & 0
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! \end{pmatrix}
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! \Leftrightarrow
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! \begin{pmatrix}
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! 1 & 2 & 3 & 4 & 5 & 6
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! \end{pmatrix}
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! \end{align*}
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! !!! Here the algorithm only work for the lower diagonal !!!
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! Input:
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! | p,q | integer | indexes of a matrix element in the lower diagonal |
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! | | | p > q, q -> column |
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! | | | p -> row, |
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! | | | q -> column |
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! Input:
|
||||
! | i | integer | corresponding index in the vector |
|
||||
|
||||
|
||||
subroutine mat_to_vec_index(p,q,i)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer, intent(in) :: p,q
|
||||
|
||||
! out
|
||||
integer, intent(out) :: i
|
||||
|
||||
! internal
|
||||
integer :: a,b
|
||||
double precision :: da
|
||||
|
||||
! Calculation
|
||||
|
||||
a = p-1
|
||||
b = a*(a-1)/2
|
||||
|
||||
i = q+b
|
||||
|
||||
end subroutine
|
||||
|
|
@ -0,0 +1,7 @@
|
|||
#!/bin/sh
|
||||
|
||||
list='ls *.org'
|
||||
for element in $list
|
||||
do
|
||||
emacs --batch $element -f org-babel-tangle
|
||||
done
|
||||
|
|
@ -0,0 +1,593 @@
|
|||
* Algorithm for the trust region
|
||||
|
||||
step_in_trust_region:
|
||||
Computes the step in the trust region (delta)
|
||||
(automatically sets at the iteration 0 and which evolves during the
|
||||
process in function of the evolution of rho). The step is computing by
|
||||
constraining its norm with a lagrange multiplier.
|
||||
Since the calculation of the step is based on the Newton method, an
|
||||
estimation of the gain in energy is given using the Taylors series
|
||||
truncated at the second order (criterion_model).
|
||||
If (DABS(criterion-criterion_model) < 1d-12) then
|
||||
must_exit = .True.
|
||||
else
|
||||
must_exit = .False.
|
||||
|
||||
This estimation of the gain in energy is used by
|
||||
is_step_cancel_trust_region to say if the step is accepted or cancelled.
|
||||
|
||||
If the step must be cancelled, the calculation restart from the same
|
||||
hessian and gradient and recomputes the step but in a smaller trust
|
||||
region and so on until the step is accepted. If the step is accepted
|
||||
the hessian and the gradient are recomputed to produce a new step.
|
||||
|
||||
Example:
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle algo_trust.irp.f
|
||||
! !### Initialization ###
|
||||
! delta = 0d0
|
||||
! nb_iter = 0 ! Must start at 0 !!!
|
||||
! rho = 0.5d0
|
||||
! not_converged = .True.
|
||||
!
|
||||
! ! ### TODO ###
|
||||
! ! Compute the criterion before the loop
|
||||
! call #your_criterion(prev_criterion)
|
||||
!
|
||||
! do while (not_converged)
|
||||
! ! ### TODO ##
|
||||
! ! Call your gradient
|
||||
! ! Call you hessian
|
||||
! call #your_gradient(v_grad) (1D array)
|
||||
! call #your_hessian(H) (2D array)
|
||||
!
|
||||
! ! ### TODO ###
|
||||
! ! Diagonalization of the hessian
|
||||
! call diagonalization_hessian(n,H,e_val,w)
|
||||
!
|
||||
! cancel_step = .True. ! To enter in the loop just after
|
||||
! ! Loop to Reduce the trust radius until the criterion decreases and rho >= thresh_rho
|
||||
! do while (cancel_step)
|
||||
!
|
||||
! ! Hessian,gradient,Criterion -> x
|
||||
! call trust_region_step_w_expected_e(tmp_n,W,e_val,v_grad,prev_criterion,rho,nb_iter,delta,criterion_model,tmp_x,must_exit)
|
||||
!
|
||||
! if (must_exit) then
|
||||
! ! ### Message ###
|
||||
! ! if step_in_trust_region sets must_exit on true for numerical reasons
|
||||
! print*,'algo_trust1 sends the message : Exit'
|
||||
! !### exit ###
|
||||
! endif
|
||||
!
|
||||
! !### TODO ###
|
||||
! ! Compute x -> m_x
|
||||
! ! Compute m_x -> R
|
||||
! ! Apply R and keep the previous MOs...
|
||||
! ! Update/touch
|
||||
! ! Compute the new criterion/energy -> criterion
|
||||
!
|
||||
! call #your_routine_1D_to_2D_antisymmetric_array(x,m_x)
|
||||
! call #your_routine_2D_antisymmetric_array_to_rotation_matrix(m_x,R)
|
||||
! call #your_routine_to_apply_the_rotation_matrix(R,prev_mos)
|
||||
!
|
||||
! TOUCH #your_variables
|
||||
!
|
||||
! call #your_criterion(criterion)
|
||||
!
|
||||
! ! Criterion -> step accepted or rejected
|
||||
! call trust_region_is_step_cancelled(nb_iter,prev_criterion, criterion, criterion_model,rho,cancel_step)
|
||||
!
|
||||
! ! ### TODO ###
|
||||
! !if (cancel_step) then
|
||||
! ! Cancel the previous step (mo_coef = prev_mos if you keep them...)
|
||||
! !endif
|
||||
! #if (cancel_step) then
|
||||
! #mo_coef = prev_mos
|
||||
! #endif
|
||||
!
|
||||
! enddo
|
||||
!
|
||||
! !call save_mos() !### depend of the time for 1 iteration
|
||||
!
|
||||
! ! To exit the external loop if must_exit = .True.
|
||||
! if (must_exit) then
|
||||
! !### exit ###
|
||||
! endif
|
||||
!
|
||||
! ! Step accepted, nb iteration + 1
|
||||
! nb_iter = nb_iter + 1
|
||||
!
|
||||
! ! ### TODO ###
|
||||
! !if (###Conditions###) then
|
||||
! ! no_converged = .False.
|
||||
! !endif
|
||||
! #if (#your_conditions) then
|
||||
! # not_converged = .False.
|
||||
! #endif
|
||||
!
|
||||
! enddo
|
||||
#+END_SRC
|
||||
|
||||
Variables:
|
||||
|
||||
Input:
|
||||
| n | integer | m*(m-1)/2 |
|
||||
| m | integer | number of mo in the mo_class |
|
||||
| H(n,n) | double precision | Hessian |
|
||||
| v_grad(n) | double precision | Gradient |
|
||||
| W(n,n) | double precision | Eigenvectors of the hessian |
|
||||
| e_val(n) | double precision | Eigenvalues of the hessian |
|
||||
| criterion | double precision | Actual criterion |
|
||||
| prev_criterion | double precision | Value of the criterion before the first iteration/after the previous iteration |
|
||||
| rho | double precision | Given by is_step_cancel_trus_region |
|
||||
| | | Agreement between the real function and the Taylor series (2nd order) |
|
||||
| nb_iter | integer | Actual number of iterations |
|
||||
|
||||
Input/output:
|
||||
| delta | double precision | Radius of the trust region |
|
||||
|
||||
Output:
|
||||
| criterion_model | double precision | Predicted criterion after the rotation |
|
||||
| x(n) | double precision | Step |
|
||||
| must_exit | logical | If the program must exit the loop |
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle algo_trust.irp.f
|
||||
subroutine trust_region_step_w_expected_e(n,n2,H,W,e_val,v_grad,prev_criterion,rho,nb_iter,delta,criterion_model,x,must_exit)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute the step and the expected criterion/energy after the step
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! in
|
||||
integer, intent(in) :: n,n2, nb_iter
|
||||
double precision, intent(in) :: H(n,n2), W(n,n2), v_grad(n)
|
||||
double precision, intent(in) :: rho, prev_criterion
|
||||
|
||||
! inout
|
||||
double precision, intent(inout) :: delta, e_val(n)
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: criterion_model, x(n)
|
||||
logical, intent(out) :: must_exit
|
||||
|
||||
! internal
|
||||
integer :: info
|
||||
|
||||
must_exit = .False.
|
||||
|
||||
call trust_region_step(n,n2,nb_iter,v_grad,rho,e_val,W,x,delta)
|
||||
|
||||
call trust_region_expected_e(n,n2,v_grad,H,x,prev_criterion,criterion_model)
|
||||
|
||||
! exit if DABS(prev_criterion - criterion_model) < 1d-12
|
||||
if (DABS(prev_criterion - criterion_model) < thresh_model) then
|
||||
print*,''
|
||||
print*,'###############################################################################'
|
||||
print*,'DABS(prev_criterion - criterion_model) <', thresh_model, 'stop the trust region'
|
||||
print*,'###############################################################################'
|
||||
print*,''
|
||||
must_exit = .True.
|
||||
endif
|
||||
|
||||
if (delta < thresh_delta) then
|
||||
print*,''
|
||||
print*,'##############################################'
|
||||
print*,'Delta <', thresh_delta, 'stop the trust region'
|
||||
print*,'##############################################'
|
||||
print*,''
|
||||
must_exit = .True.
|
||||
endif
|
||||
|
||||
! Add after the call to this subroutine, a statement:
|
||||
! "if (must_exit) then
|
||||
! exit
|
||||
! endif"
|
||||
! in order to exit the optimization loop
|
||||
|
||||
end subroutine
|
||||
#+END_SRC
|
||||
|
||||
Variables:
|
||||
|
||||
Input:
|
||||
| nb_iter | integer | actual number of iterations |
|
||||
| prev_criterion | double precision | criterion before the application of the step x |
|
||||
| criterion | double precision | criterion after the application of the step x |
|
||||
| criterion_model | double precision | predicted criterion after the application of x |
|
||||
|
||||
Output:
|
||||
| rho | double precision | Agreement between the predicted criterion and the real new criterion |
|
||||
| cancel_step | logical | If the step must be cancelled |
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle algo_trust.irp.f
|
||||
subroutine trust_region_is_step_cancelled(nb_iter,prev_criterion, criterion, criterion_model,rho,cancel_step)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute if the step should be cancelled
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! in
|
||||
double precision, intent(in) :: prev_criterion, criterion, criterion_model
|
||||
|
||||
! inout
|
||||
integer, intent(inout) :: nb_iter
|
||||
|
||||
! out
|
||||
logical, intent(out) :: cancel_step
|
||||
double precision, intent(out) :: rho
|
||||
|
||||
! Computes rho
|
||||
call trust_region_rho(prev_criterion,criterion,criterion_model,rho)
|
||||
|
||||
if (nb_iter == 0) then
|
||||
nb_iter = 1 ! in order to enable the change of delta if the first iteration is cancelled
|
||||
endif
|
||||
|
||||
! If rho < thresh_rho -> give something in output to cancel the step
|
||||
if (rho >= thresh_rho) then !0.1d0) then
|
||||
! The step is accepted
|
||||
cancel_step = .False.
|
||||
else
|
||||
! The step is rejected
|
||||
cancel_step = .True.
|
||||
print*, '***********************'
|
||||
print*, 'Step cancel : rho <', thresh_rho
|
||||
print*, '***********************'
|
||||
endif
|
||||
|
||||
end subroutine
|
||||
#+END_SRC
|
||||
|
||||
** Template for MOs
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_template_mos.txt
|
||||
subroutine algo_trust_template(tmp_n, tmp_list_size, tmp_list)
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! In
|
||||
integer, intent(in) :: tmp_n, tmp_list_size, tmp_list(tmp_list_size)
|
||||
|
||||
! Out
|
||||
! Rien ou un truc pour savoir si ça c'est bien passé
|
||||
|
||||
! Internal
|
||||
double precision, allocatable :: e_val(:), W(:,:), tmp_R(:,:), R(:,:), tmp_x(:), tmp_m_x(:,:)
|
||||
double precision, allocatable :: prev_mos(:,:)
|
||||
double precision :: criterion, prev_criterion, criterion_model
|
||||
double precision :: delta, rho
|
||||
logical :: not_converged, cancel_step, must_exit, enforce_step_cancellation
|
||||
integer :: nb_iter, info, nb_sub_iter
|
||||
integer :: i,j,tmp_i,tmp_j
|
||||
|
||||
allocate(W(tmp_n, tmp_n),e_val(tmp_n),tmp_x(tmp_n),tmp_m_x(tmp_list_size, tmp_list_size))
|
||||
allocate(tmp_R(tmp_list_size, tmp_list_size), R(mo_num, mo_num))
|
||||
allocate(prev_mos(ao_num, mo_num))
|
||||
|
||||
! Provide the criterion, but unnecessary because it's done
|
||||
! automatically
|
||||
PROVIDE C_PROVIDER H_PROVIDER g_PROVIDER cc_PROVIDER
|
||||
|
||||
! Initialization
|
||||
delta = 0d0
|
||||
nb_iter = 0 ! Must start at 0 !!!
|
||||
rho = 0.5d0 ! Must start at 0.5
|
||||
not_converged = .True. ! Must be true
|
||||
|
||||
! Compute the criterion before the loop
|
||||
prev_criterion = C_PROVIDER
|
||||
|
||||
do while (not_converged)
|
||||
|
||||
print*,''
|
||||
print*,'******************'
|
||||
print*,'Iteration', nb_iter
|
||||
print*,'******************'
|
||||
print*,''
|
||||
|
||||
! The new hessian and gradient are computed at the end of the previous iteration
|
||||
! Diagonalization of the hessian
|
||||
call diagonalization_hessian(tmp_n, H_PROVIDER, e_val, W)
|
||||
|
||||
cancel_step = .True. ! To enter in the loop just after
|
||||
nb_sub_iter = 0
|
||||
|
||||
! Loop to Reduce the trust radius until the criterion decreases and rho >= thresh_rho
|
||||
do while (cancel_step)
|
||||
|
||||
print*,'-----------------------------'
|
||||
print*,'Iteration:', nb_iter
|
||||
print*,'Sub iteration:', nb_sub_iter
|
||||
print*,'-----------------------------'
|
||||
|
||||
! Hessian,gradient,Criterion -> x
|
||||
call trust_region_step_w_expected_e(tmp_n,tmp_n, H_PROVIDER, W, e_val, g_PROVIDER, &
|
||||
prev_criterion, rho, nb_iter, delta, criterion_model, tmp_x, must_exit)
|
||||
|
||||
if (must_exit) then
|
||||
! if step_in_trust_region sets must_exit on true for numerical reasons
|
||||
print*,'trust_region_step_w_expected_e sent the message : Exit'
|
||||
exit
|
||||
endif
|
||||
|
||||
! 1D tmp -> 2D tmp
|
||||
call vec_to_mat_v2(tmp_n, tmp_list_size, tmp_x, tmp_m_x)
|
||||
|
||||
! Rotation submatrix (square matrix tmp_list_size by tmp_list_size)
|
||||
call rotation_matrix(tmp_m_x, tmp_list_size, tmp_R, tmp_list_size, tmp_list_size, info, enforce_step_cancellation)
|
||||
|
||||
if (enforce_step_cancellation) then
|
||||
print*, 'Forces the step cancellation, too large error in the rotation matrix'
|
||||
rho = 0d0
|
||||
cycle
|
||||
endif
|
||||
|
||||
! tmp_R to R, subspace to full space
|
||||
call sub_to_full_rotation_matrix(tmp_list_size, tmp_list, tmp_R, R)
|
||||
|
||||
! Rotation of the MOs
|
||||
call apply_mo_rotation(R, prev_mos)
|
||||
|
||||
! touch mo_coef
|
||||
call clear_mo_map ! Only if you are using the bi-electronic integrals
|
||||
! mo_coef becomes valid
|
||||
! And avoid the recomputation of the providers which depend of mo_coef
|
||||
TOUCH mo_coef C_PROVIDER H_PROVIDER g_PROVIDER cc_PROVIDER
|
||||
|
||||
! To update the other parameters if needed
|
||||
call #update_parameters()
|
||||
|
||||
! To enforce the program to provide new criterion after the update
|
||||
! of the parameters
|
||||
FREE C_PROVIDER
|
||||
PROVIDE C_PROVIDER
|
||||
criterion = C_PROVIDER
|
||||
|
||||
! Criterion -> step accepted or rejected
|
||||
call trust_region_is_step_cancelled(nb_iter, prev_criterion, criterion, criterion_model, rho, cancel_step)
|
||||
|
||||
! Cancellation of the step ?
|
||||
if (cancel_step) then
|
||||
! Replacement by the previous MOs
|
||||
mo_coef = prev_mos
|
||||
! call save_mos() ! depends of the time for 1 iteration
|
||||
|
||||
! No need to clear_mo_map since we don't recompute the gradient and the hessian
|
||||
! mo_coef becomes valid
|
||||
! Avoid the recomputation of the providers which depend of mo_coef
|
||||
TOUCH mo_coef H_PROVIDER g_PROVIDER C_PROVIDER cc_PROVIDER
|
||||
else
|
||||
! The step is accepted:
|
||||
! criterion -> prev criterion
|
||||
|
||||
! The replacement "criterion -> prev criterion" is already done
|
||||
! in trust_region_rho, so if the criterion does not have a reason
|
||||
! to change, it will change nothing for the criterion and will
|
||||
! force the program to provide the new hessian, gradient and
|
||||
! convergence criterion for the next iteration.
|
||||
! But in the case of orbital optimization we diagonalize the CI
|
||||
! matrix after the "FREE" statement, so the criterion will change
|
||||
|
||||
FREE C_PROVIDER H_PROVIDER g_PROVIDER cc_PROVIDER
|
||||
PROVIDE C_PROVIDER H_PROVIDER g_PROVIDER cc_PROVIDER
|
||||
prev_criterion = C_PROVIDER
|
||||
|
||||
endif
|
||||
|
||||
nb_sub_iter = nb_sub_iter + 1
|
||||
enddo
|
||||
|
||||
! call save_mos() ! depends of the time for 1 iteration
|
||||
|
||||
! To exit the external loop if must_exit = .True.
|
||||
if (must_exit) then
|
||||
exit
|
||||
endif
|
||||
|
||||
! Step accepted, nb iteration + 1
|
||||
nb_iter = nb_iter + 1
|
||||
|
||||
! Provide the convergence criterion
|
||||
! Provide the gradient and the hessian for the next iteration
|
||||
PROVIDE cc_PROVIDER
|
||||
|
||||
! To exit
|
||||
if (dabs(cc_PROVIDER) < thresh_opt_max_elem_grad) then
|
||||
not_converged = .False.
|
||||
endif
|
||||
|
||||
if (nb_iter > optimization_max_nb_iter) then
|
||||
not_converged = .False.
|
||||
endif
|
||||
|
||||
if (delta < thresh_delta) then
|
||||
not_converged = .False.
|
||||
endif
|
||||
|
||||
enddo
|
||||
|
||||
! Save the final MOs
|
||||
call save_mos()
|
||||
|
||||
! Diagonalization of the hessian
|
||||
! (To see the eigenvalues at the end of the optimization)
|
||||
call diagonalization_hessian(tmp_n, H_PROVIDER, e_val, W)
|
||||
|
||||
deallocate(e_val, W, tmp_R, R, tmp_x, prev_mos)
|
||||
|
||||
end
|
||||
#+END_SRC
|
||||
|
||||
** Cartesian version
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_template_xyz.txt
|
||||
subroutine algo_trust_cartesian_template(tmp_n)
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! In
|
||||
integer, intent(in) :: tmp_n
|
||||
|
||||
! Out
|
||||
! Rien ou un truc pour savoir si ça c'est bien passé
|
||||
|
||||
! Internal
|
||||
double precision, allocatable :: e_val(:), W(:,:), tmp_x(:)
|
||||
double precision :: criterion, prev_criterion, criterion_model
|
||||
double precision :: delta, rho
|
||||
logical :: not_converged, cancel_step, must_exit
|
||||
integer :: nb_iter, nb_sub_iter
|
||||
integer :: i,j
|
||||
|
||||
allocate(W(tmp_n, tmp_n),e_val(tmp_n),tmp_x(tmp_n))
|
||||
|
||||
PROVIDE C_PROVIDER X_PROVIDER H_PROVIDER g_PROVIDER
|
||||
|
||||
! Initialization
|
||||
delta = 0d0
|
||||
nb_iter = 0 ! Must start at 0 !!!
|
||||
rho = 0.5d0 ! Must start at 0.5
|
||||
not_converged = .True. ! Must be true
|
||||
|
||||
! Compute the criterion before the loop
|
||||
prev_criterion = C_PROVIDER
|
||||
|
||||
do while (not_converged)
|
||||
|
||||
print*,''
|
||||
print*,'******************'
|
||||
print*,'Iteration', nb_iter
|
||||
print*,'******************'
|
||||
print*,''
|
||||
|
||||
if (nb_iter > 0) then
|
||||
PROVIDE H_PROVIDER g_PROVIDER
|
||||
endif
|
||||
|
||||
! Diagonalization of the hessian
|
||||
call diagonalization_hessian(tmp_n, H_PROVIDER, e_val, W)
|
||||
|
||||
cancel_step = .True. ! To enter in the loop just after
|
||||
nb_sub_iter = 0
|
||||
|
||||
! Loop to Reduce the trust radius until the criterion decreases and rho >= thresh_rho
|
||||
do while (cancel_step)
|
||||
|
||||
print*,'-----------------------------'
|
||||
print*,'Iteration:', nb_iter
|
||||
print*,'Sub iteration:', nb_sub_iter
|
||||
print*,'-----------------------------'
|
||||
|
||||
! Hessian,gradient,Criterion -> x
|
||||
call trust_region_step_w_expected_e(tmp_n,tmp_n, H_PROVIDER, W, e_val, g_PROVIDER, &
|
||||
prev_criterion, rho, nb_iter, delta, criterion_model, tmp_x, must_exit)
|
||||
|
||||
if (must_exit) then
|
||||
! if step_in_trust_region sets must_exit on true for numerical reasons
|
||||
print*,'trust_region_step_w_expected_e sent the message : Exit'
|
||||
exit
|
||||
endif
|
||||
|
||||
! New coordinates, check the sign
|
||||
X_PROVIDER = X_PROVIDER - tmp_x
|
||||
|
||||
! touch X_PROVIDER
|
||||
TOUCH X_PROVIDER H_PROVIDER g_PROVIDER cc_PROVIDER
|
||||
|
||||
! To update the other parameters if needed
|
||||
call #update_parameters()
|
||||
|
||||
! New criterion
|
||||
PROVIDE C_PROVIDER ! Unnecessary
|
||||
criterion = C_PROVIDER
|
||||
|
||||
! Criterion -> step accepted or rejected
|
||||
call trust_region_is_step_cancelled(nb_iter, prev_criterion, criterion, criterion_model, rho, cancel_step)
|
||||
|
||||
! Cancel the previous step
|
||||
if (cancel_step) then
|
||||
! Replacement by the previous coordinates, check the sign
|
||||
X_PROVIDER = X_PROVIDER + tmp_x
|
||||
|
||||
! Avoid the recomputation of the hessian and the gradient
|
||||
TOUCH X_PROVIDER H_PROVIDER g_PROVIDER C_PROVIDER cc_PROVIDER
|
||||
endif
|
||||
|
||||
nb_sub_iter = nb_sub_iter + 1
|
||||
enddo
|
||||
|
||||
! To exit the external loop if must_exit = .True.
|
||||
if (must_exit) then
|
||||
exit
|
||||
endif
|
||||
|
||||
! Step accepted, nb iteration + 1
|
||||
nb_iter = nb_iter + 1
|
||||
|
||||
PROVIDE cc_PROVIDER
|
||||
|
||||
! To exit
|
||||
if (dabs(cc_PROVIDER) < thresh_opt_max_elem_grad) then
|
||||
not_converged = .False.
|
||||
endif
|
||||
|
||||
if (nb_iter > optimization_max_nb_iter) then
|
||||
not_converged = .False.
|
||||
endif
|
||||
|
||||
if (delta < thresh_delta) then
|
||||
not_converged = .False.
|
||||
endif
|
||||
|
||||
enddo
|
||||
|
||||
deallocate(e_val, W, tmp_x)
|
||||
|
||||
end
|
||||
#+END_SRC
|
||||
|
||||
** Script template
|
||||
#+BEGIN_SRC bash :tangle script_template_mos.sh
|
||||
#!/bin/bash
|
||||
|
||||
your_file=
|
||||
|
||||
your_C_PROVIDER=
|
||||
your_H_PROVIDER=
|
||||
your_g_PROVIDER=
|
||||
your_cc_PROVIDER=
|
||||
|
||||
sed "s/C_PROVIDER/$your_C_PROVIDER/g" trust_region_template_mos.txt > $your_file
|
||||
sed -i "s/H_PROVIDER/$your_H_PROVIDER/g" $your_file
|
||||
sed -i "s/g_PROVIDER/$your_g_PROVIDER/g" $your_file
|
||||
sed -i "s/cc_PROVIDER/$your_cc_PROVIDER/g" $your_file
|
||||
#+END_SRC
|
||||
|
||||
#+BEGIN_SRC bash :tangle script_template_xyz.sh
|
||||
#!/bin/bash
|
||||
|
||||
your_file=
|
||||
|
||||
your_C_PROVIDER=
|
||||
your_X_PROVIDER=
|
||||
your_H_PROVIDER=
|
||||
your_g_PROVIDER=
|
||||
your_cc_PROVIDER=
|
||||
|
||||
sed "s/C_PROVIDER/$your_C_PROVIDER/g" trust_region_template_xyz.txt > $your_file
|
||||
sed -i "s/X_PROVIDER/$your_X_PROVIDER/g" $your_file
|
||||
sed -i "s/H_PROVIDER/$your_H_PROVIDER/g" $your_file
|
||||
sed -i "s/g_PROVIDER/$your_g_PROVIDER/g" $your_file
|
||||
sed -i "s/cc_PROVIDER/$your_cc_PROVIDER/g" $your_file
|
||||
#+END_SRC
|
||||
|
||||
|
|
@ -0,0 +1,86 @@
|
|||
* Apply MO rotation
|
||||
Subroutine to apply the rotation matrix to the coefficients of the
|
||||
MOs.
|
||||
|
||||
New MOs = Old MOs . Rotation matrix
|
||||
|
||||
*Compute the new MOs with the previous MOs and a rotation matrix*
|
||||
|
||||
Provided:
|
||||
| mo_num | integer | number of MOs |
|
||||
| ao_num | integer | number of AOs |
|
||||
| mo_coef(ao_num,mo_num) | double precision | coefficients of the MOs |
|
||||
|
||||
Intent in:
|
||||
| R(mo_num,mo_num) | double precision | rotation matrix |
|
||||
|
||||
Intent out:
|
||||
| prev_mos(ao_num,mo_num) | double precision | MOs before the rotation |
|
||||
|
||||
Internal:
|
||||
| new_mos(ao_num,mo_num) | double precision | MOs after the rotation |
|
||||
| i,j | integer | indexes |
|
||||
#+BEGIN_SRC f90 :comments org :tangle apply_mo_rotation.irp.f
|
||||
subroutine apply_mo_rotation(R,prev_mos)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute the new MOs knowing the rotation matrix
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
double precision, intent(in) :: R(mo_num,mo_num)
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: prev_mos(ao_num,mo_num)
|
||||
|
||||
! internal
|
||||
double precision, allocatable :: new_mos(:,:)
|
||||
integer :: i,j
|
||||
double precision :: t1,t2,t3
|
||||
|
||||
print*,''
|
||||
print*,'---apply_mo_rotation---'
|
||||
|
||||
call wall_time(t1)
|
||||
|
||||
! Allocation
|
||||
allocate(new_mos(ao_num,mo_num))
|
||||
|
||||
! Calculation
|
||||
|
||||
! Product of old MOs (mo_coef) by Rotation matrix (R)
|
||||
call dgemm('N','N',ao_num,mo_num,mo_num,1d0,mo_coef,size(mo_coef,1),R,size(R,1),0d0,new_mos,size(new_mos,1))
|
||||
|
||||
prev_mos = mo_coef
|
||||
mo_coef = new_mos
|
||||
|
||||
if (debug) then
|
||||
print*,'New mo_coef : '
|
||||
do i = 1, mo_num
|
||||
write(*,'(100(F10.5))') mo_coef(i,:)
|
||||
enddo
|
||||
endif
|
||||
|
||||
! Save the new MOs and change the label
|
||||
mo_label = 'MCSCF'
|
||||
!call save_mos
|
||||
call ezfio_set_determinants_mo_label(mo_label)
|
||||
|
||||
!print*,'Done, MOs saved'
|
||||
|
||||
! Deallocation, end
|
||||
deallocate(new_mos)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2 - t1
|
||||
print*,'Time in apply mo rotation:', t3
|
||||
print*,'---End apply_mo_rotation---'
|
||||
|
||||
end subroutine
|
||||
#+END_SRC
|
||||
|
|
@ -0,0 +1,63 @@
|
|||
* Matrix to vector index
|
||||
|
||||
*Compute the index i of a vector element from the indexes p,q of a
|
||||
matrix element*
|
||||
|
||||
Lower diagonal matrix (p,q), p > q -> vector (i)
|
||||
|
||||
If a matrix is antisymmetric it can be reshaped as a vector. And the
|
||||
vector can be reshaped as an antisymmetric matrix
|
||||
|
||||
\begin{align*}
|
||||
\begin{pmatrix}
|
||||
0 & -1 & -2 & -4 \\
|
||||
1 & 0 & -3 & -5 \\
|
||||
2 & 3 & 0 & -6 \\
|
||||
4 & 5 & 6 & 0
|
||||
\end{pmatrix}
|
||||
\Leftrightarrow
|
||||
\begin{pmatrix}
|
||||
1 & 2 & 3 & 4 & 5 & 6
|
||||
\end{pmatrix}
|
||||
\end{align*}
|
||||
|
||||
!!! Here the algorithm only work for the lower diagonal !!!
|
||||
|
||||
Input:
|
||||
| p,q | integer | indexes of a matrix element in the lower diagonal |
|
||||
| | | p > q, q -> column |
|
||||
| | | p -> row, |
|
||||
| | | q -> column |
|
||||
|
||||
Input:
|
||||
| i | integer | corresponding index in the vector |
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle mat_to_vec_index.irp.f
|
||||
subroutine mat_to_vec_index(p,q,i)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer, intent(in) :: p,q
|
||||
|
||||
! out
|
||||
integer, intent(out) :: i
|
||||
|
||||
! internal
|
||||
integer :: a,b
|
||||
double precision :: da
|
||||
|
||||
! Calculation
|
||||
|
||||
a = p-1
|
||||
b = a*(a-1)/2
|
||||
|
||||
i = q+b
|
||||
|
||||
end subroutine
|
||||
#+END_SRC
|
||||
|
||||
|
|
@ -0,0 +1,452 @@
|
|||
* Rotation matrix
|
||||
|
||||
*Build a rotation matrix from an antisymmetric matrix*
|
||||
|
||||
Compute a rotation matrix $\textbf{R}$ from an antisymmetric matrix $$\textbf{A}$$ such as :
|
||||
$$
|
||||
\textbf{R}=\exp(\textbf{A})
|
||||
$$
|
||||
|
||||
So :
|
||||
\begin{align*}
|
||||
\textbf{R}=& \exp(\textbf{A}) \\
|
||||
=& \sum_k^{\infty} \frac{1}{k!}\textbf{A}^k \\
|
||||
=& \textbf{W} \cdot \cos(\tau) \cdot \textbf{W}^{\dagger} + \textbf{W} \cdot \tau^{-1} \cdot \sin(\tau) \cdot \textbf{W}^{\dagger} \cdot \textbf{A}
|
||||
\end{align*}
|
||||
|
||||
With :
|
||||
$\textbf{W}$ : eigenvectors of $\textbf{A}^2$
|
||||
$\tau$ : $\sqrt{-x}$
|
||||
$x$ : eigenvalues of $\textbf{A}^2$
|
||||
|
||||
Input:
|
||||
| A(n,n) | double precision | antisymmetric matrix |
|
||||
| n | integer | number of columns of the A matrix |
|
||||
| LDA | integer | specifies the leading dimension of A, must be at least max(1,n) |
|
||||
| LDR | integer | specifies the leading dimension of R, must be at least max(1,n) |
|
||||
|
||||
Output:
|
||||
| R(n,n) | double precision | Rotation matrix |
|
||||
| info | integer | if info = 0, the execution is successful |
|
||||
| | | if info = k, the k-th parameter has an illegal value |
|
||||
| | | if info = -k, the algorithm failed |
|
||||
|
||||
Internal:
|
||||
| B(n,n) | double precision | B = A.A |
|
||||
| work(lwork,n) | double precision | work matrix for dysev, dimension max(1,lwork) |
|
||||
| lwork | integer | dimension of the syev work array >= max(1, 3n-1) |
|
||||
| W(n,n) | double precision | eigenvectors of B |
|
||||
| e_val(n) | double precision | eigenvalues of B |
|
||||
| m_diag(n,n) | double precision | diagonal matrix with the eigenvalues of B |
|
||||
| cos_tau(n,n) | double precision | diagonal matrix with cos(tau) values |
|
||||
| sin_tau(n,n) | double precision | diagonal matrix with sin cos(tau) values |
|
||||
| tau_m1(n,n) | double precision | diagonal matrix with (tau)^-1 values |
|
||||
| part_1(n,n) | double precision | matrix W.cos_tau.W^t |
|
||||
| part_1a(n,n) | double precision | matrix cos_tau.W^t |
|
||||
| part_2(n,n) | double precision | matrix W.tau_m1.sin_tau.W^t.A |
|
||||
| part_2a(n,n) | double precision | matrix W^t.A |
|
||||
| part_2b(n,n) | double precision | matrix sin_tau.W^t.A |
|
||||
| part_2c(n,n) | double precision | matrix tau_m1.sin_tau.W^t.A |
|
||||
| RR_t(n,n) | double precision | R.R^t must be equal to the identity<=> R.R^t-1=0 <=> norm = 0 |
|
||||
| norm | integer | norm of R.R^t-1, must be equal to 0 |
|
||||
| i,j | integer | indexes |
|
||||
|
||||
Functions:
|
||||
| dnrm2 | double precision | Lapack function, compute the norm of a matrix |
|
||||
| disnan | logical | Lapack function, check if an element is NaN |
|
||||
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix.irp.f
|
||||
subroutine rotation_matrix(A,LDA,R,LDR,n,info,enforce_step_cancellation)
|
||||
|
||||
implicit none
|
||||
|
||||
!BEGIN_DOC
|
||||
! Rotation matrix to rotate the molecular orbitals.
|
||||
! If the rotation is too large the transformation is not unitary and must be cancelled.
|
||||
!END_DOC
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer, intent(in) :: n,LDA,LDR
|
||||
double precision, intent(inout) :: A(LDA,n)
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: R(LDR,n)
|
||||
integer, intent(out) :: info
|
||||
logical, intent(out) :: enforce_step_cancellation
|
||||
|
||||
! internal
|
||||
double precision, allocatable :: B(:,:)
|
||||
double precision, allocatable :: work(:,:)
|
||||
double precision, allocatable :: W(:,:), e_val(:)
|
||||
double precision, allocatable :: m_diag(:,:),cos_tau(:,:),sin_tau(:,:),tau_m1(:,:)
|
||||
double precision, allocatable :: part_1(:,:),part_1a(:,:)
|
||||
double precision, allocatable :: part_2(:,:),part_2a(:,:),part_2b(:,:),part_2c(:,:)
|
||||
double precision, allocatable :: RR_t(:,:)
|
||||
integer :: i,j
|
||||
integer :: info2, lwork ! for dsyev
|
||||
double precision :: norm, max_elem, max_elem_A, t1,t2,t3
|
||||
|
||||
! function
|
||||
double precision :: dnrm2
|
||||
logical :: disnan
|
||||
|
||||
print*,''
|
||||
print*,'---rotation_matrix---'
|
||||
|
||||
call wall_time(t1)
|
||||
|
||||
! Allocation
|
||||
allocate(B(n,n))
|
||||
allocate(m_diag(n,n),cos_tau(n,n),sin_tau(n,n),tau_m1(n,n))
|
||||
allocate(W(n,n),e_val(n))
|
||||
allocate(part_1(n,n),part_1a(n,n))
|
||||
allocate(part_2(n,n),part_2a(n,n),part_2b(n,n),part_2c(n,n))
|
||||
allocate(RR_t(n,n))
|
||||
#+END_SRC
|
||||
|
||||
** Pre-conditions
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix.irp.f
|
||||
! Initialization
|
||||
info=0
|
||||
enforce_step_cancellation = .False.
|
||||
|
||||
! Size of matrix A must be at least 1 by 1
|
||||
if (n<1) then
|
||||
info = 3
|
||||
print*, 'WARNING: invalid parameter 5'
|
||||
print*, 'n<1'
|
||||
return
|
||||
endif
|
||||
|
||||
! Leading dimension of A must be >= n
|
||||
if (LDA < n) then
|
||||
info = 25
|
||||
print*, 'WARNING: invalid parameter 2 or 5'
|
||||
print*, 'LDA < n'
|
||||
return
|
||||
endif
|
||||
|
||||
! Leading dimension of A must be >= n
|
||||
if (LDR < n) then
|
||||
info = 4
|
||||
print*, 'WARNING: invalid parameter 4'
|
||||
print*, 'LDR < n'
|
||||
return
|
||||
endif
|
||||
|
||||
! Matrix elements of A must by non-NaN
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (disnan(A(i,j))) then
|
||||
info=1
|
||||
print*, 'WARNING: invalid parameter 1'
|
||||
print*, 'NaN element in A matrix'
|
||||
return
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
do i = 1, n
|
||||
if (A(i,i) /= 0d0) then
|
||||
print*, 'WARNING: matrix A is not antisymmetric'
|
||||
print*, 'Non 0 element on the diagonal', i, A(i,i)
|
||||
call ABORT
|
||||
endif
|
||||
enddo
|
||||
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (A(i,j)+A(j,i)>1d-16) then
|
||||
print*, 'WANRING: matrix A is not antisymmetric'
|
||||
print*, 'A(i,j) /= - A(j,i):', i,j,A(i,j), A(j,i)
|
||||
print*, 'diff:', A(i,j)+A(j,i)
|
||||
call ABORT
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! Fix for too big elements ! bad idea better to cancel if the error is too big
|
||||
!do j = 1, n
|
||||
! do i = 1, n
|
||||
! A(i,j) = mod(A(i,j),2d0*pi)
|
||||
! if (dabs(A(i,j)) > pi) then
|
||||
! A(i,j) = 0d0
|
||||
! endif
|
||||
! enddo
|
||||
!enddo
|
||||
|
||||
max_elem_A = 0d0
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (ABS(A(i,j)) > ABS(max_elem_A)) then
|
||||
max_elem_A = A(i,j)
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
!print*,'max element in A', max_elem_A
|
||||
|
||||
if (ABS(max_elem_A) > 2 * pi) then
|
||||
print*,''
|
||||
print*,'WARNING: ABS(max_elem_A) > 2 pi '
|
||||
print*,''
|
||||
endif
|
||||
|
||||
#+END_SRC
|
||||
|
||||
** Calculations
|
||||
|
||||
*** B=A.A
|
||||
- Calculation of the matrix $\textbf{B} = \textbf{A}^2$
|
||||
- Diagonalization of $\textbf{B}$
|
||||
W, the eigenvectors
|
||||
e_val, the eigenvalues
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix.irp.f
|
||||
! Compute B=A.A
|
||||
|
||||
call dgemm('N','N',n,n,n,1d0,A,size(A,1),A,size(A,1),0d0,B,size(B,1))
|
||||
|
||||
! Copy B in W, diagonalization will put the eigenvectors in W
|
||||
W=B
|
||||
|
||||
! Diagonalization of B
|
||||
! Eigenvalues -> e_val
|
||||
! Eigenvectors -> W
|
||||
lwork = 3*n-1
|
||||
allocate(work(lwork,n))
|
||||
|
||||
!print*,'Starting diagonalization ...'
|
||||
|
||||
call dsyev('V','U',n,W,size(W,1),e_val,work,lwork,info2)
|
||||
|
||||
deallocate(work)
|
||||
|
||||
if (info2 < 0) then
|
||||
print*, 'WARNING: error in the diagonalization'
|
||||
print*, 'Illegal value of the ', info2,'-th parameter'
|
||||
elseif (info2 >0) then
|
||||
print*, "WARNING: Diagonalization failed to converge"
|
||||
endif
|
||||
#+END_SRC
|
||||
|
||||
*** Tau^-1, cos(tau), sin(tau)
|
||||
$$\tau = \sqrt{-x}$$
|
||||
- Calculation of $\cos(\tau)$ $\Leftrightarrow$ $\cos(\sqrt{-x})$
|
||||
- Calculation of $\sin(\tau)$ $\Leftrightarrow$ $\sin(\sqrt{-x})$
|
||||
- Calculation of $\tau^{-1}$ $\Leftrightarrow$ $(\sqrt{-x})^{-1}$
|
||||
These matrices are diagonals
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix.irp.f
|
||||
! Diagonal matrix m_diag
|
||||
do j = 1, n
|
||||
if (e_val(j) >= -1d-12) then !0.d0) then !!! e_avl(i) must be < -1d-12 to avoid numerical problems
|
||||
e_val(j) = 0.d0
|
||||
else
|
||||
e_val(j) = - e_val(j)
|
||||
endif
|
||||
enddo
|
||||
|
||||
m_diag = 0.d0
|
||||
do i = 1, n
|
||||
m_diag(i,i) = e_val(i)
|
||||
enddo
|
||||
|
||||
! cos_tau
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (i==j) then
|
||||
cos_tau(i,j) = dcos(dsqrt(e_val(i)))
|
||||
else
|
||||
cos_tau(i,j) = 0d0
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! sin_tau
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (i==j) then
|
||||
sin_tau(i,j) = dsin(dsqrt(e_val(i)))
|
||||
else
|
||||
sin_tau(i,j) = 0d0
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! Debug, display the cos_tau and sin_tau matrix
|
||||
!if (debug) then
|
||||
! print*, 'cos_tau'
|
||||
! do i = 1, n
|
||||
! print*, cos_tau(i,:)
|
||||
! enddo
|
||||
! print*, 'sin_tau'
|
||||
! do i = 1, n
|
||||
! print*, sin_tau(i,:)
|
||||
! enddo
|
||||
!endif
|
||||
|
||||
! tau^-1
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if ((i==j) .and. (e_val(i) > 1d-16)) then!0d0)) then !!! Convergence problem can come from here if the threshold is too big/small
|
||||
tau_m1(i,j) = 1d0/(dsqrt(e_val(i)))
|
||||
else
|
||||
tau_m1(i,j) = 0d0
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
max_elem = 0d0
|
||||
do i = 1, n
|
||||
if (ABS(tau_m1(i,i)) > ABS(max_elem)) then
|
||||
max_elem = tau_m1(i,i)
|
||||
endif
|
||||
enddo
|
||||
!print*,'max elem tau^-1:', max_elem
|
||||
|
||||
! Debug
|
||||
!print*,'eigenvalues:'
|
||||
!do i = 1, n
|
||||
! print*, e_val(i)
|
||||
!enddo
|
||||
|
||||
!Debug, display tau^-1
|
||||
!if (debug) then
|
||||
! print*, 'tau^-1'
|
||||
! do i = 1, n
|
||||
! print*,tau_m1(i,:)
|
||||
! enddo
|
||||
!endif
|
||||
#+END_SRC
|
||||
|
||||
*** Rotation matrix
|
||||
\begin{align*}
|
||||
\textbf{R} = \textbf{W} \cos(\tau) \textbf{W}^{\dagger} + \textbf{W} \tau^{-1} \sin(\tau) \textbf{W}^{\dagger} \textbf{A}
|
||||
\end{align*}
|
||||
\begin{align*}
|
||||
\textbf{Part1} = \textbf{W} \cos(\tau) \textbf{W}^{\dagger}
|
||||
\end{align*}
|
||||
\begin{align*}
|
||||
\textbf{Part2} = \textbf{W} \tau^{-1} \sin(\tau) \textbf{W}^{\dagger} \textbf{A}
|
||||
\end{align*}
|
||||
|
||||
First:
|
||||
part_1 = dgemm(W, dgemm(cos_tau, W^t))
|
||||
part_1a = dgemm(cos_tau, W^t)
|
||||
part_1 = dgemm(W, part_1a)
|
||||
And:
|
||||
part_2= dgemm(W, dgemm(tau_m1, dgemm(sin_tau, dgemm(W^t, A))))
|
||||
part_2a = dgemm(W^t, A)
|
||||
part_2b = dgemm(sin_tau, part_2a)
|
||||
part_2c = dgemm(tau_m1, part_2b)
|
||||
part_2 = dgemm(W, part_2c)
|
||||
Finally:
|
||||
Rotation matrix, R = part_1+part_2
|
||||
|
||||
If $R$ is a rotation matrix:
|
||||
$R.R^T=R^T.R=\textbf{1}$
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix.irp.f
|
||||
! part_1
|
||||
call dgemm('N','T',n,n,n,1d0,cos_tau,size(cos_tau,1),W,size(W,1),0d0,part_1a,size(part_1a,1))
|
||||
call dgemm('N','N',n,n,n,1d0,W,size(W,1),part_1a,size(part_1a,1),0d0,part_1,size(part_1,1))
|
||||
|
||||
! part_2
|
||||
call dgemm('T','N',n,n,n,1d0,W,size(W,1),A,size(A,1),0d0,part_2a,size(part_2a,1))
|
||||
call dgemm('N','N',n,n,n,1d0,sin_tau,size(sin_tau,1),part_2a,size(part_2a,1),0d0,part_2b,size(part_2b,1))
|
||||
call dgemm('N','N',n,n,n,1d0,tau_m1,size(tau_m1,1),part_2b,size(part_2b,1),0d0,part_2c,size(part_2c,1))
|
||||
call dgemm('N','N',n,n,n,1d0,W,size(W,1),part_2c,size(part_2c,1),0d0,part_2,size(part_2,1))
|
||||
|
||||
! Rotation matrix R
|
||||
R = part_1 + part_2
|
||||
|
||||
! Matrix check
|
||||
! R.R^t and R^t.R must be equal to identity matrix
|
||||
do j = 1, n
|
||||
do i=1,n
|
||||
if (i==j) then
|
||||
RR_t(i,j) = 1d0
|
||||
else
|
||||
RR_t(i,j) = 0d0
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
call dgemm('N','T',n,n,n,1d0,R,size(R,1),R,size(R,1),-1d0,RR_t,size(RR_t,1))
|
||||
|
||||
norm = dnrm2(n*n,RR_t,1)
|
||||
!print*, 'Rotation matrix check, norm R.R^T = ', norm
|
||||
|
||||
! Debug
|
||||
!if (debug) then
|
||||
! print*, 'RR_t'
|
||||
! do i = 1, n
|
||||
! print*, RR_t(i,:)
|
||||
! enddo
|
||||
!endif
|
||||
#+END_SRC
|
||||
|
||||
*** Post conditions
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix.irp.f
|
||||
! Check if R.R^T=1
|
||||
max_elem = 0d0
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (ABS(RR_t(i,j)) > ABS(max_elem)) then
|
||||
max_elem = RR_t(i,j)
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
print*, 'Max error in R.R^T:', max_elem
|
||||
!print*, 'e_val(1):', e_val(1)
|
||||
!print*, 'e_val(n):', e_val(n)
|
||||
!print*, 'max elem in A:', max_elem_A
|
||||
|
||||
if (ABS(max_elem) > 1d-12) then
|
||||
print*, 'WARNING: max error in R.R^T > 1d-12'
|
||||
print*, 'Enforce the step cancellation'
|
||||
enforce_step_cancellation = .True.
|
||||
endif
|
||||
|
||||
! Matrix elements of R must by non-NaN
|
||||
do j = 1,n
|
||||
do i = 1,LDR
|
||||
if (disnan(R(i,j))) then
|
||||
info = 666
|
||||
print*, 'NaN in rotation matrix'
|
||||
call ABORT
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! Display
|
||||
!if (debug) then
|
||||
! print*,'Rotation matrix :'
|
||||
! do i = 1, n
|
||||
! write(*,'(100(F10.5))') R(i,:)
|
||||
! enddo
|
||||
!endif
|
||||
#+END_SRC
|
||||
|
||||
** Deallocation, end
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix.irp.f
|
||||
deallocate(B)
|
||||
deallocate(m_diag,cos_tau,sin_tau,tau_m1)
|
||||
deallocate(W,e_val)
|
||||
deallocate(part_1,part_1a)
|
||||
deallocate(part_2,part_2a,part_2b,part_2c)
|
||||
deallocate(RR_t)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2-t1
|
||||
print*,'Time in rotation matrix:', t3
|
||||
|
||||
print*,'---End rotation_matrix---'
|
||||
|
||||
end subroutine
|
||||
#+END_SRC
|
||||
|
||||
|
|
@ -0,0 +1,136 @@
|
|||
* Rotation matrix with the iterative method
|
||||
|
||||
\begin{align*}
|
||||
\textbf{R} = \sum_{k=0}^{\infty} \frac{1}{k!} \textbf{X}^k
|
||||
\end{align*}
|
||||
|
||||
!!! Doesn't work !!!
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix_iterative.irp.f
|
||||
subroutine rotation_matrix_iterative(m,X,R)
|
||||
|
||||
implicit none
|
||||
|
||||
! in
|
||||
integer, intent(in) :: m
|
||||
double precision, intent(in) :: X(m,m)
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: R(m,m)
|
||||
|
||||
! internal
|
||||
double precision :: max_elem, pre_factor
|
||||
double precision :: t1,t2,t3
|
||||
integer :: k,l,i,j
|
||||
logical :: not_converged
|
||||
double precision, allocatable :: RRT(:,:), A(:,:), B(:,:)
|
||||
|
||||
! Functions
|
||||
integer :: factorial
|
||||
|
||||
print*,'---rotation_matrix_iterative---'
|
||||
call wall_time(t1)
|
||||
|
||||
allocate(RRT(m,m),A(m,m),B(m,m))
|
||||
|
||||
! k = 0
|
||||
R = 0d0
|
||||
do i = 1, m
|
||||
R(i,i) = 1d0
|
||||
enddo
|
||||
|
||||
! k = 1
|
||||
R = R + X
|
||||
|
||||
k = 2
|
||||
|
||||
not_converged = .True.
|
||||
|
||||
do while (not_converged)
|
||||
|
||||
pre_factor = 1d0/DBLE(factorial(k))
|
||||
if (pre_factor < 1d-15) then
|
||||
print*,'pre factor=', pre_factor,'< 1d-15, exit'
|
||||
exit
|
||||
endif
|
||||
|
||||
A = X
|
||||
B = 0d0
|
||||
do l = 1, k-1
|
||||
call dgemm('N','N',m,m,m,1d0,X,size(X,1),A,size(A,1),0d0,B,size(B,1))
|
||||
A = B
|
||||
enddo
|
||||
|
||||
!print*,'B'
|
||||
!do i = 1, m
|
||||
! print*,B(i,:) * 1d0/DBLE(factorial(k))
|
||||
!enddo
|
||||
|
||||
R = R + pre_factor * B
|
||||
|
||||
k = k + 1
|
||||
call dgemm('T','N',m,m,m,1d0,R,size(R,1),R,size(R,1),0d0,RRT,size(RRT,1))
|
||||
|
||||
!print*,'R'
|
||||
!do i = 1, m
|
||||
! write(*,'(10(E12.5))') R(i,:)
|
||||
!enddo
|
||||
|
||||
do i = 1, m
|
||||
RRT(i,i) = RRT(i,i) - 1d0
|
||||
enddo
|
||||
|
||||
!print*,'RRT'
|
||||
!do i = 1, m
|
||||
! write(*,'(10(E12.5))') RRT(i,:)
|
||||
!enddo
|
||||
|
||||
max_elem = 0d0
|
||||
do j = 1, m
|
||||
do i = 1, m
|
||||
if (dabs(RRT(i,j)) > max_elem) then
|
||||
max_elem = dabs(RRT(i,j))
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
print*, 'Iteration:', k
|
||||
print*, 'Max error in R:', max_elem
|
||||
|
||||
if (max_elem < 1d-12) then
|
||||
not_converged = .False.
|
||||
endif
|
||||
|
||||
enddo
|
||||
|
||||
deallocate(RRT,A,B)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2 - t1
|
||||
print*,'Time in rotation matrix iterative:', t3
|
||||
print*,'---End roration_matrix_iterative---'
|
||||
|
||||
|
||||
print*,'Does not work yet, abort'
|
||||
call abort
|
||||
|
||||
end
|
||||
#+END_SRC
|
||||
|
||||
** Factorial
|
||||
#+BEGIN_SRC f90 :comments org :tangle rotation_matrix_iterative.irp.f
|
||||
function factorial(n)
|
||||
|
||||
implicit none
|
||||
|
||||
integer, intent(in) :: n
|
||||
integer :: factorial, k
|
||||
|
||||
factorial = 1
|
||||
|
||||
do k = 1, n
|
||||
factorial = factorial * k
|
||||
enddo
|
||||
|
||||
end
|
||||
#+END_SRC
|
||||
|
|
@ -0,0 +1,65 @@
|
|||
* Rotation matrix in a subspace to rotation matrix in the full space
|
||||
|
||||
Usually, we are using a list of MOs, for exemple the active ones. When
|
||||
we compute a rotation matrix to rotate the MOs, we just compute a
|
||||
rotation matrix for these MOs in order to reduce the size of the
|
||||
matrix which has to be computed. Since the computation of a rotation
|
||||
matrix scale in $O(N^3)$ with $N$ the number of MOs, it's better to
|
||||
reuce the number of MOs involved.
|
||||
After that we replace the rotation matrix in the full space by
|
||||
building the elements of the rotation matrix in the full space from
|
||||
the elements of the rotation matrix in the subspace and adding some 0
|
||||
on the extradiagonal elements and some 1 on the diagonal elements,
|
||||
for the MOs that are not involved in the rotation.
|
||||
|
||||
Provided:
|
||||
| mo_num | integer | Number of MOs |
|
||||
|
||||
Input:
|
||||
| m | integer | Size of tmp_list, m <= mo_num |
|
||||
| tmp_list(m) | integer | List of MOs |
|
||||
| tmp_R(m,m) | double precision | Rotation matrix in the space of |
|
||||
| | | the MOs containing by tmp_list |
|
||||
|
||||
Output:
|
||||
| R(mo_num,mo_num | double precision | Rotation matrix in the space |
|
||||
| | | of all the MOs |
|
||||
|
||||
Internal:
|
||||
| i,j | integer | indexes in the full space |
|
||||
| tmp_i,tmp_j | integer | indexes in the subspace |
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle sub_to_full_rotation_matrix.irp.f
|
||||
subroutine sub_to_full_rotation_matrix(m,tmp_list,tmp_R,R)
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute the full rotation matrix from a smaller one
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! in
|
||||
integer, intent(in) :: m, tmp_list(m)
|
||||
double precision, intent(in) :: tmp_R(m,m)
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: R(mo_num,mo_num)
|
||||
|
||||
! internal
|
||||
integer :: i,j,tmp_i,tmp_j
|
||||
|
||||
! tmp_R to R, subspace to full space
|
||||
R = 0d0
|
||||
do i = 1, mo_num
|
||||
R(i,i) = 1d0 ! 1 on the diagonal because it is a rotation matrix, 1 = nothing change for the corresponding orbital
|
||||
enddo
|
||||
do tmp_j = 1, m
|
||||
j = tmp_list(tmp_j)
|
||||
do tmp_i = 1, m
|
||||
i = tmp_list(tmp_i)
|
||||
R(i,j) = tmp_R(tmp_i,tmp_j)
|
||||
enddo
|
||||
enddo
|
||||
|
||||
end
|
||||
#+END_SRC
|
||||
|
|
@ -0,0 +1,128 @@
|
|||
* Predicted energy : e_model
|
||||
|
||||
*Compute the energy predicted by the Taylor series*
|
||||
|
||||
The energy is predicted using a Taylor expansion truncated at te 2nd
|
||||
order :
|
||||
|
||||
\begin{align*}
|
||||
E_{k+1} = E_{k} + \textbf{g}_k^{T} \cdot \textbf{x}_{k+1} + \frac{1}{2} \cdot \textbf{x}_{k+1}^T \cdot \textbf{H}_{k} \cdot \textbf{x}_{k+1} + \mathcal{O}(\textbf{x}_{k+1}^2)
|
||||
\end{align*}
|
||||
|
||||
Input:
|
||||
| n | integer | m*(m-1)/2 |
|
||||
| n2 | integer | m*(m-1)/2 or 1 if the hessian is diagonal |
|
||||
| v_grad(n) | double precision | gradient |
|
||||
| H(n,n) | double precision | hessian |
|
||||
| x(n) | double precision | Step in the trust region |
|
||||
| prev_energy | double precision | previous energy |
|
||||
|
||||
Output:
|
||||
| e_model | double precision | predicted energy after the rotation of the MOs |
|
||||
|
||||
Internal:
|
||||
| part_1 | double precision | v_grad^T.x |
|
||||
| part_2 | double precision | 1/2 . x^T.H.x |
|
||||
| part_2a | double precision | H.x |
|
||||
| i,j | integer | indexes |
|
||||
|
||||
Function:
|
||||
| ddot | double precision | dot product (Lapack) |
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_expected_e.irp.f
|
||||
subroutine trust_region_expected_e(n,n2,v_grad,H,x,prev_energy,e_model)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute the expected criterion/energy after the application of the step x
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer, intent(in) :: n,n2
|
||||
double precision, intent(in) :: v_grad(n),H(n,n2),x(n)
|
||||
double precision, intent(in) :: prev_energy
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: e_model
|
||||
|
||||
! internal
|
||||
double precision :: part_1, part_2, t1,t2,t3
|
||||
double precision, allocatable :: part_2a(:)
|
||||
|
||||
integer :: i,j
|
||||
|
||||
!Function
|
||||
double precision :: ddot
|
||||
|
||||
print*,''
|
||||
print*,'---Trust_e_model---'
|
||||
|
||||
call wall_time(t1)
|
||||
|
||||
! Allocation
|
||||
allocate(part_2a(n))
|
||||
#+END_SRC
|
||||
|
||||
** Calculations
|
||||
|
||||
part_1 corresponds to the product g.x
|
||||
part_2a corresponds to the product H.x
|
||||
part_2 corresponds to the product 0.5*(x^T.H.x)
|
||||
|
||||
TODO: remove the dot products
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_expected_e.irp.f
|
||||
! Product v_grad.x
|
||||
part_1 = ddot(n,v_grad,1,x,1)
|
||||
|
||||
!if (debug) then
|
||||
! print*,'g.x : ', part_1
|
||||
!endif
|
||||
|
||||
! Product H.x
|
||||
if (n == n2) then
|
||||
call dgemv('N',n,n,1d0,H,size(H,1),x,1,0d0,part_2a,1)
|
||||
else
|
||||
! If the hessian is diagonal
|
||||
do i = 1, n
|
||||
part_2a(i) = H(i,1) * x(i)
|
||||
enddo
|
||||
endif
|
||||
|
||||
! Product 1/2 . x^T.H.x
|
||||
part_2 = 0.5d0 * ddot(n,x,1,part_2a,1)
|
||||
|
||||
!if (debug) then
|
||||
! print*,'1/2*x^T.H.x : ', part_2
|
||||
!endif
|
||||
|
||||
|
||||
! Sum
|
||||
e_model = prev_energy + part_1 + part_2
|
||||
|
||||
! Writing the predicted energy
|
||||
print*, 'prev_energy: ', prev_energy
|
||||
print*, 'Predicted energy after the rotation:', e_model
|
||||
print*, 'Previous energy - predicted energy: ', prev_energy - e_model
|
||||
|
||||
! Can be deleted, already in another subroutine
|
||||
if (DABS(prev_energy - e_model) < 1d-12 ) then
|
||||
print*,'WARNING: ABS(prev_energy - e_model) < 1d-12'
|
||||
endif
|
||||
|
||||
! Deallocation
|
||||
deallocate(part_2a)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2 - t1
|
||||
print*,'Time in trust e model:', t3
|
||||
|
||||
print*,'---End trust_e_model---'
|
||||
|
||||
end subroutine
|
||||
#+END_SRC
|
||||
File diff suppressed because it is too large
Load Diff
|
|
@ -0,0 +1,122 @@
|
|||
* Agreement with the model: Rho
|
||||
|
||||
*Compute the ratio : rho = (prev_energy - energy) / (prev_energy - e_model)*
|
||||
|
||||
Rho represents the agreement between the model (the predicted energy
|
||||
by the Taylor expansion truncated at the 2nd order) and the real
|
||||
energy :
|
||||
|
||||
\begin{equation}
|
||||
\rho^{k+1} = \frac{E^{k} - E^{k+1}}{E^{k} - m^{k+1}}
|
||||
\end{equation}
|
||||
With :
|
||||
$E^{k}$ the energy at the previous iteration
|
||||
$E^{k+1}$ the energy at the actual iteration
|
||||
$m^{k+1}$ the predicted energy for the actual iteration
|
||||
(cf. trust_e_model)
|
||||
|
||||
If $\rho \approx 1$, the agreement is good, contrary to $\rho \approx 0$.
|
||||
If $\rho \leq 0$ the previous energy is lower than the actual
|
||||
energy. We have to cancel the last step and use a smaller trust
|
||||
region.
|
||||
Here we cancel the last step if $\rho < 0.1$, because even if
|
||||
the energy decreases, the agreement is bad, i.e., the Taylor expansion
|
||||
truncated at the second order doesn't represent correctly the energy
|
||||
landscape. So it's better to cancel the step and restart with a
|
||||
smaller trust region.
|
||||
|
||||
Provided in qp_edit:
|
||||
| thresh_rho |
|
||||
|
||||
Input:
|
||||
| prev_energy | double precision | previous energy (energy before the rotation) |
|
||||
| e_model | double precision | predicted energy after the rotation |
|
||||
|
||||
Output:
|
||||
| rho | double precision | the agreement between the model (predicted) and the real energy |
|
||||
| prev_energy | double precision | if rho >= 0.1 the actual energy becomes the previous energy |
|
||||
| | | else the previous energy doesn't change |
|
||||
|
||||
Internal:
|
||||
| energy | double precision | energy (real) after the rotation |
|
||||
| i | integer | index |
|
||||
| t* | double precision | time |
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_rho.irp.f
|
||||
subroutine trust_region_rho(prev_energy, energy,e_model,rho)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute rho, the agreement between the predicted criterion/energy and the real one
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! In
|
||||
double precision, intent(inout) :: prev_energy
|
||||
double precision, intent(in) :: e_model, energy
|
||||
|
||||
! Out
|
||||
double precision, intent(out) :: rho
|
||||
|
||||
! Internal
|
||||
double precision :: t1, t2, t3
|
||||
integer :: i
|
||||
|
||||
print*,''
|
||||
print*,'---Rho_model---'
|
||||
|
||||
!call wall_time(t1)
|
||||
#+END_SRC
|
||||
|
||||
** Rho
|
||||
\begin{equation}
|
||||
\rho^{k+1} = \frac{E^{k} - E^{k+1}}{E^{k} - m^{k+1}}
|
||||
\end{equation}
|
||||
|
||||
In function of $\rho$ th step can be accepted or cancelled.
|
||||
|
||||
If we cancel the last step (k+1), the previous energy (k) doesn't
|
||||
change!
|
||||
If the step (k+1) is accepted, then the "previous energy" becomes E(k+1)
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_rho.irp.f
|
||||
! Already done in an other subroutine
|
||||
!if (ABS(prev_energy - e_model) < 1d-12) then
|
||||
! print*,'WARNING: prev_energy - e_model < 1d-12'
|
||||
! print*,'=> rho will tend toward infinity'
|
||||
! print*,'Check you convergence criterion !'
|
||||
!endif
|
||||
|
||||
rho = (prev_energy - energy) / (prev_energy - e_model)
|
||||
|
||||
!print*, 'previous energy, prev_energy:', prev_energy
|
||||
!print*, 'predicted energy, e_model:', e_model
|
||||
!print*, 'real energy, energy:', energy
|
||||
!print*, 'prev_energy - energy:', prev_energy - energy
|
||||
!print*, 'prev_energy - e_model:', prev_energy - e_model
|
||||
print*, 'Rho:', rho
|
||||
!print*, 'Threshold for rho:', thresh_rho
|
||||
|
||||
! Modification of prev_energy in function of rho
|
||||
if (rho < thresh_rho) then !0.1) then
|
||||
! the step is cancelled
|
||||
print*, 'Rho <', thresh_rho,', the previous energy does not changed'
|
||||
!print*, 'prev_energy :', prev_energy
|
||||
else
|
||||
! the step is accepted
|
||||
prev_energy = energy
|
||||
print*, 'Rho >=', thresh_rho,', energy -> prev_energy:', energy
|
||||
endif
|
||||
|
||||
!call wall_time(t2)
|
||||
!t3 = t2 - t1
|
||||
!print*,'Time in rho model:', t3
|
||||
|
||||
print*,'---End rho_model---'
|
||||
|
||||
end subroutine
|
||||
#+END_SRC
|
||||
|
|
@ -0,0 +1,759 @@
|
|||
* Trust region
|
||||
|
||||
*Compute the next step with the trust region algorithm*
|
||||
|
||||
The Newton method is an iterative method to find a minimum of a given
|
||||
function. It uses a Taylor series truncated at the second order of the
|
||||
targeted function and gives its minimizer. The minimizer is taken as
|
||||
the new position and the same thing is done. And by doing so
|
||||
iteratively the method find a minimum, a local or global one depending
|
||||
of the starting point and the convexity/nonconvexity of the targeted
|
||||
function.
|
||||
|
||||
The goal of the trust region is to constrain the step size of the
|
||||
Newton method in a certain area around the actual position, where the
|
||||
Taylor series is a good approximation of the targeted function. This
|
||||
area is called the "trust region".
|
||||
|
||||
In addition, in function of the agreement between the Taylor
|
||||
development of the energy and the real energy, the size of the trust
|
||||
region will be updated at each iteration. By doing so, the step sizes
|
||||
are not too larges. In addition, since we add a criterion to cancel the
|
||||
step if the energy increases (more precisely if rho < 0.1), so it's
|
||||
impossible to diverge. \newline
|
||||
|
||||
References: \newline
|
||||
Nocedal & Wright, Numerical Optimization, chapter 4 (1999), \newline
|
||||
https://link.springer.com/book/10.1007/978-0-387-40065-5, \newline
|
||||
ISBN: 978-0-387-40065-5 \newline
|
||||
|
||||
By using the first and the second derivatives, the Newton method gives
|
||||
a step:
|
||||
\begin{align*}
|
||||
\textbf{x}_{(k+1)}^{\text{Newton}} = - \textbf{H}_{(k)}^{-1} \cdot
|
||||
\textbf{g}_{(k)}
|
||||
\end{align*}
|
||||
which leads to the minimizer of the Taylor series.
|
||||
!!! Warning: the Newton method gives the minimizer if and only if
|
||||
$\textbf{H}$ is positive definite, else it leads to a saddle point !!!
|
||||
But we want a step $\textbf{x}_{(k+1)}$ with a constraint on its (euclidian) norm:
|
||||
\begin{align*}
|
||||
||\textbf{x}_{(k+1)}|| \leq \Delta_{(k+1)}
|
||||
\end{align*}
|
||||
which is equivalent to
|
||||
\begin{align*}
|
||||
\textbf{x}_{(k+1)}^T \cdot \textbf{x}_{(k+1)} \leq \Delta_{(k+1)}^2
|
||||
\end{align*}
|
||||
|
||||
with: \newline
|
||||
$\textbf{x}_{(k+1)}$ is the step for the k+1-th iteration (vector of
|
||||
size n) \newline
|
||||
$\textbf{H}_{(k)}$ is the hessian at the k-th iteration (n by n
|
||||
matrix) \newline
|
||||
$\textbf{g}_{(k)}$ is the gradient at the k-th iteration (vector of
|
||||
size n) \newline
|
||||
$\Delta_{(k+1)}$ is the trust radius for the (k+1)-th iteration
|
||||
\newline
|
||||
|
||||
Thus we want to constrain the step size $\textbf{x}_{(k+1)}$ into a
|
||||
hypersphere of radius $\Delta_{(k+1)}$.\newline
|
||||
|
||||
So, if $||\textbf{x}_{(k+1)}^{\text{Newton}}|| \leq \Delta_{(k)}$ and
|
||||
$\textbf{H}$ is positive definite, the
|
||||
solution is the step given by the Newton method
|
||||
$\textbf{x}_{(k+1)} = \textbf{x}_{(k+1)}^{\text{Newton}}$.
|
||||
Else we have to constrain the step size. For simplicity we will remove
|
||||
the index $_{(k)}$ and $_{(k+1)}$. To restict the step size, we have
|
||||
to put a constraint on $\textbf{x}$ with a Lagrange multiplier.
|
||||
Starting from the Taylor series of a function E (here, the energy)
|
||||
truncated at the 2nd order, we have:
|
||||
\begin{align*}
|
||||
E(\textbf{x}) = E +\textbf{g}^T \cdot \textbf{x} + \frac{1}{2}
|
||||
\cdot \textbf{x}^T \cdot \textbf{H} \cdot \textbf{x} +
|
||||
\mathcal{O}(\textbf{x}^2)
|
||||
\end{align*}
|
||||
|
||||
With the constraint on the norm of $\textbf{x}$ we can write the
|
||||
Lagrangian
|
||||
\begin{align*}
|
||||
\mathcal{L}(\textbf{x},\lambda) = E + \textbf{g}^T \cdot \textbf{x}
|
||||
+ \frac{1}{2} \cdot \textbf{x}^T \cdot \textbf{H} \cdot \textbf{x}
|
||||
+ \frac{1}{2} \lambda (\textbf{x}^T \cdot \textbf{x} - \Delta^2)
|
||||
\end{align*}
|
||||
Where: \newline
|
||||
$\lambda$ is the Lagrange multiplier \newline
|
||||
$E$ is the energy at the k-th iteration $\Leftrightarrow
|
||||
E(\textbf{x} = \textbf{0})$ \newline
|
||||
|
||||
To solve this equation, we search a stationary point where the first
|
||||
derivative of $\mathcal{L}$ with respect to $\textbf{x}$ becomes 0, i.e.
|
||||
\begin{align*}
|
||||
\frac{\partial \mathcal{L}(\textbf{x},\lambda)}{\partial \textbf{x}}=0
|
||||
\end{align*}
|
||||
|
||||
The derivative is:
|
||||
\begin{align*}
|
||||
\frac{\partial \mathcal{L}(\textbf{x},\lambda)}{\partial \textbf{x}}
|
||||
= \textbf{g} + \textbf{H} \cdot \textbf{x} + \lambda \cdot \textbf{x}
|
||||
\end{align*}
|
||||
|
||||
So, we search $\textbf{x}$ such as:
|
||||
\begin{align*}
|
||||
\frac{\partial \mathcal{L}(\textbf{x},\lambda)}{\partial \textbf{x}}
|
||||
= \textbf{g} + \textbf{H} \cdot \textbf{x} + \lambda \cdot \textbf{x} = 0
|
||||
\end{align*}
|
||||
|
||||
We can rewrite that as:
|
||||
\begin{align*}
|
||||
\textbf{g} + \textbf{H} \cdot \textbf{x} + \lambda \cdot \textbf{x}
|
||||
= \textbf{g} + (\textbf{H} +\textbf{I} \lambda) \cdot \textbf{x} = 0
|
||||
\end{align*}
|
||||
with $\textbf{I}$ is the identity matrix.
|
||||
|
||||
By doing so, the solution is:
|
||||
\begin{align*}
|
||||
(\textbf{H} +\textbf{I} \lambda) \cdot \textbf{x}= -\textbf{g}
|
||||
\end{align*}
|
||||
\begin{align*}
|
||||
\textbf{x}= - (\textbf{H} + \textbf{I} \lambda)^{-1} \cdot \textbf{g}
|
||||
\end{align*}
|
||||
with $\textbf{x}^T \textbf{x} = \Delta^2$.
|
||||
|
||||
We have to solve this previous equation to find this $\textbf{x}$ in the
|
||||
trust region, i.e. $||\textbf{x}|| = \Delta$. Now, this problem is
|
||||
just a one dimension problem because we can express $\textbf{x}$ as a
|
||||
function of $\lambda$:
|
||||
\begin{align*}
|
||||
\textbf{x}(\lambda) = - (\textbf{H} + \textbf{I} \lambda)^{-1} \cdot \textbf{g}
|
||||
\end{align*}
|
||||
|
||||
We start from the fact that the hessian is diagonalizable. So we have:
|
||||
\begin{align*}
|
||||
\textbf{H} = \textbf{W} \cdot \textbf{h} \cdot \textbf{W}^T
|
||||
\end{align*}
|
||||
with: \newline
|
||||
$\textbf{H}$, the hessian matrix \newline
|
||||
$\textbf{W}$, the matrix containing the eigenvectors \newline
|
||||
$\textbf{w}_i$, the i-th eigenvector, i.e. i-th column of $\textbf{W}$ \newline
|
||||
$\textbf{h}$, the matrix containing the eigenvalues in ascending order \newline
|
||||
$h_i$, the i-th eigenvalue in ascending order \newline
|
||||
|
||||
Now we use the fact that adding a constant on the diagonal just shifts
|
||||
the eigenvalues:
|
||||
\begin{align*}
|
||||
\textbf{H} + \textbf{I} \lambda = \textbf{W} \cdot (\textbf{h}
|
||||
+\textbf{I} \lambda) \cdot \textbf{W}^T
|
||||
\end{align*}
|
||||
|
||||
By doing so we can express $\textbf{x}$ as a function of $\lambda$
|
||||
\begin{align*}
|
||||
\textbf{x}(\lambda) = - \sum_{i=1}^n \frac{\textbf{w}_i^T \cdot
|
||||
\textbf{g}}{h_i + \lambda} \cdot \textbf{w}_i
|
||||
\end{align*}
|
||||
with $\lambda \neq - h_i$.
|
||||
|
||||
An interesting thing in our case is the norm of $\textbf{x}$,
|
||||
because we want $||\textbf{x}|| = \Delta$. Due to the orthogonality of
|
||||
the eigenvectors $\left\{\textbf{w} \right\} _{i=1}^n$ we have:
|
||||
\begin{align*}
|
||||
||\textbf{x}(\lambda)||^2 = \sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot
|
||||
\textbf{g})^2}{(h_i + \lambda)^2}
|
||||
\end{align*}
|
||||
|
||||
So the $||\textbf{x}(\lambda)||^2$ is just a function of $\lambda$.
|
||||
And if we study the properties of this function we see that:
|
||||
\begin{align*}
|
||||
\lim_{\lambda\to\infty} ||\textbf{x}(\lambda)|| = 0
|
||||
\end{align*}
|
||||
and if $\textbf{w}_i^T \cdot \textbf{g} \neq 0$:
|
||||
\begin{align*}
|
||||
\lim_{\lambda\to -h_i} ||\textbf{x}(\lambda)|| = + \infty
|
||||
\end{align*}
|
||||
|
||||
From these limits and knowing that $h_1$ is the lowest eigenvalue, we
|
||||
can conclude that $||\textbf{x}(\lambda)||$ is a continuous and
|
||||
strictly decreasing function on the interval $\lambda \in
|
||||
(-h_1;\infty)$. Thus, there is one $\lambda$ in this interval which
|
||||
gives $||\textbf{x}(\lambda)|| = \Delta$, consequently there is one
|
||||
solution.
|
||||
|
||||
Since $\textbf{x} = - (\textbf{H} + \lambda \textbf{I})^{-1} \cdot
|
||||
\textbf{g}$ and we want to reduce the norm of $\textbf{x}$, clearly,
|
||||
$\lambda > 0$ ($\lambda = 0$ is the unconstraint solution). But the
|
||||
Newton method is only defined for a positive definite hessian matrix,
|
||||
so $(\textbf{H} + \textbf{I} \lambda)$ must be positive
|
||||
definite. Consequently, in the case where $\textbf{H}$ is not positive
|
||||
definite, to ensure the positive definiteness, $\lambda$ must be
|
||||
greater than $- h_1$.
|
||||
\begin{align*}
|
||||
\lambda > 0 \quad \text{and} \quad \lambda \geq - h_1
|
||||
\end{align*}
|
||||
|
||||
From that there are five cases:
|
||||
- if $\textbf{H}$ is positive definite, $-h_1 < 0$, $\lambda \in (0,\infty)$
|
||||
- if $\textbf{H}$ is not positive definite and $\textbf{w}_1^T \cdot
|
||||
\textbf{g} \neq 0$, $(\textbf{H} + \textbf{I}
|
||||
\lambda)$
|
||||
must be positve definite, $-h_1 > 0$, $\lambda \in (-h_1, \infty)$
|
||||
- if $\textbf{H}$ is not positive definite , $\textbf{w}_1^T \cdot
|
||||
\textbf{g} = 0$ and $||\textbf{x}(-h_1)|| > \Delta$ by removing
|
||||
$j=1$ in the sum, $(\textbf{H} + \textbf{I} \lambda)$ must be
|
||||
positive definite, $-h_1 > 0$, $\lambda \in (-h_1, \infty$)
|
||||
- if $\textbf{H}$ is not positive definite , $\textbf{w}_1^T \cdot
|
||||
\textbf{g} = 0$ and $||\textbf{x}(-h_1)|| \leq \Delta$ by removing
|
||||
$j=1$ in the sum, $(\textbf{H} + \textbf{I} \lambda)$ must be
|
||||
positive definite, $-h_1 > 0$, $\lambda = -h_1$). This case is
|
||||
similar to the case where $\textbf{H}$ and $||\textbf{x}(\lambda =
|
||||
0)|| \leq \Delta$
|
||||
but we can also add to $\textbf{x}$, the first eigenvector $\textbf{W}_1$
|
||||
time a constant to ensure the condition $||\textbf{x}(\lambda =
|
||||
-h_1)|| = \Delta$ and escape from the saddle point
|
||||
|
||||
Thus to find the solution, we can write:
|
||||
\begin{align*}
|
||||
||\textbf{x}(\lambda)|| = \Delta
|
||||
\end{align*}
|
||||
\begin{align*}
|
||||
||\textbf{x}(\lambda)|| - \Delta = 0
|
||||
\end{align*}
|
||||
|
||||
Taking the square of this equation
|
||||
\begin{align*}
|
||||
(||\textbf{x}(\lambda)|| - \Delta)^2 = 0
|
||||
\end{align*}
|
||||
we have a function with one minimum for the optimal $\lambda$.
|
||||
Since we have the formula of $||\textbf{x}(\lambda)||^2$, we solve
|
||||
\begin{align*}
|
||||
(||\textbf{x}(\lambda)||^2 - \Delta^2)^2 = 0
|
||||
\end{align*}
|
||||
|
||||
But in practice, it is more effective to solve:
|
||||
\begin{align*}
|
||||
(\frac{1}{||\textbf{x}(\lambda)||^2} - \frac{1}{\Delta^2})^2 = 0
|
||||
\end{align*}
|
||||
|
||||
To do that, we just use the Newton method with "trust_newton" using
|
||||
first and second derivative of $(||\textbf{x}(\lambda)||^2 -
|
||||
\Delta^2)^2$ with respect to $\textbf{x}$.
|
||||
This will give the optimal $\lambda$ to compute the
|
||||
solution $\textbf{x}$ with the formula seen previously:
|
||||
\begin{align*}
|
||||
\textbf{x}(\lambda) = - \sum_{i=1}^n \frac{\textbf{w}_i^T \cdot
|
||||
\textbf{g}}{h_i + \lambda} \cdot \textbf{w}_i
|
||||
\end{align*}
|
||||
|
||||
The solution $\textbf{x}(\lambda)$ with the optimal $\lambda$ is our
|
||||
step to go from the (k)-th to the (k+1)-th iteration, is noted $\textbf{x}^*$.
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
#+END_SRC
|
||||
|
||||
** Evolution of the trust region
|
||||
|
||||
We initialize the trust region at the first iteration using a radius
|
||||
\begin{align*}
|
||||
\Delta = ||\textbf{x}(\lambda=0)||
|
||||
\end{align*}
|
||||
|
||||
And for the next iteration the trust region will evolves depending of
|
||||
the agreement of the energy prediction based on the Taylor series
|
||||
truncated at the 2nd order and the real energy. If the Taylor series
|
||||
truncated at the 2nd order represents correctly the energy landscape
|
||||
the trust region will be extent else it will be reduced. In order to
|
||||
mesure this agreement we use the ratio rho cf. "rho_model" and
|
||||
"trust_e_model". From that we use the following values:
|
||||
- if $\rho \geq 0.75$, then $\Delta = 2 \Delta$,
|
||||
- if $0.5 \geq \rho < 0.75$, then $\Delta = \Delta$,
|
||||
- if $0.25 \geq \rho < 0.5$, then $\Delta = 0.5 \Delta$,
|
||||
- if $\rho < 0.25$, then $\Delta = 0.25 \Delta$.
|
||||
|
||||
In addition, if $\rho < 0.1$ the iteration is cancelled, so it
|
||||
restarts with a smaller trust region until the energy decreases.
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
#+END_SRC
|
||||
|
||||
** Summary
|
||||
|
||||
To summarize, knowing the hessian (eigenvectors and eigenvalues), the
|
||||
gradient and the radius of the trust region we can compute the norm of
|
||||
the Newton step
|
||||
\begin{align*}
|
||||
||\textbf{x}(\lambda = 0)||^2 = ||- \textbf{H}^{-1} \cdot \textbf{g}||^2 = \sum_{i=1}^n
|
||||
\frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2}, \quad h_i \neq 0
|
||||
\end{align*}
|
||||
|
||||
- if $h_1 \geq 0$, $||\textbf{x}(\lambda = 0)|| \leq \Delta$ and
|
||||
$\textbf{x}(\lambda=0)$ is in the trust region and it is not
|
||||
necessary to put a constraint on $\textbf{x}$, the solution is the
|
||||
unconstrained one, $\textbf{x}^* = \textbf{x}(\lambda = 0)$.
|
||||
- else if $h_1 < 0$, $\textbf{w}_1^T \cdot \textbf{g} = 0$ and
|
||||
$||\textbf{x}(\lambda = -h_1)|| \leq \Delta$ (by removing $j=1$ in
|
||||
the sum), the solution is $\textbf{x}^* = \textbf{x}(\lambda =
|
||||
-h_1)$, similarly to the previous case.
|
||||
But we can add to $\textbf{x}$, the first eigenvector $\textbf{W}_1$
|
||||
time a constant to ensure the condition $||\textbf{x}(\lambda =
|
||||
-h_1)|| = \Delta$ and escape from the saddle point
|
||||
- else if $h_1 < 0$ and $\textbf{w}_1^T \cdot \textbf{g} \neq 0$ we
|
||||
have to search $\lambda \in (-h_1, \infty)$ such as
|
||||
$\textbf{x}(\lambda) = \Delta$ by solving with the Newton method
|
||||
\begin{align*}
|
||||
(||\textbf{x}(\lambda)||^2 - \Delta^2)^2 = 0
|
||||
\end{align*}
|
||||
or
|
||||
\begin{align*}
|
||||
(\frac{1}{||\textbf{x}(\lambda)||^2} - \frac{1}{\Delta^2})^2 = 0
|
||||
\end{align*}
|
||||
which is numerically more stable. And finally compute
|
||||
\begin{align*}
|
||||
\textbf{x}^* = \textbf{x}(\lambda) = - \sum_{i=1}^n \frac{\textbf{w}_i^T \cdot
|
||||
\textbf{g}}{h_i + \lambda} \cdot \textbf{w}_i
|
||||
\end{align*}
|
||||
- else if $h_1 \geq 0$ and $||\textbf{x}(\lambda = 0)|| > \Delta$ we
|
||||
do exactly the same thing that the previous case but we search
|
||||
$\lambda \in (0, \infty)$
|
||||
- else if $h_1 < 0$ and $\textbf{w}_1^T \cdot \textbf{g} = 0$ and
|
||||
$||\textbf{x}(\lambda = -h_1)|| > \Delta$ (by removing $j=1$ in the
|
||||
sum), again we do exactly the same thing that the previous case
|
||||
searching $\lambda \in (-h_1, \infty)$.
|
||||
|
||||
|
||||
For the cases where $\textbf{w}_1^T \cdot \textbf{g} = 0$ it is not
|
||||
necessary in fact to remove the $j = 1$ in the sum since the term
|
||||
where $h_i - \lambda < 10^{-6}$ are not computed.
|
||||
|
||||
After that, we take this vector $\textbf{x}^*$, called "x", and we do
|
||||
the transformation to an antisymmetric matrix $\textbf{X}$, called
|
||||
m_x. This matrix $\textbf{X}$ will be used to compute a rotation
|
||||
matrix $\textbf{R}= \exp(\textbf{X})$ in "rotation_matrix".
|
||||
|
||||
NB:
|
||||
An improvement can be done using a elleptical trust region.
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
#+END_SRC
|
||||
|
||||
** Code
|
||||
|
||||
Provided:
|
||||
| mo_num | integer | number of MOs |
|
||||
|
||||
Cf. qp_edit in orbital optimization section, for some constants/thresholds
|
||||
|
||||
Input:
|
||||
| m | integer | number of MOs |
|
||||
| n | integer | m*(m-1)/2 |
|
||||
| n2 | integer | m*(m-1)/2 or 1 if the hessian is diagonal |
|
||||
| H(n,n2) | double precision | hessian |
|
||||
| v_grad(n) | double precision | gradient |
|
||||
| e_val(n) | double precision | eigenvalues of the hessian |
|
||||
| W(n, n) | double precision | eigenvectors of the hessian |
|
||||
| rho | double precision | agreement between the model and the reality, |
|
||||
| | | represents the quality of the energy prediction |
|
||||
| nb_iter | integer | number of iteration |
|
||||
|
||||
Input/Ouput:
|
||||
| delta | double precision | radius of the trust region |
|
||||
|
||||
Output:
|
||||
| x(n) | double precision | vector containing the step |
|
||||
|
||||
Internal:
|
||||
| accu | double precision | temporary variable to compute the step |
|
||||
| lambda | double precision | lagrange multiplier |
|
||||
| trust_radius2 | double precision | square of the radius of the trust region |
|
||||
| norm2_x | double precision | norm^2 of the vector x |
|
||||
| norm2_g | double precision | norm^2 of the vector containing the gradient |
|
||||
| tmp_wtg(n) | double precision | tmp_wtg(i) = w_i^T . g |
|
||||
| i, j, k | integer | indexes |
|
||||
|
||||
Function:
|
||||
| dnrm2 | double precision | Blas function computing the norm |
|
||||
| f_norm_trust_region_omp | double precision | compute the value of norm(x(lambda)^2) |
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
subroutine trust_region_step(n,n2,nb_iter,v_grad,rho,e_val,w,x,delta)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compuet the step in the trust region
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer, intent(in) :: n,n2
|
||||
double precision, intent(in) :: v_grad(n), rho
|
||||
integer, intent(inout) :: nb_iter
|
||||
double precision, intent(in) :: e_val(n), w(n,n2)
|
||||
|
||||
! inout
|
||||
double precision, intent(inout) :: delta
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: x(n)
|
||||
|
||||
! Internal
|
||||
double precision :: accu, lambda, trust_radius2
|
||||
double precision :: norm2_x, norm2_g
|
||||
double precision, allocatable :: tmp_wtg(:)
|
||||
integer :: i,j,k
|
||||
double precision :: t1,t2,t3
|
||||
integer :: n_neg_eval
|
||||
|
||||
|
||||
! Functions
|
||||
double precision :: ddot, dnrm2
|
||||
double precision :: f_norm_trust_region_omp
|
||||
|
||||
print*,''
|
||||
print*,'=================='
|
||||
print*,'---Trust_region---'
|
||||
print*,'=================='
|
||||
|
||||
call wall_time(t1)
|
||||
|
||||
! Allocation
|
||||
allocate(tmp_wtg(n))
|
||||
#+END_SRC
|
||||
|
||||
|
||||
*** Initialization and norm
|
||||
|
||||
The norm of the step size will be useful for the trust region
|
||||
algorithm. We start from a first guess and the radius of the trust
|
||||
region will evolve during the optimization.
|
||||
|
||||
avoid_saddle is actually a test to avoid saddle points
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
! Initialization of the Lagrange multiplier
|
||||
lambda = 0d0
|
||||
|
||||
! List of w^T.g, to avoid the recomputation
|
||||
tmp_wtg = 0d0
|
||||
if (n == n2) then
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
tmp_wtg(j) = tmp_wtg(j) + w(i,j) * v_grad(i)
|
||||
enddo
|
||||
enddo
|
||||
else
|
||||
! For the diagonal case
|
||||
do j = 1, n
|
||||
k = int(w(j,1)+1d-15)
|
||||
tmp_wtg(j) = v_grad(k)
|
||||
enddo
|
||||
endif
|
||||
|
||||
! Replacement of the small tmp_wtg corresponding to a negative eigenvalue
|
||||
! in the case of avoid_saddle
|
||||
if (avoid_saddle .and. e_val(1) < - thresh_eig) then
|
||||
i = 2
|
||||
! Number of negative eigenvalues
|
||||
do while (e_val(i) < - thresh_eig)
|
||||
if (tmp_wtg(i) < thresh_wtg2) then
|
||||
if (version_avoid_saddle == 1) then
|
||||
tmp_wtg(i) = 1d0
|
||||
elseif (version_avoid_saddle == 2) then
|
||||
tmp_wtg(i) = DABS(e_val(i))
|
||||
elseif (version_avoid_saddle == 3) then
|
||||
tmp_wtg(i) = dsqrt(DABS(e_val(i)))
|
||||
else
|
||||
tmp_wtg(i) = thresh_wtg2
|
||||
endif
|
||||
endif
|
||||
i = i + 1
|
||||
enddo
|
||||
|
||||
! For the fist one it's a little bit different
|
||||
if (tmp_wtg(1) < thresh_wtg2) then
|
||||
tmp_wtg(1) = 0d0
|
||||
endif
|
||||
|
||||
endif
|
||||
|
||||
! Norm^2 of x, ||x||^2
|
||||
norm2_x = f_norm_trust_region_omp(n,e_val,tmp_wtg,0d0)
|
||||
! We just use this norm for the nb_iter = 0 in order to initialize the trust radius delta
|
||||
! We don't care about the sign of the eigenvalue we just want the size of the step in a normal Newton-Raphson algorithm
|
||||
! Anyway if the step is too big it will be reduced
|
||||
!print*,'||x||^2 :', norm2_x
|
||||
|
||||
! Norm^2 of the gradient, ||v_grad||^2
|
||||
norm2_g = (dnrm2(n,v_grad,1))**2
|
||||
!print*,'||grad||^2 :', norm2_g
|
||||
#+END_SRC
|
||||
|
||||
*** Trust radius initialization
|
||||
|
||||
At the first iteration (nb_iter = 0) we initialize the trust region
|
||||
with the norm of the step generate by the Newton's method ($\textbf{x}_1 =
|
||||
(\textbf{H}_0)^{-1} \cdot \textbf{g}_0$,
|
||||
we compute this norm using f_norm_trust_region_omp as explain just
|
||||
below)
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
! trust radius
|
||||
if (nb_iter == 0) then
|
||||
trust_radius2 = norm2_x
|
||||
! To avoid infinite loop of cancellation of this first step
|
||||
! without changing delta
|
||||
nb_iter = 1
|
||||
|
||||
! Compute delta, delta = sqrt(trust_radius)
|
||||
delta = dsqrt(trust_radius2)
|
||||
endif
|
||||
#+END_SRC
|
||||
|
||||
*** Modification of the trust radius
|
||||
|
||||
In function of rho (which represents the agreement between the model
|
||||
and the reality, cf. rho_model) the trust region evolves. We update
|
||||
delta (the radius of the trust region).
|
||||
|
||||
To avoid too big trust region we put a maximum size.
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
! Modification of the trust radius in function of rho
|
||||
if (rho >= 0.75d0) then
|
||||
delta = 2d0 * delta
|
||||
elseif (rho >= 0.5d0) then
|
||||
delta = delta
|
||||
elseif (rho >= 0.25d0) then
|
||||
delta = 0.5d0 * delta
|
||||
else
|
||||
delta = 0.25d0 * delta
|
||||
endif
|
||||
|
||||
! Maximum size of the trust region
|
||||
!if (delta > 0.5d0 * n * pi) then
|
||||
! delta = 0.5d0 * n * pi
|
||||
! print*,'Delta > delta_max, delta = 0.5d0 * n * pi'
|
||||
!endif
|
||||
|
||||
if (delta > 1d10) then
|
||||
delta = 1d10
|
||||
endif
|
||||
|
||||
!print*, 'Delta :', delta
|
||||
#+END_SRC
|
||||
|
||||
*** Calculation of the optimal lambda
|
||||
|
||||
We search the solution of $(||x||^2 - \Delta^2)^2 = 0$
|
||||
- If $||\textbf{x}|| > \Delta$ or $h_1 < 0$ we have to add a constant
|
||||
$\lambda > 0 \quad \text{and} \quad \lambda > -h_1$
|
||||
- If $||\textbf{x}|| \leq \Delta$ and $h_1 \geq 0$ the solution is the
|
||||
unconstrained one, $\lambda = 0$
|
||||
|
||||
You will find more details at the beginning
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
! By giving delta, we search (||x||^2 - delta^2)^2 = 0
|
||||
! and not (||x||^2 - delta)^2 = 0
|
||||
|
||||
! Research of lambda to solve ||x(lambda)|| = Delta
|
||||
|
||||
! Display
|
||||
!print*, 'e_val(1) = ', e_val(1)
|
||||
!print*, 'w_1^T.g =', tmp_wtg(1)
|
||||
|
||||
! H positive definite
|
||||
if (e_val(1) > - thresh_eig) then
|
||||
norm2_x = f_norm_trust_region_omp(n,e_val,tmp_wtg,0d0)
|
||||
!print*, '||x(0)||=', dsqrt(norm2_x)
|
||||
!print*, 'Delta=', delta
|
||||
|
||||
! H positive definite, ||x(lambda = 0)|| <= Delta
|
||||
if (dsqrt(norm2_x) <= delta) then
|
||||
!print*, 'H positive definite, ||x(lambda = 0)|| <= Delta'
|
||||
!print*, 'lambda = 0, no lambda optimization'
|
||||
lambda = 0d0
|
||||
|
||||
! H positive definite, ||x(lambda = 0)|| > Delta
|
||||
else
|
||||
! Constraint solution
|
||||
!print*, 'H positive definite, ||x(lambda = 0)|| > Delta'
|
||||
!print*,'Computation of the optimal lambda...'
|
||||
call trust_region_optimal_lambda(n,e_val,tmp_wtg,delta,lambda)
|
||||
endif
|
||||
|
||||
! H indefinite
|
||||
else
|
||||
if (DABS(tmp_wtg(1)) < thresh_wtg) then
|
||||
norm2_x = f_norm_trust_region_omp(n,e_val,tmp_wtg, - e_val(1))
|
||||
!print*, 'w_1^T.g <', thresh_wtg,', ||x(lambda = -e_val(1))|| =', dsqrt(norm2_x)
|
||||
endif
|
||||
|
||||
! H indefinite, w_1^T.g = 0, ||x(lambda = -e_val(1))|| <= Delta
|
||||
if (dsqrt(norm2_x) <= delta .and. DABS(tmp_wtg(1)) < thresh_wtg) then
|
||||
! Add e_val(1) in order to have (H - e_val(1) I) positive definite
|
||||
!print*, 'H indefinite, w_1^T.g = 0, ||x(lambda = -e_val(1))|| <= Delta'
|
||||
!print*, 'lambda = -e_val(1), no lambda optimization'
|
||||
lambda = - e_val(1)
|
||||
|
||||
! H indefinite, w_1^T.g = 0, ||x(lambda = -e_val(1))|| > Delta
|
||||
! and
|
||||
! H indefinite, w_1^T.g =/= 0
|
||||
else
|
||||
! Constraint solution/ add lambda
|
||||
!if (DABS(tmp_wtg(1)) < thresh_wtg) then
|
||||
! print*, 'H indefinite, w_1^T.g = 0, ||x(lambda = -e_val(1))|| > Delta'
|
||||
!else
|
||||
! print*, 'H indefinite, w_1^T.g =/= 0'
|
||||
!endif
|
||||
!print*, 'Computation of the optimal lambda...'
|
||||
call trust_region_optimal_lambda(n,e_val,tmp_wtg,delta,lambda)
|
||||
endif
|
||||
|
||||
endif
|
||||
|
||||
! Recomputation of the norm^2 of the step x
|
||||
norm2_x = f_norm_trust_region_omp(n,e_val,tmp_wtg,lambda)
|
||||
print*,''
|
||||
print*,'Summary after the trust region:'
|
||||
print*,'lambda:', lambda
|
||||
print*,'||x||:', dsqrt(norm2_x)
|
||||
print*,'delta:', delta
|
||||
#+END_SRC
|
||||
|
||||
*** Calculation of the step x
|
||||
|
||||
x refers to $\textbf{x}^*$
|
||||
We compute x in function of lambda using its formula :
|
||||
\begin{align*}
|
||||
\textbf{x}^* = \textbf{x}(\lambda) = - \sum_{i=1}^n \frac{\textbf{w}_i^T \cdot \textbf{g}}{h_i
|
||||
+ \lambda} \cdot \textbf{w}_i
|
||||
\end{align*}
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
! Initialisation
|
||||
x = 0d0
|
||||
|
||||
! Calculation of the step x
|
||||
|
||||
if (n == n2) then
|
||||
! Normal version
|
||||
if (.not. absolute_eig) then
|
||||
|
||||
do i = 1, n
|
||||
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
|
||||
do j = 1, n
|
||||
x(j) = x(j) - tmp_wtg(i) * W(j,i) / (e_val(i) + lambda)
|
||||
enddo
|
||||
endif
|
||||
enddo
|
||||
|
||||
! Version to use the absolute value of the eigenvalues
|
||||
else
|
||||
|
||||
do i = 1, n
|
||||
if (DABS(e_val(i)) > thresh_eig) then
|
||||
do j = 1, n
|
||||
x(j) = x(j) - tmp_wtg(i) * W(j,i) / (DABS(e_val(i)) + lambda)
|
||||
enddo
|
||||
endif
|
||||
enddo
|
||||
|
||||
endif
|
||||
else
|
||||
! If the hessian is diagonal
|
||||
! Normal version
|
||||
if (.not. absolute_eig) then
|
||||
|
||||
do i = 1, n
|
||||
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
|
||||
j = int(w(i,1) + 1d-15)
|
||||
x(j) = - tmp_wtg(i) * 1d0 / (e_val(i) + lambda)
|
||||
endif
|
||||
enddo
|
||||
|
||||
! Version to use the absolute value of the eigenvalues
|
||||
else
|
||||
|
||||
do i = 1, n
|
||||
if (DABS(e_val(i)) > thresh_eig) then
|
||||
j = int(w(i,1) + 1d-15)
|
||||
x(j) = - tmp_wtg(i) * 1d0 / (DABS(e_val(i)) + lambda)
|
||||
endif
|
||||
enddo
|
||||
|
||||
endif
|
||||
endif
|
||||
|
||||
double precision :: beta, norm_x
|
||||
|
||||
! Test
|
||||
! If w_1^T.g = 0, the lim of ||x(lambda)|| when lambda tend to -e_val(1)
|
||||
! is not + infinity. So ||x(lambda=-e_val(1))|| < delta, we add the first
|
||||
! eigenvectors multiply by a constant to ensure the condition
|
||||
! ||x(lambda=-e_val(1))|| = delta and escape the saddle point
|
||||
if (avoid_saddle .and. e_val(1) < - thresh_eig) then
|
||||
if (tmp_wtg(1) < 1d-15 .and. (1d0 - dsqrt(norm2_x)/delta) > 1d-3 ) then
|
||||
|
||||
! norm of x
|
||||
norm_x = dnrm2(n,x,1)
|
||||
|
||||
! Computes the coefficient for the w_1
|
||||
beta = delta**2 - norm_x**2
|
||||
|
||||
! Updates the step x
|
||||
x = x + W(:,1) * dsqrt(beta)
|
||||
|
||||
! Recomputes the norm to check
|
||||
norm_x = dnrm2(n,x,1)
|
||||
|
||||
print*, 'Add w_1 * dsqrt(delta^2 - ||x||^2):'
|
||||
print*, '||x||', norm_x
|
||||
endif
|
||||
endif
|
||||
#+END_SRC
|
||||
|
||||
*** Transformation of x
|
||||
|
||||
x is a vector of size n, so it can be write as a m by m
|
||||
antisymmetric matrix m_x cf. "mat_to_vec_index" and "vec_to_mat_index".
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
! ! Step transformation vector -> matrix
|
||||
! ! Vector with n element -> mo_num by mo_num matrix
|
||||
! do j = 1, m
|
||||
! do i = 1, m
|
||||
! if (i>j) then
|
||||
! call mat_to_vec_index(i,j,k)
|
||||
! m_x(i,j) = x(k)
|
||||
! else
|
||||
! m_x(i,j) = 0d0
|
||||
! endif
|
||||
! enddo
|
||||
! enddo
|
||||
!
|
||||
! ! Antisymmetrization of the previous matrix
|
||||
! do j = 1, m
|
||||
! do i = 1, m
|
||||
! if (i<j) then
|
||||
! m_x(i,j) = - m_x(j,i)
|
||||
! endif
|
||||
! enddo
|
||||
! enddo
|
||||
#+END_SRC
|
||||
|
||||
*** Deallocation, end
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle trust_region_step.irp.f
|
||||
deallocate(tmp_wtg)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2 - t1
|
||||
print*,'Time in trust_region:', t3
|
||||
print*,'======================'
|
||||
print*,'---End trust_region---'
|
||||
print*,'======================'
|
||||
|
||||
end
|
||||
#+END_SRC
|
||||
|
||||
|
|
@ -0,0 +1,72 @@
|
|||
* Vector to matrix indexes
|
||||
|
||||
*Compute the indexes p,q of a matrix element with the vector index i*
|
||||
|
||||
Vector (i) -> lower diagonal matrix (p,q), p > q
|
||||
|
||||
If a matrix is antisymmetric it can be reshaped as a vector. And the
|
||||
vector can be reshaped as an antisymmetric matrix
|
||||
|
||||
\begin{align*}
|
||||
\begin{pmatrix}
|
||||
0 & -1 & -2 & -4 \\
|
||||
1 & 0 & -3 & -5 \\
|
||||
2 & 3 & 0 & -6 \\
|
||||
4 & 5 & 6 & 0
|
||||
\end{pmatrix}
|
||||
\Leftrightarrow
|
||||
\begin{pmatrix}
|
||||
1 & 2 & 3 & 4 & 5 & 6
|
||||
\end{pmatrix}
|
||||
\end{align*}
|
||||
|
||||
!!! Here the algorithm only work for the lower diagonal !!!
|
||||
|
||||
Input:
|
||||
| i | integer | index in the vector |
|
||||
|
||||
Ouput:
|
||||
| p,q | integer | corresponding indexes in the lower diagonal of a matrix |
|
||||
| | | p > q, |
|
||||
| | | p -> row, |
|
||||
| | | q -> column |
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle vec_to_mat_index.irp.f
|
||||
subroutine vec_to_mat_index(i,p,q)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute the indexes (p,q) of the element in the lower diagonal matrix knowing
|
||||
! its index i a vector
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer,intent(in) :: i
|
||||
|
||||
! out
|
||||
integer, intent(out) :: p,q
|
||||
|
||||
! internal
|
||||
integer :: a,b
|
||||
double precision :: da
|
||||
|
||||
da = 0.5d0*(1+ sqrt(1d0+8d0*DBLE(i)))
|
||||
a = INT(da)
|
||||
if ((a*(a-1))/2==i) then
|
||||
p = a-1
|
||||
else
|
||||
p = a
|
||||
endif
|
||||
b = p*(p-1)/2
|
||||
|
||||
! Matrix element indexes
|
||||
p = p + 1
|
||||
q = i - b
|
||||
|
||||
end subroutine
|
||||
#+END_SRC
|
||||
|
|
@ -0,0 +1,40 @@
|
|||
* Vect to antisymmetric matrix using mat_to_vec_index
|
||||
|
||||
Vector to antisymmetric matrix transformation using mat_to_vec_index
|
||||
subroutine.
|
||||
|
||||
Can be done in OMP (for the first part and with omp critical for the second)
|
||||
|
||||
#+BEGIN_SRC f90 :comments org :tangle vec_to_mat_v2.irp.f
|
||||
subroutine vec_to_mat_v2(n,m,v_x,m_x)
|
||||
|
||||
!BEGIN_DOC
|
||||
! Vector to antisymmetric matrix
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
integer, intent(in) :: n,m
|
||||
double precision, intent(in) :: v_x(n)
|
||||
double precision, intent(out) :: m_x(m,m)
|
||||
|
||||
integer :: i,j,k
|
||||
|
||||
! 1D -> 2D lower diagonal
|
||||
m_x = 0d0
|
||||
do j = 1, m - 1
|
||||
do i = j + 1, m
|
||||
call mat_to_vec_index(i,j,k)
|
||||
m_x(i,j) = v_x(k)
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! Antisym
|
||||
do i = 1, m - 1
|
||||
do j = i + 1, m
|
||||
m_x(i,j) = - m_x(j,i)
|
||||
enddo
|
||||
enddo
|
||||
|
||||
end
|
||||
#+END_SRC
|
||||
|
|
@ -0,0 +1,2 @@
|
|||
logical, parameter :: debug=.False.
|
||||
double precision, parameter :: pi = 3.1415926535897932d0
|
||||
|
|
@ -0,0 +1,441 @@
|
|||
! Rotation matrix
|
||||
|
||||
! *Build a rotation matrix from an antisymmetric matrix*
|
||||
|
||||
! Compute a rotation matrix $\textbf{R}$ from an antisymmetric matrix $$\textbf{A}$$ such as :
|
||||
! $$
|
||||
! \textbf{R}=\exp(\textbf{A})
|
||||
! $$
|
||||
|
||||
! So :
|
||||
! \begin{align*}
|
||||
! \textbf{R}=& \exp(\textbf{A}) \\
|
||||
! =& \sum_k^{\infty} \frac{1}{k!}\textbf{A}^k \\
|
||||
! =& \textbf{W} \cdot \cos(\tau) \cdot \textbf{W}^{\dagger} + \textbf{W} \cdot \tau^{-1} \cdot \sin(\tau) \cdot \textbf{W}^{\dagger} \cdot \textbf{A}
|
||||
! \end{align*}
|
||||
|
||||
! With :
|
||||
! $\textbf{W}$ : eigenvectors of $\textbf{A}^2$
|
||||
! $\tau$ : $\sqrt{-x}$
|
||||
! $x$ : eigenvalues of $\textbf{A}^2$
|
||||
|
||||
! Input:
|
||||
! | A(n,n) | double precision | antisymmetric matrix |
|
||||
! | n | integer | number of columns of the A matrix |
|
||||
! | LDA | integer | specifies the leading dimension of A, must be at least max(1,n) |
|
||||
! | LDR | integer | specifies the leading dimension of R, must be at least max(1,n) |
|
||||
|
||||
! Output:
|
||||
! | R(n,n) | double precision | Rotation matrix |
|
||||
! | info | integer | if info = 0, the execution is successful |
|
||||
! | | | if info = k, the k-th parameter has an illegal value |
|
||||
! | | | if info = -k, the algorithm failed |
|
||||
|
||||
! Internal:
|
||||
! | B(n,n) | double precision | B = A.A |
|
||||
! | work(lwork,n) | double precision | work matrix for dysev, dimension max(1,lwork) |
|
||||
! | lwork | integer | dimension of the syev work array >= max(1, 3n-1) |
|
||||
! | W(n,n) | double precision | eigenvectors of B |
|
||||
! | e_val(n) | double precision | eigenvalues of B |
|
||||
! | m_diag(n,n) | double precision | diagonal matrix with the eigenvalues of B |
|
||||
! | cos_tau(n,n) | double precision | diagonal matrix with cos(tau) values |
|
||||
! | sin_tau(n,n) | double precision | diagonal matrix with sin cos(tau) values |
|
||||
! | tau_m1(n,n) | double precision | diagonal matrix with (tau)^-1 values |
|
||||
! | part_1(n,n) | double precision | matrix W.cos_tau.W^t |
|
||||
! | part_1a(n,n) | double precision | matrix cos_tau.W^t |
|
||||
! | part_2(n,n) | double precision | matrix W.tau_m1.sin_tau.W^t.A |
|
||||
! | part_2a(n,n) | double precision | matrix W^t.A |
|
||||
! | part_2b(n,n) | double precision | matrix sin_tau.W^t.A |
|
||||
! | part_2c(n,n) | double precision | matrix tau_m1.sin_tau.W^t.A |
|
||||
! | RR_t(n,n) | double precision | R.R^t must be equal to the identity<=> R.R^t-1=0 <=> norm = 0 |
|
||||
! | norm | integer | norm of R.R^t-1, must be equal to 0 |
|
||||
! | i,j | integer | indexes |
|
||||
|
||||
! Functions:
|
||||
! | dnrm2 | double precision | Lapack function, compute the norm of a matrix |
|
||||
! | disnan | logical | Lapack function, check if an element is NaN |
|
||||
|
||||
|
||||
|
||||
subroutine rotation_matrix(A,LDA,R,LDR,n,info,enforce_step_cancellation)
|
||||
|
||||
implicit none
|
||||
|
||||
!BEGIN_DOC
|
||||
! Rotation matrix to rotate the molecular orbitals.
|
||||
! If the rotation is too large the transformation is not unitary and must be cancelled.
|
||||
!END_DOC
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer, intent(in) :: n,LDA,LDR
|
||||
double precision, intent(inout) :: A(LDA,n)
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: R(LDR,n)
|
||||
integer, intent(out) :: info
|
||||
logical, intent(out) :: enforce_step_cancellation
|
||||
|
||||
! internal
|
||||
double precision, allocatable :: B(:,:)
|
||||
double precision, allocatable :: work(:,:)
|
||||
double precision, allocatable :: W(:,:), e_val(:)
|
||||
double precision, allocatable :: m_diag(:,:),cos_tau(:,:),sin_tau(:,:),tau_m1(:,:)
|
||||
double precision, allocatable :: part_1(:,:),part_1a(:,:)
|
||||
double precision, allocatable :: part_2(:,:),part_2a(:,:),part_2b(:,:),part_2c(:,:)
|
||||
double precision, allocatable :: RR_t(:,:)
|
||||
integer :: i,j
|
||||
integer :: info2, lwork ! for dsyev
|
||||
double precision :: norm, max_elem, max_elem_A, t1,t2,t3
|
||||
|
||||
! function
|
||||
double precision :: dnrm2
|
||||
logical :: disnan
|
||||
|
||||
print*,''
|
||||
print*,'---rotation_matrix---'
|
||||
|
||||
call wall_time(t1)
|
||||
|
||||
! Allocation
|
||||
allocate(B(n,n))
|
||||
allocate(m_diag(n,n),cos_tau(n,n),sin_tau(n,n),tau_m1(n,n))
|
||||
allocate(W(n,n),e_val(n))
|
||||
allocate(part_1(n,n),part_1a(n,n))
|
||||
allocate(part_2(n,n),part_2a(n,n),part_2b(n,n),part_2c(n,n))
|
||||
allocate(RR_t(n,n))
|
||||
|
||||
! Pre-conditions
|
||||
|
||||
! Initialization
|
||||
info=0
|
||||
enforce_step_cancellation = .False.
|
||||
|
||||
! Size of matrix A must be at least 1 by 1
|
||||
if (n<1) then
|
||||
info = 3
|
||||
print*, 'WARNING: invalid parameter 5'
|
||||
print*, 'n<1'
|
||||
return
|
||||
endif
|
||||
|
||||
! Leading dimension of A must be >= n
|
||||
if (LDA < n) then
|
||||
info = 25
|
||||
print*, 'WARNING: invalid parameter 2 or 5'
|
||||
print*, 'LDA < n'
|
||||
return
|
||||
endif
|
||||
|
||||
! Leading dimension of A must be >= n
|
||||
if (LDR < n) then
|
||||
info = 4
|
||||
print*, 'WARNING: invalid parameter 4'
|
||||
print*, 'LDR < n'
|
||||
return
|
||||
endif
|
||||
|
||||
! Matrix elements of A must by non-NaN
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (disnan(A(i,j))) then
|
||||
info=1
|
||||
print*, 'WARNING: invalid parameter 1'
|
||||
print*, 'NaN element in A matrix'
|
||||
return
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
do i = 1, n
|
||||
if (A(i,i) /= 0d0) then
|
||||
print*, 'WARNING: matrix A is not antisymmetric'
|
||||
print*, 'Non 0 element on the diagonal', i, A(i,i)
|
||||
call ABORT
|
||||
endif
|
||||
enddo
|
||||
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (A(i,j)+A(j,i)>1d-16) then
|
||||
print*, 'WANRING: matrix A is not antisymmetric'
|
||||
print*, 'A(i,j) /= - A(j,i):', i,j,A(i,j), A(j,i)
|
||||
print*, 'diff:', A(i,j)+A(j,i)
|
||||
call ABORT
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! Fix for too big elements ! bad idea better to cancel if the error is too big
|
||||
!do j = 1, n
|
||||
! do i = 1, n
|
||||
! A(i,j) = mod(A(i,j),2d0*pi)
|
||||
! if (dabs(A(i,j)) > pi) then
|
||||
! A(i,j) = 0d0
|
||||
! endif
|
||||
! enddo
|
||||
!enddo
|
||||
|
||||
max_elem_A = 0d0
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (ABS(A(i,j)) > ABS(max_elem_A)) then
|
||||
max_elem_A = A(i,j)
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
!print*,'max element in A', max_elem_A
|
||||
|
||||
if (ABS(max_elem_A) > 2 * pi) then
|
||||
print*,''
|
||||
print*,'WARNING: ABS(max_elem_A) > 2 pi '
|
||||
print*,''
|
||||
endif
|
||||
|
||||
! B=A.A
|
||||
! - Calculation of the matrix $\textbf{B} = \textbf{A}^2$
|
||||
! - Diagonalization of $\textbf{B}$
|
||||
! W, the eigenvectors
|
||||
! e_val, the eigenvalues
|
||||
|
||||
|
||||
! Compute B=A.A
|
||||
|
||||
call dgemm('N','N',n,n,n,1d0,A,size(A,1),A,size(A,1),0d0,B,size(B,1))
|
||||
|
||||
! Copy B in W, diagonalization will put the eigenvectors in W
|
||||
W=B
|
||||
|
||||
! Diagonalization of B
|
||||
! Eigenvalues -> e_val
|
||||
! Eigenvectors -> W
|
||||
lwork = 3*n-1
|
||||
allocate(work(lwork,n))
|
||||
|
||||
!print*,'Starting diagonalization ...'
|
||||
|
||||
call dsyev('V','U',n,W,size(W,1),e_val,work,lwork,info2)
|
||||
|
||||
deallocate(work)
|
||||
|
||||
if (info2 < 0) then
|
||||
print*, 'WARNING: error in the diagonalization'
|
||||
print*, 'Illegal value of the ', info2,'-th parameter'
|
||||
elseif (info2 >0) then
|
||||
print*, "WARNING: Diagonalization failed to converge"
|
||||
endif
|
||||
|
||||
! Tau^-1, cos(tau), sin(tau)
|
||||
! $$\tau = \sqrt{-x}$$
|
||||
! - Calculation of $\cos(\tau)$ $\Leftrightarrow$ $\cos(\sqrt{-x})$
|
||||
! - Calculation of $\sin(\tau)$ $\Leftrightarrow$ $\sin(\sqrt{-x})$
|
||||
! - Calculation of $\tau^{-1}$ $\Leftrightarrow$ $(\sqrt{-x})^{-1}$
|
||||
! These matrices are diagonals
|
||||
|
||||
! Diagonal matrix m_diag
|
||||
do j = 1, n
|
||||
if (e_val(j) >= -1d-12) then !0.d0) then !!! e_avl(i) must be < -1d-12 to avoid numerical problems
|
||||
e_val(j) = 0.d0
|
||||
else
|
||||
e_val(j) = - e_val(j)
|
||||
endif
|
||||
enddo
|
||||
|
||||
m_diag = 0.d0
|
||||
do i = 1, n
|
||||
m_diag(i,i) = e_val(i)
|
||||
enddo
|
||||
|
||||
! cos_tau
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (i==j) then
|
||||
cos_tau(i,j) = dcos(dsqrt(e_val(i)))
|
||||
else
|
||||
cos_tau(i,j) = 0d0
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! sin_tau
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (i==j) then
|
||||
sin_tau(i,j) = dsin(dsqrt(e_val(i)))
|
||||
else
|
||||
sin_tau(i,j) = 0d0
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! Debug, display the cos_tau and sin_tau matrix
|
||||
!if (debug) then
|
||||
! print*, 'cos_tau'
|
||||
! do i = 1, n
|
||||
! print*, cos_tau(i,:)
|
||||
! enddo
|
||||
! print*, 'sin_tau'
|
||||
! do i = 1, n
|
||||
! print*, sin_tau(i,:)
|
||||
! enddo
|
||||
!endif
|
||||
|
||||
! tau^-1
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if ((i==j) .and. (e_val(i) > 1d-16)) then!0d0)) then !!! Convergence problem can come from here if the threshold is too big/small
|
||||
tau_m1(i,j) = 1d0/(dsqrt(e_val(i)))
|
||||
else
|
||||
tau_m1(i,j) = 0d0
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
max_elem = 0d0
|
||||
do i = 1, n
|
||||
if (ABS(tau_m1(i,i)) > ABS(max_elem)) then
|
||||
max_elem = tau_m1(i,i)
|
||||
endif
|
||||
enddo
|
||||
!print*,'max elem tau^-1:', max_elem
|
||||
|
||||
! Debug
|
||||
!print*,'eigenvalues:'
|
||||
!do i = 1, n
|
||||
! print*, e_val(i)
|
||||
!enddo
|
||||
|
||||
!Debug, display tau^-1
|
||||
!if (debug) then
|
||||
! print*, 'tau^-1'
|
||||
! do i = 1, n
|
||||
! print*,tau_m1(i,:)
|
||||
! enddo
|
||||
!endif
|
||||
|
||||
! Rotation matrix
|
||||
! \begin{align*}
|
||||
! \textbf{R} = \textbf{W} \cos(\tau) \textbf{W}^{\dagger} + \textbf{W} \tau^{-1} \sin(\tau) \textbf{W}^{\dagger} \textbf{A}
|
||||
! \end{align*}
|
||||
! \begin{align*}
|
||||
! \textbf{Part1} = \textbf{W} \cos(\tau) \textbf{W}^{\dagger}
|
||||
! \end{align*}
|
||||
! \begin{align*}
|
||||
! \textbf{Part2} = \textbf{W} \tau^{-1} \sin(\tau) \textbf{W}^{\dagger} \textbf{A}
|
||||
! \end{align*}
|
||||
|
||||
! First:
|
||||
! part_1 = dgemm(W, dgemm(cos_tau, W^t))
|
||||
! part_1a = dgemm(cos_tau, W^t)
|
||||
! part_1 = dgemm(W, part_1a)
|
||||
! And:
|
||||
! part_2= dgemm(W, dgemm(tau_m1, dgemm(sin_tau, dgemm(W^t, A))))
|
||||
! part_2a = dgemm(W^t, A)
|
||||
! part_2b = dgemm(sin_tau, part_2a)
|
||||
! part_2c = dgemm(tau_m1, part_2b)
|
||||
! part_2 = dgemm(W, part_2c)
|
||||
! Finally:
|
||||
! Rotation matrix, R = part_1+part_2
|
||||
|
||||
! If $R$ is a rotation matrix:
|
||||
! $R.R^T=R^T.R=\textbf{1}$
|
||||
|
||||
! part_1
|
||||
call dgemm('N','T',n,n,n,1d0,cos_tau,size(cos_tau,1),W,size(W,1),0d0,part_1a,size(part_1a,1))
|
||||
call dgemm('N','N',n,n,n,1d0,W,size(W,1),part_1a,size(part_1a,1),0d0,part_1,size(part_1,1))
|
||||
|
||||
! part_2
|
||||
call dgemm('T','N',n,n,n,1d0,W,size(W,1),A,size(A,1),0d0,part_2a,size(part_2a,1))
|
||||
call dgemm('N','N',n,n,n,1d0,sin_tau,size(sin_tau,1),part_2a,size(part_2a,1),0d0,part_2b,size(part_2b,1))
|
||||
call dgemm('N','N',n,n,n,1d0,tau_m1,size(tau_m1,1),part_2b,size(part_2b,1),0d0,part_2c,size(part_2c,1))
|
||||
call dgemm('N','N',n,n,n,1d0,W,size(W,1),part_2c,size(part_2c,1),0d0,part_2,size(part_2,1))
|
||||
|
||||
! Rotation matrix R
|
||||
R = part_1 + part_2
|
||||
|
||||
! Matrix check
|
||||
! R.R^t and R^t.R must be equal to identity matrix
|
||||
do j = 1, n
|
||||
do i=1,n
|
||||
if (i==j) then
|
||||
RR_t(i,j) = 1d0
|
||||
else
|
||||
RR_t(i,j) = 0d0
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
call dgemm('N','T',n,n,n,1d0,R,size(R,1),R,size(R,1),-1d0,RR_t,size(RR_t,1))
|
||||
|
||||
norm = dnrm2(n*n,RR_t,1)
|
||||
!print*, 'Rotation matrix check, norm R.R^T = ', norm
|
||||
|
||||
! Debug
|
||||
!if (debug) then
|
||||
! print*, 'RR_t'
|
||||
! do i = 1, n
|
||||
! print*, RR_t(i,:)
|
||||
! enddo
|
||||
!endif
|
||||
|
||||
! Post conditions
|
||||
|
||||
! Check if R.R^T=1
|
||||
max_elem = 0d0
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
if (ABS(RR_t(i,j)) > ABS(max_elem)) then
|
||||
max_elem = RR_t(i,j)
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
print*, 'Max error in R.R^T:', max_elem
|
||||
!print*, 'e_val(1):', e_val(1)
|
||||
!print*, 'e_val(n):', e_val(n)
|
||||
!print*, 'max elem in A:', max_elem_A
|
||||
|
||||
if (ABS(max_elem) > 1d-12) then
|
||||
print*, 'WARNING: max error in R.R^T > 1d-12'
|
||||
print*, 'Enforce the step cancellation'
|
||||
enforce_step_cancellation = .True.
|
||||
endif
|
||||
|
||||
! Matrix elements of R must by non-NaN
|
||||
do j = 1,n
|
||||
do i = 1,LDR
|
||||
if (disnan(R(i,j))) then
|
||||
info = 666
|
||||
print*, 'NaN in rotation matrix'
|
||||
call ABORT
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! Display
|
||||
!if (debug) then
|
||||
! print*,'Rotation matrix :'
|
||||
! do i = 1, n
|
||||
! write(*,'(100(F10.5))') R(i,:)
|
||||
! enddo
|
||||
!endif
|
||||
|
||||
! Deallocation, end
|
||||
|
||||
deallocate(B)
|
||||
deallocate(m_diag,cos_tau,sin_tau,tau_m1)
|
||||
deallocate(W,e_val)
|
||||
deallocate(part_1,part_1a)
|
||||
deallocate(part_2,part_2a,part_2b,part_2c)
|
||||
deallocate(RR_t)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2-t1
|
||||
print*,'Time in rotation matrix:', t3
|
||||
|
||||
print*,'---End rotation_matrix---'
|
||||
|
||||
end subroutine
|
||||
|
|
@ -0,0 +1,134 @@
|
|||
! Rotation matrix with the iterative method
|
||||
|
||||
! \begin{align*}
|
||||
! \textbf{R} = \sum_{k=0}^{\infty} \frac{1}{k!} \textbf{X}^k
|
||||
! \end{align*}
|
||||
|
||||
! !!! Doesn't work !!!
|
||||
|
||||
|
||||
subroutine rotation_matrix_iterative(m,X,R)
|
||||
|
||||
implicit none
|
||||
|
||||
! in
|
||||
integer, intent(in) :: m
|
||||
double precision, intent(in) :: X(m,m)
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: R(m,m)
|
||||
|
||||
! internal
|
||||
double precision :: max_elem, pre_factor
|
||||
double precision :: t1,t2,t3
|
||||
integer :: k,l,i,j
|
||||
logical :: not_converged
|
||||
double precision, allocatable :: RRT(:,:), A(:,:), B(:,:)
|
||||
|
||||
! Functions
|
||||
integer :: factorial
|
||||
|
||||
print*,'---rotation_matrix_iterative---'
|
||||
call wall_time(t1)
|
||||
|
||||
allocate(RRT(m,m),A(m,m),B(m,m))
|
||||
|
||||
! k = 0
|
||||
R = 0d0
|
||||
do i = 1, m
|
||||
R(i,i) = 1d0
|
||||
enddo
|
||||
|
||||
! k = 1
|
||||
R = R + X
|
||||
|
||||
k = 2
|
||||
|
||||
not_converged = .True.
|
||||
|
||||
do while (not_converged)
|
||||
|
||||
pre_factor = 1d0/DBLE(factorial(k))
|
||||
if (pre_factor < 1d-15) then
|
||||
print*,'pre factor=', pre_factor,'< 1d-15, exit'
|
||||
exit
|
||||
endif
|
||||
|
||||
A = X
|
||||
B = 0d0
|
||||
do l = 1, k-1
|
||||
call dgemm('N','N',m,m,m,1d0,X,size(X,1),A,size(A,1),0d0,B,size(B,1))
|
||||
A = B
|
||||
enddo
|
||||
|
||||
!print*,'B'
|
||||
!do i = 1, m
|
||||
! print*,B(i,:) * 1d0/DBLE(factorial(k))
|
||||
!enddo
|
||||
|
||||
R = R + pre_factor * B
|
||||
|
||||
k = k + 1
|
||||
call dgemm('T','N',m,m,m,1d0,R,size(R,1),R,size(R,1),0d0,RRT,size(RRT,1))
|
||||
|
||||
!print*,'R'
|
||||
!do i = 1, m
|
||||
! write(*,'(10(E12.5))') R(i,:)
|
||||
!enddo
|
||||
|
||||
do i = 1, m
|
||||
RRT(i,i) = RRT(i,i) - 1d0
|
||||
enddo
|
||||
|
||||
!print*,'RRT'
|
||||
!do i = 1, m
|
||||
! write(*,'(10(E12.5))') RRT(i,:)
|
||||
!enddo
|
||||
|
||||
max_elem = 0d0
|
||||
do j = 1, m
|
||||
do i = 1, m
|
||||
if (dabs(RRT(i,j)) > max_elem) then
|
||||
max_elem = dabs(RRT(i,j))
|
||||
endif
|
||||
enddo
|
||||
enddo
|
||||
|
||||
print*, 'Iteration:', k
|
||||
print*, 'Max error in R:', max_elem
|
||||
|
||||
if (max_elem < 1d-12) then
|
||||
not_converged = .False.
|
||||
endif
|
||||
|
||||
enddo
|
||||
|
||||
deallocate(RRT,A,B)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2 - t1
|
||||
print*,'Time in rotation matrix iterative:', t3
|
||||
print*,'---End roration_matrix_iterative---'
|
||||
|
||||
|
||||
print*,'Does not work yet, abort'
|
||||
call abort
|
||||
|
||||
end
|
||||
|
||||
! Factorial
|
||||
|
||||
function factorial(n)
|
||||
|
||||
implicit none
|
||||
|
||||
integer, intent(in) :: n
|
||||
integer :: factorial, k
|
||||
|
||||
factorial = 1
|
||||
|
||||
do k = 1, n
|
||||
factorial = factorial * k
|
||||
enddo
|
||||
|
||||
end
|
||||
|
|
@ -0,0 +1,64 @@
|
|||
! Rotation matrix in a subspace to rotation matrix in the full space
|
||||
|
||||
! Usually, we are using a list of MOs, for exemple the active ones. When
|
||||
! we compute a rotation matrix to rotate the MOs, we just compute a
|
||||
! rotation matrix for these MOs in order to reduce the size of the
|
||||
! matrix which has to be computed. Since the computation of a rotation
|
||||
! matrix scale in $O(N^3)$ with $N$ the number of MOs, it's better to
|
||||
! reuce the number of MOs involved.
|
||||
! After that we replace the rotation matrix in the full space by
|
||||
! building the elements of the rotation matrix in the full space from
|
||||
! the elements of the rotation matrix in the subspace and adding some 0
|
||||
! on the extradiagonal elements and some 1 on the diagonal elements,
|
||||
! for the MOs that are not involved in the rotation.
|
||||
|
||||
! Provided:
|
||||
! | mo_num | integer | Number of MOs |
|
||||
|
||||
! Input:
|
||||
! | m | integer | Size of tmp_list, m <= mo_num |
|
||||
! | tmp_list(m) | integer | List of MOs |
|
||||
! | tmp_R(m,m) | double precision | Rotation matrix in the space of |
|
||||
! | | | the MOs containing by tmp_list |
|
||||
|
||||
! Output:
|
||||
! | R(mo_num,mo_num | double precision | Rotation matrix in the space |
|
||||
! | | | of all the MOs |
|
||||
|
||||
! Internal:
|
||||
! | i,j | integer | indexes in the full space |
|
||||
! | tmp_i,tmp_j | integer | indexes in the subspace |
|
||||
|
||||
|
||||
subroutine sub_to_full_rotation_matrix(m,tmp_list,tmp_R,R)
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute the full rotation matrix from a smaller one
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! in
|
||||
integer, intent(in) :: m, tmp_list(m)
|
||||
double precision, intent(in) :: tmp_R(m,m)
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: R(mo_num,mo_num)
|
||||
|
||||
! internal
|
||||
integer :: i,j,tmp_i,tmp_j
|
||||
|
||||
! tmp_R to R, subspace to full space
|
||||
R = 0d0
|
||||
do i = 1, mo_num
|
||||
R(i,i) = 1d0 ! 1 on the diagonal because it is a rotation matrix, 1 = nothing change for the corresponding orbital
|
||||
enddo
|
||||
do tmp_j = 1, m
|
||||
j = tmp_list(tmp_j)
|
||||
do tmp_i = 1, m
|
||||
i = tmp_list(tmp_i)
|
||||
R(i,j) = tmp_R(tmp_i,tmp_j)
|
||||
enddo
|
||||
enddo
|
||||
|
||||
end
|
||||
|
|
@ -0,0 +1,126 @@
|
|||
! Predicted energy : e_model
|
||||
|
||||
! *Compute the energy predicted by the Taylor series*
|
||||
|
||||
! The energy is predicted using a Taylor expansion truncated at te 2nd
|
||||
! order :
|
||||
|
||||
! \begin{align*}
|
||||
! E_{k+1} = E_{k} + \textbf{g}_k^{T} \cdot \textbf{x}_{k+1} + \frac{1}{2} \cdot \textbf{x}_{k+1}^T \cdot \textbf{H}_{k} \cdot \textbf{x}_{k+1} + \mathcal{O}(\textbf{x}_{k+1}^2)
|
||||
! \end{align*}
|
||||
|
||||
! Input:
|
||||
! | n | integer | m*(m-1)/2 |
|
||||
! | n2 | integer | m*(m-1)/2 or 1 if the hessian is diagonal |
|
||||
! | v_grad(n) | double precision | gradient |
|
||||
! | H(n,n) | double precision | hessian |
|
||||
! | x(n) | double precision | Step in the trust region |
|
||||
! | prev_energy | double precision | previous energy |
|
||||
|
||||
! Output:
|
||||
! | e_model | double precision | predicted energy after the rotation of the MOs |
|
||||
|
||||
! Internal:
|
||||
! | part_1 | double precision | v_grad^T.x |
|
||||
! | part_2 | double precision | 1/2 . x^T.H.x |
|
||||
! | part_2a | double precision | H.x |
|
||||
! | i,j | integer | indexes |
|
||||
|
||||
! Function:
|
||||
! | ddot | double precision | dot product (Lapack) |
|
||||
|
||||
|
||||
subroutine trust_region_expected_e(n,n2,v_grad,H,x,prev_energy,e_model)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute the expected criterion/energy after the application of the step x
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer, intent(in) :: n,n2
|
||||
double precision, intent(in) :: v_grad(n),H(n,n2),x(n)
|
||||
double precision, intent(in) :: prev_energy
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: e_model
|
||||
|
||||
! internal
|
||||
double precision :: part_1, part_2, t1,t2,t3
|
||||
double precision, allocatable :: part_2a(:)
|
||||
|
||||
integer :: i,j
|
||||
|
||||
!Function
|
||||
double precision :: ddot
|
||||
|
||||
print*,''
|
||||
print*,'---Trust_e_model---'
|
||||
|
||||
call wall_time(t1)
|
||||
|
||||
! Allocation
|
||||
allocate(part_2a(n))
|
||||
|
||||
! Calculations
|
||||
|
||||
! part_1 corresponds to the product g.x
|
||||
! part_2a corresponds to the product H.x
|
||||
! part_2 corresponds to the product 0.5*(x^T.H.x)
|
||||
|
||||
! TODO: remove the dot products
|
||||
|
||||
|
||||
! Product v_grad.x
|
||||
part_1 = ddot(n,v_grad,1,x,1)
|
||||
|
||||
!if (debug) then
|
||||
! print*,'g.x : ', part_1
|
||||
!endif
|
||||
|
||||
! Product H.x
|
||||
if (n == n2) then
|
||||
call dgemv('N',n,n,1d0,H,size(H,1),x,1,0d0,part_2a,1)
|
||||
else
|
||||
! If the hessian is diagonal
|
||||
do i = 1, n
|
||||
part_2a(i) = H(i,1) * x(i)
|
||||
enddo
|
||||
endif
|
||||
|
||||
! Product 1/2 . x^T.H.x
|
||||
part_2 = 0.5d0 * ddot(n,x,1,part_2a,1)
|
||||
|
||||
!if (debug) then
|
||||
! print*,'1/2*x^T.H.x : ', part_2
|
||||
!endif
|
||||
|
||||
|
||||
! Sum
|
||||
e_model = prev_energy + part_1 + part_2
|
||||
|
||||
! Writing the predicted energy
|
||||
print*, 'prev_energy: ', prev_energy
|
||||
print*, 'Predicted energy after the rotation:', e_model
|
||||
print*, 'Previous energy - predicted energy: ', prev_energy - e_model
|
||||
|
||||
! Can be deleted, already in another subroutine
|
||||
if (DABS(prev_energy - e_model) < 1d-12 ) then
|
||||
print*,'WARNING: ABS(prev_energy - e_model) < 1d-12'
|
||||
endif
|
||||
|
||||
! Deallocation
|
||||
deallocate(part_2a)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2 - t1
|
||||
print*,'Time in trust e model:', t3
|
||||
|
||||
print*,'---End trust_e_model---'
|
||||
|
||||
end subroutine
|
||||
File diff suppressed because it is too large
Load Diff
|
|
@ -0,0 +1,120 @@
|
|||
! Agreement with the model: Rho
|
||||
|
||||
! *Compute the ratio : rho = (prev_energy - energy) / (prev_energy - e_model)*
|
||||
|
||||
! Rho represents the agreement between the model (the predicted energy
|
||||
! by the Taylor expansion truncated at the 2nd order) and the real
|
||||
! energy :
|
||||
|
||||
! \begin{equation}
|
||||
! \rho^{k+1} = \frac{E^{k} - E^{k+1}}{E^{k} - m^{k+1}}
|
||||
! \end{equation}
|
||||
! With :
|
||||
! $E^{k}$ the energy at the previous iteration
|
||||
! $E^{k+1}$ the energy at the actual iteration
|
||||
! $m^{k+1}$ the predicted energy for the actual iteration
|
||||
! (cf. trust_e_model)
|
||||
|
||||
! If $\rho \approx 1$, the agreement is good, contrary to $\rho \approx 0$.
|
||||
! If $\rho \leq 0$ the previous energy is lower than the actual
|
||||
! energy. We have to cancel the last step and use a smaller trust
|
||||
! region.
|
||||
! Here we cancel the last step if $\rho < 0.1$, because even if
|
||||
! the energy decreases, the agreement is bad, i.e., the Taylor expansion
|
||||
! truncated at the second order doesn't represent correctly the energy
|
||||
! landscape. So it's better to cancel the step and restart with a
|
||||
! smaller trust region.
|
||||
|
||||
! Provided in qp_edit:
|
||||
! | thresh_rho |
|
||||
|
||||
! Input:
|
||||
! | prev_energy | double precision | previous energy (energy before the rotation) |
|
||||
! | e_model | double precision | predicted energy after the rotation |
|
||||
|
||||
! Output:
|
||||
! | rho | double precision | the agreement between the model (predicted) and the real energy |
|
||||
! | prev_energy | double precision | if rho >= 0.1 the actual energy becomes the previous energy |
|
||||
! | | | else the previous energy doesn't change |
|
||||
|
||||
! Internal:
|
||||
! | energy | double precision | energy (real) after the rotation |
|
||||
! | i | integer | index |
|
||||
! | t* | double precision | time |
|
||||
|
||||
|
||||
subroutine trust_region_rho(prev_energy, energy,e_model,rho)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute rho, the agreement between the predicted criterion/energy and the real one
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! In
|
||||
double precision, intent(inout) :: prev_energy
|
||||
double precision, intent(in) :: e_model, energy
|
||||
|
||||
! Out
|
||||
double precision, intent(out) :: rho
|
||||
|
||||
! Internal
|
||||
double precision :: t1, t2, t3
|
||||
integer :: i
|
||||
|
||||
print*,''
|
||||
print*,'---Rho_model---'
|
||||
|
||||
!call wall_time(t1)
|
||||
|
||||
! Rho
|
||||
! \begin{equation}
|
||||
! \rho^{k+1} = \frac{E^{k} - E^{k+1}}{E^{k} - m^{k+1}}
|
||||
! \end{equation}
|
||||
|
||||
! In function of $\rho$ th step can be accepted or cancelled.
|
||||
|
||||
! If we cancel the last step (k+1), the previous energy (k) doesn't
|
||||
! change!
|
||||
! If the step (k+1) is accepted, then the "previous energy" becomes E(k+1)
|
||||
|
||||
|
||||
! Already done in an other subroutine
|
||||
!if (ABS(prev_energy - e_model) < 1d-12) then
|
||||
! print*,'WARNING: prev_energy - e_model < 1d-12'
|
||||
! print*,'=> rho will tend toward infinity'
|
||||
! print*,'Check you convergence criterion !'
|
||||
!endif
|
||||
|
||||
rho = (prev_energy - energy) / (prev_energy - e_model)
|
||||
|
||||
!print*, 'previous energy, prev_energy:', prev_energy
|
||||
!print*, 'predicted energy, e_model:', e_model
|
||||
!print*, 'real energy, energy:', energy
|
||||
!print*, 'prev_energy - energy:', prev_energy - energy
|
||||
!print*, 'prev_energy - e_model:', prev_energy - e_model
|
||||
print*, 'Rho:', rho
|
||||
!print*, 'Threshold for rho:', thresh_rho
|
||||
|
||||
! Modification of prev_energy in function of rho
|
||||
if (rho < thresh_rho) then !0.1) then
|
||||
! the step is cancelled
|
||||
print*, 'Rho <', thresh_rho,', the previous energy does not changed'
|
||||
!print*, 'prev_energy :', prev_energy
|
||||
else
|
||||
! the step is accepted
|
||||
prev_energy = energy
|
||||
print*, 'Rho >=', thresh_rho,', energy -> prev_energy:', energy
|
||||
endif
|
||||
|
||||
!call wall_time(t2)
|
||||
!t3 = t2 - t1
|
||||
!print*,'Time in rho model:', t3
|
||||
|
||||
print*,'---End rho_model---'
|
||||
|
||||
end subroutine
|
||||
|
|
@ -0,0 +1,749 @@
|
|||
! Trust region
|
||||
|
||||
! *Compute the next step with the trust region algorithm*
|
||||
|
||||
! The Newton method is an iterative method to find a minimum of a given
|
||||
! function. It uses a Taylor series truncated at the second order of the
|
||||
! targeted function and gives its minimizer. The minimizer is taken as
|
||||
! the new position and the same thing is done. And by doing so
|
||||
! iteratively the method find a minimum, a local or global one depending
|
||||
! of the starting point and the convexity/nonconvexity of the targeted
|
||||
! function.
|
||||
|
||||
! The goal of the trust region is to constrain the step size of the
|
||||
! Newton method in a certain area around the actual position, where the
|
||||
! Taylor series is a good approximation of the targeted function. This
|
||||
! area is called the "trust region".
|
||||
|
||||
! In addition, in function of the agreement between the Taylor
|
||||
! development of the energy and the real energy, the size of the trust
|
||||
! region will be updated at each iteration. By doing so, the step sizes
|
||||
! are not too larges. In addition, since we add a criterion to cancel the
|
||||
! step if the energy increases (more precisely if rho < 0.1), so it's
|
||||
! impossible to diverge. \newline
|
||||
|
||||
! References: \newline
|
||||
! Nocedal & Wright, Numerical Optimization, chapter 4 (1999), \newline
|
||||
! https://link.springer.com/book/10.1007/978-0-387-40065-5, \newline
|
||||
! ISBN: 978-0-387-40065-5 \newline
|
||||
|
||||
! By using the first and the second derivatives, the Newton method gives
|
||||
! a step:
|
||||
! \begin{align*}
|
||||
! \textbf{x}_{(k+1)}^{\text{Newton}} = - \textbf{H}_{(k)}^{-1} \cdot
|
||||
! \textbf{g}_{(k)}
|
||||
! \end{align*}
|
||||
! which leads to the minimizer of the Taylor series.
|
||||
! !!! Warning: the Newton method gives the minimizer if and only if
|
||||
! $\textbf{H}$ is positive definite, else it leads to a saddle point !!!
|
||||
! But we want a step $\textbf{x}_{(k+1)}$ with a constraint on its (euclidian) norm:
|
||||
! \begin{align*}
|
||||
! ||\textbf{x}_{(k+1)}|| \leq \Delta_{(k+1)}
|
||||
! \end{align*}
|
||||
! which is equivalent to
|
||||
! \begin{align*}
|
||||
! \textbf{x}_{(k+1)}^T \cdot \textbf{x}_{(k+1)} \leq \Delta_{(k+1)}^2
|
||||
! \end{align*}
|
||||
|
||||
! with: \newline
|
||||
! $\textbf{x}_{(k+1)}$ is the step for the k+1-th iteration (vector of
|
||||
! size n) \newline
|
||||
! $\textbf{H}_{(k)}$ is the hessian at the k-th iteration (n by n
|
||||
! matrix) \newline
|
||||
! $\textbf{g}_{(k)}$ is the gradient at the k-th iteration (vector of
|
||||
! size n) \newline
|
||||
! $\Delta_{(k+1)}$ is the trust radius for the (k+1)-th iteration
|
||||
! \newline
|
||||
|
||||
! Thus we want to constrain the step size $\textbf{x}_{(k+1)}$ into a
|
||||
! hypersphere of radius $\Delta_{(k+1)}$.\newline
|
||||
|
||||
! So, if $||\textbf{x}_{(k+1)}^{\text{Newton}}|| \leq \Delta_{(k)}$ and
|
||||
! $\textbf{H}$ is positive definite, the
|
||||
! solution is the step given by the Newton method
|
||||
! $\textbf{x}_{(k+1)} = \textbf{x}_{(k+1)}^{\text{Newton}}$.
|
||||
! Else we have to constrain the step size. For simplicity we will remove
|
||||
! the index $_{(k)}$ and $_{(k+1)}$. To restict the step size, we have
|
||||
! to put a constraint on $\textbf{x}$ with a Lagrange multiplier.
|
||||
! Starting from the Taylor series of a function E (here, the energy)
|
||||
! truncated at the 2nd order, we have:
|
||||
! \begin{align*}
|
||||
! E(\textbf{x}) = E +\textbf{g}^T \cdot \textbf{x} + \frac{1}{2}
|
||||
! \cdot \textbf{x}^T \cdot \textbf{H} \cdot \textbf{x} +
|
||||
! \mathcal{O}(\textbf{x}^2)
|
||||
! \end{align*}
|
||||
|
||||
! With the constraint on the norm of $\textbf{x}$ we can write the
|
||||
! Lagrangian
|
||||
! \begin{align*}
|
||||
! \mathcal{L}(\textbf{x},\lambda) = E + \textbf{g}^T \cdot \textbf{x}
|
||||
! + \frac{1}{2} \cdot \textbf{x}^T \cdot \textbf{H} \cdot \textbf{x}
|
||||
! + \frac{1}{2} \lambda (\textbf{x}^T \cdot \textbf{x} - \Delta^2)
|
||||
! \end{align*}
|
||||
! Where: \newline
|
||||
! $\lambda$ is the Lagrange multiplier \newline
|
||||
! $E$ is the energy at the k-th iteration $\Leftrightarrow
|
||||
! E(\textbf{x} = \textbf{0})$ \newline
|
||||
|
||||
! To solve this equation, we search a stationary point where the first
|
||||
! derivative of $\mathcal{L}$ with respect to $\textbf{x}$ becomes 0, i.e.
|
||||
! \begin{align*}
|
||||
! \frac{\partial \mathcal{L}(\textbf{x},\lambda)}{\partial \textbf{x}}=0
|
||||
! \end{align*}
|
||||
|
||||
! The derivative is:
|
||||
! \begin{align*}
|
||||
! \frac{\partial \mathcal{L}(\textbf{x},\lambda)}{\partial \textbf{x}}
|
||||
! = \textbf{g} + \textbf{H} \cdot \textbf{x} + \lambda \cdot \textbf{x}
|
||||
! \end{align*}
|
||||
|
||||
! So, we search $\textbf{x}$ such as:
|
||||
! \begin{align*}
|
||||
! \frac{\partial \mathcal{L}(\textbf{x},\lambda)}{\partial \textbf{x}}
|
||||
! = \textbf{g} + \textbf{H} \cdot \textbf{x} + \lambda \cdot \textbf{x} = 0
|
||||
! \end{align*}
|
||||
|
||||
! We can rewrite that as:
|
||||
! \begin{align*}
|
||||
! \textbf{g} + \textbf{H} \cdot \textbf{x} + \lambda \cdot \textbf{x}
|
||||
! = \textbf{g} + (\textbf{H} +\textbf{I} \lambda) \cdot \textbf{x} = 0
|
||||
! \end{align*}
|
||||
! with $\textbf{I}$ is the identity matrix.
|
||||
|
||||
! By doing so, the solution is:
|
||||
! \begin{align*}
|
||||
! (\textbf{H} +\textbf{I} \lambda) \cdot \textbf{x}= -\textbf{g}
|
||||
! \end{align*}
|
||||
! \begin{align*}
|
||||
! \textbf{x}= - (\textbf{H} + \textbf{I} \lambda)^{-1} \cdot \textbf{g}
|
||||
! \end{align*}
|
||||
! with $\textbf{x}^T \textbf{x} = \Delta^2$.
|
||||
|
||||
! We have to solve this previous equation to find this $\textbf{x}$ in the
|
||||
! trust region, i.e. $||\textbf{x}|| = \Delta$. Now, this problem is
|
||||
! just a one dimension problem because we can express $\textbf{x}$ as a
|
||||
! function of $\lambda$:
|
||||
! \begin{align*}
|
||||
! \textbf{x}(\lambda) = - (\textbf{H} + \textbf{I} \lambda)^{-1} \cdot \textbf{g}
|
||||
! \end{align*}
|
||||
|
||||
! We start from the fact that the hessian is diagonalizable. So we have:
|
||||
! \begin{align*}
|
||||
! \textbf{H} = \textbf{W} \cdot \textbf{h} \cdot \textbf{W}^T
|
||||
! \end{align*}
|
||||
! with: \newline
|
||||
! $\textbf{H}$, the hessian matrix \newline
|
||||
! $\textbf{W}$, the matrix containing the eigenvectors \newline
|
||||
! $\textbf{w}_i$, the i-th eigenvector, i.e. i-th column of $\textbf{W}$ \newline
|
||||
! $\textbf{h}$, the matrix containing the eigenvalues in ascending order \newline
|
||||
! $h_i$, the i-th eigenvalue in ascending order \newline
|
||||
|
||||
! Now we use the fact that adding a constant on the diagonal just shifts
|
||||
! the eigenvalues:
|
||||
! \begin{align*}
|
||||
! \textbf{H} + \textbf{I} \lambda = \textbf{W} \cdot (\textbf{h}
|
||||
! +\textbf{I} \lambda) \cdot \textbf{W}^T
|
||||
! \end{align*}
|
||||
|
||||
! By doing so we can express $\textbf{x}$ as a function of $\lambda$
|
||||
! \begin{align*}
|
||||
! \textbf{x}(\lambda) = - \sum_{i=1}^n \frac{\textbf{w}_i^T \cdot
|
||||
! \textbf{g}}{h_i + \lambda} \cdot \textbf{w}_i
|
||||
! \end{align*}
|
||||
! with $\lambda \neq - h_i$.
|
||||
|
||||
! An interesting thing in our case is the norm of $\textbf{x}$,
|
||||
! because we want $||\textbf{x}|| = \Delta$. Due to the orthogonality of
|
||||
! the eigenvectors $\left\{\textbf{w} \right\} _{i=1}^n$ we have:
|
||||
! \begin{align*}
|
||||
! ||\textbf{x}(\lambda)||^2 = \sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot
|
||||
! \textbf{g})^2}{(h_i + \lambda)^2}
|
||||
! \end{align*}
|
||||
|
||||
! So the $||\textbf{x}(\lambda)||^2$ is just a function of $\lambda$.
|
||||
! And if we study the properties of this function we see that:
|
||||
! \begin{align*}
|
||||
! \lim_{\lambda\to\infty} ||\textbf{x}(\lambda)|| = 0
|
||||
! \end{align*}
|
||||
! and if $\textbf{w}_i^T \cdot \textbf{g} \neq 0$:
|
||||
! \begin{align*}
|
||||
! \lim_{\lambda\to -h_i} ||\textbf{x}(\lambda)|| = + \infty
|
||||
! \end{align*}
|
||||
|
||||
! From these limits and knowing that $h_1$ is the lowest eigenvalue, we
|
||||
! can conclude that $||\textbf{x}(\lambda)||$ is a continuous and
|
||||
! strictly decreasing function on the interval $\lambda \in
|
||||
! (-h_1;\infty)$. Thus, there is one $\lambda$ in this interval which
|
||||
! gives $||\textbf{x}(\lambda)|| = \Delta$, consequently there is one
|
||||
! solution.
|
||||
|
||||
! Since $\textbf{x} = - (\textbf{H} + \lambda \textbf{I})^{-1} \cdot
|
||||
! \textbf{g}$ and we want to reduce the norm of $\textbf{x}$, clearly,
|
||||
! $\lambda > 0$ ($\lambda = 0$ is the unconstraint solution). But the
|
||||
! Newton method is only defined for a positive definite hessian matrix,
|
||||
! so $(\textbf{H} + \textbf{I} \lambda)$ must be positive
|
||||
! definite. Consequently, in the case where $\textbf{H}$ is not positive
|
||||
! definite, to ensure the positive definiteness, $\lambda$ must be
|
||||
! greater than $- h_1$.
|
||||
! \begin{align*}
|
||||
! \lambda > 0 \quad \text{and} \quad \lambda \geq - h_1
|
||||
! \end{align*}
|
||||
|
||||
! From that there are five cases:
|
||||
! - if $\textbf{H}$ is positive definite, $-h_1 < 0$, $\lambda \in (0,\infty)$
|
||||
! - if $\textbf{H}$ is not positive definite and $\textbf{w}_1^T \cdot
|
||||
! \textbf{g} \neq 0$, $(\textbf{H} + \textbf{I}
|
||||
! \lambda)$
|
||||
! must be positve definite, $-h_1 > 0$, $\lambda \in (-h_1, \infty)$
|
||||
! - if $\textbf{H}$ is not positive definite , $\textbf{w}_1^T \cdot
|
||||
! \textbf{g} = 0$ and $||\textbf{x}(-h_1)|| > \Delta$ by removing
|
||||
! $j=1$ in the sum, $(\textbf{H} + \textbf{I} \lambda)$ must be
|
||||
! positive definite, $-h_1 > 0$, $\lambda \in (-h_1, \infty$)
|
||||
! - if $\textbf{H}$ is not positive definite , $\textbf{w}_1^T \cdot
|
||||
! \textbf{g} = 0$ and $||\textbf{x}(-h_1)|| \leq \Delta$ by removing
|
||||
! $j=1$ in the sum, $(\textbf{H} + \textbf{I} \lambda)$ must be
|
||||
! positive definite, $-h_1 > 0$, $\lambda = -h_1$). This case is
|
||||
! similar to the case where $\textbf{H}$ and $||\textbf{x}(\lambda =
|
||||
! 0)|| \leq \Delta$
|
||||
! but we can also add to $\textbf{x}$, the first eigenvector $\textbf{W}_1$
|
||||
! time a constant to ensure the condition $||\textbf{x}(\lambda =
|
||||
! -h_1)|| = \Delta$ and escape from the saddle point
|
||||
|
||||
! Thus to find the solution, we can write:
|
||||
! \begin{align*}
|
||||
! ||\textbf{x}(\lambda)|| = \Delta
|
||||
! \end{align*}
|
||||
! \begin{align*}
|
||||
! ||\textbf{x}(\lambda)|| - \Delta = 0
|
||||
! \end{align*}
|
||||
|
||||
! Taking the square of this equation
|
||||
! \begin{align*}
|
||||
! (||\textbf{x}(\lambda)|| - \Delta)^2 = 0
|
||||
! \end{align*}
|
||||
! we have a function with one minimum for the optimal $\lambda$.
|
||||
! Since we have the formula of $||\textbf{x}(\lambda)||^2$, we solve
|
||||
! \begin{align*}
|
||||
! (||\textbf{x}(\lambda)||^2 - \Delta^2)^2 = 0
|
||||
! \end{align*}
|
||||
|
||||
! But in practice, it is more effective to solve:
|
||||
! \begin{align*}
|
||||
! (\frac{1}{||\textbf{x}(\lambda)||^2} - \frac{1}{\Delta^2})^2 = 0
|
||||
! \end{align*}
|
||||
|
||||
! To do that, we just use the Newton method with "trust_newton" using
|
||||
! first and second derivative of $(||\textbf{x}(\lambda)||^2 -
|
||||
! \Delta^2)^2$ with respect to $\textbf{x}$.
|
||||
! This will give the optimal $\lambda$ to compute the
|
||||
! solution $\textbf{x}$ with the formula seen previously:
|
||||
! \begin{align*}
|
||||
! \textbf{x}(\lambda) = - \sum_{i=1}^n \frac{\textbf{w}_i^T \cdot
|
||||
! \textbf{g}}{h_i + \lambda} \cdot \textbf{w}_i
|
||||
! \end{align*}
|
||||
|
||||
! The solution $\textbf{x}(\lambda)$ with the optimal $\lambda$ is our
|
||||
! step to go from the (k)-th to the (k+1)-th iteration, is noted $\textbf{x}^*$.
|
||||
|
||||
|
||||
|
||||
|
||||
! Evolution of the trust region
|
||||
|
||||
! We initialize the trust region at the first iteration using a radius
|
||||
! \begin{align*}
|
||||
! \Delta = ||\textbf{x}(\lambda=0)||
|
||||
! \end{align*}
|
||||
|
||||
! And for the next iteration the trust region will evolves depending of
|
||||
! the agreement of the energy prediction based on the Taylor series
|
||||
! truncated at the 2nd order and the real energy. If the Taylor series
|
||||
! truncated at the 2nd order represents correctly the energy landscape
|
||||
! the trust region will be extent else it will be reduced. In order to
|
||||
! mesure this agreement we use the ratio rho cf. "rho_model" and
|
||||
! "trust_e_model". From that we use the following values:
|
||||
! - if $\rho \geq 0.75$, then $\Delta = 2 \Delta$,
|
||||
! - if $0.5 \geq \rho < 0.75$, then $\Delta = \Delta$,
|
||||
! - if $0.25 \geq \rho < 0.5$, then $\Delta = 0.5 \Delta$,
|
||||
! - if $\rho < 0.25$, then $\Delta = 0.25 \Delta$.
|
||||
|
||||
! In addition, if $\rho < 0.1$ the iteration is cancelled, so it
|
||||
! restarts with a smaller trust region until the energy decreases.
|
||||
|
||||
|
||||
|
||||
|
||||
! Summary
|
||||
|
||||
! To summarize, knowing the hessian (eigenvectors and eigenvalues), the
|
||||
! gradient and the radius of the trust region we can compute the norm of
|
||||
! the Newton step
|
||||
! \begin{align*}
|
||||
! ||\textbf{x}(\lambda = 0)||^2 = ||- \textbf{H}^{-1} \cdot \textbf{g}||^2 = \sum_{i=1}^n
|
||||
! \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2}, \quad h_i \neq 0
|
||||
! \end{align*}
|
||||
|
||||
! - if $h_1 \geq 0$, $||\textbf{x}(\lambda = 0)|| \leq \Delta$ and
|
||||
! $\textbf{x}(\lambda=0)$ is in the trust region and it is not
|
||||
! necessary to put a constraint on $\textbf{x}$, the solution is the
|
||||
! unconstrained one, $\textbf{x}^* = \textbf{x}(\lambda = 0)$.
|
||||
! - else if $h_1 < 0$, $\textbf{w}_1^T \cdot \textbf{g} = 0$ and
|
||||
! $||\textbf{x}(\lambda = -h_1)|| \leq \Delta$ (by removing $j=1$ in
|
||||
! the sum), the solution is $\textbf{x}^* = \textbf{x}(\lambda =
|
||||
! -h_1)$, similarly to the previous case.
|
||||
! But we can add to $\textbf{x}$, the first eigenvector $\textbf{W}_1$
|
||||
! time a constant to ensure the condition $||\textbf{x}(\lambda =
|
||||
! -h_1)|| = \Delta$ and escape from the saddle point
|
||||
! - else if $h_1 < 0$ and $\textbf{w}_1^T \cdot \textbf{g} \neq 0$ we
|
||||
! have to search $\lambda \in (-h_1, \infty)$ such as
|
||||
! $\textbf{x}(\lambda) = \Delta$ by solving with the Newton method
|
||||
! \begin{align*}
|
||||
! (||\textbf{x}(\lambda)||^2 - \Delta^2)^2 = 0
|
||||
! \end{align*}
|
||||
! or
|
||||
! \begin{align*}
|
||||
! (\frac{1}{||\textbf{x}(\lambda)||^2} - \frac{1}{\Delta^2})^2 = 0
|
||||
! \end{align*}
|
||||
! which is numerically more stable. And finally compute
|
||||
! \begin{align*}
|
||||
! \textbf{x}^* = \textbf{x}(\lambda) = - \sum_{i=1}^n \frac{\textbf{w}_i^T \cdot
|
||||
! \textbf{g}}{h_i + \lambda} \cdot \textbf{w}_i
|
||||
! \end{align*}
|
||||
! - else if $h_1 \geq 0$ and $||\textbf{x}(\lambda = 0)|| > \Delta$ we
|
||||
! do exactly the same thing that the previous case but we search
|
||||
! $\lambda \in (0, \infty)$
|
||||
! - else if $h_1 < 0$ and $\textbf{w}_1^T \cdot \textbf{g} = 0$ and
|
||||
! $||\textbf{x}(\lambda = -h_1)|| > \Delta$ (by removing $j=1$ in the
|
||||
! sum), again we do exactly the same thing that the previous case
|
||||
! searching $\lambda \in (-h_1, \infty)$.
|
||||
|
||||
|
||||
! For the cases where $\textbf{w}_1^T \cdot \textbf{g} = 0$ it is not
|
||||
! necessary in fact to remove the $j = 1$ in the sum since the term
|
||||
! where $h_i - \lambda < 10^{-6}$ are not computed.
|
||||
|
||||
! After that, we take this vector $\textbf{x}^*$, called "x", and we do
|
||||
! the transformation to an antisymmetric matrix $\textbf{X}$, called
|
||||
! m_x. This matrix $\textbf{X}$ will be used to compute a rotation
|
||||
! matrix $\textbf{R}= \exp(\textbf{X})$ in "rotation_matrix".
|
||||
|
||||
! NB:
|
||||
! An improvement can be done using a elleptical trust region.
|
||||
|
||||
|
||||
|
||||
|
||||
! Code
|
||||
|
||||
! Provided:
|
||||
! | mo_num | integer | number of MOs |
|
||||
|
||||
! Cf. qp_edit in orbital optimization section, for some constants/thresholds
|
||||
|
||||
! Input:
|
||||
! | m | integer | number of MOs |
|
||||
! | n | integer | m*(m-1)/2 |
|
||||
! | n2 | integer | m*(m-1)/2 or 1 if the hessian is diagonal |
|
||||
! | H(n,n2) | double precision | hessian |
|
||||
! | v_grad(n) | double precision | gradient |
|
||||
! | e_val(n) | double precision | eigenvalues of the hessian |
|
||||
! | W(n, n) | double precision | eigenvectors of the hessian |
|
||||
! | rho | double precision | agreement between the model and the reality, |
|
||||
! | | | represents the quality of the energy prediction |
|
||||
! | nb_iter | integer | number of iteration |
|
||||
|
||||
! Input/Ouput:
|
||||
! | delta | double precision | radius of the trust region |
|
||||
|
||||
! Output:
|
||||
! | x(n) | double precision | vector containing the step |
|
||||
|
||||
! Internal:
|
||||
! | accu | double precision | temporary variable to compute the step |
|
||||
! | lambda | double precision | lagrange multiplier |
|
||||
! | trust_radius2 | double precision | square of the radius of the trust region |
|
||||
! | norm2_x | double precision | norm^2 of the vector x |
|
||||
! | norm2_g | double precision | norm^2 of the vector containing the gradient |
|
||||
! | tmp_wtg(n) | double precision | tmp_wtg(i) = w_i^T . g |
|
||||
! | i, j, k | integer | indexes |
|
||||
|
||||
! Function:
|
||||
! | dnrm2 | double precision | Blas function computing the norm |
|
||||
! | f_norm_trust_region_omp | double precision | compute the value of norm(x(lambda)^2) |
|
||||
|
||||
|
||||
subroutine trust_region_step(n,n2,nb_iter,v_grad,rho,e_val,w,x,delta)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compuet the step in the trust region
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer, intent(in) :: n,n2
|
||||
double precision, intent(in) :: v_grad(n), rho
|
||||
integer, intent(inout) :: nb_iter
|
||||
double precision, intent(in) :: e_val(n), w(n,n2)
|
||||
|
||||
! inout
|
||||
double precision, intent(inout) :: delta
|
||||
|
||||
! out
|
||||
double precision, intent(out) :: x(n)
|
||||
|
||||
! Internal
|
||||
double precision :: accu, lambda, trust_radius2
|
||||
double precision :: norm2_x, norm2_g
|
||||
double precision, allocatable :: tmp_wtg(:)
|
||||
integer :: i,j,k
|
||||
double precision :: t1,t2,t3
|
||||
integer :: n_neg_eval
|
||||
|
||||
|
||||
! Functions
|
||||
double precision :: ddot, dnrm2
|
||||
double precision :: f_norm_trust_region_omp
|
||||
|
||||
print*,''
|
||||
print*,'=================='
|
||||
print*,'---Trust_region---'
|
||||
print*,'=================='
|
||||
|
||||
call wall_time(t1)
|
||||
|
||||
! Allocation
|
||||
allocate(tmp_wtg(n))
|
||||
|
||||
! Initialization and norm
|
||||
|
||||
! The norm of the step size will be useful for the trust region
|
||||
! algorithm. We start from a first guess and the radius of the trust
|
||||
! region will evolve during the optimization.
|
||||
|
||||
! avoid_saddle is actually a test to avoid saddle points
|
||||
|
||||
|
||||
! Initialization of the Lagrange multiplier
|
||||
lambda = 0d0
|
||||
|
||||
! List of w^T.g, to avoid the recomputation
|
||||
tmp_wtg = 0d0
|
||||
if (n == n2) then
|
||||
do j = 1, n
|
||||
do i = 1, n
|
||||
tmp_wtg(j) = tmp_wtg(j) + w(i,j) * v_grad(i)
|
||||
enddo
|
||||
enddo
|
||||
else
|
||||
! For the diagonal case
|
||||
do j = 1, n
|
||||
k = int(w(j,1)+1d-15)
|
||||
tmp_wtg(j) = v_grad(k)
|
||||
enddo
|
||||
endif
|
||||
|
||||
! Replacement of the small tmp_wtg corresponding to a negative eigenvalue
|
||||
! in the case of avoid_saddle
|
||||
if (avoid_saddle .and. e_val(1) < - thresh_eig) then
|
||||
i = 2
|
||||
! Number of negative eigenvalues
|
||||
do while (e_val(i) < - thresh_eig)
|
||||
if (tmp_wtg(i) < thresh_wtg2) then
|
||||
if (version_avoid_saddle == 1) then
|
||||
tmp_wtg(i) = 1d0
|
||||
elseif (version_avoid_saddle == 2) then
|
||||
tmp_wtg(i) = DABS(e_val(i))
|
||||
elseif (version_avoid_saddle == 3) then
|
||||
tmp_wtg(i) = dsqrt(DABS(e_val(i)))
|
||||
else
|
||||
tmp_wtg(i) = thresh_wtg2
|
||||
endif
|
||||
endif
|
||||
i = i + 1
|
||||
enddo
|
||||
|
||||
! For the fist one it's a little bit different
|
||||
if (tmp_wtg(1) < thresh_wtg2) then
|
||||
tmp_wtg(1) = 0d0
|
||||
endif
|
||||
|
||||
endif
|
||||
|
||||
! Norm^2 of x, ||x||^2
|
||||
norm2_x = f_norm_trust_region_omp(n,e_val,tmp_wtg,0d0)
|
||||
! We just use this norm for the nb_iter = 0 in order to initialize the trust radius delta
|
||||
! We don't care about the sign of the eigenvalue we just want the size of the step in a normal Newton-Raphson algorithm
|
||||
! Anyway if the step is too big it will be reduced
|
||||
!print*,'||x||^2 :', norm2_x
|
||||
|
||||
! Norm^2 of the gradient, ||v_grad||^2
|
||||
norm2_g = (dnrm2(n,v_grad,1))**2
|
||||
!print*,'||grad||^2 :', norm2_g
|
||||
|
||||
! Trust radius initialization
|
||||
|
||||
! At the first iteration (nb_iter = 0) we initialize the trust region
|
||||
! with the norm of the step generate by the Newton's method ($\textbf{x}_1 =
|
||||
! (\textbf{H}_0)^{-1} \cdot \textbf{g}_0$,
|
||||
! we compute this norm using f_norm_trust_region_omp as explain just
|
||||
! below)
|
||||
|
||||
|
||||
! trust radius
|
||||
if (nb_iter == 0) then
|
||||
trust_radius2 = norm2_x
|
||||
! To avoid infinite loop of cancellation of this first step
|
||||
! without changing delta
|
||||
nb_iter = 1
|
||||
|
||||
! Compute delta, delta = sqrt(trust_radius)
|
||||
delta = dsqrt(trust_radius2)
|
||||
endif
|
||||
|
||||
! Modification of the trust radius
|
||||
|
||||
! In function of rho (which represents the agreement between the model
|
||||
! and the reality, cf. rho_model) the trust region evolves. We update
|
||||
! delta (the radius of the trust region).
|
||||
|
||||
! To avoid too big trust region we put a maximum size.
|
||||
|
||||
|
||||
! Modification of the trust radius in function of rho
|
||||
if (rho >= 0.75d0) then
|
||||
delta = 2d0 * delta
|
||||
elseif (rho >= 0.5d0) then
|
||||
delta = delta
|
||||
elseif (rho >= 0.25d0) then
|
||||
delta = 0.5d0 * delta
|
||||
else
|
||||
delta = 0.25d0 * delta
|
||||
endif
|
||||
|
||||
! Maximum size of the trust region
|
||||
!if (delta > 0.5d0 * n * pi) then
|
||||
! delta = 0.5d0 * n * pi
|
||||
! print*,'Delta > delta_max, delta = 0.5d0 * n * pi'
|
||||
!endif
|
||||
|
||||
if (delta > 1d10) then
|
||||
delta = 1d10
|
||||
endif
|
||||
|
||||
!print*, 'Delta :', delta
|
||||
|
||||
! Calculation of the optimal lambda
|
||||
|
||||
! We search the solution of $(||x||^2 - \Delta^2)^2 = 0$
|
||||
! - If $||\textbf{x}|| > \Delta$ or $h_1 < 0$ we have to add a constant
|
||||
! $\lambda > 0 \quad \text{and} \quad \lambda > -h_1$
|
||||
! - If $||\textbf{x}|| \leq \Delta$ and $h_1 \geq 0$ the solution is the
|
||||
! unconstrained one, $\lambda = 0$
|
||||
|
||||
! You will find more details at the beginning
|
||||
|
||||
|
||||
! By giving delta, we search (||x||^2 - delta^2)^2 = 0
|
||||
! and not (||x||^2 - delta)^2 = 0
|
||||
|
||||
! Research of lambda to solve ||x(lambda)|| = Delta
|
||||
|
||||
! Display
|
||||
!print*, 'e_val(1) = ', e_val(1)
|
||||
!print*, 'w_1^T.g =', tmp_wtg(1)
|
||||
|
||||
! H positive definite
|
||||
if (e_val(1) > - thresh_eig) then
|
||||
norm2_x = f_norm_trust_region_omp(n,e_val,tmp_wtg,0d0)
|
||||
!print*, '||x(0)||=', dsqrt(norm2_x)
|
||||
!print*, 'Delta=', delta
|
||||
|
||||
! H positive definite, ||x(lambda = 0)|| <= Delta
|
||||
if (dsqrt(norm2_x) <= delta) then
|
||||
!print*, 'H positive definite, ||x(lambda = 0)|| <= Delta'
|
||||
!print*, 'lambda = 0, no lambda optimization'
|
||||
lambda = 0d0
|
||||
|
||||
! H positive definite, ||x(lambda = 0)|| > Delta
|
||||
else
|
||||
! Constraint solution
|
||||
!print*, 'H positive definite, ||x(lambda = 0)|| > Delta'
|
||||
!print*,'Computation of the optimal lambda...'
|
||||
call trust_region_optimal_lambda(n,e_val,tmp_wtg,delta,lambda)
|
||||
endif
|
||||
|
||||
! H indefinite
|
||||
else
|
||||
if (DABS(tmp_wtg(1)) < thresh_wtg) then
|
||||
norm2_x = f_norm_trust_region_omp(n,e_val,tmp_wtg, - e_val(1))
|
||||
!print*, 'w_1^T.g <', thresh_wtg,', ||x(lambda = -e_val(1))|| =', dsqrt(norm2_x)
|
||||
endif
|
||||
|
||||
! H indefinite, w_1^T.g = 0, ||x(lambda = -e_val(1))|| <= Delta
|
||||
if (dsqrt(norm2_x) <= delta .and. DABS(tmp_wtg(1)) < thresh_wtg) then
|
||||
! Add e_val(1) in order to have (H - e_val(1) I) positive definite
|
||||
!print*, 'H indefinite, w_1^T.g = 0, ||x(lambda = -e_val(1))|| <= Delta'
|
||||
!print*, 'lambda = -e_val(1), no lambda optimization'
|
||||
lambda = - e_val(1)
|
||||
|
||||
! H indefinite, w_1^T.g = 0, ||x(lambda = -e_val(1))|| > Delta
|
||||
! and
|
||||
! H indefinite, w_1^T.g =/= 0
|
||||
else
|
||||
! Constraint solution/ add lambda
|
||||
!if (DABS(tmp_wtg(1)) < thresh_wtg) then
|
||||
! print*, 'H indefinite, w_1^T.g = 0, ||x(lambda = -e_val(1))|| > Delta'
|
||||
!else
|
||||
! print*, 'H indefinite, w_1^T.g =/= 0'
|
||||
!endif
|
||||
!print*, 'Computation of the optimal lambda...'
|
||||
call trust_region_optimal_lambda(n,e_val,tmp_wtg,delta,lambda)
|
||||
endif
|
||||
|
||||
endif
|
||||
|
||||
! Recomputation of the norm^2 of the step x
|
||||
norm2_x = f_norm_trust_region_omp(n,e_val,tmp_wtg,lambda)
|
||||
print*,''
|
||||
print*,'Summary after the trust region:'
|
||||
print*,'lambda:', lambda
|
||||
print*,'||x||:', dsqrt(norm2_x)
|
||||
print*,'delta:', delta
|
||||
|
||||
! Calculation of the step x
|
||||
|
||||
! x refers to $\textbf{x}^*$
|
||||
! We compute x in function of lambda using its formula :
|
||||
! \begin{align*}
|
||||
! \textbf{x}^* = \textbf{x}(\lambda) = - \sum_{i=1}^n \frac{\textbf{w}_i^T \cdot \textbf{g}}{h_i
|
||||
! + \lambda} \cdot \textbf{w}_i
|
||||
! \end{align*}
|
||||
|
||||
|
||||
! Initialisation
|
||||
x = 0d0
|
||||
|
||||
! Calculation of the step x
|
||||
|
||||
if (n == n2) then
|
||||
! Normal version
|
||||
if (.not. absolute_eig) then
|
||||
|
||||
do i = 1, n
|
||||
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
|
||||
do j = 1, n
|
||||
x(j) = x(j) - tmp_wtg(i) * W(j,i) / (e_val(i) + lambda)
|
||||
enddo
|
||||
endif
|
||||
enddo
|
||||
|
||||
! Version to use the absolute value of the eigenvalues
|
||||
else
|
||||
|
||||
do i = 1, n
|
||||
if (DABS(e_val(i)) > thresh_eig) then
|
||||
do j = 1, n
|
||||
x(j) = x(j) - tmp_wtg(i) * W(j,i) / (DABS(e_val(i)) + lambda)
|
||||
enddo
|
||||
endif
|
||||
enddo
|
||||
|
||||
endif
|
||||
else
|
||||
! If the hessian is diagonal
|
||||
! Normal version
|
||||
if (.not. absolute_eig) then
|
||||
|
||||
do i = 1, n
|
||||
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
|
||||
j = int(w(i,1) + 1d-15)
|
||||
x(j) = - tmp_wtg(i) * 1d0 / (e_val(i) + lambda)
|
||||
endif
|
||||
enddo
|
||||
|
||||
! Version to use the absolute value of the eigenvalues
|
||||
else
|
||||
|
||||
do i = 1, n
|
||||
if (DABS(e_val(i)) > thresh_eig) then
|
||||
j = int(w(i,1) + 1d-15)
|
||||
x(j) = - tmp_wtg(i) * 1d0 / (DABS(e_val(i)) + lambda)
|
||||
endif
|
||||
enddo
|
||||
|
||||
endif
|
||||
endif
|
||||
|
||||
double precision :: beta, norm_x
|
||||
|
||||
! Test
|
||||
! If w_1^T.g = 0, the lim of ||x(lambda)|| when lambda tend to -e_val(1)
|
||||
! is not + infinity. So ||x(lambda=-e_val(1))|| < delta, we add the first
|
||||
! eigenvectors multiply by a constant to ensure the condition
|
||||
! ||x(lambda=-e_val(1))|| = delta and escape the saddle point
|
||||
if (avoid_saddle .and. e_val(1) < - thresh_eig) then
|
||||
if (tmp_wtg(1) < 1d-15 .and. (1d0 - dsqrt(norm2_x)/delta) > 1d-3 ) then
|
||||
|
||||
! norm of x
|
||||
norm_x = dnrm2(n,x,1)
|
||||
|
||||
! Computes the coefficient for the w_1
|
||||
beta = delta**2 - norm_x**2
|
||||
|
||||
! Updates the step x
|
||||
x = x + W(:,1) * dsqrt(beta)
|
||||
|
||||
! Recomputes the norm to check
|
||||
norm_x = dnrm2(n,x,1)
|
||||
|
||||
print*, 'Add w_1 * dsqrt(delta^2 - ||x||^2):'
|
||||
print*, '||x||', norm_x
|
||||
endif
|
||||
endif
|
||||
|
||||
! Transformation of x
|
||||
|
||||
! x is a vector of size n, so it can be write as a m by m
|
||||
! antisymmetric matrix m_x cf. "mat_to_vec_index" and "vec_to_mat_index".
|
||||
|
||||
|
||||
! ! Step transformation vector -> matrix
|
||||
! ! Vector with n element -> mo_num by mo_num matrix
|
||||
! do j = 1, m
|
||||
! do i = 1, m
|
||||
! if (i>j) then
|
||||
! call mat_to_vec_index(i,j,k)
|
||||
! m_x(i,j) = x(k)
|
||||
! else
|
||||
! m_x(i,j) = 0d0
|
||||
! endif
|
||||
! enddo
|
||||
! enddo
|
||||
!
|
||||
! ! Antisymmetrization of the previous matrix
|
||||
! do j = 1, m
|
||||
! do i = 1, m
|
||||
! if (i<j) then
|
||||
! m_x(i,j) = - m_x(j,i)
|
||||
! endif
|
||||
! enddo
|
||||
! enddo
|
||||
|
||||
! Deallocation, end
|
||||
|
||||
|
||||
deallocate(tmp_wtg)
|
||||
|
||||
call wall_time(t2)
|
||||
t3 = t2 - t1
|
||||
print*,'Time in trust_region:', t3
|
||||
print*,'======================'
|
||||
print*,'---End trust_region---'
|
||||
print*,'======================'
|
||||
|
||||
end
|
||||
|
|
@ -0,0 +1,71 @@
|
|||
! Vector to matrix indexes
|
||||
|
||||
! *Compute the indexes p,q of a matrix element with the vector index i*
|
||||
|
||||
! Vector (i) -> lower diagonal matrix (p,q), p > q
|
||||
|
||||
! If a matrix is antisymmetric it can be reshaped as a vector. And the
|
||||
! vector can be reshaped as an antisymmetric matrix
|
||||
|
||||
! \begin{align*}
|
||||
! \begin{pmatrix}
|
||||
! 0 & -1 & -2 & -4 \\
|
||||
! 1 & 0 & -3 & -5 \\
|
||||
! 2 & 3 & 0 & -6 \\
|
||||
! 4 & 5 & 6 & 0
|
||||
! \end{pmatrix}
|
||||
! \Leftrightarrow
|
||||
! \begin{pmatrix}
|
||||
! 1 & 2 & 3 & 4 & 5 & 6
|
||||
! \end{pmatrix}
|
||||
! \end{align*}
|
||||
|
||||
! !!! Here the algorithm only work for the lower diagonal !!!
|
||||
|
||||
! Input:
|
||||
! | i | integer | index in the vector |
|
||||
|
||||
! Ouput:
|
||||
! | p,q | integer | corresponding indexes in the lower diagonal of a matrix |
|
||||
! | | | p > q, |
|
||||
! | | | p -> row, |
|
||||
! | | | q -> column |
|
||||
|
||||
|
||||
subroutine vec_to_mat_index(i,p,q)
|
||||
|
||||
include 'pi.h'
|
||||
|
||||
!BEGIN_DOC
|
||||
! Compute the indexes (p,q) of the element in the lower diagonal matrix knowing
|
||||
! its index i a vector
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
! Variables
|
||||
|
||||
! in
|
||||
integer,intent(in) :: i
|
||||
|
||||
! out
|
||||
integer, intent(out) :: p,q
|
||||
|
||||
! internal
|
||||
integer :: a,b
|
||||
double precision :: da
|
||||
|
||||
da = 0.5d0*(1+ sqrt(1d0+8d0*DBLE(i)))
|
||||
a = INT(da)
|
||||
if ((a*(a-1))/2==i) then
|
||||
p = a-1
|
||||
else
|
||||
p = a
|
||||
endif
|
||||
b = p*(p-1)/2
|
||||
|
||||
! Matrix element indexes
|
||||
p = p + 1
|
||||
q = i - b
|
||||
|
||||
end subroutine
|
||||
|
|
@ -0,0 +1,39 @@
|
|||
! Vect to antisymmetric matrix using mat_to_vec_index
|
||||
|
||||
! Vector to antisymmetric matrix transformation using mat_to_vec_index
|
||||
! subroutine.
|
||||
|
||||
! Can be done in OMP (for the first part and with omp critical for the second)
|
||||
|
||||
|
||||
subroutine vec_to_mat_v2(n,m,v_x,m_x)
|
||||
|
||||
!BEGIN_DOC
|
||||
! Vector to antisymmetric matrix
|
||||
!END_DOC
|
||||
|
||||
implicit none
|
||||
|
||||
integer, intent(in) :: n,m
|
||||
double precision, intent(in) :: v_x(n)
|
||||
double precision, intent(out) :: m_x(m,m)
|
||||
|
||||
integer :: i,j,k
|
||||
|
||||
! 1D -> 2D lower diagonal
|
||||
m_x = 0d0
|
||||
do j = 1, m - 1
|
||||
do i = j + 1, m
|
||||
call mat_to_vec_index(i,j,k)
|
||||
m_x(i,j) = v_x(k)
|
||||
enddo
|
||||
enddo
|
||||
|
||||
! Antisym
|
||||
do i = 1, m - 1
|
||||
do j = i + 1, m
|
||||
m_x(i,j) = - m_x(j,i)
|
||||
enddo
|
||||
enddo
|
||||
|
||||
end
|
||||
Loading…
Reference in New Issue