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qp2/src/utils_trust_region/trust_region_optimal_lambda.irp.f
Anthony Scemama d84092a29d
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Added utils_trust_region directory
2023-02-08 16:45:04 +01:00

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! Newton's method to find the optimal lambda
! *Compute the lambda value for the trust region*
! This subroutine uses the Newton method in order to find the optimal
! lambda. This constant is added on the diagonal of the hessian to shift
! the eiganvalues. It has a double role:
! - ensure that the resulting hessian is positive definite for the
! Newton method
! - constrain the step in the trust region, i.e.,
! $||\textbf{x}(\lambda)|| \leq \Delta$, where $\Delta$ is the radius
! of the trust region.
! We search $\lambda$ which minimizes
! \begin{align*}
! f(\lambda) = (||\textbf{x}_{(k+1)}(\lambda)||^2 -\Delta^2)^2
! \end{align*}
! or
! \begin{align*}
! \tilde{f}(\lambda) = (\frac{1}{||\textbf{x}_{(k+1)}(\lambda)||^2}-\frac{1}{\Delta^2})^2
! \end{align*}
! and gives obviously 0 in both cases. \newline
! There are several cases:
! - If $\textbf{H}$ is positive definite the interval containing the
! solution is $\lambda \in (0, \infty)$ (and $-h_1 < 0$).
! - If $\textbf{H}$ is indefinite ($h_1 < 0$) and $\textbf{w}_1^T \cdot
! \textbf{g} \neq 0$ then the interval containing
! the solution is $\lambda \in (-h_1, \infty)$.
! - If $\textbf{H}$ is indefinite ($h_1 < 0$) and $\textbf{w}_1^T \cdot
! \textbf{g} = 0$ then the interval containing the solution is
! $\lambda \in (-h_1, \infty)$. The terms where $|h_i - \lambda| <
! 10^{-12}$ are not computed, so the term where $i = 1$ is
! automatically removed and this case becomes similar to the previous one.
! So to avoid numerical problems (cf. trust_region) we start the
! algorithm at $\lambda=\max(0 + \epsilon,-h_1 + \epsilon)$,
! with $\epsilon$ a little constant.
! The research must be restricted to the interval containing the
! solution. For that reason a little trust region in 1D is used.
! The Newton method to find the optimal $\lambda$ is :
! \begin{align*}
! \lambda_{(l+1)} &= \lambda_{(l)} - f^{''}(\lambda)_{(l)}^{-1} f^{'}(\lambda)_{(l)}^{} \\
! \end{align*}
! $f^{'}(\lambda)_{(l)}$: the first derivative of $f$ with respect to
! $\lambda$ at the l-th iteration,
! $f^{''}(\lambda)_{(l)}$: the second derivative of $f$ with respect to
! $\lambda$ at the l-th iteration.\newline
! Noting the Newton step $y = - f^{''}(\lambda)_{(l)}^{-1}
! f^{'}(\lambda)_{(l)}^{}$ we constrain $y$ such as
! \begin{align*}
! y \leq \alpha
! \end{align*}
! with $\alpha$ a scalar representing the trust length (trust region in
! 1D) where the function $f$ or $\tilde{f}$ is correctly describe by the
! Taylor series truncated at the second order. Thus, if $y > \alpha$,
! the constraint is applied as
! \begin{align*}
! y^* = \alpha \frac{y}{|y|}
! \end{align*}
! with $y^*$ the solution in the trust region.
! The size of the trust region evolves in function of $\rho$ as for the
! trust region seen previously cf. trust_region, rho_model.
! The prediction of the value of $f$ or $\tilde{f}$ is done using the
! Taylor series truncated at the second order cf. "trust_region",
! "trust_e_model".
! The first and second derivatives of $f(\lambda) = (||\textbf{x}(\lambda)||^2 -
! \Delta^2)^2$ with respect to $\lambda$ are:
! \begin{align*}
! \frac{\partial }{\partial \lambda} (||\textbf{x}(\lambda)||^2 - \Delta^2)^2
! = 2 \left(\sum_{i=1}^n \frac{-2(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \right)
! \left( - \Delta^2 + \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i+ \lambda)^2} \right)
! \end{align*}
! \begin{align*}
! \frac{\partial^2}{\partial \lambda^2} (||\textbf{x}(\lambda)||^2 - \Delta^2)^2
! = 2 \left[ \left( \sum_{i=1}^n 6 \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^4} \right) \left( - \Delta^2 + \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \right) + \left( \sum_{i=1}^n -2 \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \right)^2 \right]
! \end{align*}
! The first and second derivatives of $\tilde{f}(\lambda) = (1/||\textbf{x}(\lambda)||^2 -
! 1/\Delta^2)^2$ with respect to $\lambda$ are:
! \begin{align*}
! \frac{\partial}{\partial \lambda} (1/||\textbf{x}(\lambda)||^2 - 1/\Delta^2)^2
! &= 4 \frac{\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3}}
! {(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - \frac{4}{\Delta^2} \frac{\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3)}}
! {(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \\
! &= 4 \sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3}
! \left( \frac{1}{(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - \frac{1}{\Delta^2 (\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \right)
! \end{align*}
! \begin{align*}
! \frac{\partial^2}{\partial \lambda^2} (1/||\textbf{x}(\lambda)||^2 - 1/\Delta^2)^2
! &= 4 \left[ \frac{(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3)})^2}
! {(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^4}
! - 3 \frac{\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^4}}
! {(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3} \right] \\
! &- \frac{4}{\Delta^2} \left[ \frac{(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}
! {(h_i + \lambda)^3)})^2}{(\sum_ {i=1}^n\frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - 3 \frac{\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^4}}
! {(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \right]
! \end{align*}
! Provided in qp_edit:
! | thresh_rho_2 |
! | thresh_cc |
! | nb_it_max_lambda |
! | version_lambda_search |
! | nb_it_max_pre_search |
! see qp_edit for more details
! Input:
! | n | integer | m*(m-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | tmp_wtg(n) | double precision | w_i^T.v_grad(i) |
! | delta | double precision | delta for the trust region |
! Output:
! | lambda | double precision | Lagrange multiplier to constrain the norm of the size of the Newton step |
! | | | lambda > 0 |
! Internal:
! | d1_N | double precision | value of d1_norm_trust_region |
! | d2_N | double precision | value of d2_norm_trust_region |
! | f_N | double precision | value of f_norm_trust_region |
! | prev_f_N | double precision | previous value of f_norm_trust_region |
! | f_R | double precision | (norm(x)^2 - delta^2)^2 or (1/norm(x)^2 - 1/delta^2)^2 |
! | prev_f_R | double precision | previous value of f_R |
! | model | double precision | predicted value of f_R from prev_f_R and y |
! | d_1 | double precision | value of the first derivative |
! | d_2 | double precision | value of the second derivative |
! | y | double precision | Newton's step, y = -f''^-1 . f' = lambda - prev_lambda |
! | prev_lambda | double precision | previous value of lambda |
! | t1,t2,t3 | double precision | wall time |
! | i | integer | index |
! | epsilon | double precision | little constant to avoid numerical problem |
! | rho_2 | double precision | (prev_f_R - f_R)/(prev_f_R - model), agreement between model and f_R |
! | version | integer | version of the root finding method |
! Function:
! | d1_norm_trust_region | double precision | first derivative with respect to lambda of (norm(x)^2 - Delta^2)^2 |
! | d2_norm_trust_region | double precision | first derivative with respect to lambda of (norm(x)^2 - Delta^2)^2 |
! | d1_norm_inverse_trust_region | double precision | first derivative with respect to lambda of (1/norm(x)^2 - 1/Delta^2)^2 |
! | d2_norm_inverse_trust_region | double precision | second derivative with respect to lambda of (1/norm(x)^2 - 1/Delta^2)^2 |
! | f_norm_trust_region | double precision | value of norm(x)^2 |
subroutine trust_region_optimal_lambda(n,e_val,tmp_wtg,delta,lambda)
include 'pi.h'
BEGIN_DOC
! Research the optimal lambda to constrain the step size in the trust region
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(inout) :: e_val(n)
double precision, intent(in) :: delta
double precision, intent(in) :: tmp_wtg(n)
! out
double precision, intent(out) :: lambda
! Internal
double precision :: d1_N, d2_N, f_N, prev_f_N
double precision :: prev_f_R, f_R
double precision :: model
double precision :: d_1, d_2
double precision :: t1,t2,t3
integer :: i
double precision :: epsilon
double precision :: y
double precision :: prev_lambda
double precision :: rho_2
double precision :: alpha
integer :: version
! Functions
double precision :: d1_norm_trust_region,d1_norm_trust_region_omp
double precision :: d2_norm_trust_region, d2_norm_trust_region_omp
double precision :: f_norm_trust_region, f_norm_trust_region_omp
double precision :: d1_norm_inverse_trust_region
double precision :: d2_norm_inverse_trust_region
double precision :: d1_norm_inverse_trust_region_omp
double precision :: d2_norm_inverse_trust_region_omp
print*,''
print*,'---Trust_newton---'
print*,''
call wall_time(t1)
! version_lambda_search
! 1 -> ||x||^2 - delta^2 = 0,
! 2 -> 1/||x||^2 - 1/delta^2 = 0 (better)
if (version_lambda_search == 1) then
print*, 'Research of the optimal lambda by solving ||x||^2 - delta^2 = 0'
else
print*, 'Research of the optimal lambda by solving 1/||x||^2 - 1/delta^2 = 0'
endif
! Version 2 is normally better
! Resolution with the Newton method:
! Initialization
epsilon = 1d-4
lambda =MAX(0d0, -e_val(1))
! Pre research of lambda to start near the optimal lambda
! by adding a constant epsilon and changing the constant to
! have ||x(lambda + epsilon)|| ~ delta, before setting
! lambda = lambda + epsilon
print*, 'Pre research of lambda:'
print*,'Initial lambda =', lambda
f_N = f_norm_trust_region_omp(n,e_val,tmp_wtg,lambda + epsilon)
print*,'||x(lambda)||=', dsqrt(f_N),'delta=',delta
i = 1
! To increase lambda
if (f_N > delta**2) then
print*,'Increasing lambda...'
do while (f_N > delta**2 .and. i <= nb_it_max_pre_search)
! Update the previous norm
prev_f_N = f_N
! New epsilon
epsilon = epsilon * 2d0
! New norm
f_N = f_norm_trust_region_omp(n,e_val,tmp_wtg,lambda + epsilon)
print*, 'lambda', lambda + epsilon, '||x||', dsqrt(f_N), 'delta', delta
! Security
if (prev_f_N < f_N) then
print*,'WARNING, error: prev_f_N < f_N, exit'
epsilon = epsilon * 0.5d0
i = nb_it_max_pre_search + 1
endif
i = i + 1
enddo
! To reduce lambda
else
print*,'Reducing lambda...'
do while (f_N < delta**2 .and. i <= nb_it_max_pre_search)
! Update the previous norm
prev_f_N = f_N
! New epsilon
epsilon = epsilon * 0.5d0
! New norm
f_N = f_norm_trust_region_omp(n,e_val,tmp_wtg,lambda + epsilon)
print*, 'lambda', lambda + epsilon, '||x||', dsqrt(f_N), 'delta', delta
! Security
if (prev_f_N > f_N) then
print*,'WARNING, error: prev_f_N > f_N, exit'
epsilon = epsilon * 2d0
i = nb_it_max_pre_search + 1
endif
i = i + 1
enddo
endif
print*,'End of the pre research of lambda'
! New value of lambda
lambda = lambda + epsilon
print*, 'e_val(1):', e_val(1)
print*, 'Staring point, lambda =', lambda
! thresh_cc, threshold for the research of the optimal lambda
! Leaves the loop when ABS(1d0-||x||^2/delta^2) > thresh_cc
! thresh_rho_2, threshold to cancel the step in the research
! of the optimal lambda, the step is cancelled if rho_2 < thresh_rho_2
print*,'Threshold for the CC:', thresh_cc
print*,'Threshold for rho_2:', thresh_rho_2
print*, 'w_1^T . g =', tmp_wtg(1)
! Debug
!if (debug) then
! print*, 'Iteration rho_2 lambda delta ||x|| |1-(||x||^2/delta^2)|'
!endif
! Initialization
i = 1
f_N = f_norm_trust_region_omp(n,e_val,tmp_wtg,lambda) ! Value of the ||x(lambda)||^2
model = 0d0 ! predicted value of (||x||^2 - delta^2)^2
prev_f_N = 0d0 ! previous value of ||x||^2
prev_f_R = 0d0 ! previous value of (||x||^2 - delta^2)^2
f_R = 0d0 ! value of (||x||^2 - delta^2)^2
rho_2 = 0d0 ! (prev_f_R - f_R)/(prev_f_R - m)
y = 0d0 ! step size
prev_lambda = 0d0 ! previous lambda
! Derivatives
if (version_lambda_search == 1) then
d_1 = d1_norm_trust_region_omp(n,e_val,tmp_wtg,lambda,delta) ! first derivative of (||x(lambda)||^2 - delta^2)^2
d_2 = d2_norm_trust_region_omp(n,e_val,tmp_wtg,lambda,delta) ! second derivative of (||x(lambda)||^2 - delta^2)^2
else
d_1 = d1_norm_inverse_trust_region_omp(n,e_val,tmp_wtg,lambda,delta) ! first derivative of (1/||x(lambda)||^2 - 1/delta^2)^2
d_2 = d2_norm_inverse_trust_region_omp(n,e_val,tmp_wtg,lambda,delta) ! second derivative of (1/||x(lambda)||^2 - 1/delta^2)^2
endif
! Trust length
alpha = DABS((1d0/d_2)*d_1)
! Newton's method
do while (i <= 100 .and. DABS(1d0-f_N/delta**2) > thresh_cc)
print*,'--------------------------------------'
print*,'Research of lambda, iteration:', i
print*,'--------------------------------------'
! Update of f_N, f_R and the derivatives
prev_f_N = f_N
if (version_lambda_search == 1) then
prev_f_R = (prev_f_N - delta**2)**2
d_1 = d1_norm_trust_region_omp(n,e_val,tmp_wtg,lambda,delta) ! first derivative of (||x(lambda)||^2 - delta^2)^2
d_2 = d2_norm_trust_region_omp(n,e_val,tmp_wtg,lambda,delta) ! second derivative of (||x(lambda)||^2 - delta^2)^2
else
prev_f_R = (1d0/prev_f_N - 1d0/delta**2)**2
d_1 = d1_norm_inverse_trust_region_omp(n,e_val,tmp_wtg,lambda,delta) ! first derivative of (1/||x(lambda)||^2 - 1/delta^2)^2
d_2 = d2_norm_inverse_trust_region_omp(n,e_val,tmp_wtg,lambda,delta) ! second derivative of (1/||x(lambda)||^2 - 1/delta^2)^2
endif
write(*,'(a,E12.5,a,E12.5)') ' 1st and 2nd derivative: ', d_1,', ', d_2
! Newton's step
y = -(1d0/DABS(d_2))*d_1
! Constraint on y (the newton step)
if (DABS(y) > alpha) then
y = alpha * (y/DABS(y)) ! preservation of the sign of y
endif
write(*,'(a,E12.5)') ' Step length: ', y
! Predicted value of (||x(lambda)||^2 - delta^2)^2, Taylor series
model = prev_f_R + d_1 * y + 0.5d0 * d_2 * y**2
! Updates lambda
prev_lambda = lambda
lambda = prev_lambda + y
print*,'prev lambda:', prev_lambda
print*,'new lambda:', lambda
! Checks if lambda is in (-h_1, \infty)
if (lambda > MAX(0d0, -e_val(1))) then
! New value of ||x(lambda)||^2
f_N = f_norm_trust_region_omp(n,e_val,tmp_wtg,lambda)
! New f_R
if (version_lambda_search == 1) then
f_R = (f_N - delta**2)**2 ! new value of (||x(lambda)||^2 - delta^2)^2
else
f_R = (1d0/f_N - 1d0/delta**2)**2 ! new value of (1/||x(lambda)||^2 -1/delta^2)^2
endif
if (version_lambda_search == 1) then
print*,'Previous value of (||x(lambda)||^2 - delta^2)^2:', prev_f_R
print*,'Actual value of (||x(lambda)||^2 - delta^2)^2:', f_R
print*,'Predicted value of (||x(lambda)||^2 - delta^2)^2:', model
else
print*,'Previous value of (1/||x(lambda)||^2 - 1/delta^2)^2:', prev_f_R
print*,'Actual value of (1/||x(lambda)||^2 - 1/delta^2)^2:', f_R
print*,'Predicted value of (1/||x(lambda)||^2 - 1/delta^2)^2:', model
endif
print*,'previous - actual:', prev_f_R - f_R
print*,'previous - model:', prev_f_R - model
! Check the gain
if (DABS(prev_f_R - model) < thresh_model_2) then
print*,''
print*,'WARNING: ABS(previous - model) <', thresh_model_2, 'rho_2 will tend toward infinity'
print*,''
endif
! Will be deleted
!if (prev_f_R - f_R <= 1d-16 .or. prev_f_R - model <= 1d-16) then
! print*,''
! print*,'WARNING: ABS(previous - model) <= 1d-16, exit'
! print*,''
! exit
!endif
! Computes rho_2
rho_2 = (prev_f_R - f_R)/(prev_f_R - model)
print*,'rho_2:', rho_2
else
rho_2 = 0d0 ! in order to reduce the size of the trust region, alpha, until lambda is in (-h_1, \infty)
print*,'lambda < -e_val(1) ===> rho_2 = 0'
endif
! Evolution of the trust length, alpha
if (rho_2 >= 0.75d0) then
alpha = 2d0 * alpha
elseif (rho_2 >= 0.5d0) then
alpha = alpha
elseif (rho_2 >= 0.25d0) then
alpha = 0.5d0 * alpha
else
alpha = 0.25d0 * alpha
endif
write(*,'(a,E12.5)') ' New trust length alpha: ', alpha
! cancellaion of the step if rho < 0.1
if (rho_2 < thresh_rho_2) then !0.1d0) then
lambda = prev_lambda
f_N = prev_f_N
print*,'Rho_2 <', thresh_rho_2,', cancellation of the step: lambda = prev_lambda'
endif
print*,''
print*,'lambda, ||x||, delta:'
print*, lambda, dsqrt(f_N), delta
print*,'CC:', DABS(1d0 - f_N/delta**2)
print*,''
i = i + 1
enddo
! if trust newton failed
if (i > nb_it_max_lambda) then
print*,''
print*,'######################################################'
print*,'WARNING: i >', nb_it_max_lambda,'for the trust Newton'
print*,'The research of the optimal lambda has failed'
print*,'######################################################'
print*,''
endif
print*,'Number of iterations :', i
print*,'Value of lambda :', lambda
print*,'Error on the trust region (1d0-f_N/delta**2) (Convergence criterion) :', 1d0-f_N/delta**2
print*,'Error on the trust region (||x||^2 - delta^2)^2) :', (f_N - delta**2)**2
print*,'Error on the trust region (1/||x||^2 - 1/delta^2)^2)', (1d0/f_N - 1d0/delta**2)**2
! Time
call wall_time(t2)
t3 = t2 - t1
print*,'Time in trust_newton:', t3
print*,''
print*,'---End trust_newton---'
print*,''
end subroutine
! OMP: First derivative of (||x||^2 - Delta^2)^2
! *Function to compute the first derivative of (||x||^2 - Delta^2)^2*
! This function computes the first derivative of (||x||^2 - Delta^2)^2
! with respect to lambda.
! \begin{align*}
! \frac{\partial }{\partial \lambda} (||\textbf{x}(\lambda)||^2 - \Delta^2)^2
! = -4 \left(\sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3} \right)
! \left( - \Delta^2 + \sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i+ \lambda)^2} \right)
! \end{align*}
! \begin{align*}
! \text{accu1} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2} \\
! \text{accu2} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3}
! \end{align*}
! Provided:
! | mo_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | accu1 | double precision | first sum of the formula |
! | accu2 | double precision | second sum of the formula |
! | tmp_accu1 | double precision | temporary array for the first sum |
! | tmp_accu2 | double precision | temporary array for the second sum |
! | tmp_wtg(n) | double precision | temporary array for W^t.v_grad |
! | i,j | integer | indexes |
! Function:
! | d1_norm_trust_region | double precision | first derivative with respect to lambda of (norm(x)^2 - Delta^2)^2 |
function d1_norm_trust_region_omp(n,e_val,tmp_wtg,lambda,delta)
use omp_lib
include 'pi.h'
BEGIN_DOC
! Compute the first derivative with respect to lambda of (||x(lambda)||^2 - Delta^2)^2
END_DOC
implicit none
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: tmp_wtg(n)
double precision, intent(in) :: lambda
double precision, intent(in) :: delta
! Internal
double precision :: wtg,accu1,accu2
integer :: i,j
double precision, allocatable :: tmp_accu1(:), tmp_accu2(:)
! Functions
double precision :: d1_norm_trust_region_omp
! Allocation
allocate(tmp_accu1(n), tmp_accu2(n))
! OMP
call omp_set_max_active_levels(1)
! OMP
!$OMP PARALLEL &
!$OMP PRIVATE(i,j) &
!$OMP SHARED(n,lambda, e_val, thresh_eig,&
!$OMP tmp_accu1, tmp_accu2, tmp_wtg, accu1,accu2) &
!$OMP DEFAULT(NONE)
!$OMP MASTER
accu1 = 0d0
accu2 = 0d0
!$OMP END MASTER
!$OMP DO
do i = 1, n
tmp_accu1(i) = 0d0
enddo
!$OMP END DO
!$OMP DO
do i = 1, n
tmp_accu2(i) = 0d0
enddo
!$OMP END DO
!$OMP DO
do i = 1, n
if (ABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu1(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**2
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu1 = accu1 + tmp_accu1(i)
enddo
!$OMP END MASTER
!$OMP DO
do i = 1, n
if (ABS(e_val(i)) > thresh_eig) then
tmp_accu2(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**3
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu2 = accu2 + tmp_accu2(i)
enddo
!$OMP END MASTER
!$OMP END PARALLEL
call omp_set_max_active_levels(4)
d1_norm_trust_region_omp = -4d0 * accu2 * (accu1 - delta**2)
deallocate(tmp_accu1, tmp_accu2)
end function
! OMP: Second derivative of (||x||^2 - Delta^2)^2
! *Function to compute the second derivative of (||x||^2 - Delta^2)^2*
! This function computes the second derivative of (||x||^2 - Delta^2)^2
! with respect to lambda.
! \begin{align*}
! \frac{\partial^2 }{\partial \lambda^2} (||\textbf{x}(\lambda)||^2 - \Delta^2)^2
! = 2 \left[ \left( \sum_{i=1}^n 6 \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^4} \right) \left( - \Delta^2 + \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \right) + \left( \sum_{i=1}^n -2 \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \right)^2 \right]
! \end{align*}
! \begin{align*}
! \text{accu1} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \\
! \text{accu2} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \\
! \text{accu3} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^4}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | accu1 | double precision | first sum of the formula |
! | accu2 | double precision | second sum of the formula |
! | accu3 | double precision | third sum of the formula |
! | tmp_accu1 | double precision | temporary array for the first sum |
! | tmp_accu2 | double precision | temporary array for the second sum |
! | tmp_accu2 | double precision | temporary array for the third sum |
! | tmp_wtg(n) | double precision | temporary array for W^t.v_grad |
! | i,j | integer | indexes |
! Function:
! | d2_norm_trust_region | double precision | second derivative with respect to lambda of (norm(x)^2 - Delta^2)^2 |
function d2_norm_trust_region_omp(n,e_val,tmp_wtg,lambda,delta)
use omp_lib
include 'pi.h'
BEGIN_DOC
! Compute the second derivative with respect to lambda of (||x(lambda)||^2 - Delta^2)^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: tmp_wtg(n)
double precision, intent(in) :: lambda
double precision, intent(in) :: delta
! Functions
double precision :: d2_norm_trust_region_omp
double precision :: ddot
! Internal
double precision :: accu1,accu2,accu3
double precision, allocatable :: tmp_accu1(:), tmp_accu2(:), tmp_accu3(:)
integer :: i, j
! Allocation
allocate(tmp_accu1(n), tmp_accu2(n), tmp_accu3(n))
call omp_set_max_active_levels(1)
! OMP
!$OMP PARALLEL &
!$OMP PRIVATE(i,j) &
!$OMP SHARED(n,lambda, e_val, thresh_eig,&
!$OMP tmp_accu1, tmp_accu2, tmp_accu3, tmp_wtg, &
!$OMP accu1, accu2, accu3) &
!$OMP DEFAULT(NONE)
! Initialization
!$OMP MASTER
accu1 = 0d0
accu2 = 0d0
accu3 = 0d0
!$OMP END MASTER
!$OMP DO
do i = 1, n
tmp_accu1(i) = 0d0
enddo
!$OMP END DO
!$OMP DO
do i = 1, n
tmp_accu2(i) = 0d0
enddo
!$OMP END DO
!$OMP DO
do i = 1, n
tmp_accu3(i) = 0d0
enddo
!$OMP END DO
! Calculations
! accu1
!$OMP DO
do i = 1, n
if (ABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu1(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**2
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu1 = accu1 + tmp_accu1(i)
enddo
!$OMP END MASTER
! accu2
!$OMP DO
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu2(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**3
endif
enddo
!$OMP END DO
! accu3
!$OMP MASTER
do i = 1, n
accu2 = accu2 + tmp_accu2(i)
enddo
!$OMP END MASTER
!$OMP DO
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu3(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**4
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu3 = accu3 + tmp_accu3(i)
enddo
!$OMP END MASTER
!$OMP END PARALLEL
d2_norm_trust_region_omp = 2d0 * (6d0 * accu3 * (- delta**2 + accu1) + (-2d0 * accu2)**2)
deallocate(tmp_accu1, tmp_accu2, tmp_accu3)
end function
! OMP: Function value of ||x||^2
! *Compute the value of ||x||^2*
! This function computes the value of ||x(lambda)||^2
! \begin{align*}
! ||\textbf{x}(\lambda)||^2 = \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! Internal:
! | tmp_wtg(n) | double precision | temporary array for W^T.v_grad |
! | tmp_fN | double precision | temporary array for the function |
! | i,j | integer | indexes |
function f_norm_trust_region_omp(n,e_val,tmp_wtg,lambda)
use omp_lib
include 'pi.h'
BEGIN_DOC
! Compute ||x(lambda)||^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: tmp_wtg(n)
double precision, intent(in) :: lambda
! functions
double precision :: f_norm_trust_region_omp
! internal
double precision, allocatable :: tmp_fN(:)
integer :: i,j
! Allocation
allocate(tmp_fN(n))
call omp_set_max_active_levels(1)
! OMP
!$OMP PARALLEL &
!$OMP PRIVATE(i,j) &
!$OMP SHARED(n,lambda, e_val, thresh_eig,&
!$OMP tmp_fN, tmp_wtg, f_norm_trust_region_omp) &
!$OMP DEFAULT(NONE)
! Initialization
!$OMP MASTER
f_norm_trust_region_omp = 0d0
!$OMP END MASTER
!$OMP DO
do i = 1, n
tmp_fN(i) = 0d0
enddo
!$OMP END DO
! Calculations
!$OMP DO
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_fN(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**2
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
f_norm_trust_region_omp = f_norm_trust_region_omp + tmp_fN(i)
enddo
!$OMP END MASTER
!$OMP END PARALLEL
deallocate(tmp_fN)
end function
! First derivative of (||x||^2 - Delta^2)^2
! Version without omp
! *Function to compute the first derivative of ||x||^2 - Delta*
! This function computes the first derivative of (||x||^2 - Delta^2)^2
! with respect to lambda.
! \begin{align*}
! \frac{\partial }{\partial \lambda} (||\textbf{x}(\lambda)||^2 - \Delta^2)^2
! = 2 \left(-2\sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \right)
! \left( - \Delta^2 + \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i+ \lambda)^2} \right)
! \end{align*}
! \begin{align*}
! \text{accu1} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \\
! \text{accu2} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | accu1 | double precision | first sum of the formula |
! | accu2 | double precision | second sum of the formula |
! | wtg | double precision | temporary variable to store W^T.v_grad |
! | i,j | integer | indexes |
! Function:
! | d1_norm_trust_region | double precision | first derivative with respect to lambda of (norm(x)^2 - Delta^2)^2 |
! | ddot | double precision | blas dot product |
function d1_norm_trust_region(n,e_val,w,v_grad,lambda,delta)
include 'pi.h'
BEGIN_DOC
! Compute the first derivative with respect to lambda of (||x(lambda)||^2 - Delta^2)^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: w(n,n)
double precision, intent(in) :: v_grad(n)
double precision, intent(in) :: lambda
double precision, intent(in) :: delta
! Internal
double precision :: wtg, accu1, accu2
integer :: i, j
! Functions
double precision :: d1_norm_trust_region
double precision :: ddot
! Initialization
accu1 = 0d0
accu2 = 0d0
do i = 1, n
wtg = 0d0
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
!wtg = ddot(n,w(:,i),1,v_grad,1)
accu1 = accu1 + wtg**2 / (e_val(i) + lambda)**2
endif
enddo
do i = 1, n
wtg = 0d0
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
!wtg = ddot(n,w(:,i),1,v_grad,1)
accu2 = accu2 - 2d0 * wtg**2 / (e_val(i) + lambda)**3
endif
enddo
d1_norm_trust_region = 2d0 * accu2 * (accu1 - delta**2)
end function
! Second derivative of (||x||^2 - Delta^2)^2
! Version without OMP
! *Function to compute the second derivative of ||x||^2 - Delta*
! \begin{equation}
! \frac{\partial^2 }{\partial \lambda^2} (||\textbf{x}(\lambda)||^2 - \Delta^2)^2
! = 2 \left[ \left( \sum_{i=1}^n 6 \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^4} \right) \left( - \Delta^2 + \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \right) + \left( \sum_{i=1}^n -2 \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \right)^2 \right]
! \end{equation}
! \begin{align*}
! \text{accu1} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \\
! \text{accu2} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \\
! \text{accu3} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^4}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | accu1 | double precision | first sum of the formula |
! | accu2 | double precision | second sum of the formula |
! | accu3 | double precision | third sum of the formula |
! | wtg | double precision | temporary variable to store W^T.v_grad |
! | i,j | integer | indexes |
! Function:
! | d2_norm_trust_region | double precision | second derivative with respect to lambda of norm(x)^2 - Delta^2 |
! | ddot | double precision | blas dot product |
function d2_norm_trust_region(n,e_val,w,v_grad,lambda,delta)
include 'pi.h'
BEGIN_DOC
! Compute the second derivative with respect to lambda of (||x(lambda)||^2 - Delta^2)^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: w(n,n)
double precision, intent(in) :: v_grad(n)
double precision, intent(in) :: lambda
double precision, intent(in) :: delta
! Functions
double precision :: d2_norm_trust_region
double precision :: ddot
! Internal
double precision :: wtg,accu1,accu2,accu3
integer :: i, j
! Initialization
accu1 = 0d0
accu2 = 0d0
accu3 = 0d0
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
wtg = 0d0
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
!wtg = ddot(n,w(:,i),1,v_grad,1)
accu1 = accu1 + wtg**2 / (e_val(i) + lambda)**2 !4
endif
enddo
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
wtg = 0d0
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
!wtg = ddot(n,w(:,i),1,v_grad,1)
accu2 = accu2 - 2d0 * wtg**2 / (e_val(i) + lambda)**3 !2
endif
enddo
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
wtg = 0d0
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
!wtg = ddot(n,w(:,i),1,v_grad,1)
accu3 = accu3 + 6d0 * wtg**2 / (e_val(i) + lambda)**4 !3
endif
enddo
d2_norm_trust_region = 2d0 * (accu3 * (- delta**2 + accu1) + accu2**2)
end function
! Function value of ||x||^2
! Version without OMP
! *Compute the value of ||x||^2*
! This function computes the value of ||x(lambda)||^2
! \begin{align*}
! ||\textbf{x}(\lambda)||^2 = \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | wtg | double precision | temporary variable to store W^T.v_grad |
! | i,j | integer | indexes |
! Function:
! | f_norm_trust_region | double precision | value of norm(x)^2 |
! | ddot | double precision | blas dot product |
function f_norm_trust_region(n,e_val,tmp_wtg,lambda)
include 'pi.h'
BEGIN_DOC
! Compute ||x(lambda)||^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: tmp_wtg(n)
double precision, intent(in) :: lambda
! function
double precision :: f_norm_trust_region
double precision :: ddot
! internal
integer :: i,j
! Initialization
f_norm_trust_region = 0d0
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
f_norm_trust_region = f_norm_trust_region + tmp_wtg(i)**2 / (e_val(i) + lambda)**2
endif
enddo
end function
! OMP: First derivative of (1/||x||^2 - 1/Delta^2)^2
! Version with OMP
! *Compute the first derivative of (1/||x||^2 - 1/Delta^2)^2*
! This function computes the value of (1/||x(lambda)||^2 - 1/Delta^2)^2
! \begin{align*}
! \frac{\partial}{\partial \lambda} (1/||\textbf{x}(\lambda)||^2 - 1/\Delta^2)^2
! &= 4 \frac{\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3}}
! {(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - \frac{4}{\Delta^2} \frac{\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3)}}
! {(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \\
! &= 4 \sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3}
! \left( \frac{1}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - \frac{1}{\Delta^2 (\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \right)
! \end{align*}
! \begin{align*}
! \text{accu1} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \\
! \text{accu2} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | wtg | double precision | temporary variable to store W^T.v_grad |
! | tmp_accu1 | double precision | temporary array for the first sum |
! | tmp_accu2 | double precision | temporary array for the second sum |
! | tmp_wtg(n) | double precision | temporary array for W^t.v_grad |
! | i,j | integer | indexes |
! Function:
! | d1_norm_inverse_trust_region | double precision | value of the first derivative |
function d1_norm_inverse_trust_region_omp(n,e_val,tmp_wtg,lambda,delta)
use omp_lib
include 'pi.h'
BEGIN_DOC
! Compute the first derivative of (1/||x||^2 - 1/Delta^2)^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: tmp_wtg(n)
double precision, intent(in) :: lambda
double precision, intent(in) :: delta
! Internal
double precision :: accu1, accu2
integer :: i,j
double precision, allocatable :: tmp_accu1(:), tmp_accu2(:)
! Functions
double precision :: d1_norm_inverse_trust_region_omp
! Allocation
allocate(tmp_accu1(n), tmp_accu2(n))
! OMP
call omp_set_max_active_levels(1)
! OMP
!$OMP PARALLEL &
!$OMP PRIVATE(i,j) &
!$OMP SHARED(n,lambda, e_val, thresh_eig,&
!$OMP tmp_accu1, tmp_accu2, tmp_wtg, accu1, accu2) &
!$OMP DEFAULT(NONE)
!$OMP MASTER
accu1 = 0d0
accu2 = 0d0
!$OMP END MASTER
!$OMP DO
do i = 1, n
tmp_accu1(i) = 0d0
enddo
!$OMP END DO
!$OMP DO
do i = 1, n
tmp_accu2(i) = 0d0
enddo
!$OMP END DO
! !$OMP MASTER
! do i = 1, n
! if (ABS(e_val(i)+lambda) > 1d-12) then
! tmp_accu1(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**2
! endif
! enddo
! !$OMP END MASTER
!$OMP DO
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu1(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**2
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu1 = accu1 + tmp_accu1(i)
enddo
!$OMP END MASTER
! !$OMP MASTER
! do i = 1, n
! if (ABS(e_val(i)+lambda) > 1d-12) then
! tmp_accu2(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**3
! endif
! enddo
! !$OMP END MASTER
!$OMP DO
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu2(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**3
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu2 = accu2 + tmp_accu2(i)
enddo
!$OMP END MASTER
!$OMP END PARALLEL
call omp_set_max_active_levels(4)
d1_norm_inverse_trust_region_omp = 4d0 * accu2 * (1d0/accu1**3 - 1d0/(delta**2 * accu1**2))
deallocate(tmp_accu1, tmp_accu2)
end
! OMP: Second derivative of (1/||x||^2 - 1/Delta^2)^2
! Version with OMP
! *Compute the first derivative of (1/||x||^2 - 1/Delta^2)^2*
! This function computes the value of (1/||x(lambda)||^2 - 1/Delta^2)^2
! \begin{align*}
! \frac{\partial^2}{\partial \lambda^2} (1/||\textbf{x}(\lambda)||^2 - 1/\Delta^2)^2
! &= 4 \left[ \frac{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3)})^2}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^4}
! - 3 \frac{\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^4}}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3} \right] \\
! &- \frac{4}{\Delta^2} \left[ \frac{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3)})^2}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - 3 \frac{\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^4}}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \right]
! \end{align*}
! \begin{align*}
! \text{accu1} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \\
! \text{accu2} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \\
! \text{accu3} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^4}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | wtg | double precision | temporary variable to store W^T.v_grad |
! | tmp_accu1 | double precision | temporary array for the first sum |
! | tmp_accu2 | double precision | temporary array for the second sum |
! | tmp_wtg(n) | double precision | temporary array for W^t.v_grad |
! | i,j | integer | indexes |
! Function:
! | d1_norm_inverse_trust_region | double precision | value of the first derivative |
function d2_norm_inverse_trust_region_omp(n,e_val,tmp_wtg,lambda,delta)
use omp_lib
include 'pi.h'
BEGIN_DOC
! Compute the second derivative of (1/||x||^2 - 1/Delta^2)^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: tmp_wtg(n)
double precision, intent(in) :: lambda
double precision, intent(in) :: delta
! Internal
double precision :: accu1, accu2, accu3
integer :: i,j
double precision, allocatable :: tmp_accu1(:), tmp_accu2(:), tmp_accu3(:)
! Functions
double precision :: d2_norm_inverse_trust_region_omp
! Allocation
allocate(tmp_accu1(n), tmp_accu2(n), tmp_accu3(n))
! OMP
call omp_set_max_active_levels(1)
! OMP
!$OMP PARALLEL &
!$OMP PRIVATE(i,j) &
!$OMP SHARED(n,lambda, e_val, thresh_eig,&
!$OMP tmp_accu1, tmp_accu2, tmp_accu3, tmp_wtg, &
!$OMP accu1, accu2, accu3) &
!$OMP DEFAULT(NONE)
!$OMP MASTER
accu1 = 0d0
accu2 = 0d0
accu3 = 0d0
!$OMP END MASTER
!$OMP DO
do i = 1, n
tmp_accu1(i) = 0d0
enddo
!$OMP END DO
!$OMP DO
do i = 1, n
tmp_accu2(i) = 0d0
enddo
!$OMP END DO
!$OMP DO
do i = 1, n
tmp_accu3(i) = 0d0
enddo
!$OMP END DO
!$OMP DO
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu1(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**2
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu1 = accu1 + tmp_accu1(i)
enddo
!$OMP END MASTER
!$OMP DO
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu2(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**3
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu2 = accu2 + tmp_accu2(i)
enddo
!$OMP END MASTER
!$OMP DO
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
tmp_accu3(i) = tmp_wtg(i)**2 / (e_val(i) + lambda)**4
endif
enddo
!$OMP END DO
!$OMP MASTER
do i = 1, n
accu3 = accu3 + tmp_accu3(i)
enddo
!$OMP END MASTER
!$OMP END PARALLEL
call omp_set_max_active_levels(4)
d2_norm_inverse_trust_region_omp = 4d0 * (6d0 * accu2**2/accu1**4 - 3d0 * accu3/accu1**3) &
- 4d0/delta**2 * (4d0 * accu2**2/accu1**3 - 3d0 * accu3/accu1**2)
deallocate(tmp_accu1,tmp_accu2,tmp_accu3)
end
! First derivative of (1/||x||^2 - 1/Delta^2)^2
! Version without OMP
! *Compute the first derivative of (1/||x||^2 - 1/Delta^2)^2*
! This function computes the value of (1/||x(lambda)||^2 - 1/Delta^2)^2
! \begin{align*}
! \frac{\partial}{\partial \lambda} (1/||\textbf{x}(\lambda)||^2 - 1/\Delta^2)^2
! &= 4 \frac{\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3}}
! {(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - \frac{4}{\Delta^2} \frac{\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3)}}
! {(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \\
! &= 4 \sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3}
! \left( \frac{1}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - \frac{1}{\Delta^2 (\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \right)
! \end{align*}
! \begin{align*}
! \text{accu1} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \\
! \text{accu2} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | wtg | double precision | temporary variable to store W^T.v_grad |
! | i,j | integer | indexes |
! Function:
! | d1_norm_inverse_trust_region | double precision | value of the first derivative |
function d1_norm_inverse_trust_region(n,e_val,w,v_grad,lambda,delta)
include 'pi.h'
BEGIN_DOC
! Compute the first derivative of (1/||x||^2 - 1/Delta^2)^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: w(n,n)
double precision, intent(in) :: v_grad(n)
double precision, intent(in) :: lambda
double precision, intent(in) :: delta
! Internal
double precision :: wtg, accu1, accu2
integer :: i,j
! Functions
double precision :: d1_norm_inverse_trust_region
accu1 = 0d0
accu2 = 0d0
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
wtg = 0d0
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
accu1 = accu1 + wtg**2 / (e_val(i) + lambda)**2
endif
enddo
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
wtg = 0d0
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
accu2 = accu2 + wtg**2 / (e_val(i) + lambda)**3
endif
enddo
d1_norm_inverse_trust_region = 4d0 * accu2 * (1d0/accu1**3 - 1d0/(delta**2 * accu1**2))
end
! Second derivative of (1/||x||^2 - 1/Delta^2)^2
! Version without OMP
! *Compute the second derivative of (1/||x||^2 - 1/Delta^2)^2*
! This function computes the value of (1/||x(lambda)||^2 - 1/Delta^2)^2
! \begin{align*}
! \frac{\partial^2}{\partial \lambda^2} (1/||\textbf{x}(\lambda)||^2 - 1/\Delta^2)^2
! &= 4 \left[ \frac{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3)})^2}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^4}
! - 3 \frac{\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^4}}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3} \right] \\
! &- \frac{4}{\Delta^2} \left[ \frac{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^3)})^2}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^3}
! - 3 \frac{\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^4}}{(\sum_i \frac{(\textbf{w}_i^T \cdot \textbf{g})^2}{(h_i + \lambda)^2})^2} \right]
! \end{align*}
! \begin{align*}
! \text{accu1} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^2} \\
! \text{accu2} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^3} \\
! \text{accu3} &= \sum_{i=1}^n \frac{(\textbf{w}_i^T \textbf{g})^2}{(h_i + \lambda)^4}
! \end{align*}
! Provided:
! | m_num | integer | number of MOs |
! Input:
! | n | integer | mo_num*(mo_num-1)/2 |
! | e_val(n) | double precision | eigenvalues of the hessian |
! | W(n,n) | double precision | eigenvectors of the hessian |
! | v_grad(n) | double precision | gradient |
! | lambda | double precision | Lagrange multiplier |
! | delta | double precision | Delta of the trust region |
! Internal:
! | wtg | double precision | temporary variable to store W^T.v_grad |
! | i,j | integer | indexes |
! Function:
! | d2_norm_inverse_trust_region | double precision | value of the first derivative |
function d2_norm_inverse_trust_region(n,e_val,w,v_grad,lambda,delta)
include 'pi.h'
BEGIN_DOC
! Compute the second derivative of (1/||x||^2 - 1/Delta^2)^2
END_DOC
implicit none
! Variables
! in
integer, intent(in) :: n
double precision, intent(in) :: e_val(n)
double precision, intent(in) :: w(n,n)
double precision, intent(in) :: v_grad(n)
double precision, intent(in) :: lambda
double precision, intent(in) :: delta
! Internal
double precision :: wtg, accu1, accu2, accu3
integer :: i,j
! Functions
double precision :: d2_norm_inverse_trust_region
accu1 = 0d0
accu2 = 0d0
accu3 = 0d0
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
wtg = 0d0
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
accu1 = accu1 + wtg**2 / (e_val(i) + lambda)**2
endif
enddo
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
wtg = 0d0
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
accu2 = accu2 + wtg**2 / (e_val(i) + lambda)**3
endif
enddo
do i = 1, n
if (DABS(e_val(i)) > thresh_eig .and. DABS(e_val(i)+lambda) > thresh_eig) then
wtg = 0d0
do j = 1, n
wtg = wtg + w(j,i) * v_grad(j)
enddo
accu3 = accu3 + wtg**2 / (e_val(i) + lambda)**4
endif
enddo
d2_norm_inverse_trust_region = 4d0 * (6d0 * accu2**2/accu1**4 - 3d0 * accu3/accu1**3) &
- 4d0/delta**2 * (4d0 * accu2**2/accu1**3 - 3d0 * accu3/accu1**2)
end
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