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Added exponential of matrix
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@ -1,5 +1,5 @@
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let integrals_cutoff = 1.e-15
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let epsilon = 1.e-20
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let epsilon = 2.e-15
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let integrals_cutoff = epsilon
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(** Constants *)
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let pi = acos (-1.)
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@ -1,4 +1,5 @@
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open Lacaml.D
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open Common
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type ('a, 'b) t = Mat.t
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@ -352,6 +353,16 @@ let scale_cols a v =
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a'
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let scale_rows_inplace v a =
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let v' = Vector.to_bigarray_inplace v in
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Mat.scal_rows v' a
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let scale_rows v a =
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let a' = copy a in
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let v' = Vector.to_bigarray_inplace v in
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Mat.scal_rows v' a' ;
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a'
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let svd a =
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let d, u, vt =
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gesvd (lacpy a)
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@ -382,24 +393,48 @@ let qr a =
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q, r
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let exponential a =
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let exponential_iterative a =
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assert (dim1 a = dim2 a);
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let a = to_bigarray_inplace a in
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let n = Mat.dim1 a in
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let a2 = Lacaml.D.gemm a a in
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let (lv, wr, _wi, rv) = Lacaml.D.geev a2 in
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let tau = Vec.map (fun x -> -. sqrt x) wr in
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let cos_tau =
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Mat.init_cols n n (fun i j ->
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if i<>j then 0. else cos tau.{i})
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let rec loop result accu n =
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let b = scale (1./.n) a in
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let new_accu = gemm accu b in
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let residual =
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sub new_accu accu
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|> amax
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|> abs_float
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in
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let sin_tau =
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Mat.init_cols n n (fun i j ->
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if i<>j then 0. else (sin tau.{i}) /. tau.{i})
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let result = add result new_accu in
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if residual > Constants.epsilon then
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loop result new_accu (n+.1.)
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else
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result
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in
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let g = Lacaml.D.gemm in
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Mat.add (g lv @@ g cos_tau rv) (g lv @@ g sin_tau @@ g rv a)
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|> of_bigarray_inplace
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let id = identity (dim1 a) in
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loop id id 1.
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let exponential a =
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let n = dim1 a in
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assert (n = dim2 a);
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let a2 = gemm a a in
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let (u, w, vt) = svd a2 in
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let tau = Vector.map (fun x -> -. sqrt x) w in
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let cos_tau = Vector.cos tau in
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let sin_tau_tau = Vector.mul (Vector.sin tau) (Vector.reci tau) in
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let result =
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add (gemm (scale_cols u cos_tau) vt) (gemm (scale_cols u sin_tau_tau) @@ gemm vt a)
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in
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(* Post-condition: Check if exp(-A) * exp(A) = I *)
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let id = identity n in
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let test =
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gemm_tn ~beta:(-.1.0) ~c:id result result
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|> amax
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|> abs_float
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in
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assert (test < Constants.epsilon);
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result
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let to_file ~filename ?(sym=false) ?(cutoff=0.) t =
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@ -171,6 +171,12 @@ val scale_cols: ('a,'b) t -> 'b Vector.t -> ('a,'b) t
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val scale_cols_inplace: ('a,'b) t -> 'b Vector.t -> unit
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(** Multiplies the matrix by a constant *)
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val scale_rows: 'a Vector.t -> ('a,'b) t -> ('a,'b) t
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(** Multiplies the matrix by a constant *)
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val scale_rows_inplace: 'a Vector.t -> ('a,'b) t -> unit
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(** Multiplies the matrix by a constant *)
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val sycon: ('a,'b) t -> float
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(** Returns the condition number of a matrix *)
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@ -300,6 +306,9 @@ val diagonalize_symm : ('a,'a) t -> ('a,'a) t * 'a Vector.t
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val exponential : ('a,'a) t -> ('a,'a) t
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(** Computes the exponential of a square matrix. *)
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val exponential_iterative : ('a,'a) t -> ('a,'a) t
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(** Computes the exponential of a square matrix with an iteratve algorithm. *)
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val xt_o_x : o:('a,'a) t -> x:('a,'b) t -> ('b,'b) t
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(** Computes {% $\mathbf{X^\dag\, O\, X}$ %} *)
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