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------------------------------------------------------------------------------ -- -- -- GNAT RUN-TIME COMPONENTS -- -- -- -- ADA.NUMERICS.GENERIC_REAL_ARRAYS -- -- -- -- B o d y -- -- -- -- Copyright (C) 2006-2009, Free Software Foundation, Inc. -- -- -- -- GNAT is free software; you can redistribute it and/or modify it under -- -- terms of the GNU General Public License as published by the Free Soft- -- -- ware Foundation; either version 3, or (at your option) any later ver- -- -- sion. GNAT is distributed in the hope that it will be useful, but WITH- -- -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY -- -- or FITNESS FOR A PARTICULAR PURPOSE. -- -- -- -- As a special exception under Section 7 of GPL version 3, you are granted -- -- additional permissions described in the GCC Runtime Library Exception, -- -- version 3.1, as published by the Free Software Foundation. -- -- -- -- You should have received a copy of the GNU General Public License and -- -- a copy of the GCC Runtime Library Exception along with this program; -- -- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see -- -- <http://www.gnu.org/licenses/>. -- -- -- -- GNAT was originally developed by the GNAT team at New York University. -- -- Extensive contributions were provided by Ada Core Technologies Inc. -- -- -- ------------------------------------------------------------------------------ with System; use System; with System.Generic_Real_BLAS; with System.Generic_Real_LAPACK; with System.Generic_Array_Operations; use System.Generic_Array_Operations; package body Ada.Numerics.Generic_Real_Arrays is -- Operations involving inner products use BLAS library implementations. -- This allows larger matrices and vectors to be computed efficiently, -- taking into account memory hierarchy issues and vector instructions -- that vary widely between machines. -- Operations that are defined in terms of operations on the type Real, -- such as addition, subtraction and scaling, are computed in the canonical -- way looping over all elements. -- Operations for solving linear systems and computing determinant, -- eigenvalues, eigensystem and inverse, are implemented using the -- LAPACK library. package BLAS is new Generic_Real_BLAS (Real'Base, Real_Vector, Real_Matrix); package LAPACK is new Generic_Real_LAPACK (Real'Base, Real_Vector, Real_Matrix); use BLAS, LAPACK; -- Procedure versions of functions returning unconstrained values. -- This allows for inlining the function wrapper. procedure Eigenvalues (A : Real_Matrix; Values : out Real_Vector); procedure Inverse (A : Real_Matrix; R : out Real_Matrix); procedure Solve (A : Real_Matrix; X : Real_Vector; B : out Real_Vector); procedure Solve (A : Real_Matrix; X : Real_Matrix; B : out Real_Matrix); procedure Transpose is new Generic_Array_Operations.Transpose (Scalar => Real'Base, Matrix => Real_Matrix); -- Helper function that raises a Constraint_Error is the argument is -- not a square matrix, and otherwise returns its length. function Length is new Square_Matrix_Length (Real'Base, Real_Matrix); -- Instantiating the following subprograms directly would lead to -- name clashes, so use a local package. package Instantiations is function "+" is new Vector_Elementwise_Operation (X_Scalar => Real'Base, Result_Scalar => Real'Base, X_Vector => Real_Vector, Result_Vector => Real_Vector, Operation => "+"); function "+" is new Matrix_Elementwise_Operation (X_Scalar => Real'Base, Result_Scalar => Real'Base, X_Matrix => Real_Matrix, Result_Matrix => Real_Matrix, Operation => "+"); function "+" is new Vector_Vector_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Vector => Real_Vector, Right_Vector => Real_Vector, Result_Vector => Real_Vector, Operation => "+"); function "+" is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Matrix => Real_Matrix, Right_Matrix => Real_Matrix, Result_Matrix => Real_Matrix, Operation => "+"); function "-" is new Vector_Elementwise_Operation (X_Scalar => Real'Base, Result_Scalar => Real'Base, X_Vector => Real_Vector, Result_Vector => Real_Vector, Operation => "-"); function "-" is new Matrix_Elementwise_Operation (X_Scalar => Real'Base, Result_Scalar => Real'Base, X_Matrix => Real_Matrix, Result_Matrix => Real_Matrix, Operation => "-"); function "-" is new Vector_Vector_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Vector => Real_Vector, Right_Vector => Real_Vector, Result_Vector => Real_Vector, Operation => "-"); function "-" is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Matrix => Real_Matrix, Right_Matrix => Real_Matrix, Result_Matrix => Real_Matrix, Operation => "-"); function "*" is new Scalar_Vector_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Right_Vector => Real_Vector, Result_Vector => Real_Vector, Operation => "*"); function "*" is new Scalar_Matrix_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Right_Matrix => Real_Matrix, Result_Matrix => Real_Matrix, Operation => "*"); function "*" is new Vector_Scalar_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Vector => Real_Vector, Result_Vector => Real_Vector, Operation => "*"); function "*" is new Matrix_Scalar_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Matrix => Real_Matrix, Result_Matrix => Real_Matrix, Operation => "*"); function "*" is new Outer_Product (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Vector => Real_Vector, Right_Vector => Real_Vector, Matrix => Real_Matrix); function "/" is new Vector_Scalar_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Vector => Real_Vector, Result_Vector => Real_Vector, Operation => "/"); function "/" is new Matrix_Scalar_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Matrix => Real_Matrix, Result_Matrix => Real_Matrix, Operation => "/"); function "abs" is new Vector_Elementwise_Operation (X_Scalar => Real'Base, Result_Scalar => Real'Base, X_Vector => Real_Vector, Result_Vector => Real_Vector, Operation => "abs"); function "abs" is new Matrix_Elementwise_Operation (X_Scalar => Real'Base, Result_Scalar => Real'Base, X_Matrix => Real_Matrix, Result_Matrix => Real_Matrix, Operation => "abs"); function Unit_Matrix is new Generic_Array_Operations.Unit_Matrix (Scalar => Real'Base, Matrix => Real_Matrix, Zero => 0.0, One => 1.0); function Unit_Vector is new Generic_Array_Operations.Unit_Vector (Scalar => Real'Base, Vector => Real_Vector, Zero => 0.0, One => 1.0); end Instantiations; --------- -- "+" -- --------- function "+" (Right : Real_Vector) return Real_Vector renames Instantiations."+"; function "+" (Right : Real_Matrix) return Real_Matrix renames Instantiations."+"; function "+" (Left, Right : Real_Vector) return Real_Vector renames Instantiations."+"; function "+" (Left, Right : Real_Matrix) return Real_Matrix renames Instantiations."+"; --------- -- "-" -- --------- function "-" (Right : Real_Vector) return Real_Vector renames Instantiations."-"; function "-" (Right : Real_Matrix) return Real_Matrix renames Instantiations."-"; function "-" (Left, Right : Real_Vector) return Real_Vector renames Instantiations."-"; function "-" (Left, Right : Real_Matrix) return Real_Matrix renames Instantiations."-"; --------- -- "*" -- --------- -- Scalar multiplication function "*" (Left : Real'Base; Right : Real_Vector) return Real_Vector renames Instantiations."*"; function "*" (Left : Real_Vector; Right : Real'Base) return Real_Vector renames Instantiations."*"; function "*" (Left : Real'Base; Right : Real_Matrix) return Real_Matrix renames Instantiations."*"; function "*" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix renames Instantiations."*"; -- Vector multiplication function "*" (Left, Right : Real_Vector) return Real'Base is begin if Left'Length /= Right'Length then raise Constraint_Error with "vectors are of different length in inner product"; end if; return dot (Left'Length, X => Left, Y => Right); end "*"; function "*" (Left, Right : Real_Vector) return Real_Matrix renames Instantiations."*"; function "*" (Left : Real_Vector; Right : Real_Matrix) return Real_Vector is R : Real_Vector (Right'Range (2)); begin if Left'Length /= Right'Length (1) then raise Constraint_Error with "incompatible dimensions in vector-matrix multiplication"; end if; gemv (Trans => No_Trans'Access, M => Right'Length (2), N => Right'Length (1), A => Right, Ld_A => Right'Length (2), X => Left, Y => R); return R; end "*"; function "*" (Left : Real_Matrix; Right : Real_Vector) return Real_Vector is R : Real_Vector (Left'Range (1)); begin if Left'Length (2) /= Right'Length then raise Constraint_Error with "incompatible dimensions in matrix-vector multiplication"; end if; gemv (Trans => Trans'Access, M => Left'Length (2), N => Left'Length (1), A => Left, Ld_A => Left'Length (2), X => Right, Y => R); return R; end "*"; -- Matrix Multiplication function "*" (Left, Right : Real_Matrix) return Real_Matrix is R : Real_Matrix (Left'Range (1), Right'Range (2)); begin if Left'Length (2) /= Right'Length (1) then raise Constraint_Error with "incompatible dimensions in matrix-matrix multiplication"; end if; gemm (Trans_A => No_Trans'Access, Trans_B => No_Trans'Access, M => Right'Length (2), N => Left'Length (1), K => Right'Length (1), A => Right, Ld_A => Right'Length (2), B => Left, Ld_B => Left'Length (2), C => R, Ld_C => R'Length (2)); return R; end "*"; --------- -- "/" -- --------- function "/" (Left : Real_Vector; Right : Real'Base) return Real_Vector renames Instantiations."/"; function "/" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix renames Instantiations."/"; ----------- -- "abs" -- ----------- function "abs" (Right : Real_Vector) return Real'Base is begin return nrm2 (Right'Length, Right); end "abs"; function "abs" (Right : Real_Vector) return Real_Vector renames Instantiations."abs"; function "abs" (Right : Real_Matrix) return Real_Matrix renames Instantiations."abs"; ----------------- -- Determinant -- ----------------- function Determinant (A : Real_Matrix) return Real'Base is N : constant Integer := Length (A); LU : Real_Matrix (1 .. N, 1 .. N) := A; Piv : Integer_Vector (1 .. N); Info : aliased Integer := -1; Det : Real := 1.0; begin getrf (M => N, N => N, A => LU, Ld_A => N, I_Piv => Piv, Info => Info'Access); if Info /= 0 then raise Constraint_Error with "ill-conditioned matrix"; end if; for J in 1 .. N loop Det := (if Piv (J) /= J then -Det * LU (J, J) else Det * LU (J, J)); end loop; return Det; end Determinant; ----------------- -- Eigensystem -- ----------------- procedure Eigensystem (A : Real_Matrix; Values : out Real_Vector; Vectors : out Real_Matrix) is N : constant Natural := Length (A); Tau : Real_Vector (1 .. N); L_Work : Real_Vector (1 .. 1); Info : aliased Integer; E : Real_Vector (1 .. N); pragma Warnings (Off, E); begin if Values'Length /= N then raise Constraint_Error with "wrong length for output vector"; end if; if N = 0 then return; end if; -- Initialize working matrix and check for symmetric input matrix Transpose (A, Vectors); if A /= Vectors then raise Argument_Error with "matrix not symmetric"; end if; -- Compute size of additional working space sytrd (Uplo => Lower'Access, N => N, A => Vectors, Ld_A => N, D => Values, E => E, Tau => Tau, Work => L_Work, L_Work => -1, Info => Info'Access); declare Work : Real_Vector (1 .. Integer'Max (Integer (L_Work (1)), 2 * N)); pragma Warnings (Off, Work); Comp_Z : aliased constant Character := 'V'; begin -- Reduce matrix to tridiagonal form sytrd (Uplo => Lower'Access, N => N, A => Vectors, Ld_A => A'Length (1), D => Values, E => E, Tau => Tau, Work => Work, L_Work => Work'Length, Info => Info'Access); if Info /= 0 then raise Program_Error; end if; -- Generate the real orthogonal matrix determined by sytrd orgtr (Uplo => Lower'Access, N => N, A => Vectors, Ld_A => N, Tau => Tau, Work => Work, L_Work => Work'Length, Info => Info'Access); if Info /= 0 then raise Program_Error; end if; -- Compute all eigenvalues and eigenvectors using QR algorithm steqr (Comp_Z => Comp_Z'Access, N => N, D => Values, E => E, Z => Vectors, Ld_Z => N, Work => Work, Info => Info'Access); if Info /= 0 then raise Constraint_Error with "eigensystem computation failed to converge"; end if; end; end Eigensystem; ----------------- -- Eigenvalues -- ----------------- procedure Eigenvalues (A : Real_Matrix; Values : out Real_Vector) is N : constant Natural := Length (A); L_Work : Real_Vector (1 .. 1); Info : aliased Integer; B : Real_Matrix (1 .. N, 1 .. N); Tau : Real_Vector (1 .. N); E : Real_Vector (1 .. N); pragma Warnings (Off, B); pragma Warnings (Off, Tau); pragma Warnings (Off, E); begin if Values'Length /= N then raise Constraint_Error with "wrong length for output vector"; end if; if N = 0 then return; end if; -- Initialize working matrix and check for symmetric input matrix Transpose (A, B); if A /= B then raise Argument_Error with "matrix not symmetric"; end if; -- Find size of work area sytrd (Uplo => Lower'Access, N => N, A => B, Ld_A => N, D => Values, E => E, Tau => Tau, Work => L_Work, L_Work => -1, Info => Info'Access); declare Work : Real_Vector (1 .. Integer'Min (Integer (L_Work (1)), 4 * N)); pragma Warnings (Off, Work); begin -- Reduce matrix to tridiagonal form sytrd (Uplo => Lower'Access, N => N, A => B, Ld_A => A'Length (1), D => Values, E => E, Tau => Tau, Work => Work, L_Work => Work'Length, Info => Info'Access); if Info /= 0 then raise Constraint_Error; end if; -- Compute all eigenvalues using QR algorithm sterf (N => N, D => Values, E => E, Info => Info'Access); if Info /= 0 then raise Constraint_Error with "eigenvalues computation failed to converge"; end if; end; end Eigenvalues; function Eigenvalues (A : Real_Matrix) return Real_Vector is R : Real_Vector (A'Range (1)); begin Eigenvalues (A, R); return R; end Eigenvalues; ------------- -- Inverse -- ------------- procedure Inverse (A : Real_Matrix; R : out Real_Matrix) is N : constant Integer := Length (A); Piv : Integer_Vector (1 .. N); L_Work : Real_Vector (1 .. 1); Info : aliased Integer := -1; begin -- All computations are done using column-major order, but this works -- out fine, because Transpose (Inverse (Transpose (A))) = Inverse (A). R := A; -- Compute LU decomposition getrf (M => N, N => N, A => R, Ld_A => N, I_Piv => Piv, Info => Info'Access); if Info /= 0 then raise Constraint_Error with "inverting singular matrix"; end if; -- Determine size of work area getri (N => N, A => R, Ld_A => N, I_Piv => Piv, Work => L_Work, L_Work => -1, Info => Info'Access); if Info /= 0 then raise Constraint_Error; end if; declare Work : Real_Vector (1 .. Integer (L_Work (1))); pragma Warnings (Off, Work); begin -- Compute inverse from LU decomposition getri (N => N, A => R, Ld_A => N, I_Piv => Piv, Work => Work, L_Work => Work'Length, Info => Info'Access); if Info /= 0 then raise Constraint_Error with "inverting singular matrix"; end if; -- ??? Should iterate with gerfs, based on implementation advice end; end Inverse; function Inverse (A : Real_Matrix) return Real_Matrix is R : Real_Matrix (A'Range (2), A'Range (1)); begin Inverse (A, R); return R; end Inverse; ----------- -- Solve -- ----------- procedure Solve (A : Real_Matrix; X : Real_Vector; B : out Real_Vector) is begin if Length (A) /= X'Length then raise Constraint_Error with "incompatible matrix and vector dimensions"; end if; -- ??? Should solve directly, is faster and more accurate B := Inverse (A) * X; end Solve; procedure Solve (A : Real_Matrix; X : Real_Matrix; B : out Real_Matrix) is begin if Length (A) /= X'Length (1) then raise Constraint_Error with "incompatible matrix dimensions"; end if; -- ??? Should solve directly, is faster and more accurate B := Inverse (A) * X; end Solve; function Solve (A : Real_Matrix; X : Real_Vector) return Real_Vector is B : Real_Vector (A'Range (2)); begin Solve (A, X, B); return B; end Solve; function Solve (A, X : Real_Matrix) return Real_Matrix is B : Real_Matrix (A'Range (2), X'Range (2)); begin Solve (A, X, B); return B; end Solve; --------------- -- Transpose -- --------------- function Transpose (X : Real_Matrix) return Real_Matrix is R : Real_Matrix (X'Range (2), X'Range (1)); begin Transpose (X, R); return R; end Transpose; ----------------- -- Unit_Matrix -- ----------------- function Unit_Matrix (Order : Positive; First_1 : Integer := 1; First_2 : Integer := 1) return Real_Matrix renames Instantiations.Unit_Matrix; ----------------- -- Unit_Vector -- ----------------- function Unit_Vector (Index : Integer; Order : Positive; First : Integer := 1) return Real_Vector renames Instantiations.Unit_Vector; end Ada.Numerics.Generic_Real_Arrays;
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