------------------------------------------------------------------------------ -- -- -- GNAT COMPILER COMPONENTS -- -- -- -- ADA.NUMERICS.GENERIC_COMPLEX_ARRAYS -- -- -- -- B o d y -- -- -- -- Copyright (C) 2006-2018, 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 -- -- . -- -- -- -- GNAT was originally developed by the GNAT team at New York University. -- -- Extensive contributions were provided by Ada Core Technologies Inc. -- -- -- ------------------------------------------------------------------------------ with System.Generic_Array_Operations; use System.Generic_Array_Operations; package body Ada.Numerics.Generic_Complex_Arrays is -- 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. package Ops renames System.Generic_Array_Operations; subtype Real is Real_Arrays.Real; -- Work around visibility bug ??? function Is_Non_Zero (X : Complex) return Boolean is (X /= (0.0, 0.0)); -- Needed by Back_Substitute procedure Back_Substitute is new Ops.Back_Substitute (Scalar => Complex, Matrix => Complex_Matrix, Is_Non_Zero => Is_Non_Zero); procedure Forward_Eliminate is new Ops.Forward_Eliminate (Scalar => Complex, Real => Real'Base, Matrix => Complex_Matrix, Zero => (0.0, 0.0), One => (1.0, 0.0)); procedure Transpose is new Ops.Transpose (Scalar => Complex, Matrix => Complex_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 (Complex, Complex_Matrix); -- Instant a generic square root implementation here, in order to avoid -- instantiating a complete copy of Generic_Elementary_Functions. -- Speed of the square root is not a big concern here. function Sqrt is new Ops.Sqrt (Real'Base); -- Instantiating the following subprograms directly would lead to -- name clashes, so use a local package. package Instantiations is --------- -- "*" -- --------- function "*" is new Vector_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "*"); function "*" is new Vector_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "*"); function "*" is new Scalar_Vector_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Right_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "*"); function "*" is new Scalar_Vector_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Right_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "*"); function "*" is new Inner_Product (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Complex_Vector, Right_Vector => Real_Vector, Zero => (0.0, 0.0)); function "*" is new Inner_Product (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Real_Vector, Right_Vector => Complex_Vector, Zero => (0.0, 0.0)); function "*" is new Inner_Product (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Complex_Vector, Right_Vector => Complex_Vector, Zero => (0.0, 0.0)); function "*" is new Outer_Product (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Complex_Vector, Right_Vector => Complex_Vector, Matrix => Complex_Matrix); function "*" is new Outer_Product (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Real_Vector, Right_Vector => Complex_Vector, Matrix => Complex_Matrix); function "*" is new Outer_Product (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Complex_Vector, Right_Vector => Real_Vector, Matrix => Complex_Matrix); function "*" is new Matrix_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "*"); function "*" is new Matrix_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "*"); function "*" is new Scalar_Matrix_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Right_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "*"); function "*" is new Scalar_Matrix_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Right_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "*"); function "*" is new Matrix_Vector_Product (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Matrix => Real_Matrix, Right_Vector => Complex_Vector, Result_Vector => Complex_Vector, Zero => (0.0, 0.0)); function "*" is new Matrix_Vector_Product (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Matrix => Complex_Matrix, Right_Vector => Real_Vector, Result_Vector => Complex_Vector, Zero => (0.0, 0.0)); function "*" is new Matrix_Vector_Product (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Matrix => Complex_Matrix, Right_Vector => Complex_Vector, Result_Vector => Complex_Vector, Zero => (0.0, 0.0)); function "*" is new Vector_Matrix_Product (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Real_Vector, Matrix => Complex_Matrix, Result_Vector => Complex_Vector, Zero => (0.0, 0.0)); function "*" is new Vector_Matrix_Product (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Complex_Vector, Matrix => Real_Matrix, Result_Vector => Complex_Vector, Zero => (0.0, 0.0)); function "*" is new Vector_Matrix_Product (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Complex_Vector, Matrix => Complex_Matrix, Result_Vector => Complex_Vector, Zero => (0.0, 0.0)); function "*" is new Matrix_Matrix_Product (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Right_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Zero => (0.0, 0.0)); function "*" is new Matrix_Matrix_Product (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Left_Matrix => Real_Matrix, Right_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Zero => (0.0, 0.0)); function "*" is new Matrix_Matrix_Product (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Right_Matrix => Real_Matrix, Result_Matrix => Complex_Matrix, Zero => (0.0, 0.0)); --------- -- "+" -- --------- function "+" is new Vector_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Complex, X_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "+"); function "+" is new Vector_Vector_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Complex_Vector, Right_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "+"); function "+" is new Vector_Vector_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Real_Vector, Right_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "+"); function "+" is new Vector_Vector_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Complex_Vector, Right_Vector => Real_Vector, Result_Vector => Complex_Vector, Operation => "+"); function "+" is new Matrix_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Complex, X_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "+"); function "+" is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Right_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "+"); function "+" is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Left_Matrix => Real_Matrix, Right_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "+"); function "+" is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Right_Matrix => Real_Matrix, Result_Matrix => Complex_Matrix, Operation => "+"); --------- -- "-" -- --------- function "-" is new Vector_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Complex, X_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "-"); function "-" is new Vector_Vector_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Complex_Vector, Right_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "-"); function "-" is new Vector_Vector_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Real_Vector, Right_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "-"); function "-" is new Vector_Vector_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Complex_Vector, Right_Vector => Real_Vector, Result_Vector => Complex_Vector, Operation => "-"); function "-" is new Matrix_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Complex, X_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "-"); function "-" is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Right_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "-"); function "-" is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Complex, Result_Scalar => Complex, Left_Matrix => Real_Matrix, Right_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "-"); function "-" is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Right_Matrix => Real_Matrix, Result_Matrix => Complex_Matrix, Operation => "-"); --------- -- "/" -- --------- function "/" is new Vector_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "/"); function "/" is new Vector_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => "/"); function "/" is new Matrix_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Complex, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "/"); function "/" is new Matrix_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => "/"); ----------- -- "abs" -- ----------- function "abs" is new L2_Norm (X_Scalar => Complex, Result_Real => Real'Base, X_Vector => Complex_Vector); -------------- -- Argument -- -------------- function Argument is new Vector_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Real'Base, X_Vector => Complex_Vector, Result_Vector => Real_Vector, Operation => Argument); function Argument is new Vector_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Vector => Complex_Vector, Result_Vector => Real_Vector, Operation => Argument); function Argument is new Matrix_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Real'Base, X_Matrix => Complex_Matrix, Result_Matrix => Real_Matrix, Operation => Argument); function Argument is new Matrix_Scalar_Elementwise_Operation (Left_Scalar => Complex, Right_Scalar => Real'Base, Result_Scalar => Real'Base, Left_Matrix => Complex_Matrix, Result_Matrix => Real_Matrix, Operation => Argument); ---------------------------- -- Compose_From_Cartesian -- ---------------------------- function Compose_From_Cartesian is new Vector_Elementwise_Operation (X_Scalar => Real'Base, Result_Scalar => Complex, X_Vector => Real_Vector, Result_Vector => Complex_Vector, Operation => Compose_From_Cartesian); function Compose_From_Cartesian is new Vector_Vector_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Real_Vector, Right_Vector => Real_Vector, Result_Vector => Complex_Vector, Operation => Compose_From_Cartesian); function Compose_From_Cartesian is new Matrix_Elementwise_Operation (X_Scalar => Real'Base, Result_Scalar => Complex, X_Matrix => Real_Matrix, Result_Matrix => Complex_Matrix, Operation => Compose_From_Cartesian); function Compose_From_Cartesian is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Matrix => Real_Matrix, Right_Matrix => Real_Matrix, Result_Matrix => Complex_Matrix, Operation => Compose_From_Cartesian); ------------------------ -- Compose_From_Polar -- ------------------------ function Compose_From_Polar is new Vector_Vector_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Vector => Real_Vector, Right_Vector => Real_Vector, Result_Vector => Complex_Vector, Operation => Compose_From_Polar); function Compose_From_Polar is new Vector_Vector_Scalar_Elementwise_Operation (X_Scalar => Real'Base, Y_Scalar => Real'Base, Z_Scalar => Real'Base, Result_Scalar => Complex, X_Vector => Real_Vector, Y_Vector => Real_Vector, Result_Vector => Complex_Vector, Operation => Compose_From_Polar); function Compose_From_Polar is new Matrix_Matrix_Elementwise_Operation (Left_Scalar => Real'Base, Right_Scalar => Real'Base, Result_Scalar => Complex, Left_Matrix => Real_Matrix, Right_Matrix => Real_Matrix, Result_Matrix => Complex_Matrix, Operation => Compose_From_Polar); function Compose_From_Polar is new Matrix_Matrix_Scalar_Elementwise_Operation (X_Scalar => Real'Base, Y_Scalar => Real'Base, Z_Scalar => Real'Base, Result_Scalar => Complex, X_Matrix => Real_Matrix, Y_Matrix => Real_Matrix, Result_Matrix => Complex_Matrix, Operation => Compose_From_Polar); --------------- -- Conjugate -- --------------- function Conjugate is new Vector_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Complex, X_Vector => Complex_Vector, Result_Vector => Complex_Vector, Operation => Conjugate); function Conjugate is new Matrix_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Complex, X_Matrix => Complex_Matrix, Result_Matrix => Complex_Matrix, Operation => Conjugate); -------- -- Im -- -------- function Im is new Vector_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Real'Base, X_Vector => Complex_Vector, Result_Vector => Real_Vector, Operation => Im); function Im is new Matrix_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Real'Base, X_Matrix => Complex_Matrix, Result_Matrix => Real_Matrix, Operation => Im); ------------- -- Modulus -- ------------- function Modulus is new Vector_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Real'Base, X_Vector => Complex_Vector, Result_Vector => Real_Vector, Operation => Modulus); function Modulus is new Matrix_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Real'Base, X_Matrix => Complex_Matrix, Result_Matrix => Real_Matrix, Operation => Modulus); -------- -- Re -- -------- function Re is new Vector_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Real'Base, X_Vector => Complex_Vector, Result_Vector => Real_Vector, Operation => Re); function Re is new Matrix_Elementwise_Operation (X_Scalar => Complex, Result_Scalar => Real'Base, X_Matrix => Complex_Matrix, Result_Matrix => Real_Matrix, Operation => Re); ------------ -- Set_Im -- ------------ procedure Set_Im is new Update_Vector_With_Vector (X_Scalar => Complex, Y_Scalar => Real'Base, X_Vector => Complex_Vector, Y_Vector => Real_Vector, Update => Set_Im); procedure Set_Im is new Update_Matrix_With_Matrix (X_Scalar => Complex, Y_Scalar => Real'Base, X_Matrix => Complex_Matrix, Y_Matrix => Real_Matrix, Update => Set_Im); ------------ -- Set_Re -- ------------ procedure Set_Re is new Update_Vector_With_Vector (X_Scalar => Complex, Y_Scalar => Real'Base, X_Vector => Complex_Vector, Y_Vector => Real_Vector, Update => Set_Re); procedure Set_Re is new Update_Matrix_With_Matrix (X_Scalar => Complex, Y_Scalar => Real'Base, X_Matrix => Complex_Matrix, Y_Matrix => Real_Matrix, Update => Set_Re); ----------- -- Solve -- ----------- function Solve is new Matrix_Vector_Solution (Complex, (0.0, 0.0), Complex_Vector, Complex_Matrix); function Solve is new Matrix_Matrix_Solution (Complex, (0.0, 0.0), Complex_Matrix); ----------------- -- Unit_Matrix -- ----------------- function Unit_Matrix is new System.Generic_Array_Operations.Unit_Matrix (Scalar => Complex, Matrix => Complex_Matrix, Zero => (0.0, 0.0), One => (1.0, 0.0)); function Unit_Vector is new System.Generic_Array_Operations.Unit_Vector (Scalar => Complex, Vector => Complex_Vector, Zero => (0.0, 0.0), One => (1.0, 0.0)); end Instantiations; --------- -- "*" -- --------- function "*" (Left : Complex_Vector; Right : Complex_Vector) return Complex renames Instantiations."*"; function "*" (Left : Real_Vector; Right : Complex_Vector) return Complex renames Instantiations."*"; function "*" (Left : Complex_Vector; Right : Real_Vector) return Complex renames Instantiations."*"; function "*" (Left : Complex; Right : Complex_Vector) return Complex_Vector renames Instantiations."*"; function "*" (Left : Complex_Vector; Right : Complex) return Complex_Vector renames Instantiations."*"; function "*" (Left : Real'Base; Right : Complex_Vector) return Complex_Vector renames Instantiations."*"; function "*" (Left : Complex_Vector; Right : Real'Base) return Complex_Vector renames Instantiations."*"; function "*" (Left : Complex_Matrix; Right : Complex_Matrix) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Complex_Vector; Right : Complex_Vector) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Complex_Vector; Right : Complex_Matrix) return Complex_Vector renames Instantiations."*"; function "*" (Left : Complex_Matrix; Right : Complex_Vector) return Complex_Vector renames Instantiations."*"; function "*" (Left : Real_Matrix; Right : Complex_Matrix) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Complex_Matrix; Right : Real_Matrix) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Real_Vector; Right : Complex_Vector) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Complex_Vector; Right : Real_Vector) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Real_Vector; Right : Complex_Matrix) return Complex_Vector renames Instantiations."*"; function "*" (Left : Complex_Vector; Right : Real_Matrix) return Complex_Vector renames Instantiations."*"; function "*" (Left : Real_Matrix; Right : Complex_Vector) return Complex_Vector renames Instantiations."*"; function "*" (Left : Complex_Matrix; Right : Real_Vector) return Complex_Vector renames Instantiations."*"; function "*" (Left : Complex; Right : Complex_Matrix) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Complex_Matrix; Right : Complex) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Real'Base; Right : Complex_Matrix) return Complex_Matrix renames Instantiations."*"; function "*" (Left : Complex_Matrix; Right : Real'Base) return Complex_Matrix renames Instantiations."*"; --------- -- "+" -- --------- function "+" (Right : Complex_Vector) return Complex_Vector renames Instantiations."+"; function "+" (Left : Complex_Vector; Right : Complex_Vector) return Complex_Vector renames Instantiations."+"; function "+" (Left : Real_Vector; Right : Complex_Vector) return Complex_Vector renames Instantiations."+"; function "+" (Left : Complex_Vector; Right : Real_Vector) return Complex_Vector renames Instantiations."+"; function "+" (Right : Complex_Matrix) return Complex_Matrix renames Instantiations."+"; function "+" (Left : Complex_Matrix; Right : Complex_Matrix) return Complex_Matrix renames Instantiations."+"; function "+" (Left : Real_Matrix; Right : Complex_Matrix) return Complex_Matrix renames Instantiations."+"; function "+" (Left : Complex_Matrix; Right : Real_Matrix) return Complex_Matrix renames Instantiations."+"; --------- -- "-" -- --------- function "-" (Right : Complex_Vector) return Complex_Vector renames Instantiations."-"; function "-" (Left : Complex_Vector; Right : Complex_Vector) return Complex_Vector renames Instantiations."-"; function "-" (Left : Real_Vector; Right : Complex_Vector) return Complex_Vector renames Instantiations."-"; function "-" (Left : Complex_Vector; Right : Real_Vector) return Complex_Vector renames Instantiations."-"; function "-" (Right : Complex_Matrix) return Complex_Matrix renames Instantiations."-"; function "-" (Left : Complex_Matrix; Right : Complex_Matrix) return Complex_Matrix renames Instantiations."-"; function "-" (Left : Real_Matrix; Right : Complex_Matrix) return Complex_Matrix renames Instantiations."-"; function "-" (Left : Complex_Matrix; Right : Real_Matrix) return Complex_Matrix renames Instantiations."-"; --------- -- "/" -- --------- function "/" (Left : Complex_Vector; Right : Complex) return Complex_Vector renames Instantiations."/"; function "/" (Left : Complex_Vector; Right : Real'Base) return Complex_Vector renames Instantiations."/"; function "/" (Left : Complex_Matrix; Right : Complex) return Complex_Matrix renames Instantiations."/"; function "/" (Left : Complex_Matrix; Right : Real'Base) return Complex_Matrix renames Instantiations."/"; ----------- -- "abs" -- ----------- function "abs" (Right : Complex_Vector) return Real'Base renames Instantiations."abs"; -------------- -- Argument -- -------------- function Argument (X : Complex_Vector) return Real_Vector renames Instantiations.Argument; function Argument (X : Complex_Vector; Cycle : Real'Base) return Real_Vector renames Instantiations.Argument; function Argument (X : Complex_Matrix) return Real_Matrix renames Instantiations.Argument; function Argument (X : Complex_Matrix; Cycle : Real'Base) return Real_Matrix renames Instantiations.Argument; ---------------------------- -- Compose_From_Cartesian -- ---------------------------- function Compose_From_Cartesian (Re : Real_Vector) return Complex_Vector renames Instantiations.Compose_From_Cartesian; function Compose_From_Cartesian (Re : Real_Vector; Im : Real_Vector) return Complex_Vector renames Instantiations.Compose_From_Cartesian; function Compose_From_Cartesian (Re : Real_Matrix) return Complex_Matrix renames Instantiations.Compose_From_Cartesian; function Compose_From_Cartesian (Re : Real_Matrix; Im : Real_Matrix) return Complex_Matrix renames Instantiations.Compose_From_Cartesian; ------------------------ -- Compose_From_Polar -- ------------------------ function Compose_From_Polar (Modulus : Real_Vector; Argument : Real_Vector) return Complex_Vector renames Instantiations.Compose_From_Polar; function Compose_From_Polar (Modulus : Real_Vector; Argument : Real_Vector; Cycle : Real'Base) return Complex_Vector renames Instantiations.Compose_From_Polar; function Compose_From_Polar (Modulus : Real_Matrix; Argument : Real_Matrix) return Complex_Matrix renames Instantiations.Compose_From_Polar; function Compose_From_Polar (Modulus : Real_Matrix; Argument : Real_Matrix; Cycle : Real'Base) return Complex_Matrix renames Instantiations.Compose_From_Polar; --------------- -- Conjugate -- --------------- function Conjugate (X : Complex_Vector) return Complex_Vector renames Instantiations.Conjugate; function Conjugate (X : Complex_Matrix) return Complex_Matrix renames Instantiations.Conjugate; ----------------- -- Determinant -- ----------------- function Determinant (A : Complex_Matrix) return Complex is M : Complex_Matrix := A; B : Complex_Matrix (A'Range (1), 1 .. 0); R : Complex; begin Forward_Eliminate (M, B, R); return R; end Determinant; ----------------- -- Eigensystem -- ----------------- procedure Eigensystem (A : Complex_Matrix; Values : out Real_Vector; Vectors : out Complex_Matrix) is N : constant Natural := Length (A); -- For a Hermitian matrix C, we convert the eigenvalue problem to a -- real symmetric one: if C = A + i * B, then the (N, N) complex -- eigenvalue problem: -- (A + i * B) * (u + i * v) = Lambda * (u + i * v) -- -- is equivalent to the (2 * N, 2 * N) real eigenvalue problem: -- [ A, B ] [ u ] = Lambda * [ u ] -- [ -B, A ] [ v ] [ v ] -- -- Note that the (2 * N, 2 * N) matrix above is symmetric, as -- Transpose (A) = A and Transpose (B) = -B if C is Hermitian. -- We solve this eigensystem using the real-valued algorithms. The final -- result will have every eigenvalue twice, so in the sorted output we -- just pick every second value, with associated eigenvector u + i * v. M : Real_Matrix (1 .. 2 * N, 1 .. 2 * N); Vals : Real_Vector (1 .. 2 * N); Vecs : Real_Matrix (1 .. 2 * N, 1 .. 2 * N); begin for J in 1 .. N loop for K in 1 .. N loop declare C : constant Complex := (A (A'First (1) + (J - 1), A'First (2) + (K - 1))); begin M (J, K) := Re (C); M (J + N, K + N) := Re (C); M (J + N, K) := Im (C); M (J, K + N) := -Im (C); end; end loop; end loop; Eigensystem (M, Vals, Vecs); for J in 1 .. N loop declare Col : constant Integer := Values'First + (J - 1); begin Values (Col) := Vals (2 * J); for K in 1 .. N loop declare Row : constant Integer := Vectors'First (2) + (K - 1); begin Vectors (Row, Col) := (Vecs (J * 2, Col), Vecs (J * 2, Col + N)); end; end loop; end; end loop; end Eigensystem; ----------------- -- Eigenvalues -- ----------------- function Eigenvalues (A : Complex_Matrix) return Real_Vector is -- See Eigensystem for a description of the algorithm N : constant Natural := Length (A); R : Real_Vector (A'Range (1)); M : Real_Matrix (1 .. 2 * N, 1 .. 2 * N); Vals : Real_Vector (1 .. 2 * N); begin for J in 1 .. N loop for K in 1 .. N loop declare C : constant Complex := (A (A'First (1) + (J - 1), A'First (2) + (K - 1))); begin M (J, K) := Re (C); M (J + N, K + N) := Re (C); M (J + N, K) := Im (C); M (J, K + N) := -Im (C); end; end loop; end loop; Vals := Eigenvalues (M); for J in 1 .. N loop R (A'First (1) + (J - 1)) := Vals (2 * J); end loop; return R; end Eigenvalues; -------- -- Im -- -------- function Im (X : Complex_Vector) return Real_Vector renames Instantiations.Im; function Im (X : Complex_Matrix) return Real_Matrix renames Instantiations.Im; ------------- -- Inverse -- ------------- function Inverse (A : Complex_Matrix) return Complex_Matrix is (Solve (A, Unit_Matrix (Length (A), First_1 => A'First (2), First_2 => A'First (1)))); ------------- -- Modulus -- ------------- function Modulus (X : Complex_Vector) return Real_Vector renames Instantiations.Modulus; function Modulus (X : Complex_Matrix) return Real_Matrix renames Instantiations.Modulus; -------- -- Re -- -------- function Re (X : Complex_Vector) return Real_Vector renames Instantiations.Re; function Re (X : Complex_Matrix) return Real_Matrix renames Instantiations.Re; ------------ -- Set_Im -- ------------ procedure Set_Im (X : in out Complex_Matrix; Im : Real_Matrix) renames Instantiations.Set_Im; procedure Set_Im (X : in out Complex_Vector; Im : Real_Vector) renames Instantiations.Set_Im; ------------ -- Set_Re -- ------------ procedure Set_Re (X : in out Complex_Matrix; Re : Real_Matrix) renames Instantiations.Set_Re; procedure Set_Re (X : in out Complex_Vector; Re : Real_Vector) renames Instantiations.Set_Re; ----------- -- Solve -- ----------- function Solve (A : Complex_Matrix; X : Complex_Vector) return Complex_Vector renames Instantiations.Solve; function Solve (A : Complex_Matrix; X : Complex_Matrix) return Complex_Matrix renames Instantiations.Solve; --------------- -- Transpose -- --------------- function Transpose (X : Complex_Matrix) return Complex_Matrix is R : Complex_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 Complex_Matrix renames Instantiations.Unit_Matrix; ----------------- -- Unit_Vector -- ----------------- function Unit_Vector (Index : Integer; Order : Positive; First : Integer := 1) return Complex_Vector renames Instantiations.Unit_Vector; end Ada.Numerics.Generic_Complex_Arrays;