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------------------------------------------------------------------------------
-- --
-- GNAT RUN-TIME COMPONENTS --
-- --
-- ADA.NUMERICS.GENERIC_REAL_ARRAYS --
-- --
-- B o d y --
-- --
-- Copyright (C) 2006-2022, 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. --
-- --
------------------------------------------------------------------------------
-- This version of Generic_Real_Arrays avoids the use of BLAS and LAPACK. One
-- reason for this is new Ada 2012 requirements that prohibit algorithms such
-- as Strassen's algorithm, which may be used by some BLAS implementations. In
-- addition, some platforms lacked suitable compilers to compile the reference
-- BLAS/LAPACK implementation. Finally, on some platforms there are more
-- floating point types than supported by BLAS/LAPACK.
-- Preconditions, postconditions, ghost code, loop invariants and assertions
-- in this unit are meant for analysis only, not for run-time checking, as it
-- would be too costly otherwise. This is enforced by setting the assertion
-- policy to Ignore.
pragma Assertion_Policy (Pre => Ignore,
Post => Ignore,
Ghost => Ignore,
Loop_Invariant => Ignore,
Assert => Ignore);
with Ada.Containers.Generic_Anonymous_Array_Sort; use Ada.Containers;
with System; use System;
with System.Generic_Array_Operations; use System.Generic_Array_Operations;
package body Ada.Numerics.Generic_Real_Arrays is
package Ops renames System.Generic_Array_Operations;
function Is_Non_Zero (X : Real'Base) return Boolean is (X /= 0.0);
procedure Back_Substitute is new Ops.Back_Substitute
(Scalar => Real'Base,
Matrix => Real_Matrix,
Is_Non_Zero => Is_Non_Zero);
function Diagonal is new Ops.Diagonal
(Scalar => Real'Base,
Vector => Real_Vector,
Matrix => Real_Matrix);
procedure Forward_Eliminate is new Ops.Forward_Eliminate
(Scalar => Real'Base,
Real => Real'Base,
Matrix => Real_Matrix,
Zero => 0.0,
One => 1.0);
procedure Swap_Column is new Ops.Swap_Column
(Scalar => Real'Base,
Matrix => Real_Matrix);
procedure Transpose is new Ops.Transpose
(Scalar => Real'Base,
Matrix => Real_Matrix);
function Is_Symmetric (A : Real_Matrix) return Boolean is
(Transpose (A) = A);
-- Return True iff A is symmetric, see RM G.3.1 (90).
function Is_Tiny (Value, Compared_To : Real) return Boolean is
(abs Compared_To + 100.0 * abs (Value) = abs Compared_To);
-- Return True iff the Value is much smaller in magnitude than the least
-- significant digit of Compared_To.
procedure Jacobi
(A : Real_Matrix;
Values : out Real_Vector;
Vectors : out Real_Matrix;
Compute_Vectors : Boolean := True);
-- Perform Jacobi's eigensystem algorithm on real symmetric matrix A
function Length is new Square_Matrix_Length (Real'Base, Real_Matrix);
-- Helper function that raises a Constraint_Error is the argument is
-- not a square matrix, and otherwise returns its length.
procedure Rotate (X, Y : in out Real; Sin, Tau : Real);
-- Perform a Givens rotation
procedure Sort_Eigensystem
(Values : in out Real_Vector;
Vectors : in out Real_Matrix);
-- Sort Values and associated Vectors by decreasing absolute value
procedure Swap (Left, Right : in out Real);
-- Exchange Left and Right
function Sqrt is new Ops.Sqrt (Real);
-- 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.
------------
-- Rotate --
------------
procedure Rotate (X, Y : in out Real; Sin, Tau : Real) is
Old_X : constant Real := X;
Old_Y : constant Real := Y;
begin
X := Old_X - Sin * (Old_Y + Old_X * Tau);
Y := Old_Y + Sin * (Old_X - Old_Y * Tau);
end Rotate;
----------
-- Swap --
----------
procedure Swap (Left, Right : in out Real) is
Temp : constant Real := Left;
begin
Left := Right;
Right := Temp;
end Swap;
-- 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
Inner_Product
(Left_Scalar => Real'Base,
Right_Scalar => Real'Base,
Result_Scalar => Real'Base,
Left_Vector => Real_Vector,
Right_Vector => Real_Vector,
Zero => 0.0);
function "*" is new
Matrix_Vector_Product
(Left_Scalar => Real'Base,
Right_Scalar => Real'Base,
Result_Scalar => Real'Base,
Matrix => Real_Matrix,
Right_Vector => Real_Vector,
Result_Vector => Real_Vector,
Zero => 0.0);
function "*" is new
Vector_Matrix_Product
(Left_Scalar => Real'Base,
Right_Scalar => Real'Base,
Result_Scalar => Real'Base,
Left_Vector => Real_Vector,
Matrix => Real_Matrix,
Result_Vector => Real_Vector,
Zero => 0.0);
function "*" is new
Matrix_Matrix_Product
(Left_Scalar => Real'Base,
Right_Scalar => Real'Base,
Result_Scalar => Real'Base,
Left_Matrix => Real_Matrix,
Right_Matrix => Real_Matrix,
Result_Matrix => Real_Matrix,
Zero => 0.0);
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
L2_Norm
(X_Scalar => Real'Base,
Result_Real => Real'Base,
X_Vector => Real_Vector,
"abs" => "+");
-- While the L2_Norm by definition uses the absolute values of the
-- elements of X_Vector, for real values the subsequent squaring
-- makes this unnecessary, so we substitute the "+" identity function
-- instead.
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 Solve is new
Matrix_Vector_Solution (Real'Base, 0.0, Real_Vector, Real_Matrix);
function Solve is new
Matrix_Matrix_Solution (Real'Base, 0.0, Real_Matrix);
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
renames Instantiations."*";
function "*" (Left, Right : Real_Vector) return Real_Matrix
renames Instantiations."*";
function "*" (Left : Real_Vector; Right : Real_Matrix) return Real_Vector
renames Instantiations."*";
function "*" (Left : Real_Matrix; Right : Real_Vector) return Real_Vector
renames Instantiations."*";
-- Matrix Multiplication
function "*" (Left, Right : Real_Matrix) return Real_Matrix
renames Instantiations."*";
---------
-- "/" --
---------
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
renames Instantiations."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
M : Real_Matrix := A;
B : Real_Matrix (A'Range (1), 1 .. 0);
R : Real'Base;
begin
Forward_Eliminate (M, B, R);
return R;
end Determinant;
-----------------
-- Eigensystem --
-----------------
procedure Eigensystem
(A : Real_Matrix;
Values : out Real_Vector;
Vectors : out Real_Matrix)
is
begin
Jacobi (A, Values, Vectors, Compute_Vectors => True);
Sort_Eigensystem (Values, Vectors);
end Eigensystem;
-----------------
-- Eigenvalues --
-----------------
function Eigenvalues (A : Real_Matrix) return Real_Vector is
begin
return Values : Real_Vector (A'Range (1)) do
declare
Vectors : Real_Matrix (1 .. 0, 1 .. 0);
begin
Jacobi (A, Values, Vectors, Compute_Vectors => False);
Sort_Eigensystem (Values, Vectors);
end;
end return;
end Eigenvalues;
-------------
-- Inverse --
-------------
function Inverse (A : Real_Matrix) return Real_Matrix is
(Solve (A, Unit_Matrix (Length (A),
First_1 => A'First (2),
First_2 => A'First (1))));
------------
-- Jacobi --
------------
procedure Jacobi
(A : Real_Matrix;
Values : out Real_Vector;
Vectors : out Real_Matrix;
Compute_Vectors : Boolean := True)
is
-- This subprogram uses Carl Gustav Jacob Jacobi's iterative method
-- for computing eigenvalues and eigenvectors and is based on
-- Rutishauser's implementation.
-- The given real symmetric matrix is transformed iteratively to
-- diagonal form through a sequence of appropriately chosen elementary
-- orthogonal transformations, called Jacobi rotations here.
-- The Jacobi method produces a systematic decrease of the sum of the
-- squares of off-diagonal elements. Convergence to zero is quadratic,
-- both for this implementation, as for the classic method that doesn't
-- use row-wise scanning for pivot selection.
-- The numerical stability and accuracy of Jacobi's method make it the
-- best choice here, even though for large matrices other methods will
-- be significantly more efficient in both time and space.
-- While the eigensystem computations are absolutely foolproof for all
-- real symmetric matrices, in presence of invalid values, or similar
-- exceptional situations it might not. In such cases the results cannot
-- be trusted and Constraint_Error is raised.
-- Note: this implementation needs temporary storage for 2 * N + N**2
-- values of type Real.
Max_Iterations : constant := 50;
N : constant Natural := Length (A);
subtype Square_Matrix is Real_Matrix (1 .. N, 1 .. N);
-- In order to annihilate the M (Row, Col) element, the
-- rotation parameters Cos and Sin are computed as
-- follows:
-- Theta = Cot (2.0 * Phi)
-- = (Diag (Col) - Diag (Row)) / (2.0 * M (Row, Col))
-- Then Tan (Phi) as the smaller root (in modulus) of
-- T**2 + 2 * T * Theta = 1 (or 0.5 / Theta, if Theta is large)
function Compute_Tan (Theta : Real) return Real is
(Real'Copy_Sign (1.0 / (abs Theta + Sqrt (1.0 + Theta**2)), Theta));
function Compute_Tan (P, H : Real) return Real is
(if Is_Tiny (P, Compared_To => H) then P / H
else Compute_Tan (Theta => H / (2.0 * P)));
pragma Annotate
(CodePeer, False_Positive, "divide by zero", "H, P /= 0");
function Sum_Strict_Upper (M : Square_Matrix) return Real;
-- Return the sum of all elements in the strict upper triangle of M
----------------------
-- Sum_Strict_Upper --
----------------------
function Sum_Strict_Upper (M : Square_Matrix) return Real is
Sum : Real := 0.0;
begin
for Row in 1 .. N - 1 loop
for Col in Row + 1 .. N loop
Sum := Sum + abs M (Row, Col);
end loop;
end loop;
return Sum;
end Sum_Strict_Upper;
M : Square_Matrix := A; -- Work space for solving eigensystem
Threshold : Real;
Sum : Real;
Diag : Real_Vector (1 .. N);
Diag_Adj : Real_Vector (1 .. N);
-- The vector Diag_Adj indicates the amount of change in each value,
-- while Diag tracks the value itself and Values holds the values as
-- they were at the beginning. As the changes typically will be small
-- compared to the absolute value of Diag, at the end of each iteration
-- Diag is computed as Diag + Diag_Adj thus avoiding accumulating
-- rounding errors. This technique is due to Rutishauser.
begin
if Compute_Vectors
and then (Vectors'Length (1) /= N or else Vectors'Length (2) /= N)
then
raise Constraint_Error with "incompatible matrix dimensions";
elsif Values'Length /= N then
raise Constraint_Error with "incompatible vector length";
elsif not Is_Symmetric (M) then
raise Constraint_Error with "matrix not symmetric";
end if;
-- Note: Only the locally declared matrix M and vectors (Diag, Diag_Adj)
-- have lower bound equal to 1. The Vectors matrix may have
-- different bounds, so take care indexing elements. Assignment
-- as a whole is fine as sliding is automatic in that case.
Vectors := (if not Compute_Vectors then [1 .. 0 => [1 .. 0 => 0.0]]
else Unit_Matrix (Vectors'Length (1), Vectors'Length (2)));
Values := Diagonal (M);
Sweep : for Iteration in 1 .. Max_Iterations loop
-- The first three iterations, perform rotation for any non-zero
-- element. After this, rotate only for those that are not much
-- smaller than the average off-diagnal element. After the fifth
-- iteration, additionally zero out off-diagonal elements that are
-- very small compared to elements on the diagonal with the same
-- column or row index.
Sum := Sum_Strict_Upper (M);
exit Sweep when Sum = 0.0;
Threshold := (if Iteration < 4 then 0.2 * Sum / Real (N**2) else 0.0);
-- Iterate over all off-diagonal elements, rotating any that have
-- an absolute value that exceeds the threshold.
Diag := Values;
Diag_Adj := [others => 0.0]; -- Accumulates adjustments to Diag
for Row in 1 .. N - 1 loop
for Col in Row + 1 .. N loop
-- If, before the rotation M (Row, Col) is tiny compared to
-- Diag (Row) and Diag (Col), rotation is skipped. This is
-- meaningful, as it produces no larger error than would be
-- produced anyhow if the rotation had been performed.
-- Suppress this optimization in the first four sweeps, so
-- that this procedure can be used for computing eigenvectors
-- of perturbed diagonal matrices.
if Iteration > 4
and then Is_Tiny (M (Row, Col), Compared_To => Diag (Row))
and then Is_Tiny (M (Row, Col), Compared_To => Diag (Col))
then
M (Row, Col) := 0.0;
elsif abs M (Row, Col) > Threshold then
Perform_Rotation : declare
Tan : constant Real := Compute_Tan (M (Row, Col),
Diag (Col) - Diag (Row));
Cos : constant Real := 1.0 / Sqrt (1.0 + Tan**2);
Sin : constant Real := Tan * Cos;
Tau : constant Real := Sin / (1.0 + Cos);
Adj : constant Real := Tan * M (Row, Col);
begin
Diag_Adj (Row) := Diag_Adj (Row) - Adj;
Diag_Adj (Col) := Diag_Adj (Col) + Adj;
Diag (Row) := Diag (Row) - Adj;
Diag (Col) := Diag (Col) + Adj;
M (Row, Col) := 0.0;
for J in 1 .. Row - 1 loop -- 1 <= J < Row
Rotate (M (J, Row), M (J, Col), Sin, Tau);
end loop;
for J in Row + 1 .. Col - 1 loop -- Row < J < Col
Rotate (M (Row, J), M (J, Col), Sin, Tau);
end loop;
for J in Col + 1 .. N loop -- Col < J <= N
Rotate (M (Row, J), M (Col, J), Sin, Tau);
end loop;
for J in Vectors'Range (1) loop
Rotate (Vectors (J, Row - 1 + Vectors'First (2)),
Vectors (J, Col - 1 + Vectors'First (2)),
Sin, Tau);
end loop;
end Perform_Rotation;
end if;
end loop;
end loop;
Values := Values + Diag_Adj;
end loop Sweep;
-- All normal matrices with valid values should converge perfectly.
if Sum /= 0.0 then
raise Constraint_Error with "eigensystem solution does not converge";
end if;
end Jacobi;
-----------
-- Solve --
-----------
function Solve (A : Real_Matrix; X : Real_Vector) return Real_Vector
renames Instantiations.Solve;
function Solve (A, X : Real_Matrix) return Real_Matrix
renames Instantiations.Solve;
----------------------
-- Sort_Eigensystem --
----------------------
procedure Sort_Eigensystem
(Values : in out Real_Vector;
Vectors : in out Real_Matrix)
is
procedure Swap (Left, Right : Integer);
-- Swap Values (Left) with Values (Right), and also swap the
-- corresponding eigenvectors. Note that lowerbounds may differ.
function Less (Left, Right : Integer) return Boolean is
(Values (Left) > Values (Right));
-- Sort by decreasing eigenvalue, see RM G.3.1 (76).
procedure Sort is new Generic_Anonymous_Array_Sort (Integer);
-- Sorts eigenvalues and eigenvectors by decreasing value
procedure Swap (Left, Right : Integer) is
begin
Swap (Values (Left), Values (Right));
Swap_Column (Vectors, Left - Values'First + Vectors'First (2),
Right - Values'First + Vectors'First (2));
end Swap;
begin
Sort (Values'First, Values'Last);
end Sort_Eigensystem;
---------------
-- Transpose --
---------------
function Transpose (X : Real_Matrix) return Real_Matrix is
begin
return R : Real_Matrix (X'Range (2), X'Range (1)) do
Transpose (X, R);
end return;
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;