1 ------------------------------------------------------------------------------
3 -- GNAT RUN-TIME COMPONENTS --
5 -- ADA.NUMERICS.GENERIC_REAL_ARRAYS --
9 -- Copyright (C) 2006-2011, Free Software Foundation, Inc. --
11 -- GNAT is free software; you can redistribute it and/or modify it under --
12 -- terms of the GNU General Public License as published by the Free Soft- --
13 -- ware Foundation; either version 3, or (at your option) any later ver- --
14 -- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
15 -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
16 -- or FITNESS FOR A PARTICULAR PURPOSE. --
18 -- As a special exception under Section 7 of GPL version 3, you are granted --
19 -- additional permissions described in the GCC Runtime Library Exception, --
20 -- version 3.1, as published by the Free Software Foundation. --
22 -- You should have received a copy of the GNU General Public License and --
23 -- a copy of the GCC Runtime Library Exception along with this program; --
24 -- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
25 -- <http://www.gnu.org/licenses/>. --
27 -- GNAT was originally developed by the GNAT team at New York University. --
28 -- Extensive contributions were provided by Ada Core Technologies Inc. --
30 ------------------------------------------------------------------------------
32 -- This version of Generic_Real_Arrays avoids the use of BLAS and LAPACK. One
33 -- reason for this is new Ada 2012 requirements that prohibit algorithms such
34 -- as Strassen's algorithm, which may be used by some BLAS implementations. In
35 -- addition, some platforms lacked suitable compilers to compile the reference
36 -- BLAS/LAPACK implementation. Finally, on some platforms there are more
37 -- floating point types than supported by BLAS/LAPACK.
39 with Ada.Containers.Generic_Anonymous_Array_Sort; use Ada.Containers;
41 with System; use System;
42 with System.Generic_Array_Operations; use System.Generic_Array_Operations;
44 package body Ada.Numerics.Generic_Real_Arrays is
46 package Ops renames System.Generic_Array_Operations;
48 function Is_Non_Zero (X : Real'Base) return Boolean is (X /= 0.0);
50 procedure Back_Substitute is new Ops.Back_Substitute
52 Matrix => Real_Matrix,
53 Is_Non_Zero => Is_Non_Zero);
55 function Diagonal is new Ops.Diagonal
57 Vector => Real_Vector,
58 Matrix => Real_Matrix);
60 procedure Forward_Eliminate is new Ops.Forward_Eliminate
63 Matrix => Real_Matrix,
67 procedure Swap_Column is new Ops.Swap_Column
69 Matrix => Real_Matrix);
71 procedure Transpose is new Ops.Transpose
73 Matrix => Real_Matrix);
75 function Is_Symmetric (A : Real_Matrix) return Boolean is
77 -- Return True iff A is symmetric, see RM G.3.1 (90).
79 function Is_Tiny (Value, Compared_To : Real) return Boolean is
80 (abs Compared_To + 100.0 * abs (Value) = abs Compared_To);
81 -- Return True iff the Value is much smaller in magnitude than the least
82 -- significant digit of Compared_To.
86 Values : out Real_Vector;
87 Vectors : out Real_Matrix;
88 Compute_Vectors : Boolean := True);
89 -- Perform Jacobi's eigensystem algorithm on real symmetric matrix A
91 function Length is new Square_Matrix_Length (Real'Base, Real_Matrix);
92 -- Helper function that raises a Constraint_Error is the argument is
93 -- not a square matrix, and otherwise returns its length.
95 procedure Rotate (X, Y : in out Real; Sin, Tau : Real);
96 -- Perform a Givens rotation
98 procedure Sort_Eigensystem
99 (Values : in out Real_Vector;
100 Vectors : in out Real_Matrix);
101 -- Sort Values and associated Vectors by decreasing absolute value
103 procedure Swap (Left, Right : in out Real);
104 -- Exchange Left and Right
106 function Sqrt is new Ops.Sqrt (Real);
107 -- Instant a generic square root implementation here, in order to avoid
108 -- instantiating a complete copy of Generic_Elementary_Functions.
109 -- Speed of the square root is not a big concern here.
115 procedure Rotate (X, Y : in out Real; Sin, Tau : Real) is
116 Old_X : constant Real := X;
117 Old_Y : constant Real := Y;
119 X := Old_X - Sin * (Old_Y + Old_X * Tau);
120 Y := Old_Y + Sin * (Old_X - Old_Y * Tau);
127 procedure Swap (Left, Right : in out Real) is
128 Temp : constant Real := Left;
134 -- Instantiating the following subprograms directly would lead to
135 -- name clashes, so use a local package.
137 package Instantiations is
140 Vector_Elementwise_Operation
141 (X_Scalar => Real'Base,
142 Result_Scalar => Real'Base,
143 X_Vector => Real_Vector,
144 Result_Vector => Real_Vector,
148 Matrix_Elementwise_Operation
149 (X_Scalar => Real'Base,
150 Result_Scalar => Real'Base,
151 X_Matrix => Real_Matrix,
152 Result_Matrix => Real_Matrix,
156 Vector_Vector_Elementwise_Operation
157 (Left_Scalar => Real'Base,
158 Right_Scalar => Real'Base,
159 Result_Scalar => Real'Base,
160 Left_Vector => Real_Vector,
161 Right_Vector => Real_Vector,
162 Result_Vector => Real_Vector,
166 Matrix_Matrix_Elementwise_Operation
167 (Left_Scalar => Real'Base,
168 Right_Scalar => Real'Base,
169 Result_Scalar => Real'Base,
170 Left_Matrix => Real_Matrix,
171 Right_Matrix => Real_Matrix,
172 Result_Matrix => Real_Matrix,
176 Vector_Elementwise_Operation
177 (X_Scalar => Real'Base,
178 Result_Scalar => Real'Base,
179 X_Vector => Real_Vector,
180 Result_Vector => Real_Vector,
184 Matrix_Elementwise_Operation
185 (X_Scalar => Real'Base,
186 Result_Scalar => Real'Base,
187 X_Matrix => Real_Matrix,
188 Result_Matrix => Real_Matrix,
192 Vector_Vector_Elementwise_Operation
193 (Left_Scalar => Real'Base,
194 Right_Scalar => Real'Base,
195 Result_Scalar => Real'Base,
196 Left_Vector => Real_Vector,
197 Right_Vector => Real_Vector,
198 Result_Vector => Real_Vector,
202 Matrix_Matrix_Elementwise_Operation
203 (Left_Scalar => Real'Base,
204 Right_Scalar => Real'Base,
205 Result_Scalar => Real'Base,
206 Left_Matrix => Real_Matrix,
207 Right_Matrix => Real_Matrix,
208 Result_Matrix => Real_Matrix,
212 Scalar_Vector_Elementwise_Operation
213 (Left_Scalar => Real'Base,
214 Right_Scalar => Real'Base,
215 Result_Scalar => Real'Base,
216 Right_Vector => Real_Vector,
217 Result_Vector => Real_Vector,
221 Scalar_Matrix_Elementwise_Operation
222 (Left_Scalar => Real'Base,
223 Right_Scalar => Real'Base,
224 Result_Scalar => Real'Base,
225 Right_Matrix => Real_Matrix,
226 Result_Matrix => Real_Matrix,
230 Vector_Scalar_Elementwise_Operation
231 (Left_Scalar => Real'Base,
232 Right_Scalar => Real'Base,
233 Result_Scalar => Real'Base,
234 Left_Vector => Real_Vector,
235 Result_Vector => Real_Vector,
239 Matrix_Scalar_Elementwise_Operation
240 (Left_Scalar => Real'Base,
241 Right_Scalar => Real'Base,
242 Result_Scalar => Real'Base,
243 Left_Matrix => Real_Matrix,
244 Result_Matrix => Real_Matrix,
249 (Left_Scalar => Real'Base,
250 Right_Scalar => Real'Base,
251 Result_Scalar => Real'Base,
252 Left_Vector => Real_Vector,
253 Right_Vector => Real_Vector,
254 Matrix => Real_Matrix);
258 (Left_Scalar => Real'Base,
259 Right_Scalar => Real'Base,
260 Result_Scalar => Real'Base,
261 Left_Vector => Real_Vector,
262 Right_Vector => Real_Vector,
266 Matrix_Vector_Product
267 (Left_Scalar => Real'Base,
268 Right_Scalar => Real'Base,
269 Result_Scalar => Real'Base,
270 Matrix => Real_Matrix,
271 Right_Vector => Real_Vector,
272 Result_Vector => Real_Vector,
276 Vector_Matrix_Product
277 (Left_Scalar => Real'Base,
278 Right_Scalar => Real'Base,
279 Result_Scalar => Real'Base,
280 Left_Vector => Real_Vector,
281 Matrix => Real_Matrix,
282 Result_Vector => Real_Vector,
286 Matrix_Matrix_Product
287 (Left_Scalar => Real'Base,
288 Right_Scalar => Real'Base,
289 Result_Scalar => Real'Base,
290 Left_Matrix => Real_Matrix,
291 Right_Matrix => Real_Matrix,
292 Result_Matrix => Real_Matrix,
296 Vector_Scalar_Elementwise_Operation
297 (Left_Scalar => Real'Base,
298 Right_Scalar => Real'Base,
299 Result_Scalar => Real'Base,
300 Left_Vector => Real_Vector,
301 Result_Vector => Real_Vector,
305 Matrix_Scalar_Elementwise_Operation
306 (Left_Scalar => Real'Base,
307 Right_Scalar => Real'Base,
308 Result_Scalar => Real'Base,
309 Left_Matrix => Real_Matrix,
310 Result_Matrix => Real_Matrix,
313 function "abs" is new
315 (X_Scalar => Real'Base,
316 Result_Real => Real'Base,
317 X_Vector => Real_Vector,
319 -- While the L2_Norm by definition uses the absolute values of the
320 -- elements of X_Vector, for real values the subsequent squaring
321 -- makes this unnecessary, so we substitute the "+" identity function
324 function "abs" is new
325 Vector_Elementwise_Operation
326 (X_Scalar => Real'Base,
327 Result_Scalar => Real'Base,
328 X_Vector => Real_Vector,
329 Result_Vector => Real_Vector,
332 function "abs" is new
333 Matrix_Elementwise_Operation
334 (X_Scalar => Real'Base,
335 Result_Scalar => Real'Base,
336 X_Matrix => Real_Matrix,
337 Result_Matrix => Real_Matrix,
341 new Matrix_Vector_Solution (Real'Base, Real_Vector, Real_Matrix);
343 function Solve is new Matrix_Matrix_Solution (Real'Base, Real_Matrix);
345 function Unit_Matrix is new
346 Generic_Array_Operations.Unit_Matrix
347 (Scalar => Real'Base,
348 Matrix => Real_Matrix,
352 function Unit_Vector is new
353 Generic_Array_Operations.Unit_Vector
354 (Scalar => Real'Base,
355 Vector => Real_Vector,
365 function "+" (Right : Real_Vector) return Real_Vector
366 renames Instantiations."+";
368 function "+" (Right : Real_Matrix) return Real_Matrix
369 renames Instantiations."+";
371 function "+" (Left, Right : Real_Vector) return Real_Vector
372 renames Instantiations."+";
374 function "+" (Left, Right : Real_Matrix) return Real_Matrix
375 renames Instantiations."+";
381 function "-" (Right : Real_Vector) return Real_Vector
382 renames Instantiations."-";
384 function "-" (Right : Real_Matrix) return Real_Matrix
385 renames Instantiations."-";
387 function "-" (Left, Right : Real_Vector) return Real_Vector
388 renames Instantiations."-";
390 function "-" (Left, Right : Real_Matrix) return Real_Matrix
391 renames Instantiations."-";
397 -- Scalar multiplication
399 function "*" (Left : Real'Base; Right : Real_Vector) return Real_Vector
400 renames Instantiations."*";
402 function "*" (Left : Real_Vector; Right : Real'Base) return Real_Vector
403 renames Instantiations."*";
405 function "*" (Left : Real'Base; Right : Real_Matrix) return Real_Matrix
406 renames Instantiations."*";
408 function "*" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix
409 renames Instantiations."*";
411 -- Vector multiplication
413 function "*" (Left, Right : Real_Vector) return Real'Base
414 renames Instantiations."*";
416 function "*" (Left, Right : Real_Vector) return Real_Matrix
417 renames Instantiations."*";
419 function "*" (Left : Real_Vector; Right : Real_Matrix) return Real_Vector
420 renames Instantiations."*";
422 function "*" (Left : Real_Matrix; Right : Real_Vector) return Real_Vector
423 renames Instantiations."*";
425 -- Matrix Multiplication
427 function "*" (Left, Right : Real_Matrix) return Real_Matrix
428 renames Instantiations."*";
434 function "/" (Left : Real_Vector; Right : Real'Base) return Real_Vector
435 renames Instantiations."/";
437 function "/" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix
438 renames Instantiations."/";
444 function "abs" (Right : Real_Vector) return Real'Base
445 renames Instantiations."abs";
447 function "abs" (Right : Real_Vector) return Real_Vector
448 renames Instantiations."abs";
450 function "abs" (Right : Real_Matrix) return Real_Matrix
451 renames Instantiations."abs";
457 function Determinant (A : Real_Matrix) return Real'Base is
458 M : Real_Matrix := A;
459 B : Real_Matrix (A'Range (1), 1 .. 0);
462 Forward_Eliminate (M, B, R);
470 procedure Eigensystem
472 Values : out Real_Vector;
473 Vectors : out Real_Matrix)
476 Jacobi (A, Values, Vectors, Compute_Vectors => True);
477 Sort_Eigensystem (Values, Vectors);
484 function Eigenvalues (A : Real_Matrix) return Real_Vector is
485 Values : Real_Vector (A'Range (1));
486 Vectors : Real_Matrix (1 .. 0, 1 .. 0);
488 Jacobi (A, Values, Vectors, Compute_Vectors => False);
489 Sort_Eigensystem (Values, Vectors);
497 function Inverse (A : Real_Matrix) return Real_Matrix is
498 (Solve (A, Unit_Matrix (Length (A))));
506 Values : out Real_Vector;
507 Vectors : out Real_Matrix;
508 Compute_Vectors : Boolean := True)
510 -- This subprogram uses Carl Gustav Jacob Jacobi's iterative method
511 -- for computing eigenvalues and eigenvectors and is based on
512 -- Rutishauser's implementation.
514 -- The given real symmetric matrix is transformed iteratively to
515 -- diagonal form through a sequence of appropriately chosen elementary
516 -- orthogonal transformations, called Jacobi rotations here.
518 -- The Jacobi method produces a systematic decrease of the sum of the
519 -- squares of off-diagonal elements. Convergence to zero is quadratic,
520 -- both for this implementation, as for the classic method that doesn't
521 -- use row-wise scanning for pivot selection.
523 -- The numerical stability and accuracy of Jacobi's method make it the
524 -- best choice here, even though for large matrices other methods will
525 -- be significantly more efficient in both time and space.
527 -- While the eigensystem computations are absolutely foolproof for all
528 -- real symmetric matrices, in presence of invalid values, or similar
529 -- exceptional situations it might not. In such cases the results cannot
530 -- be trusted and Constraint_Error is raised.
532 -- Note: this implementation needs temporary storage for 2 * N + N**2
533 -- values of type Real.
535 Max_Iterations : constant := 50;
536 N : constant Natural := Length (A);
538 subtype Square_Matrix is Real_Matrix (1 .. N, 1 .. N);
540 -- In order to annihilate the M (Row, Col) element, the
541 -- rotation parameters Cos and Sin are computed as
544 -- Theta = Cot (2.0 * Phi)
545 -- = (Diag (Col) - Diag (Row)) / (2.0 * M (Row, Col))
547 -- Then Tan (Phi) as the smaller root (in modulus) of
549 -- T**2 + 2 * T * Theta = 1 (or 0.5 / Theta, if Theta is large)
551 function Compute_Tan (Theta : Real) return Real is
552 (Real'Copy_Sign (1.0 / (abs Theta + Sqrt (1.0 + Theta**2)), Theta));
554 function Compute_Tan (P, H : Real) return Real is
555 (if Is_Tiny (P, Compared_To => H) then P / H
556 else Compute_Tan (Theta => H / (2.0 * P)));
558 function Sum_Strict_Upper (M : Square_Matrix) return Real;
559 -- Return the sum of all elements in the strict upper triangle of M
561 ----------------------
562 -- Sum_Strict_Upper --
563 ----------------------
565 function Sum_Strict_Upper (M : Square_Matrix) return Real is
569 for Row in 1 .. N - 1 loop
570 for Col in Row + 1 .. N loop
571 Sum := Sum + abs M (Row, Col);
576 end Sum_Strict_Upper;
578 M : Square_Matrix := A; -- Work space for solving eigensystem
581 Diag : Real_Vector (1 .. N);
582 Diag_Adj : Real_Vector (1 .. N);
584 -- The vector Diag_Adj indicates the amount of change in each value,
585 -- while Diag tracks the value itself and Values holds the values as
586 -- they were at the beginning. As the changes typically will be small
587 -- compared to the absolute value of Diag, at the end of each iteration
588 -- Diag is computed as Diag + Diag_Adj thus avoiding accumulating
589 -- rounding errors. This technique is due to Rutishauser.
593 and then (Vectors'Length (1) /= N or else Vectors'Length (2) /= N)
595 raise Constraint_Error with "incompatible matrix dimensions";
597 elsif Values'Length /= N then
598 raise Constraint_Error with "incompatible vector length";
600 elsif not Is_Symmetric (M) then
601 raise Constraint_Error with "matrix not symmetric";
604 -- Note: Only the locally declared matrix M and vectors (Diag, Diag_Adj)
605 -- have lower bound equal to 1. The Vectors matrix may have
606 -- different bounds, so take care indexing elements. Assignment
607 -- as a whole is fine as sliding is automatic in that case.
609 Vectors := (if not Compute_Vectors then (1 .. 0 => (1 .. 0 => 0.0))
610 else Unit_Matrix (Vectors'Length (1), Vectors'Length (2)));
611 Values := Diagonal (M);
613 Sweep : for Iteration in 1 .. Max_Iterations loop
615 -- The first three iterations, perform rotation for any non-zero
616 -- element. After this, rotate only for those that are not much
617 -- smaller than the average off-diagnal element. After the fifth
618 -- iteration, additionally zero out off-diagonal elements that are
619 -- very small compared to elements on the diagonal with the same
620 -- column or row index.
622 Sum := Sum_Strict_Upper (M);
624 exit Sweep when Sum = 0.0;
626 Threshold := (if Iteration < 4 then 0.2 * Sum / Real (N**2) else 0.0);
628 -- Iterate over all off-diagonal elements, rotating any that have
629 -- an absolute value that exceeds the threshold.
632 Diag_Adj := (others => 0.0); -- Accumulates adjustments to Diag
634 for Row in 1 .. N - 1 loop
635 for Col in Row + 1 .. N loop
637 -- If, before the rotation M (Row, Col) is tiny compared to
638 -- Diag (Row) and Diag (Col), rotation is skipped. This is
639 -- meaningful, as it produces no larger error than would be
640 -- produced anyhow if the rotation had been performed.
641 -- Suppress this optimization in the first four sweeps, so
642 -- that this procedure can be used for computing eigenvectors
643 -- of perturbed diagonal matrices.
646 and then Is_Tiny (M (Row, Col), Compared_To => Diag (Row))
647 and then Is_Tiny (M (Row, Col), Compared_To => Diag (Col))
651 elsif abs M (Row, Col) > Threshold then
652 Perform_Rotation : declare
653 Tan : constant Real := Compute_Tan (M (Row, Col),
654 Diag (Col) - Diag (Row));
655 Cos : constant Real := 1.0 / Sqrt (1.0 + Tan**2);
656 Sin : constant Real := Tan * Cos;
657 Tau : constant Real := Sin / (1.0 + Cos);
658 Adj : constant Real := Tan * M (Row, Col);
661 Diag_Adj (Row) := Diag_Adj (Row) - Adj;
662 Diag_Adj (Col) := Diag_Adj (Col) + Adj;
663 Diag (Row) := Diag (Row) - Adj;
664 Diag (Col) := Diag (Col) + Adj;
668 for J in 1 .. Row - 1 loop -- 1 <= J < Row
669 Rotate (M (J, Row), M (J, Col), Sin, Tau);
672 for J in Row + 1 .. Col - 1 loop -- Row < J < Col
673 Rotate (M (Row, J), M (J, Col), Sin, Tau);
676 for J in Col + 1 .. N loop -- Col < J <= N
677 Rotate (M (Row, J), M (Col, J), Sin, Tau);
680 for J in Vectors'Range (1) loop
681 Rotate (Vectors (J, Row - 1 + Vectors'First (2)),
682 Vectors (J, Col - 1 + Vectors'First (2)),
685 end Perform_Rotation;
690 Values := Values + Diag_Adj;
693 -- All normal matrices with valid values should converge perfectly.
696 raise Constraint_Error with "eigensystem solution does not converge";
704 function Solve (A : Real_Matrix; X : Real_Vector) return Real_Vector
705 renames Instantiations.Solve;
707 function Solve (A, X : Real_Matrix) return Real_Matrix
708 renames Instantiations.Solve;
710 ----------------------
711 -- Sort_Eigensystem --
712 ----------------------
714 procedure Sort_Eigensystem
715 (Values : in out Real_Vector;
716 Vectors : in out Real_Matrix)
718 procedure Swap (Left, Right : Integer);
719 -- Swap Values (Left) with Values (Right), and also swap the
720 -- corresponding eigenvectors. Note that lowerbounds may differ.
722 function Less (Left, Right : Integer) return Boolean is
723 (Values (Left) > Values (Right));
724 -- Sort by decreasing eigenvalue, see RM G.3.1 (76).
726 procedure Sort is new Generic_Anonymous_Array_Sort (Integer);
727 -- Sorts eigenvalues and eigenvectors by decreasing value
729 procedure Swap (Left, Right : Integer) is
731 Swap (Values (Left), Values (Right));
732 Swap_Column (Vectors, Left - Values'First + Vectors'First (2),
733 Right - Values'First + Vectors'First (2));
737 Sort (Values'First, Values'Last);
738 end Sort_Eigensystem;
744 function Transpose (X : Real_Matrix) return Real_Matrix is
745 R : Real_Matrix (X'Range (2), X'Range (1));
757 First_1 : Integer := 1;
758 First_2 : Integer := 1) return Real_Matrix
759 renames Instantiations.Unit_Matrix;
768 First : Integer := 1) return Real_Vector
769 renames Instantiations.Unit_Vector;
771 end Ada.Numerics.Generic_Real_Arrays;