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author | John W. Eaton <jwe@octave.org> |
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date | Tue, 28 Dec 2021 18:22:40 -0500 |
parents | 7854d5752dd2 |
children | e1788b1a315f |
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######################################################################## ## ## Copyright (C) 2006-2022 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or <https://octave.org/copyright/>. ## ## This file is part of Octave. ## ## Octave is free software: you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <https://www.gnu.org/licenses/>. ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {@var{x} =} bicg (@var{A}, @var{b}) ## @deftypefnx {} {@var{x} =} bicg (@var{A}, @var{b}, @var{tol}) ## @deftypefnx {} {@var{x} =} bicg (@var{A}, @var{b}, @var{tol}, @var{maxit}) ## @deftypefnx {} {@var{x} =} bicg (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M}) ## @deftypefnx {} {@var{x} =} bicg (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M1}, @var{M2}) ## @deftypefnx {} {@var{x} =} bicg (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M}, [], @var{x0}) ## @deftypefnx {} {@var{x} =} bicg (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0}) ## @deftypefnx {} {@var{x} =} bicg (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M}, [], @var{x0}, @dots{}) ## @deftypefnx {} {@var{x} =} bicg (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0}, @dots{}) ## @deftypefnx {} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} bicg (@var{A}, @var{b}, @dots{}) ## Solve the linear system of equations @w{@code{@var{A} * @var{x} = @var{b}}} ## by means of the Bi-Conjugate Gradient iterative method. ## ## The input arguments are: ## ## @itemize ## ## @item @var{A} is the matrix of the linear system and it must be square. ## @var{A} can be passed as a matrix, function handle, or inline function ## @code{Afun} such that @w{@code{Afun (x, "notransp") = A * x}} and ## @w{@code{Afun (x, "transp") = A' * x}}. Additional parameters to ## @code{Afun} may be passed after @var{x0}. ## ## @item @var{b} is the right-hand side vector. It must be a column vector ## with the same number of rows as @var{A}. ## ## @item ## @var{tol} is the required relative tolerance for the residual error, ## @w{@code{@var{b} - @var{A} * @var{x}}}. The iteration stops if ## @w{@code{norm (@var{b} - @var{A} * @var{x})} @leq{} ## @w{@code{@var{tol} * norm (@var{b})}}}. ## If @var{tol} is omitted or empty, then a tolerance of 1e-6 is used. ## ## @item ## @var{maxit} is the maximum allowed number of iterations; if @var{maxit} ## is omitted or empty then a value of 20 is used. ## ## @item ## @var{M1}, @var{M2} are the preconditioners. The preconditioner @var{M} is ## given as @code{@var{M} = @var{M1} * @var{M2}}. Both @var{M1} and @var{M2} ## can be passed as a matrix or as a function handle or inline function ## @code{g} such that @w{@code{g (@var{x}, "notransp") = @var{M1} \ @var{x}}} ## or @w{@code{g (@var{x}, "notransp") = @var{M2} \ @var{x}}} and ## @w{@code{g (@var{x}, "transp") = @var{M1}' \ @var{x}}} or ## @w{@code{g (@var{x}, "transp") = @var{M2}' \ @var{x}}}. ## If @var{M1} is omitted or empty, then preconditioning is not applied. ## The preconditioned system is theoretically equivalent to applying the ## @code{bicg} method to the linear system ## @code{inv (@var{M1}) * A * inv (@var{M2}) * @var{y} = inv ## (@var{M1}) * @var{b}} and ## @code{inv (@var{M2'}) * A' * inv (@var{M1'}) * @var{z} = ## inv (@var{M2'}) * @var{b}} and then setting ## @code{@var{x} = inv (@var{M2}) * @var{y}}. ## ## @item ## @var{x0} is the initial guess. If @var{x0} is omitted or empty then the ## function sets @var{x0} to a zero vector by default. ## @end itemize ## ## Any arguments which follow @var{x0} are treated as parameters, and passed in ## an appropriate manner to any of the functions (@var{Afun} or @var{Mfun}) or ## that have been given to @code{bicg}. ## ## The output parameters are: ## ## @itemize ## ## @item ## @var{x} is the computed approximation to the solution of ## @w{@code{@var{A} * @var{x} = @var{b}}}. If the algorithm did not converge, ## then @var{x} is the iteration which has the minimum residual. ## ## @item ## @var{flag} indicates the exit status: ## ## @itemize ## @item 0: The algorithm converged to within the prescribed tolerance. ## ## @item 1: The algorithm did not converge and it reached the maximum number of ## iterations. ## ## @item 2: The preconditioner matrix is singular. ## ## @item 3: The algorithm stagnated, i.e., the absolute value of the ## difference between the current iteration @var{x} and the previous is less ## than @code{eps * norm (@var{x},2)}. ## ## @item 4: The algorithm could not continue because intermediate values ## became too small or too large for reliable computation. ## @end itemize ## ## @item ## @var{relres} is the ratio of the final residual to its initial value, ## measured in the Euclidean norm. ## ## @item ## @var{iter} is the iteration which @var{x} is computed. ## ## @item ## @var{resvec} is a vector containing the residual at each iteration. ## The total number of iterations performed is given by ## @code{length (@var{resvec}) - 1}. ## @end itemize ## ## Consider a trivial problem with a tridiagonal matrix ## ## @example ## @group ## n = 20; ## A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ... ## toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ... ## sparse (1, 2, 1, 1, n) * n / 2); ## b = A * ones (n, 1); ## restart = 5; ## [M1, M2] = ilu (A); # in this tridiag case, it corresponds to lu (A) ## M = M1 * M2; ## Afun = @@(x, string) strcmp (string, "notransp") * (A * x) + ... ## strcmp (string, "transp") * (A' * x); ## Mfun = @@(x, string) strcmp (string, "notransp") * (M \ x) + ... ## strcmp (string, "transp") * (M' \ x); ## M1fun = @@(x, string) strcmp (string, "notransp") * (M1 \ x) + ... ## strcmp (string, "transp") * (M1' \ x); ## M2fun = @@(x, string) strcmp (string, "notransp") * (M2 \ x) + ... ## strcmp (string, "transp") * (M2' \ x); ## @end group ## @end example ## ## @sc{Example 1:} simplest usage of @code{bicg} ## ## @example ## x = bicg (A, b) ## @end example ## ## @sc{Example 2:} @code{bicg} with a function that computes ## @code{@var{A}*@var{x}} and @code{@var{A'}*@var{x}} ## ## @example ## x = bicg (Afun, b, [], n) ## @end example ## ## @sc{Example 3:} @code{bicg} with a preconditioner matrix @var{M} ## ## @example ## x = bicg (A, b, 1e-6, n, M) ## @end example ## ## @sc{Example 4:} @code{bicg} with a function as preconditioner ## ## @example ## x = bicg (Afun, b, 1e-6, n, Mfun) ## @end example ## ## @sc{Example 5:} @code{bicg} with preconditioner matrices @var{M1} ## and @var{M2} ## ## @example ## x = bicg (A, b, 1e-6, n, M1, M2) ## @end example ## ## @sc{Example 6:} @code{bicg} with functions as preconditioners ## ## @example ## x = bicg (Afun, b, 1e-6, n, M1fun, M2fun) ## @end example ## ## @sc{Example 7:} @code{bicg} with as input a function requiring an argument ## ## @example ## @group ## function y = Ap (A, x, string, z) ## ## compute A^z * x or (A^z)' * x ## y = x; ## if (strcmp (string, "notransp")) ## for i = 1:z ## y = A * y; ## endfor ## elseif (strcmp (string, "transp")) ## for i = 1:z ## y = A' * y; ## endfor ## endif ## endfunction ## ## Apfun = @@(x, string, p) Ap (A, x, string, p); ## x = bicg (Apfun, b, [], [], [], [], [], 2); ## @end group ## @end example ## ## Reference: ## ## @nospell{Y. Saad}, @cite{Iterative Methods for Sparse Linear Systems}, ## Second edition, 2003, SIAM. ## ## @seealso{bicgstab, cgs, gmres, pcg, qmr, tfqmr} ## @end deftypefn function [x_min, flag, relres, iter_min, resvec] = ... bicg (A, b, tol = [], maxit = [], M1 = [], M2 = [], x0 = [], varargin) [Afun, M1fun, M2fun] = __alltohandles__ (A, b, M1, M2, "bicg"); [tol, maxit, x0] = __default__input__ ({1e-06, min(rows(b), 20), ... zeros(rows (b),1)}, tol, maxit, x0); if (columns (b) == 2) c = b(:,2); b = b(:,1); else c = b; endif norm_b = norm (b, 2); if (norm_b == 0) # the only (only iff det (A) == 0) solution is x = 0 if (nargout < 2) printf ("The right hand side vector is all zero so bicg\n") printf ("returned an all zero solution without iterating.\n") endif x_min = zeros (numel (b), 1); flag = 0; relres = 0; iter_min = 0; resvec = 0; return; endif x = x_min = x_pr = x0; iter = iter_min = 0; flag = 1; # Default flag is "maximum number of iterations reached" resvec = zeros (maxit + 1, 1); r0 = b - Afun (x, "notransp", varargin{:}); # Residual of the system s0 = c - Afun (x, "transp", varargin{:}); # Res. of the "dual system" resvec(1) = norm (r0, 2); try warning ("error", "Octave:singular-matrix", "local"); prec_r0 = M1fun (r0, "notransp", varargin{:}); # r0 preconditioned prec_s0 = s0; prec_r0 = M2fun (prec_r0, "notransp", varargin{:}); prec_s0 = M2fun (prec_s0, "transp", varargin{:}); prec_s0 = M1fun (prec_s0, "transp", varargin{:}); # s0 preconditioned p = prec_r0; # Direction of the system q = prec_s0; # Direction of the "dual system" catch flag = 2; end_try_catch while ((flag != 2) && (iter < maxit) && (resvec(iter+1) >= norm_b * tol)) v = Afun (p, "notransp", varargin{:}); prod_qv = q' * v; alpha = (s0' * prec_r0); if (abs (prod_qv) <= eps * abs (alpha)) flag = 4; break; endif alpha ./= prod_qv; x += alpha * p; prod_rs = (s0' * prec_r0); # Product between r0 and s0 r0 -= alpha * v; s0 -= conj (alpha) * Afun (q, "transp", varargin{:}); prec_r0 = M1fun (r0, "notransp", varargin{:}); prec_s0 = s0; prec_r0 = M2fun (prec_r0, "notransp", varargin{:}); beta = s0' * prec_r0; if (abs (prod_rs) <= abs (beta)) flag = 4; break; endif beta ./= prod_rs; prec_s0 = M2fun (prec_s0, "transp", varargin{:}); prec_s0 = M1fun (prec_s0, "transp", varargin{:}); iter += 1; resvec(iter+1) = norm (r0); if (resvec(iter+1) <= resvec(iter_min+1)) x_min = x; iter_min = iter; endif if (norm (x - x_pr) <= norm (x) * eps) flag = 3; break; endif p = prec_r0 + beta*p; q = prec_s0 + conj (beta) * q; endwhile resvec = resvec(1:iter+1,1); if (flag == 2) relres = 1; else relres = resvec(iter_min+1) / norm_b; endif if ((flag == 1) && (relres <= tol)) flag = 0; endif if (nargout < 2) switch (flag) case 0 printf ("bicg converged at iteration %i ", iter_min); printf ("to a solution with relative residual %e\n", relres); case 1 printf ("bicg stopped at iteration %i ", iter); printf ("without converging to the desired tolerance %e\n", tol); printf ("because the maximum number of iterations was reached. "); printf ("The iterate returned (number %i) has ", iter_min); printf ("relative residual %e\n", relres); case 2 printf ("bicg stopped at iteration %i ", iter); printf ("without converging to the desired tolerance %e\n", tol); printf ("because the preconditioner matrix is singular.\n"); printf ("The iterate returned (number %i) ", iter_min); printf ("has relative residual %e\n", relres); case 3 printf ("bicg stopped at iteration %i ", iter); printf ("without converging to the desired tolerance %e\n", tol); printf ("because the method stagnated.\n"); printf ("The iterate returned (number %i) ", iter_min); printf ("has relative residual %e\n", relres); case 4 printf ("bicg stopped at iteration %i ", iter); printf ("without converging to the desired tolerance %e\n", tol); printf ("because the method can't continue.\n"); printf ("The iterate returned (number %i) ", iter_min); printf ("has relative residual %e\n", relres); endswitch endif endfunction %!demo %! ## simplest use case %! n = 20; %! A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ... %! toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ... %! sparse (1, 2, 1, 1, n) * n / 2); %! b = A * ones (n, 1); %! [M1, M2] = ilu (A + 0.1 * eye (n)); %! M = M1 * M2; %! x = bicg (A, b, [], n); %! function y = Ap (A, x, string, z) %! ## compute A^z * x or (A^z)' * x %! y = x; %! if (strcmp (string, "notransp")) %! for i = 1:z %! y = A * y; %! endfor %! elseif (strcmp (string, "transp")) %! for i = 1:z %! y = A' * y; %! endfor %! endif %! endfunction %! %! Afun = @(x, string) Ap (A, x, string, 1); %! x = bicg (Afun, b, [], n); %! x = bicg (A, b, 1e-6, n, M); %! x = bicg (A, b, 1e-6, n, M1, M2); %! function y = Mfun (M, x, string) %! if (strcmp (string, "notransp")) %! y = M \ x; %! else %! y = M' \ x; %! endif %! endfunction %! %! M1fun = @(x, string) Mfun (M, x, string); %! x = bicg (Afun, b, 1e-6, n, M1fun); %! M1fun = @(x, string) Mfun (M1, x, string); %! M2fun = @(x, string) Mfun (M2, x, string); %! x = bicg (Afun, b, 1e-6, n, M1fun, M2fun); %! Afun = @(x, string, p) Ap (A, x, string, p); %! ## Solution of A^2 * x = b %! x = bicg (Afun, b, [], 2*n, [], [], [], 2); %!test %! ## Check that all type of inputs work %! A = sparse (toeplitz ([2, 1, 0, 0, 0], [2, -1, 0, 0, 0])); %! b = A * ones (5, 1); %! M1 = diag (sqrt (diag (A))); %! M2 = M1; %! Afun = @(z, string) strcmp (string, "notransp") * (A * z) + ... %! strcmp (string, "transp") * (A' * z); %! M1_fun = @(z, string) strcmp (string,"notransp") * (M1 \ z) + ... %! strcmp (string, "transp") * (M1' \ z); %! M2_fun = @(z, string) strcmp (string, "notransp") * (M2 \ z) + ... %! strcmp (string, "transp") * (M2' \ z); %! [x, flag] = bicg (A, b); %! assert (flag, 0); %! [x, flag] = bicg (A, b, [], [], M1, M2); %! assert (flag, 0); %! [x, flag] = bicg (A, b, [], [], M1_fun, M2_fun); %! assert (flag, 0); %! [x, flag] = bicg (A, b,[], [], M1_fun, M2); %! assert (flag, 0); %! [x, flag] = bicg (A, b,[], [], M1, M2_fun); %! assert (flag, 0); %! [x, flag] = bicg (Afun, b); %! assert (flag, 0); %! [x, flag] = bicg (Afun, b,[], [], M1, M2); %! assert (flag, 0); %! [x, flag] = bicg (Afun, b,[], [], M1_fun, M2); %! assert (flag, 0); %! [x, flag] = bicg (Afun, b,[], [], M1, M2_fun); %! assert (flag, 0); %! [x, flag] = bicg (Afun, b,[], [], M1_fun, M2_fun); %! assert (flag, 0); %!test %! n = 100; %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n); %! b = sum (A, 2); %! tol = 1e-8; %! maxit = 15; %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n); %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n); %! [x, flag, relres, iter, resvec] = bicg (A, b, tol, maxit, M1, M2); %! assert (norm (b - A*x) / norm (b), 0, tol); %!function y = afun (x, t, a) %! switch (t) %! case "notransp" %! y = a * x; %! case "transp" %! y = a' * x; %! endswitch %!endfunction %! %!test %! n = 100; %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n); %! b = sum (A, 2); %! tol = 1e-8; %! maxit = 15; %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n); %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n); %! %! [x, flag, relres, iter, resvec] = bicg (@(x, t) afun (x, t, A), %! b, tol, maxit, M1, M2); %! assert (x, ones (size (b)), 1e-7); %!test %! n = 100; %! tol = 1e-8; %! a = sprand (n, n, .1); %! A = a' * a + 100 * eye (n); %! b = sum (A, 2); %! [x, flag, relres, iter, resvec] = bicg (A, b, tol, [], diag (diag (A))); %! assert (x, ones (size (b)), 1e-7); %!test %! ## Check that if the preconditioner is singular, the method doesn't work %! A = sparse (toeplitz ([2, 1, 0, 0, 0], [2, -1, 0, 0, 0])); %! b = ones (5,1); %! M = ones (5); %! [x, flag] = bicg (A, b, [], [], M); %! assert (flag, 2); %!test %! ## If A singular, the algorithm doesn't work due to division by zero %! A = ones (5); %! b = [1:5]'; %! [x, flag] = bicg (A, b); %! assert (flag, 4); %!test %! ## test for a complex linear system %! A = sparse (toeplitz ([2, 1, 0, 0, 0], [2, -1, 0, 0, 0]) + ... %! 1i * toeplitz ([2, 1, 0, 0, 0], [2, -1, 0, 0, 0])); %! b = sum (A, 2); %! [x, flag] = bicg (A, b); %! assert (flag, 0); %!test %! A = single (1); %! b = 1; %! [x, flag] = bicg (A, b); %! assert (class (x), "single"); %!test %! A = 1; %! b = single (1); %! [x, flag] = bicg (A, b); %! assert (class (x), "single"); %!test %! A = single (1); %! b = single (1); %! [x, flag] = bicg (A, b); %! assert (class (x), "single"); %!test %!function y = Afun (x, trans) %! A = sparse (toeplitz ([2, 1, 0, 0], [2, -1, 0, 0])); %! if (strcmp (trans, "notransp")) %! y = A * x; %! else %! y = A' * x; %! endif %!endfunction %! %! [x, flag] = bicg ("Afun", [1; 2; 2; 3]); %! assert (x, ones (4, 1), 1e-6); %!test %! ## unpreconditioned residual %! A = sparse (toeplitz ([2, 1, 0, 0, 0], [2, -1, 0, 0, 0])); %! b = sum (A, 2); %! M = magic (5); %! [x, flag, relres] = bicg (A, b, [], 2, M); %! assert (norm (b - A * x) / norm (b), 0, relres + eps); ## Preconditioned technique %!testif HAVE_UMFPACK %! A = sparse (toeplitz ([2, 1, 0, 0, 0], [2, -1, 0, 0, 0])); %! b = sum (A, 2); %! warning ("off", "Octave:lu:sparse_input", "local"); %! [M1, M2] = lu (A + eye (5)); %! [x, flag] = bicg (A, b, [], 1, M1, M2); %! ## b has two columns! %! [y, flag] = bicg (M1 \ A / M2, [M1 \ b, M2' \ b], [], 1); %! assert (x, M2 \ y, 8 * eps);