Mercurial > octave
view scripts/sparse/pcr.m @ 27978:a4268efb7334
maint: Eliminate single "Author:" lines from code base.
* color-picker.cc, color-picker.h, octave-qscintilla.cc, octave-qscintilla.h,
octave-txt-lexer.cc, octave-txt-lexer.h, balance.cc, kron.cc, lookup.cc,
ls-hdf5.cc, ls-mat5.cc, ls-oct-text.cc, sylvester.cc, tsearch.cc, jit-ir.cc,
jit-ir.h, jit-typeinfo.cc, jit-typeinfo.h, jit-util.cc, jit-util.h, pt-jit.cc,
pt-jit.h, Array-jit.cc, bsxfun.h, url-transfer.cc, url-transfer.h, acosd.m,
acotd.m, acscd.m, asecd.m, asind.m, atand.m, cosd.m, cotd.m, cscd.m, secd.m,
sind.m, tand.m, cart2pol.m, cart2sph.m, curl.m, del2.m, divergence.m, flip.m,
pol2cart.m, randi.m, sortrows.m, sph2cart.m, griddata3.m, griddatan.m,
rectint.m, voronoi.m, dialog.m, guidata.m, guihandles.m, __file_filter__.m,
__fltk_file_filter__.m, __get_funcname__.m, __is_function__.m,
__uigetdir_fltk__.m, __uigetfile_fltk__.m, __uiobject_split_args__.m,
__uiputfile_fltk__.m, uibuttongroup.m, uicontextmenu.m, uicontrol.m,
uigetdir.m, uigetfile.m, uimenu.m, uipanel.m, uipushtool.m, uiputfile.m,
uiresume.m, uitoggletool.m, uitoolbar.m, uiwait.m, doc.m, autumn.m, bone.m,
cmpermute.m, cmunique.m, cool.m, copper.m, cubehelix.m, flag.m, frame2im.m,
hot.m, hsv.m, hsv2rgb.m, im2frame.m, imfinfo.m, imformats.m, iscolormap.m,
jet.m, pink.m, prism.m, __imfinfo__.m, rainbow.m, rgb2hsv.m, spinmap.m,
spring.m, viridis.m, white.m, winter.m, importdata.m, strmatch.m, bandwidth.m,
isbanded.m, isdiag.m, istril.m, istriu.m, krylov.m, linsolve.m, logm.m,
lscov.m, subspace.m, bunzip2.m, compare_versions.m, edit.m, fullfile.m,
getfield.m, gunzip.m, inputParser.m, license.m, nargchk.m, narginchk.m,
nthargout.m, python.m, setfield.m, tar.m, unpack.m, ver.m, zip.m, odeplot.m,
fminbnd.m, fminunc.m, fsolve.m, fzero.m, glpk.m, humps.m, pathdef.m,
savepath.m, diffuse.m, lighting.m, material.m, shading.m, specular.m,
contour.m, ellipsoid.m, isocaps.m, isocolors.m, isonormals.m, isosurface.m,
light.m, pcolor.m, __interp_cube__.m, __marching_cube__.m, __patch__.m,
__stem__.m, reducepatch.m, reducevolume.m, ribbon.m, shrinkfaces.m, slice.m,
stem.m, surf.m, surfl.m, tetramesh.m, waterfall.m, __actual_axis_position__.m,
__next_line_color__.m, allchild.m, figure.m, findall.m, findobj.m, gcf.m,
hggroup.m, isprop.m, ndgrid.m, __add_default_menu__.m, saveas.m,
__splinefit__.m, prefdir.m, preferences.m, profexplore.m, profexport.m,
profile.m, profshow.m, setdiff.m, freqz.m, freqz_plot.m, sinc.m, unwrap.m,
pcr.m, qmr.m, sprandn.m, betaincinv.m, cosint.m, ellipke.m, factor.m,
gammaincinv.m, legendre.m, nchoosek.m, primes.m, invhilb.m, toeplitz.m,
erase.m, mat2str.m, strtrim.m, validatestring.m, fail.m, compare_plot_demos.m,
dump_demos.m, html_compare_plot_demos.m, addtodate.m, datenum.m, jit.tst:
Eliminate single "Author:" lines from code base.
author | Rik <rik@octave.org> |
---|---|
date | Tue, 21 Jan 2020 12:53:13 -0800 |
parents | bd51beb6205e |
children | b09432b20a84 |
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######################################################################## ## ## Copyright (C) 2004-2020 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} =} pcr (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{m}, @var{x0}, @dots{}) ## @deftypefnx {} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} pcr (@dots{}) ## ## Solve the linear system of equations @code{@var{A} * @var{x} = @var{b}} by ## means of the Preconditioned Conjugate Residuals iterative method. ## ## The input arguments are ## ## @itemize ## @item ## @var{A} can be either a square (preferably sparse) matrix or a function ## handle, inline function or string containing the name of a function which ## computes @code{@var{A} * @var{x}}. In principle @var{A} should be ## symmetric and non-singular; if @code{pcr} finds @var{A} to be numerically ## singular, you will get a warning message and the @var{flag} output ## parameter will be set. ## ## @item ## @var{b} is the right hand side vector. ## ## @item ## @var{tol} is the required relative tolerance for the residual error, ## @code{@var{b} - @var{A} * @var{x}}. The iteration stops if ## @code{norm (@var{b} - @var{A} * @var{x}) <= ## @var{tol} * norm (@var{b} - @var{A} * @var{x0})}. ## If @var{tol} is empty or is omitted, the function sets ## @code{@var{tol} = 1e-6} by default. ## ## @item ## @var{maxit} is the maximum allowable number of iterations; if @code{[]} is ## supplied for @var{maxit}, or @code{pcr} has less arguments, a default ## value equal to 20 is used. ## ## @item ## @var{m} is the (left) preconditioning matrix, so that the iteration is ## (theoretically) equivalent to solving by ## @code{pcr} @code{@var{P} * @var{x} = @var{m} \ @var{b}}, with ## @code{@var{P} = @var{m} \ @var{A}}. Note that a proper choice of the ## preconditioner may dramatically improve the overall performance of the ## method. Instead of matrix @var{m}, the user may pass a function which ## returns the results of applying the inverse of @var{m} to a vector ## (usually this is the preferred way of using the preconditioner). If ## @code{[]} is supplied for @var{m}, or @var{m} is omitted, no ## preconditioning is applied. ## ## @item ## @var{x0} is the initial guess. If @var{x0} is empty or omitted, the ## function sets @var{x0} to a zero vector by default. ## @end itemize ## ## The arguments which follow @var{x0} are treated as parameters, and passed ## in a proper way to any of the functions (@var{A} or @var{m}) which are ## passed to @code{pcr}. See the examples below for further details. ## ## The output arguments are ## ## @itemize ## @item ## @var{x} is the computed approximation to the solution of ## @code{@var{A} * @var{x} = @var{b}}. ## ## @item ## @var{flag} reports on the convergence. @code{@var{flag} = 0} means the ## solution converged and the tolerance criterion given by @var{tol} is ## satisfied. @code{@var{flag} = 1} means that the @var{maxit} limit for the ## iteration count was reached. @code{@var{flag} = 3} reports a @code{pcr} ## breakdown, see [1] for details. ## ## @item ## @var{relres} is the ratio of the final residual to its initial value, ## measured in the Euclidean norm. ## ## @item ## @var{iter} is the actual number of iterations performed. ## ## @item ## @var{resvec} describes the convergence history of the method, so that ## @code{@var{resvec} (i)} contains the Euclidean norms of the residual after ## the (@var{i}-1)-th iteration, @code{@var{i} = 1,2, @dots{}, @var{iter}+1}. ## @end itemize ## ## Let us consider a trivial problem with a diagonal matrix (we exploit the ## sparsity of A) ## ## @example ## @group ## n = 10; ## A = sparse (diag (1:n)); ## b = rand (N, 1); ## @end group ## @end example ## ## @sc{Example 1:} Simplest use of @code{pcr} ## ## @example ## x = pcr (A, b) ## @end example ## ## @sc{Example 2:} @code{pcr} with a function which computes ## @code{@var{A} * @var{x}}. ## ## @example ## @group ## function y = apply_a (x) ## y = [1:10]' .* x; ## endfunction ## ## x = pcr ("apply_a", b) ## @end group ## @end example ## ## @sc{Example 3:} Preconditioned iteration, with full diagnostics. The ## preconditioner (quite strange, because even the original matrix ## @var{A} is trivial) is defined as a function ## ## @example ## @group ## function y = apply_m (x) ## k = floor (length (x) - 2); ## y = x; ## y(1:k) = x(1:k) ./ [1:k]'; ## endfunction ## ## [x, flag, relres, iter, resvec] = ... ## pcr (A, b, [], [], "apply_m") ## semilogy ([1:iter+1], resvec); ## @end group ## @end example ## ## @sc{Example 4:} Finally, a preconditioner which depends on a ## parameter @var{k}. ## ## @example ## @group ## function y = apply_m (x, varargin) ## k = varargin@{1@}; ## y = x; ## y(1:k) = x(1:k) ./ [1:k]'; ## endfunction ## ## [x, flag, relres, iter, resvec] = ... ## pcr (A, b, [], [], "apply_m"', [], 3) ## @end group ## @end example ## ## References: ## ## [1] @nospell{W. Hackbusch}, @cite{Iterative Solution of Large Sparse ## Systems of Equations}, section 9.5.4; @nospell{Springer}, 1994 ## ## @seealso{sparse, pcg} ## @end deftypefn function [x, flag, relres, iter, resvec] = pcr (A, b, tol, maxit, m, x0, varargin) breakdown = false; if (nargin < 6 || isempty (x0)) x = zeros (size (b)); else x = x0; endif if (nargin < 5) m = []; endif if (nargin < 4 || isempty (maxit)) maxit = 20; endif maxit += 2; if (nargin < 3 || isempty (tol)) tol = 1e-6; endif if (nargin < 2) print_usage (); endif ## init if (isnumeric (A)) # is A a matrix? r = b - A*x; else # then A should be a function! r = b - feval (A, x, varargin{:}); endif if (isnumeric (m)) # is M a matrix? if (isempty (m)) # if M is empty, use no precond p = r; else # otherwise, apply the precond p = m \ r; endif else # then M should be a function! p = feval (m, r, varargin{:}); endif iter = 2; b_bot_old = 1; q_old = p_old = s_old = zeros (size (x)); if (isnumeric (A)) # is A a matrix? q = A * p; else # then A should be a function! q = feval (A, p, varargin{:}); endif resvec(1) = abs (norm (r)); ## iteration while (resvec(iter-1) > tol*resvec(1) && iter < maxit) if (isnumeric (m)) # is M a matrix? if (isempty (m)) # if M is empty, use no precond s = q; else # otherwise, apply the precond s = m \ q; endif else # then M should be a function! s = feval (m, q, varargin{:}); endif b_top = r' * s; b_bot = q' * s; if (b_bot == 0.0) breakdown = true; break; endif lambda = b_top / b_bot; x += lambda*p; r -= lambda*q; if (isnumeric (A)) # is A a matrix? t = A*s; else # then A should be a function! t = feval (A, s, varargin{:}); endif alpha0 = (t'*s) / b_bot; alpha1 = (t'*s_old) / b_bot_old; p_temp = p; q_temp = q; p = s - alpha0*p - alpha1*p_old; q = t - alpha0*q - alpha1*q_old; s_old = s; p_old = p_temp; q_old = q_temp; b_bot_old = b_bot; resvec(iter) = abs (norm (r)); iter += 1; endwhile flag = 0; relres = resvec(iter-1) ./ resvec(1); iter -= 2; if (iter >= maxit-2) flag = 1; if (nargout < 2) warning ("pcr: maximum number of iterations (%d) reached\n", iter); warning ("pcr: the initial residual norm was reduced %g times\n", 1.0/relres); endif elseif (nargout < 2 && ! breakdown) fprintf (stderr, "pcr: converged in %d iterations. \n", iter); fprintf (stderr, "pcr: the initial residual norm was reduced %g times\n", 1.0 / relres); endif if (breakdown) flag = 3; if (nargout < 2) warning ("pcr: breakdown occurred:\n"); warning ("system matrix singular or preconditioner indefinite?\n"); endif endif endfunction %!demo %! ## Simplest usage of PCR (see also 'help pcr') %! %! N = 20; %! A = diag (linspace (-3.1,3,N)); b = rand (N,1); %! y = A \ b; # y is the true solution %! x = pcr (A,b); %! printf ("The solution relative error is %g\n", norm (x-y) / norm (y)); %! %! ## You shouldn't be afraid if PCR issues some warning messages in this %! ## example: watch out in the second example, why it takes N iterations %! ## of PCR to converge to (a very accurate, by the way) solution. %!demo %! ## Full output from PCR %! ## We use this output to plot the convergence history %! %! N = 20; %! A = diag (linspace (-3.1,30,N)); b = rand (N,1); %! X = A \ b; # X is the true solution %! [x, flag, relres, iter, resvec] = pcr (A,b); %! printf ("The solution relative error is %g\n", norm (x-X) / norm (X)); %! clf; %! title ("Convergence history"); %! xlabel ("Iteration"); ylabel ("log(||b-Ax||/||b||)"); %! semilogy ([0:iter], resvec/resvec(1), "o-g;relative residual;"); %!demo %! ## Full output from PCR %! ## We use indefinite matrix based on the Hilbert matrix, with one %! ## strongly negative eigenvalue %! ## Hilbert matrix is extremely ill conditioned, so is ours, %! ## and that's why PCR WILL have problems %! %! N = 10; %! A = hilb (N); A(1,1) = -A(1,1); b = rand (N,1); %! X = A \ b; # X is the true solution %! printf ("Condition number of A is %g\n", cond (A)); %! [x, flag, relres, iter, resvec] = pcr (A,b,[],200); %! if (flag == 3) %! printf ("PCR breakdown. System matrix is [close to] singular\n"); %! endif %! clf; %! title ("Convergence history"); %! xlabel ("Iteration"); ylabel ("log(||b-Ax||)"); %! semilogy ([0:iter], resvec, "o-g;absolute residual;"); %!demo %! ## Full output from PCR %! ## We use an indefinite matrix based on the 1-D Laplacian matrix for A, %! ## and here we have cond(A) = O(N^2) %! ## That's the reason we need some preconditioner; here we take %! ## a very simple and not powerful Jacobi preconditioner, %! ## which is the diagonal of A. %! %! ## Note that we use here indefinite preconditioners! %! %! N = 100; %! ## Form 1-D Laplacian matrix %! A = 2 * eye (N,N); %! A(2:(N+1):end) = -1; %! A((N+1):(N+1):end) = -1; %! %! A = [A, zeros(size(A)); zeros(size(A)), -A]; %! b = rand (2*N,1); %! X = A \ b; # X is the true solution %! maxit = 80; %! printf ("System condition number is %g\n", cond (A)); %! ## No preconditioner: the convergence is very slow! %! %! [x, flag, relres, iter, resvec] = pcr (A,b,[],maxit); %! clf; %! title ("Convergence history"); %! xlabel ("Iteration"); ylabel ("log(||b-Ax||)"); %! semilogy ([0:iter], resvec, "o-g;NO preconditioning: absolute residual;"); %! %! pause (1); %! ## Test Jacobi preconditioner: it will not help much!!! %! %! M = diag (diag (A)); # Jacobi preconditioner %! [x, flag, relres, iter, resvec] = pcr (A,b,[],maxit,M); %! hold on; %! semilogy ([0:iter],resvec,"o-r;JACOBI preconditioner: absolute residual;"); %! %! pause (1); %! ## Test nonoverlapping block Jacobi preconditioner: this one should give %! ## some convergence speedup! %! %! M = zeros (N,N); k = 4; %! for i=1:k:N # get k x k diagonal blocks of A %! M(i:i+k-1,i:i+k-1) = A(i:i+k-1,i:i+k-1); %! endfor %! M = [M, zeros(size (M)); zeros(size(M)), -M]; %! [x, flag, relres, iter, resvec] = pcr (A,b,[],maxit,M); %! semilogy ([0:iter], resvec, "o-b;BLOCK JACOBI preconditioner: absolute residual;"); %! hold off; %!test %! ## solve small indefinite diagonal system %! %! N = 10; %! A = diag (linspace (-10.1,10,N)); b = ones (N,1); %! X = A \ b; # X is the true solution %! [x, flag] = pcr (A,b,[],N+1); %! assert (norm (x-X) / norm (X) < 1e-10); %! assert (flag, 0); %!test %! ## solve tridiagonal system, do not converge in default 20 iterations %! ## should perform max allowable default number of iterations %! %! N = 100; %! ## Form 1-D Laplacian matrix %! A = 2 * eye (N,N); %! A(2:(N+1):end) = -1; %! A((N+1):(N+1):end) = -1; %! b = ones (N,1); %! X = A \ b; # X is the true solution %! [x, flag, relres, iter, resvec] = pcr (A,b,1e-12); %! assert (flag, 1); %! assert (relres > 0.6); %! assert (iter, 20); %!test %! ## solve tridiagonal system with "perfect" preconditioner %! ## converges in one iteration %! %! N = 100; %! ## Form 1-D Laplacian matrix %! A = 2 * eye (N,N); %! A(2:(N+1):end) = -1; %! A((N+1):(N+1):end) = -1; %! b = ones (N,1); %! X = A \ b; # X is the true solution %! [x, flag, relres, iter] = pcr (A,b,[],[],A,b); %! assert (norm (x-X) / norm(X) < 1e-6); %! assert (relres < 1e-6); %! assert (flag, 0); %! assert (iter, 1); # should converge in one iteration