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Use Octave coding conventions in all m-file %!test blocks
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csc.m, cscd.m, csch.m, sec.m, secd.m, sech.m, sind.m, tand.m, accumarray.m,
accumdim.m, bitcmp.m, bitget.m, bitset.m, blkdiag.m, cart2pol.m, cart2sph.m,
celldisp.m, chop.m, circshift.m, colon.m, common_size.m, cplxpair.m,
cumtrapz.m, curl.m, dblquad.m, deal.m, divergence.m, flipdim.m, fliplr.m,
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interp1q.m, interp2.m, interp3.m, interpft.m, interpn.m, isa.m, isdir.m,
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polyarea.m, postpad.m, prepad.m, profile.m, profshow.m, quadgk.m, quadv.m,
randi.m, rat.m, repmat.m, rot90.m, rotdim.m, shift.m, shiftdim.m, sph2cart.m,
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textscan.m, commutation_matrix.m, cond.m, condest.m, cross.m,
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tcdf.m, tinv.m, tpdf.m, trnd.m, unidcdf.m, unidinv.m, unidpdf.m, unidrnd.m,
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asctime.m, clock.m, ctime.m, date.m, datenum.m, datetick.m, datevec.m,
eomday.m, etime.m, is_leap_year.m, now.m:
Use Octave coding conventions in all m-file %!test blocks
author | Rik <octave@nomad.inbox5.com> |
---|---|
date | Mon, 13 Feb 2012 07:29:44 -0800 |
parents | ce2b59a6d0e5 |
children | 5d3a684236b0 |
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## Copyright (C) 2004-2012 Piotr Krzyzanowski ## ## 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 ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {Function File} {@var{x} =} pcr (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{m}, @var{x0}, @dots{}) ## @deftypefnx {Function File} {[@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 @code{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 t ## @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] W. Hackbusch, @cite{Iterative Solution of Large Sparse Systems of ## Equations}, section 9.5.4; Springer, 1994 ## ## @seealso{sparse, pcg} ## @end deftypefn ## Author: Piotr Krzyzanowski <piotr.krzyzanowski@mimuw.edu.pl> 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++; 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 ("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, "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"); %! end %! 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; %! A = zeros (N,N); %! for i=1:N-1 # form 1-D Laplacian matrix %! A(i:i+1,i:i+1) = [2 -1; -1 2]; %! endfor %! 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; %! A = zeros (N,N); %! for i=1:N-1 # form 1-D Laplacian matrix %! A(i:i+1,i:i+1) = [2 -1; -1 2]; %! endfor %! 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; %! A = zeros (N,N); %! for i=1:N-1 # form 1-D Laplacian matrix %! A(i:i+1,i:i+1) = [2 -1; -1 2]; %! endfor %! 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