Mercurial > octave-nkf
view scripts/sparse/eigs.m @ 17338:1c89599167a6
maint: End m-files with 1 blank line.
Simplifies automated grammarchecking script.
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scripts/ui/private/message_dialog.m: End m-files with 1 blank line.
author | Rik <rik@octave.org> |
---|---|
date | Wed, 28 Aug 2013 08:33:02 -0700 |
parents | bc924baa2c4e |
children | d63878346099 |
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## Copyright (C) 2005-2012 David Bateman ## ## 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{d} =} eigs (@var{A}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{k}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{k}, @var{sigma}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{k}, @var{sigma}, @var{opts}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{B}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{B}, @var{k}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{B}, @var{k}, @var{sigma}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{A}, @var{B}, @var{k}, @var{sigma}, @var{opts}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{B}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{k}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{B}, @var{k}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{k}, @var{sigma}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{B}, @var{k}, @var{sigma}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{k}, @var{sigma}, @var{opts}) ## @deftypefnx {Function File} {@var{d} =} eigs (@var{af}, @var{n}, @var{B}, @var{k}, @var{sigma}, @var{opts}) ## @deftypefnx {Function File} {[@var{V}, @var{d}] =} eigs (@var{A}, @dots{}) ## @deftypefnx {Function File} {[@var{V}, @var{d}] =} eigs (@var{af}, @var{n}, @dots{}) ## @deftypefnx {Function File} {[@var{V}, @var{d}, @var{flag}] =} eigs (@var{A}, @dots{}) ## @deftypefnx {Function File} {[@var{V}, @var{d}, @var{flag}] =} eigs (@var{af}, @var{n}, @dots{}) ## Calculate a limited number of eigenvalues and eigenvectors of @var{A}, ## based on a selection criteria. The number of eigenvalues and eigenvectors to ## calculate is given by @var{k} and defaults to 6. ## ## By default, @code{eigs} solve the equation ## @tex ## $A \nu = \lambda \nu$, ## @end tex ## @ifinfo ## @code{A * v = lambda * v}, ## @end ifinfo ## where ## @tex ## $\lambda$ is a scalar representing one of the eigenvalues, and $\nu$ ## @end tex ## @ifinfo ## @code{lambda} is a scalar representing one of the eigenvalues, and @code{v} ## @end ifinfo ## is the corresponding eigenvector. If given the positive definite matrix ## @var{B} then @code{eigs} solves the general eigenvalue equation ## @tex ## $A \nu = \lambda B \nu$. ## @end tex ## @ifinfo ## @code{A * v = lambda * B * v}. ## @end ifinfo ## ## The argument @var{sigma} determines which eigenvalues are returned. ## @var{sigma} can be either a scalar or a string. When @var{sigma} is a ## scalar, the @var{k} eigenvalues closest to @var{sigma} are returned. If ## @var{sigma} is a string, it must have one of the following values. ## ## @table @asis ## @item @qcode{"lm"} ## Largest Magnitude (default). ## ## @item @qcode{"sm"} ## Smallest Magnitude. ## ## @item @qcode{"la"} ## Largest Algebraic (valid only for real symmetric problems). ## ## @item @qcode{"sa"} ## Smallest Algebraic (valid only for real symmetric problems). ## ## @item @qcode{"be"} ## Both Ends, with one more from the high-end if @var{k} is odd (valid only for ## real symmetric problems). ## ## @item @qcode{"lr"} ## Largest Real part (valid only for complex or unsymmetric problems). ## ## @item @qcode{"sr"} ## Smallest Real part (valid only for complex or unsymmetric problems). ## ## @item @qcode{"li"} ## Largest Imaginary part (valid only for complex or unsymmetric problems). ## ## @item @qcode{"si"} ## Smallest Imaginary part (valid only for complex or unsymmetric problems). ## @end table ## ## If @var{opts} is given, it is a structure defining possible options that ## @code{eigs} should use. The fields of the @var{opts} structure are: ## ## @table @code ## @item issym ## If @var{af} is given, then flags whether the function @var{af} defines a ## symmetric problem. It is ignored if @var{A} is given. The default is false. ## ## @item isreal ## If @var{af} is given, then flags whether the function @var{af} defines a ## real problem. It is ignored if @var{A} is given. The default is true. ## ## @item tol ## Defines the required convergence tolerance, calculated as ## @code{tol * norm (A)}. The default is @code{eps}. ## ## @item maxit ## The maximum number of iterations. The default is 300. ## ## @item p ## The number of Lanzcos basis vectors to use. More vectors will result in ## faster convergence, but a greater use of memory. The optimal value of ## @code{p} is problem dependent and should be in the range @var{k} to @var{n}. ## The default value is @code{2 * @var{k}}. ## ## @item v0 ## The starting vector for the algorithm. An initial vector close to the ## final vector will speed up convergence. The default is for @sc{arpack} ## to randomly generate a starting vector. If specified, @code{v0} must be ## an @var{n}-by-1 vector where @code{@var{n} = rows (@var{A})} ## ## @item disp ## The level of diagnostic printout (0|1|2). If @code{disp} is 0 then ## diagnostics are disabled. The default value is 0. ## ## @item cholB ## Flag if @code{chol (@var{B})} is passed rather than @var{B}. The default is ## false. ## ## @item permB ## The permutation vector of the Cholesky@tie{}factorization of @var{B} if ## @code{cholB} is true. That is @code{chol (@var{B}(permB, permB))}. The ## default is @code{1:@var{n}}. ## ## @end table ## ## It is also possible to represent @var{A} by a function denoted @var{af}. ## @var{af} must be followed by a scalar argument @var{n} defining the length ## of the vector argument accepted by @var{af}. @var{af} can be ## a function handle, an inline function, or a string. When @var{af} is a ## string it holds the name of the function to use. ## ## @var{af} is a function of the form @code{y = af (x)} ## where the required return value of @var{af} is determined by ## the value of @var{sigma}. The four possible forms are ## ## @table @code ## @item A * x ## if @var{sigma} is not given or is a string other than "sm". ## ## @item A \ x ## if @var{sigma} is 0 or "sm". ## ## @item (A - sigma * I) \ x ## for the standard eigenvalue problem, where @code{I} is the identity matrix of ## the same size as @var{A}. ## ## @item (A - sigma * B) \ x ## for the general eigenvalue problem. ## @end table ## ## The return arguments of @code{eigs} depend on the number of return arguments ## requested. With a single return argument, a vector @var{d} of length @var{k} ## is returned containing the @var{k} eigenvalues that have been found. With ## two return arguments, @var{V} is a @var{n}-by-@var{k} matrix whose columns ## are the @var{k} eigenvectors corresponding to the returned eigenvalues. The ## eigenvalues themselves are returned in @var{d} in the form of a ## @var{n}-by-@var{k} matrix, where the elements on the diagonal are the ## eigenvalues. ## ## Given a third return argument @var{flag}, @code{eigs} returns the status ## of the convergence. If @var{flag} is 0 then all eigenvalues have converged. ## Any other value indicates a failure to converge. ## ## This function is based on the @sc{arpack} package, written by R. Lehoucq, ## K. Maschhoff, D. Sorensen, and C. Yang. For more information see ## @url{http://www.caam.rice.edu/software/ARPACK/}. ## ## @seealso{eig, svds} ## @end deftypefn function varargout = eigs (varargin) ## For compatibility with Matlab, handle small matrix cases here ## that ARPACK does not. if (nargin == 0) print_usage (); endif call_eig = false; offset = 0; k = 6; sigma = "lm"; if (isnumeric (varargin{1}) && issquare (varargin{1})) a = varargin{1}; if (nargin > 1 && isnumeric (varargin{2}) && issquare (varargin{2}) && size_equal (a, varargin{2})) b = varargin{2}; offset = 1; endif if (rows (a) < 9) call_eig = true; endif if (nargin > 1 + offset) tmp = varargin{2+offset}; if (isnumeric (tmp) && isscalar (tmp) && isreal (tmp) && round (tmp) == tmp) k = tmp; if (rows (a) - k < 3) call_eig = true; endif else call_eig = false; endif if (nargin > 2 + offset) tmp = varargin{3+offset}; if (ischar (tmp) || (isnumeric (tmp) && isscalar (tmp))) sigma = tmp; else call_eig = false; endif endif endif endif if (call_eig) varargout = cell (1, min (2, max (1, nargout))); if (offset) real_valued = isreal (a) && isreal (b); symmetric = issymmetric (a) && issymmetric (b); [varargout{:}] = eig (a, b); else real_valued = isreal (a); symmetric = issymmetric (a); [varargout{:}] = eig (a); endif varargout = select (varargout, k, sigma, real_valued, symmetric); if (nargout == 3) varargout{3} = 0; endif else varargout = cell (1, max (1, nargout)); [varargout{:}] = __eigs__ (varargin{:}); endif endfunction function out = select (args, k, sigma, real_valued, symmetric) if (numel (args) == 1) d = args{1}; else d = diag (args{2}); endif if (ischar (sigma)) switch (sigma) case "lm" [~, idx] = sort (abs (d), "descend"); case "sm" [~, idx] = sort (abs (d), "ascend"); case "la" if (real_valued && symmetric) [~, idx] = sort (real (d), "descend"); else error ("sigma = \"la\" requires real symmetric problem"); endif case "sa" if (real_valued && symmetric) [~, idx] = sort (real (d), "ascend"); else error ("sigma = \"sa\" requires real symmetric problem"); endif case "be" if (real_valued && symmetric) [~, idx] = sort (real (d), "ascend"); else error ("sigma = \"be\" requires real symmetric problem"); endif case "lr" if (! (real_valued || symmetric)) [~, idx] = sort (real (d), "descend"); else error ("sigma = \"lr\" requires complex or unsymmetric problem"); endif case "sr" if (! (real_valued || symmetric)) [~, idx] = sort (real (d), "ascend"); else error ("sigma = \"sr\" requires complex or unsymmetric problem"); endif case "li" if (! (real_valued || symmetric)) [~, idx] = sort (imag (d), "descend"); else error ("sigma = \"li\" requires complex or unsymmetric problem"); endif case "si" if (! (real_valued || symmetric)) [~, idx] = sort (imag (d), "ascend"); else error ("sigma = \"si\" requires complex or unsymmetric problem"); endif otherwise error ("unrecognized value for sigma: %s", sigma); endswitch endif d = d(idx); n = numel (d); k = min (k, n); if (strcmp (sigma, 'be')) tmp = k / 2; n1 = floor (tmp); n2 = n - ceil (tmp) + 1; selection = [1:floor(k/2), n2:n]; else selection = 1:k; endif d = d(selection); if (numel (args) == 1) out{1} = d; else out{2} = diag (d); v = args{1}; v = v(:,idx); out{1} = v(selection,:); endif endfunction #### SPARSE MATRIX VERSIONS #### ## Real positive definite tests, n must be even %!shared n, k, A, d0, d2 %! n = 20; %! k = 4; %! A = sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),4*ones(1,n),ones(1,n-2)]); %! d0 = eig (A); %! d2 = sort (d0); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); # initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (d1, d0(end:-1:(end-k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, "sm"); %! assert (d1, d0(k:-1:1), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "la"); %! assert (d1, d2(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sa"); %! assert (d1, d2(1:k), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "be"); %! assert (d1, d2([1:floor(k/2), (end - ceil(k/2) + 1):end]), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1, "be"); %! assert (d1, d2([1:floor((k+1)/2), (end - ceil((k+1)/2) + 1):end]), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (d1(idx1), d0(idx0(1:k)), 1e-11); %!testif HAVE_ARPACK, HAVE_CHOLMOD %! d1 = eigs (A, speye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (eigs (A, k, 4.1), eigs (A, speye (n), k, 4.1), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (eigs (A, k, 4.1), eigs (A, speye (n), k, 4.1), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (d1, d0(k:-1:1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (d1, eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! AA = speye (10); %! fn = @(x) AA * x; %! opts.issym = 1; opts.isreal = 1; %! assert (eigs (fn, 10, AA, 3, "lm", opts), [1; 1; 1], 10*eps); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK, HAVE_UMFPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "la"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sa"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "be"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor ## Real unsymmetric tests %!shared n, k, A, d0 %! n = 20; %! k = 4; %! A = sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),1:n,-ones(1,n-2)]); %! d0 = eig (A); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (abs (d1), abs (d0(end:-1:(end-k))),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, "sm"); %! assert (abs (d1), abs (d0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lr"); %! [~, idx] = sort (real (d0)); %! d2 = d0(idx); %! assert (real (d1), real (d2(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "li"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(end:-1:(end-k+1)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "si"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(1:k))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (abs (d1(idx1)), abs (d0(idx0(1:k))), 1e-11); %! assert (sort (imag (d1(idx1))), sort (imag (d0(idx0(1:k)))), 1e-11); %!testif HAVE_ARPACK, HAVE_CHOLMOD %! d1 = eigs (A, speye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (abs (eigs (A, k, 4.1)), abs (eigs (A, speye (n), k, 4.1)), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (sort (imag (eigs (A, k, 4.1))), sort (imag (eigs (A, speye (n), k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (abs (d1), d0(1:k), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK, HAVE_UMFPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "li"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "si"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor ## Complex hermitian tests %!shared n, k, A, d0 %! n = 20; %! k = 4; %! A = sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[1i*ones(1,n-2),4*ones(1,n),-1i*ones(1,n-2)]); %! d0 = eig (A); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (abs (d1), abs (d0(end:-1:(end-k))),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, "sm"); %! assert (abs (d1), abs (d0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "li"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(end:-1:(end-k+1)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "si"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(1:k))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (abs (d1(idx1)), abs (d0(idx0(1:k))), 1e-11); %! assert (sort (imag (d1(idx1))), sort (imag (d0(idx0(1:k)))), 1e-11); %!testif HAVE_ARPACK, HAVE_CHOLMOD %! d1 = eigs (A, speye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! d1 = eigs (A, speye (n), k, 4.1, opts); %! assert (abs (abs (d1)), abs (eigs (A, k, 4.1)), 1e-11); %! assert (sort (imag (abs (d1))), sort (imag (eigs (A, k, 4.1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, speye (n)(q,q), k, 4.1, opts); %! assert (abs (abs (d1)), abs (eigs (A, k, 4.1)), 1e-11); %! assert (sort (imag (abs (d1))), sort (imag (eigs (A, k, 4.1))), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (abs (eigs (A, k, 4.1)), abs (eigs (A, speye (n), k, 4.1)), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! assert (sort (imag (eigs (A, k, 4.1))), sort (imag (eigs (A, speye (n), k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (abs (d1), d0(1:k), 1e-11); %!testif HAVE_ARPACK, HAVE_UMFPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK, HAVE_UMFPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "li"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "si"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*speye (n))*v1(:,i))), 0, 1e-11); %! endfor #### FULL MATRIX VERSIONS #### ## Real positive definite tests, n must be even %!shared n, k, A, d0, d2 %! n = 20; %! k = 4; %! A = full (sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),4*ones(1,n),ones(1,n-2)])); %! d0 = eig (A); %! d2 = sort (d0); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (d1, d0(end:-1:(end-k)),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sm"); %! assert (d1, d0(k:-1:1), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "la"); %! assert (d1, d2(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sa"); %! assert (d1, d2(1:k), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "be"); %! assert (d1, d2([1:floor(k/2), (end - ceil(k/2) + 1):end]), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1, "be"); %! assert (d1, d2([1:floor((k+1)/2), (end - ceil((k+1)/2) + 1):end]), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (d1(idx1), d0(idx0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, eye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! assert (eigs (A, k, 4.1), eigs (A, eye (n), k, 4.1), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! assert (eigs (A, k, 4.1), eigs (A, eye (n), k, 4.1), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (d1, d0(end:-1:(end-k+1)), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (d1, d0(k:-1:1), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 1; opts.isreal = 1; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (d1, eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "la"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sa"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "be"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor ## Real unsymmetric tests %!shared n, k, A, d0 %! n = 20; %! k = 4; %! A = full (sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),1:n,-ones(1,n-2)])); %! d0 = eig (A); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (abs (d1), abs (d0(end:-1:(end-k))),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sm"); %! assert (abs (d1), abs (d0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lr"); %! [~, idx] = sort (real (d0)); %! d2 = d0(idx); %! assert (real (d1), real (d2(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "li"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(end:-1:(end-k+1)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "si"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(1:k))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (abs (d1(idx1)), abs (d0(idx0(1:k))), 1e-11); %! assert (sort (imag (d1(idx1))), sort (imag (d0(idx0(1:k)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, eye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! assert (abs (eigs (A, k, 4.1)), abs (eigs (A, eye (n), k, 4.1)), 1e-11); %!testif HAVE_ARPACK %! assert (sort (imag (eigs (A, k, 4.1))), sort (imag (eigs (A, eye (n), k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (abs (d1), d0(1:k), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 0; opts.isreal = 1; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "li"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "si"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor ## Complex hermitian tests %!shared n, k, A, d0 %! n = 20; %! k = 4; %! A = full (sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[1i*ones(1,n-2),4*ones(1,n),-1i*ones(1,n-2)])); %! d0 = eig (A); %! [~, idx] = sort (abs (d0)); %! d0 = d0(idx); %! rand ("state", 42); % initialize generator to make eigs behavior reproducible %!testif HAVE_ARPACK %! d1 = eigs (A, k); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k+1); %! assert (abs (d1), abs (d0(end:-1:(end-k))),1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sm"); %! assert (abs (d1), abs (d0(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "lr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "sr"); %! [~, idx] = sort (real (abs (d0))); %! d2 = d0(idx); %! assert (real (d1), real (d2(1:k)), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "li"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(end:-1:(end-k+1)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, "si"); %! [~, idx] = sort (imag (abs (d0))); %! d2 = d0(idx); %! assert (sort (imag (d1)), sort (imag (d2(1:k))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, k, 4.1); %! [~, idx0] = sort (abs (d0 - 4.1)); %! [~, idx1] = sort (abs (d1 - 4.1)); %! assert (abs (d1(idx1)), abs (d0(idx0(1:k))), 1e-11); %! assert (sort (imag (d1(idx1))), sort (imag (d0(idx0(1:k)))), 1e-11); %!testif HAVE_ARPACK %! d1 = eigs (A, eye (n), k, "lm"); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! d1 = eigs (A, eye (n), k, 4.1, opts); %! assert (abs (abs (d1)), abs (eigs (A, k, 4.1)), 1e-11); %! assert (sort (imag (abs (d1))), sort (imag (eigs (A, k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! opts.cholB = true; %! q = [2:n,1]; %! opts.permB = q; %! d1 = eigs (A, eye (n)(q,q), k, 4.1, opts); %! assert (abs (abs (d1)), abs (eigs (A, k, 4.1)), 1e-11); %! assert (sort (imag (abs (d1))), sort (imag (eigs (A, k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! assert (abs (eigs (A, k, 4.1)), abs (eigs (A, eye (n), k, 4.1)), 1e-11); %!testif HAVE_ARPACK %! assert (sort (imag (eigs (A, k, 4.1))), sort (imag (eigs (A, eye (n), k, 4.1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A * x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, "lm", opts); %! assert (abs (d1), abs (d0(end:-1:(end-k+1))), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) A \ x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, "sm", opts); %! assert (abs (d1), d0(1:k), 1e-11); %!testif HAVE_ARPACK %! fn = @(x) (A - 4.1 * eye (n)) \ x; %! opts.issym = 0; opts.isreal = 0; %! d1 = eigs (fn, n, k, 4.1, opts); %! assert (abs (d1), eigs (A, k, 4.1), 1e-11); %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sm"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "lr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "sr"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "li"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!testif HAVE_ARPACK %! [v1,d1] = eigs (A, k, "si"); %! d1 = diag (d1); %! for i=1:k %! assert (max (abs ((A - d1(i)*eye (n))*v1(:,i))), 0, 1e-11); %! endfor %!assert (eigs (diag (1:5), 5, "sa"), [1;2;3;4;5]); %!assert (eigs (diag (1:5), 5, "la"), [5;4;3;2;1]); %!assert (eigs (diag (1:5), 3, "be"), [1;4;5]);