Description of slsymeig

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slsymeig

PURPOSE

SLSYMEIG Compute the eigenvalues and eigenvectors for symmetric matrix

SYNOPSIS

function [evals, evecs] = slsymeig(A, k, ord)

DESCRIPTION

SLSYMEIG Compute the eigenvalues and eigenvectors for symmetric matrix

 $ Syntax $
   - evals = slsymeig(A)
   - [evals, evecs] = slsymeig(A)
   - ... = slsymeig(A, k)
   - ... = slsymeig(A, k, ord)

 $ Arguments $
   - A:        the target symmetrix matrix to be decomposed
   - k:        the number of eigenvalues to be solved
   - ord:      the order of the sorting ('ascend' | 'descend')
               default = 'descend';
   - evals:    the column vector of eigenvalues of A (in descending order)
   - evecs:    the matrix of eigenvectors of A

 $ Description $
   - evals = slsymeig(A) computes only the eigenvalues of symmetric 
     matrix A.

   - [evals, evecs] = slsymeig(A) computes both the eigenvalues and
     eigenvectors of matrix A. If A is a d x d matrix, then evals
     is a d x 1 column vector with the eigenvalues in descending order.
     evecs is a d x d matrix with each column vector corresponding to
     an eigenvalue in evals.

   - ... = slsymeig(A, k) the user can specify the number of eigenvalues
     to preserve, if not specified, all eigenvalues will be preserved.

   - ... = slsymeig(A, k, ord) the user can specify how to sort the
     eigenvalues. By default, they are sorted in descending order.

 $ History $
   - Created by Dahua Lin on Apr 21, 2006
   - Modified by Dahua Lin on Sep 9, 2006
       - The user now can specify the number of eigenvalues to preserve
         if necessary.  
       - An auto-selection scheme, which will use eigs when k is small
         in order to achieve higher efficiency.

CROSS-REFERENCE INFORMATION

This function calls:

This function is called by:

• slsymgeig SLSYMGEIG Solve the generalized eigen decomposition for symmetric matrices
• slkpca SLPCA Learns a Kernel PCA model from training samples
• slgembed SLGEMBED Solves the general graph-based embedding
• slkernelembed SLKERNELEMBED Finds an embedding space to preserve inner products
• sllemap SLLEMAP Solves Laplacian Eigenmap Embedding
• sllle_wg SLLLE_WG Solves the Locally Linear Embedding from weight graph
• sllocalcoordalign SLLOCALCOORDALIGN Performs optimal local coordinate alignment
• sllocaltanspace SLLOCALTANSPACE Solves the local tangent spaces
• slgaussrnd SLGAUSSRND Generates random samples from Gaussian models
• slinvcov SLINVCOV Compute the inverse of an covariance matrix
• slwhiten_from_cov SLWHITEN_FROM_COV Compute the whitening transform from covariance matrix
• slwhiten_from_samples SLWHITEN_FROM_SAMPLES Compute the whitening matrix from sample matrix
• slcovpca SLCOVPCA Trains a PCA with the covariance matrix given
• slfld SLFLD Performs Fisher Linear Discriminant Analysis
• slnlda SLNLDA Performs Nullspace-based Linear Discriminant Analysis
• slpca SLPCA Learns a PCA model from training samples
• slrangespace SLRANGESPACE Determines the subspace of the range of X
• sl2dpcaex SL2DPCAEX Learns Extended 2D PCA on a set of matrix samples
• slpartitionpca SLPARTITIONPCA Performs Partition-based PCA and saves the models
• sldrawellipse SLDRAWELLIPSE Draws an ellipse on current axis

SOURCE CODE

0001 function [evals, evecs] = slsymeig(A, k, ord)
0002 %SLSYMEIG Compute the eigenvalues and eigenvectors for symmetric matrix
0003 %
0004 % $ Syntax $
0005 %   - evals = slsymeig(A)
0006 %   - [evals, evecs] = slsymeig(A)
0007 %   - ... = slsymeig(A, k)
0008 %   - ... = slsymeig(A, k, ord)
0009 %
0010 % $ Arguments $
0011 %   - A:        the target symmetrix matrix to be decomposed
0012 %   - k:        the number of eigenvalues to be solved
0013 %   - ord:      the order of the sorting ('ascend' | 'descend')
0014 %               default = 'descend';
0015 %   - evals:    the column vector of eigenvalues of A (in descending order)
0016 %   - evecs:    the matrix of eigenvectors of A
0017 %
0018 % $ Description $
0019 %   - evals = slsymeig(A) computes only the eigenvalues of symmetric
0020 %     matrix A.
0021 %
0022 %   - [evals, evecs] = slsymeig(A) computes both the eigenvalues and
0023 %     eigenvectors of matrix A. If A is a d x d matrix, then evals
0024 %     is a d x 1 column vector with the eigenvalues in descending order.
0025 %     evecs is a d x d matrix with each column vector corresponding to
0026 %     an eigenvalue in evals.
0027 %
0028 %   - ... = slsymeig(A, k) the user can specify the number of eigenvalues
0029 %     to preserve, if not specified, all eigenvalues will be preserved.
0030 %
0031 %   - ... = slsymeig(A, k, ord) the user can specify how to sort the
0032 %     eigenvalues. By default, they are sorted in descending order.
0033 %
0034 % $ History $
0035 %   - Created by Dahua Lin on Apr 21, 2006
0036 %   - Modified by Dahua Lin on Sep 9, 2006
0037 %       - The user now can specify the number of eigenvalues to preserve
0038 %         if necessary.
0039 %       - An auto-selection scheme, which will use eigs when k is small
0040 %         in order to achieve higher efficiency.
0041 %
0042 %
0043 
0044 %% parse and verify input arguments
0045 
0046 [d, d2] = size(A);
0047 if d2 ~= d
0048     error('sltoolbox:notsquaremat', 'The matrix A should be symmetric');
0049 end
0050 
0051 if nargin < 2 || isempty(k)
0052     k = d;
0053 else
0054     if ~isscalar(k) || k > d
0055         error('sltoolbox:invalidarg', 'k should be a scalar not larger than d');
0056     end
0057 end
0058 
0059 if nargin < 3 || isempty(ord)
0060     ord = 'descend';
0061 else
0062     if ~ismember(ord, {'descend', 'ascend'})
0063         error('sltoolbox:invalidarg', ...
0064             'The order of eigenvalues is invalid: %s', ord);
0065     end
0066 end
0067 
0068 
0069 %% compute
0070 
0071 % enforce symmetry
0072 A = 0.5 * (A + A');
0073 
0074 % decide scheme and compute
0075 if k > d / 3 && ~issparse(A)
0076     [evecs, evals] = eig(A);
0077 else
0078     switch ord
0079         case 'descend'
0080             sigma = 'LM';
0081         case 'ascend'
0082             sigma = 'SM';
0083     end
0084             
0085     opts = struct('disp', 0, 'issym', true);
0086     [evecs, evals] = eigs(A, k, sigma, opts);
0087 end
0088     
0089 evals = diag(evals);
0090 
0091 % enforce real
0092 if ~isreal(evecs)
0093     evecs = real(evecs);
0094 end
0095 
0096 if ~isreal(evals)
0097     evals = real(evals);
0098 end
0099 
0100 %% sort: make the eigenvalues in descending order
0101 
0102 [evals, si] = sort(evals, 1, ord);
0103 evecs = evecs(:, si);
0104 
0105 k0 = length(evals);
0106 if k < k0
0107     evals = evals(1:k);
0108     evecs = evecs(:, 1:k);
0109 end
0110 
0111 
0112

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