f02aef

f02aef © Numerical Algorithms Group, 2002.

Purpose

F02AEF All eigenvalues and eigenvectors of generalized real symmetric-definite eigenproblem (Black Box)

Synopsis

[r,v,ifail] = f02aef(a,b<,ifail>)

Description

 
 The problem is reduced to the standard symmetric eigenproblem 
 using Cholesky's method to decompose B into triangular matrices 
     T                                                         
 B=LL , where L is lower triangular. Then Ax=(lambda)Bx implies 
   -1  -T   T             T                            
 (L  AL  )(L x)=(lambda)(L x); hence the eigenvalues of 
 Ax=(lambda)Bx are those of Py=(lambda)y, where P is the symmetric
         -1  -T                                                
 matrix L  AL  . Householder's method is used to tridiagonalise 
 the matrix P and the eigenvalues are found using the QL 
 algorithm. An eigenvector z of the derived problem is related to 
                                                T      
 an eigenvector x of the original problem by z=L x. The 
 eigenvectors z are determined using the QL algorithm and are 
                     T                                     
 normalised so that z z=1; the eigenvectors of the original 
                                         T                       
 problem are then determined by solving L x=z, and are normalised 
          T    
 so that x Bx=1.

Parameters

f02aef

Required Input Arguments:

a (:,:)                               real
b (:,:)                               real

Optional Input Arguments:                       <Default>

ifail                                 integer  -1

Output Arguments:

r (:)                                 real
v (:,:)                               real
ifail                                 integer