IMSL_EIGSYMGEN

Syntax | Return Value | Arguments | Keywords | Discussion | Examples | Errors | Version History

The IMSL_EIGSYMGEN function computes the generalized eigenexpansion of a system Ax = λBx. The matrices A and B are real and symmetric, and B is positive definite.

Note
This routine requires an IDL Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.

Syntax

Result = IMSL_EIGSYMGEN(a, b [, /DOUBLE] [, VECTORS=array])

Return Value

One-dimensional array containing the eigenvalues of the symmetric matrix.

Arguments

a

Two-dimensional matrix containing symmetric coefficient matrix A.

b

Two-dimensional matrix containing the positive definite symmetric coefficient matrix B.

Keywords

DOUBLE

If present and nonzero, double precision is used.

VECTORS

Compute eigenvectors of the problem. A two-dimensional array containing the eigenvectors is returned in the variable name specified by VECTORS.

Discussion

The IMSL_EIGSYMGEN function computes the eigenvalues of a symmetric, positive definite eigenvalue problem by a three-phase process (Martin and Wilkinson 1971). Matrix B is reduced to factored form using the Cholesky decomposition. These factors are used to form a congruence transformation that yields a symmetric real matrix whose eigenexpansion is obtained. The problem is then transformed back to the original coordinates. Eigenvectors are calculated and transformed as required.

Examples

Example 1

This example computes the generalized eigenexpansion of a system Ax = λBx, where A and B are 3-by-3 matrices.

RM, a, 3, 3 
; Define the matrix A. 
row 0: 1.1 1.2 1.4 
row 1: 1.2 1.3 1.5 
row 2: 1.4 1.5 1.6 
RM, b, 3, 3 
; Define the matrix B. 
row 0: 2 1 0 
row 1: 1 2 1 
row 2: 0 1 2 
eigval = IMSL_EIGSYMGEN(a, b) 
; Call IMSL_EIGSYMGEN to compute the eigenexpansion. 
PM, eigval, Title = 'Eigenvalues' 
; Output the results. 
Eigenvalues 
    1.38644 
   -0.0583479 
   -0.00309042 

Example 2

This example is a variation of the first example. It computes the eigenvectors as well as the eigenvalues.

RM, a, 3, 3 
; Define the matrix A. 
row 0: 1.1 1.2 1.4 
row 1: 1.2 1.3 1.5 
row 2: 1.4 1.5 1.6 
RM, b, 3, 3 
; Define the matrix B. 
row 0: 2 1 0 
row 1: 1 2 1 
row 2: 0 1 2 
eigval = IMSL_EIGSYMGEN(a, b, Vectors = eigvec) 
; Call IMSL_EIGSYMGEN with keyword Vectors to specify the named 
; variable in which the vectors are stored. 
PM, eigval, Title = 'Eigenvalues' 
; Output the eigenvalues. 
 
Eigenvalues 
    1.38644 
    -0.0583478 
    -0.00309040 
PM, eigvec, Title = 'Eigenvectors' 
; Output the eigenvectors. 
Eigenvectors 
    0.643094     -0.114730    -0.681688 
   -0.0223849    -0.687186     0.726597 
    0.765460      0.717365    -0.0857800 

Errors

Warning Errors

MATH_SLOW_CONVERGENCE_SYM—Iteration for an eigenvalue failed to converge in 100 iterations before deflating.

Fatal Errors

MATH_SUBMATRIX_NOT_POS_DEFINITE—Leading submatrix of the input matrix is not positive definite.

MATH_MATRIX_B_NOT_POS_DEFINITE—Matrix B is not positive definite.

Version History