# IMSL_EIGSYMGEN

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 Analyst license. For more information, contact your Exelis VIS sales or technical support representative.

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

For additional information on using IMSL_EIGSYMGEN, see Additional 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

## 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.

## Additional Examples

### 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.

## Version History

6.4 | Introduced