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