## EIGENQL

The EIGENQL function computes the eigenvalues and eigenvectors of an n by n real, symmetric array using Householder reductions and the QL method with implicit shifts. The result is an n -element vector containing the eigenvalues.

### Calling Sequence

Result = EIGENQL( A )

### Arguments

#### A

An n by n symmetric single- or double-precision floating-point array.

### Keywords

#### ABSOLUTE

Set this keyword to sort the eigenvalues by their absolute value (their magnitude) rather than by their signed value.

#### ASCENDING

Set this keyword to return eigenvalues in ascending order (smallest to largest). If not set or set to zero, eigenvalues are returned in descending order (largest to smallest). The eigenvectors are correspondingly reordered.

#### DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

#### EIGENVECTORS

Set this keyword equal to a named variable that will contain the computed eigenvectors in an n by n array. The i th row of the returned array contains the i th eigenvalue. This keyword must be initialized to a non- zero value before calling EIGENQL if the eigenvectors are desired. If no variable is supplied, the array will not be computed.

#### OVERWRITE

Set this keyword to use the input array for internal storage and to overwrite its previous contents.

#### RESIDUAL

Use this keyword to specify a named variable that will contain the residuals for each eigenvalue/eigenvector ( l /x) pair. The residual is based on the definition Ax - ( l )x = 0 and is an array of the same size as A and the same type as Result. The rows of this array correspond to the residuals for each eigenvalue/eigenvector pair. This keyword must be initialized to a non- zero value before calling EIGENQL if the residuals are desired.

CAUTION: If the OVERWRITE keyword is set, the RESIDUAL keyword has no effect.

### Example

Define an n by n real, symmetric array:

A = [[ 5.0, 4.0, 0.0, -3.0], \$

[ 4.0, 5.0, 0.0, -3.0], \$

[ 0.0, 0.0, 5.0, -3.0], \$

[-3.0, -3.0, -3.0, 5.0]]

residual = 1 & evecs = 1 ; The variables that will contain the residuals and eigenvectors must be initialized as nonzero values prior to calling EIGENQL.

eigenvalues = EIGENQL(A, EIGENVECTORS = evecs, RESIDUAL = residual)
; Compute the eigenvalues and eigenvectors.

PRINT, eigenvalues ; Print the eigenvalues/.

IDL prints:

12.0915     6.18661     1.00000     0.721870

Print the eigenvectors:

PRINT, evecs

IDL prints:

-0.554531 -0.554531 -0.241745 0.571446

0.342981 0.342981 -0.813186 0.321646

0.707107 -0.707107 -2.58096e-08 0.00000

0.273605 0.273605 0.529422 0.754979

The accuracy of each eigenvalue/eigenvector ( l /x) pair may be checked by printing the residual array:

PRINT, residual

The RESIDUAL array has the same dimensions as the input array and the same type as the result. The residuals are contained in the rows of the RESIDUAL array. All residual values should be floating-point zeros.