LA_EIGENPROBLEM

The LA_EIGENPROBLEM function uses the QR algorithm to compute all eigenvalues λ and eigenvectors v ≠ 0 of an n-by-n real nonsymmetric or complex non-Hermitian array A, for the eigenproblem Av = λv. The routine can also compute the left eigenvectors u ≠ 0, which satisfy u^HA = λu^H.

LA_EIGENPROBLEM may also be used for the generalized eigenproblem:

Av = λBv and u^HA = λu^HB

where A and B are square arrays, v are the right eigenvectors, and u are the left eigenvectors.

LA_EIGENPROBLEM is based on the following LAPACK routines:

Output Type	Standard LAPACK Routine	Generalized LAPACK Routine
Float	sgeevx	sggevx
Double	dgeevx	dggevx
Complex	cgeevx	cggevx
Double complex	zgeevx	zggevx

Examples

Find the eigenvalues and eigenvectors for an array using the following program:

PRO ExLA_EIGENPROBLEM

; Create a random array:

n = 4

seed = 12321

array = RANDOMN(seed, n, n)

; Compute all eigenvalues and eigenvectors:

eigenvalues = LA_EIGENPROBLEM(array, $

EIGENVECTORS = eigenvectors)

PRINT, 'LA_EIGENPROBLEM Eigenvalues:'

PRINT, eigenvalues

; Check the results using the eigenvalue equation:

maxErr = 0d

FOR i = 0, n - 1 DO BEGIN

; A*z = lambda*z

alhs = array ## eigenvectors[*,i]

arhs = eigenvalues[i]*eigenvectors[*,i]

maxErr = maxErr > MAX(ABS(alhs - arhs))

ENDFOR

PRINT, 'LA_EIGENPROBLEM Error:', maxErr

; Now try the generalized eigenproblem:

b = IDENTITY(n) + 0.01*RANDOMN(seed, n, n)

eigenvalues = LA_EIGENPROBLEM(Array, B)

PRINT, 'LA_EIGENPROBLEM Generalized Eigenvalues:'

PRINT, EIGENVALUES

END

ExLA_EIGENPROBLEM

When this program is compiled and run, IDL prints:

LA_EIGENPROBLEM Eigenvalues:

( -1.50248, 0.592900)( -1.50248, -0.592900)( 1.99779, 0.000000)( 1.06499, 0.000000)

LA_EIGENPROBLEM Error: 9.5367432e-007

LA_EIGENPROBLEM Generalized Eigenvalues:

( -1.51732, 0.591994)( -1.51732, -0.591994)( 2.02658, 0.000000)( 1.04831, 0.000000)

Syntax

Result = LA_EIGENPROBLEM( A [, B] [, ALPHA=variable] [, BALANCE=value] [, BETA=variable] [, /DOUBLE] [, EIGENVECTORS=variable] [, LEFT_EIGENVECTORS=variable] [, NORM_BALANCE = variable] [, PERMUTE_RESULT=variable] [, SCALE_RESULT=variable] [, RCOND_VALUE=variable] [, RCOND_VECTOR=variable] [, STATUS=variable] )

Return Value

The result is a complex n-element vector containing the eigenvalues.

Arguments

A

The real or complex array for which to compute eigenvalues and eigenvectors.

B

An optional real or complex n-by-n array used for the generalized eigenproblem. The elements of B are converted to the same type as A before computation.

Keywords

ALPHA

For the generalized eigenproblem with the B argument, set this keyword to a named variable in which the numerator of the eigenvalues will be returned as a complex n -element vector. For the standard eigenproblem this keyword is ignored.

Tip: The ALPHA and BETA values are useful for eigenvalues which underflow or overflow. In this case the eigenvalue problem may be rewritten as αAv = βBv.

BALANCE

Set this keyword to one of the following values:

BALANCE = 0: No balancing is applied to A.
BALANCE = 1: Both permutation and scale balancing are performed.
BALANCE = 2: Permutations are performed to make the array more nearly upper triangular.
BALANCE = 3: Diagonally scale the array to make the columns and rows more equal in norm.

The default is BALANCE = 1, which performs both permutation and scaling balances. Balancing a nonsymmetric (or non-Hermitian) array is recommended to reduce the sensitivity of eigenvalues to rounding errors.

BETA

For the generalized eigenproblem with the B argument, set this keyword to a named variable in which the denominator of the eigenvalues will be returned as a real or complex n-element vector. For the standard eigenproblem this keyword is ignored.

Tip: The ALPHA and BETA values are useful for eigenvalues which underflow or overflow. In this case, the eigenvalue problem may be rewritten as αAv = βBv.

DOUBLE

Set this keyword to use double-precision for computations and to return a double-precision (real or complex) result. Set DOUBLE = 0 to use single-precision for computations and to return a single-precision (real or complex) result. The default is /DOUBLE if A is double precision, otherwise the default is DOUBLE = 0.

EIGENVECTORS

Set this keyword to a named variable in which the eigenvectors will be returned as a set of row vectors. If this variable is omitted then eigenvectors will not be computed unless the RCOND_VALUE or RCOND_VECTOR keywords are present.

Note: For the standard eigenproblem the eigenvectors are normalized and rotated to have norm 1 and largest component real. For the generalized eigenproblem the eigenvectors are normalized so that the largest component has abs(real) + abs(imaginary) = 1.

LEFT_EIGENVECTORS

Set this keyword to a named variable in which the left eigenvectors will be returned as a set of row vectors. If this variable is omitted then left eigenvectors will not be computed unless the RCOND_VALUE or RCOND_VECTOR keywords are present.

Note: Note - For the standard eigenproblem the eigenvectors are normalized and rotated to have norm 1 and largest component real. For the generalized eigenproblem the eigenvectors are normalized so that the largest component has abs(real) + abs(imaginary) = 1.

NORM_BALANCE

Set this keyword to a named variable in which the one-norm of the balanced matrix will be returned. The one-norm is defined as the maximum value of the sum of absolute values of the columns. For the standard eigenproblem, this will be returned as a scalar value; for the generalized eigenproblem this will be returned as a two-element vector containing the A and B norms.

PERMUTE_RESULT

Set this keyword to a named variable in which the result for permutation balancing will be returned as a two-element vector [ilo, ihi]. If permute balancing is not done then the values will be ilo = 1 and ihi = n.

RCOND_VALUE

Set this keyword to a named variable in which the reciprocal condition numbers for the eigenvalues will be returned as an n-element vector. If RCOND_VALUE is present then left and right eigenvectors must be computed.

RCOND_VECTOR

Set this keyword to a named variable in which the reciprocal condition numbers for the eigenvectors will be returned as an n-element vector. If RCOND_VECTOR is present then left and right eigenvectors must be computed.

SCALE_RESULT

Set this keyword to a named variable in which the results for permute and scale balancing will be returned. For the standard eigenproblem, this will be returned as an n-element vector. For the generalized eigenproblem, this will be returned as a n-by-2 array with the first row containing the permute and scale factors for the left side of A and B and the second row containing the factors for the right side of A and B.

STATUS

Set this keyword to a named variable that will contain the status of the computation. Possible values are:

STATUS = 0: The computation was successful.
STATUS > 0: The QR algorithm failed to compute all eigenvalues; no eigenvectors or condition numbers were computed. The STATUS value indicates that eigenvalues ilo:STATUS (starting at index 1) did not converge; all other eigenvalues converged.

Note: If STATUS is not specified, any error messages will be output to the screen.

Version History

5.6	Introduced

Resources and References

For details see Anderson et al., LAPACK Users' Guide, 3rd ed., SIAM, 1999.