The LA_ELMHES function reduces a real nonsymmetric or complex non-Hermitian array to upper Hessenberg form H. If the array is real then the decomposition is A = Q H QT, where Q is orthogonal. If the array is complex Hermitian then the decomposition is A = Q H QH, where Q is unitary. The superscript T represents the transpose while superscript H represents the Hermitian, or transpose complex conjugate.
LA_ELMHES is based on the following LAPACK routines:
|
Output Type |
Balance & Reduce
|
Norm |
Optional Q |
|
Float |
sgebal, sgehrd |
slange |
sorghr |
|
Double |
dgebal, dgehrd |
dlange |
dorghr |
|
Complex |
cgebal, cgehrd |
clange |
cunghr |
|
Double complex |
zgebal, zgehrd |
zlange |
zunghr |
Examples
See LA_EIGENVEC for an example of using this procedure.
Syntax
Result = LA_ELMHES( Array [, Q] [, BALANCE=value] [, /DOUBLE] [, NORM_BALANCE=variable] [, PERMUTE_RESULT=variable] [, SCALE_RESULT=variable] )
Return Value
The result is an array of the same type as A containing the upper Hessenberg form. The Hessenberg array is stored in the upper triangle and the first subdiagonal. Elements below the subdiagonal should be ignored but are not automatically set to zero.
Arguments
Array
The n-by-n real or complex array to reduce to upper Hessenberg form.
Q
Set this optional argument to a named variable in which the array Q will be returned. The Q argument may then be input into LA_HQR to compute the Schur vectors.
Keywords
BALANCE
Set this keyword to one of the following values:
- BALANCE = 0: No balancing is applied to Array.
- 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 array is recommended to reduce the sensitivity of eigenvalues to rounding errors.
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 Array is double precision, otherwise the default is DOUBLE = 0.
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.
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.
SCALE_RESULT
Set this keyword to a named variable in which the result for permute and scale balancing will be returned as an n-element vector.
Version History
Resources and References
For details see Anderson et al., LAPACK Users' Guide, 3rd ed., SIAM, 1999.
See Also
ELMHES, LA_HQR