The function may be called by the names: f08xec, nag_lapackeig_dhgeqz or nag_dhgeqz.
3Description
f08xec implements a single-double-shift version of the method for finding the generalized eigenvalues of the real matrix pair which is in the generalized upper Hessenberg form. If the matrix pair is not in the generalized upper Hessenberg form, then the function f08wfc should be called before invoking f08xec.
This problem is mathematically equivalent to solving the equation
Note that, to avoid underflow, overflow and other arithmetic problems, the generalized eigenvalues are never computed explicitly by this function but defined as ratios between two computed values, and :
The arguments , in general, are finite complex values and are finite real non-negative values.
If desired, the matrix pair may be reduced to generalized Schur form. That is, the transformed matrix is upper triangular and the transformed matrix is block upper triangular, where the diagonal blocks are either or . The blocks provide generalized eigenvalues which are real and the blocks give complex generalized eigenvalues.
The argument job specifies two options. If then the matrix pair is simultaneously reduced to Schur form by applying one orthogonal transformation (usually called ) on the left and another (usually called ) on the right. That is,
The upper-triangular diagonal blocks of corresponding to blocks of a will be reduced to non-negative diagonal matrices. That is, if is nonzero, then and and will be non-negative.
If , then at each iteration the same transformations are computed but they are only applied to those parts of and which are needed to compute and . This option could be used if generalized eigenvalues are required but not generalized eigenvectors.
If and or , and or , then the orthogonal transformations used to reduce the pair are accumulated into the input arrays q and z. If generalized eigenvectors are required then job must be set to and if left (right) generalized eigenvectors are to be computed then compq (compz) must be set to or and not .
If , then eigenvectors are accumulated on the identity matrix and on exit the array q contains the left eigenvector matrix . However, if then the transformations are accumulated on the user-supplied matrix in array q on entry and thus on exit q contains the matrix product . A similar convention is used for compz.
4References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal.10 241–256
Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London
5Arguments
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_JobTypeInput
On entry: specifies the operations to be performed on .
The matrix pair on exit might not be in the generalized Schur form.
The matrix pair on exit will be in the generalized Schur form.
Constraint:
or .
3: – Nag_ComputeQTypeInput
On entry: specifies the operations to be performed on :
The right transformation is accumulated on the array z.
The array z is initialized to the identity matrix before the right transformation is accumulated in z.
Constraint:
, or .
5: – IntegerInput
On entry: , the order of the matrices , , and .
Constraint:
.
6: – IntegerInput
7: – IntegerInput
On entry: the indices and , respectively which define the upper triangular parts of . The submatrices and are then upper triangular. These arguments are provided by f08whc if the matrix pair was previously balanced; otherwise, and .
Constraints:
if , ;
if , and .
8: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
.
where appears in this document, it refers to the array element
when ;
when .
On entry: the upper Hessenberg matrix . The elements below the first subdiagonal must be set to zero.
On exit: if , the matrix pair will be simultaneously reduced to generalized Schur form.
If , the and diagonal blocks of the matrix pair will give generalized eigenvalues but the remaining elements will be irrelevant.
9: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint:
.
10: – doubleInput/Output
Note: the dimension, dim, of the array b
must be at least
.
where appears in this document, it refers to the array element
when ;
when .
On entry: the upper triangular matrix . The elements below the diagonal must be zero.
On exit: if , the matrix pair will be simultaneously reduced to generalized Schur form.
If , the and diagonal blocks of the matrix pair will give generalized eigenvalues but the remaining elements will be irrelevant.
11: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint:
.
12: – doubleOutput
On exit: the real parts of
, for .
13: – doubleOutput
On exit: the imaginary parts of
, for .
14: – doubleOutput
On exit: , for .
15: – doubleInput/Output
Note: the dimension, dim, of the array q
must be at least
when
or ;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: if , the matrix . The matrix is usually the matrix returned by f08wfc.
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
if ,
if or , ;
if , ;
if ,
if or ,
;
if ,
.
19: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_ENUM_INT_2
On entry, , and .
Constraint: if or ,
; if ,
.
On entry, , and .
Constraint: if or , ;
if , .
On entry, , and .
Constraint: if or ,
; if ,
.
On entry, , and .
Constraint: if or , ;
if , .
NE_INT
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INT_3
On entry, , and .
Constraint: if , ;
if , and .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
An unexpected Library error has occurred.
NE_ITERATION_QZ
The iteration did not converge and the matrix pair is not in the generalized Schur form. The computed and should be correct for .
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SCHUR
The computation of shifts failed and the matrix pair is not in the generalized Schur form. The computed and should be correct for .
Background information to multithreading can be found in the Multithreading documentation.
f08xec makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
f08xec is the fifth step in the solution of the real generalized eigenvalue problem and is called after f08wfc.
This example computes the and arguments, which defines the generalized eigenvalues, of the matrix pair given by
This requires calls to five functions: f08whc to balance the matrix, f08aec to perform the factorization of , f08agc to apply to , f08wfc to reduce the matrix pair to the generalized Hessenberg form and f08xec to compute the eigenvalues using the algorithm.