The routine may be called by the names f08xef, nagf_lapackeig_dhgeqz or its LAPACK name dhgeqz.
3Description
f08xef 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 routine f08wff should be called before invoking f08xef.
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 routine 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: – Character(1)Input
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 .
2: – Character(1)Input
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 .
4: – IntegerInput
On entry: , the order of the matrices , , and .
Constraint:
.
5: – IntegerInput
6: – 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 f08whf if the matrix pair was previously balanced; otherwise, and .
Constraints:
if , ;
if , and .
7: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
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.
8: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08xef is called.
Constraint:
.
9: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
.
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.
10: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08xef is called.
Constraint:
.
11: – Real (Kind=nag_wp) arrayOutput
On exit: the real parts of
, for .
12: – Real (Kind=nag_wp) arrayOutput
On exit: the imaginary parts of
, for .
13: – Real (Kind=nag_wp) arrayOutput
On exit: , for .
14: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array q
must be at least
if or and at least if .
On entry: if , the matrix . The matrix is usually the matrix returned by f08wff.
On entry: the first dimension of the array z as declared in the (sub)program from which f08xef is called.
Constraints:
if or , ;
if , .
18: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
19: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08xef is called.
If , a workspace query is assumed; the routine only calculates the minimum size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Constraint:
or .
20: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The iteration did not converge and the matrix pair is not in the generalized Schur form. The computed and should be correct for .
and
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.
f08xef 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
f08xef is the fifth step in the solution of the real generalized eigenvalue problem and is called after f08wff.
This example computes the and arguments, which defines the generalized eigenvalues, of the matrix pair given by
This requires calls to five routines: f08whf to balance the matrix, f08aef to perform the factorization of , f08agf to apply to , f08wff to reduce the matrix pair to the generalized Hessenberg form and f08xef to compute the eigenvalues using the algorithm.