The function may be called by the names: f08pec, nag_lapackeig_dhseqr or nag_dhseqr.
f08pec computes all the eigenvalues and, optionally, the Schur factorization of a real upper Hessenberg matrix :
where is an upper quasi-triangular matrix (the Schur form of ), and is the orthogonal matrix whose columns are the Schur vectors . See Section 9 for details of the structure of .
The function may also be used to compute the Schur factorization of a real general matrix which has been reduced to upper Hessenberg form :
In this case, after f08nec has been called to reduce to Hessenberg form, f08nfc must be called to form explicitly; is then passed to f08pec, which must be called with .
The function can also take advantage of a previous call to f08nhc which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix has the structure:
where and are upper triangular. If so, only the central diagonal block (in rows and columns to ) needs to be further reduced to Schur form (the blocks and are also affected). Therefore, the values of and can be supplied to f08pec directly. Also, f08njc must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If f08nhc has not been called however, then must be set to and to . Note that if the Schur factorization of is required, f08nhc must not be called with or , because the balancing transformation is not orthogonal.
f08pec uses a multishift form of the upper Hessenberg algorithm, due to Bai and Demmel (1989). The Schur vectors are normalized so that , but are determined only to within a factor .
Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift iteration Internat. J. High Speed Comput.1 97–112
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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.
2: – Nag_JobTypeInput
On entry: indicates whether eigenvalues only or the Schur form is required.
Eigenvalues only are required.
The Schur form is required.
3: – Nag_ComputeZTypeInput
On entry: indicates whether the Schur vectors are to be computed.
No Schur vectors are computed (and the array z is not referenced).
The Schur vectors of are computed (and the array z must contain the matrix on entry).
The Schur vectors of are computed (and the array z is initialized by the function).
, or .
4: – IntegerInput
On entry: , the order of the matrix .
5: – IntegerInput
6: – IntegerInput
On entry: if the matrix has been balanced by f08nhc, ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to and ihi to n.
7: – doubleInput/Output
Note: the dimension, dim, of the array h
must be at least
where appears in this document, it refers to the array element
On entry: the upper Hessenberg matrix , as returned by f08nec.
On exit: if , the array contains no useful information.
If , h is overwritten by the upper quasi-triangular matrix from the Schur decomposition (the Schur form) unless NE_CONVERGENCE.
8: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array h.
9: – doubleOutput
10: – doubleOutput
Note: the dimension, dim, of the arrays wr and wi
must be at least
On exit: the real and imaginary parts, respectively, of the computed eigenvalues, unless NE_CONVERGENCE (in which case see Section 6). Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having positive imaginary part first. The eigenvalues are stored in the same order as on the diagonal of the Schur form (if computed); see Section 9 for details.
11: – doubleInput/Output
Note: the dimension, dim, of the array z
must be at least
The th element of the matrix is stored in
On entry: if , z must contain the orthogonal matrix from the reduction to Hessenberg form.
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.
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.
The computed Schur factorization is the exact factorization of a nearby matrix , where
and is the machine precision.
If is an exact eigenvalue, and is the corresponding computed value, then
where is a modestly increasing function of , and is the reciprocal condition number of . The condition numbers may be computed by calling f08qlc.
8Parallelism and Performance
f08pec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pec 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.
The total number of floating-point operations depends on how rapidly the algorithm converges, but is typically about:
if only eigenvalues are computed;
if the Schur form is computed;
if the full Schur factorization is computed.
The Schur form has the following structure (referred to as canonical Schur form).
If all the computed eigenvalues are real, is upper triangular, and the diagonal elements of are the eigenvalues; , for , and .
If some of the computed eigenvalues form complex conjugate pairs, then has diagonal blocks. Each diagonal block has the form