F08PEF (DHSEQR) (PDF version)
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NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
Warning. The specification of the parameter LWORK changed at Mark 20: LWORK is no longer redundant.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08PEF (DHSEQR) computes all the eigenvalues and, optionally, the Schur factorization of a real Hessenberg matrix or a real general matrix which has been reduced to Hessenberg form.

2  Specification

REAL (KIND=nag_wp)  H(LDH,*), WR(*), WI(*), Z(LDZ,*), WORK(max(1,LWORK))
The routine may be called by its LAPACK name dhseqr.

3  Description

F08PEF (DHSEQR) computes all the eigenvalues and, optionally, the Schur factorization of a real upper Hessenberg matrix H:
H = ZTZT ,
where T is an upper quasi-triangular matrix (the Schur form of H), and Z is the orthogonal matrix whose columns are the Schur vectors zi. See Section 8 for details of the structure of T.
The routine may also be used to compute the Schur factorization of a real general matrix A which has been reduced to upper Hessenberg form H:
A = QHQT, where ​Q​ is orthogonal, = QZTQZT.
In this case, after F08NEF (DGEHRD) has been called to reduce A to Hessenberg form, F08NFF (DORGHR) must be called to form Q explicitly; Q is then passed to F08PEF (DHSEQR), which must be called with COMPZ='V'.
The routine can also take advantage of a previous call to F08NHF (DGEBAL) which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix H has the structure:
H11 H12 H13 H22 H23 H33
where H11 and H33 are upper triangular. If so, only the central diagonal block H22 (in rows and columns ilo to ihi) needs to be further reduced to Schur form (the blocks H12 and H23 are also affected). Therefore the values of ilo and ihi can be supplied to F08PEF (DHSEQR) directly. Also, F08NJF (DGEBAK) must be called after this routine to permute the Schur vectors of the balanced matrix to those of the original matrix. If F08NHF (DGEBAL) has not been called however, then ilo must be set to 1 and ihi to n. Note that if the Schur factorization of A is required, F08NHF (DGEBAL) must not be called with JOB='S' or 'B', because the balancing transformation is not orthogonal.
F08PEF (DHSEQR) uses a multishift form of the upper Hessenberg QR algorithm, due to Bai and Demmel (1989). The Schur vectors are normalized so that zi2=1, but are determined only to within a factor ±1.

4  References

Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift QR 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

5  Parameters

1:     JOB – CHARACTER(1)Input
On entry: indicates whether eigenvalues only or the Schur form T is required.
Eigenvalues only are required.
The Schur form T is required.
Constraint: JOB='E' or 'S'.
2:     COMPZ – CHARACTER(1)Input
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 H are computed (and the array Z is initialized by the routine).
The Schur vectors of A are computed (and the array Z must contain the matrix Q on entry).
Constraint: COMPZ='N', 'V' or 'I'.
3:     N – INTEGERInput
On entry: n, the order of the matrix H.
Constraint: N0.
4:     ILO – INTEGERInput
5:     IHI – INTEGERInput
On entry: if the matrix A has been balanced by F08NHF (DGEBAL), then ILO and IHI must contain the values returned by that routine. Otherwise, ILO must be set to 1 and IHI to N.
Constraint: ILO1 and minILO,N IHIN .
6:     H(LDH,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array H must be at least max1,N.
On entry: the n by n upper Hessenberg matrix H, as returned by F08NEF (DGEHRD).
On exit: if JOB='E', the array contains no useful information.
If JOB='S', H is overwritten by the upper quasi-triangular matrix T from the Schur decomposition (the Schur form) unless INFO>0.
7:     LDH – INTEGERInput
On entry: the first dimension of the array H as declared in the (sub)program from which F08PEF (DHSEQR) is called.
Constraint: LDHmax1,N.
8:     WR(*) – REAL (KIND=nag_wp) arrayOutput
9:     WI(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the arrays WR and WI must be at least max1,N.
On exit: the real and imaginary parts, respectively, of the computed eigenvalues, unless INFO>0 (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 T (if computed); see Section 8 for details.
10:   Z(LDZ,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array Z must be at least max1,N if COMPZ='V' or 'I' and at least 1 if COMPZ='N'.
On entry: if COMPZ='V', Z must contain the orthogonal matrix Q from the reduction to Hessenberg form.
If COMPZ='I', Z need not be set.
On exit: if COMPZ='V' or 'I', Z contains the orthogonal matrix of the required Schur vectors, unless INFO>0.
If COMPZ='N', Z is not referenced.
11:   LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08PEF (DHSEQR) is called.
  • if COMPZ='I' or 'V', LDZ max1,N ;
  • if COMPZ='N', LDZ1.
12:   WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if INFO=0, WORK1 contains the minimum value of LWORK required for optimal performance.
13:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08PEF (DHSEQR) is called, unless LWORK=-1, in which case a workspace query is assumed and the routine only calculates the minimum dimension of WORK.
Constraint: LWORKmax1,N or LWORK=-1.
14:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
The algorithm has failed to find all the eigenvalues after a total of 30×IHI-ILO+1 iterations. If INFO=i, elements 1,2,,ILO-1 and i+1,i+2,,n of WR and WI contain the real and imaginary parts of contain the eigenvalues which have been found.
If JOB='E', then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix H^, formed from HILO:INFOILO:INFO , i.e., the ILO through INFO rows and columns of the final output matrix H.
If JOB='S', then on exit
* Hi U = U H~
for some matrix U, where Hi is the input upper Hessenberg matrix and H~ is an upper Hessenberg matrix formed from HINFO+1:IHIINFO+1:IHI .
If COMPZ='V', then on exit
Zout = Zin U
where U is defined in * (regardless of the value of JOB).
If COMPZ='I', then on exit
Zout = U
where U is defined in * (regardless of the value of JOB).
If INFO>0 and COMPZ='N', then Z is not accessed.

7  Accuracy

The computed Schur factorization is the exact factorization of a nearby matrix H+E, where
E2 = Oε H2 ,
and ε is the machine precision.
If λi is an exact eigenvalue, and λ~i is the corresponding computed value, then
λ~i - λi c n ε H2 si ,
where cn is a modestly increasing function of n, and si is the reciprocal condition number of λi. The condition numbers si may be computed by calling F08QLF (DTRSNA).

8  Further Comments

The total number of floating point operations depends on how rapidly the algorithm converges, but is typically about:
The Schur form T has the following structure (referred to as canonical Schur form).
If all the computed eigenvalues are real, T is upper triangular, and the diagonal elements of T are the eigenvalues; WRi=tii, for i=1,2,,n, and WIi=0.0.
If some of the computed eigenvalues form complex conjugate pairs, then T has 2 by 2 diagonal blocks. Each diagonal block has the form
tii ti,i+1 ti+1,i ti+1,i+1 = α β γ α
where βγ<0. The corresponding eigenvalues are α±βγ; WRi=WRi+1=α; WIi=+βγ; WIi+1=-WIi.
The complex analogue of this routine is F08PSF (ZHSEQR).

9  Example

This example computes all the eigenvalues and the Schur factorization of the upper Hessenberg matrix H, where
H = 0.3500 -0.1160 -0.3886 -0.2942 -0.5140 0.1225 0.1004 0.1126 0.0000 0.6443 -0.1357 -0.0977 0.0000 0.0000 0.4262 0.1632 .
See also Section 9 in F08NFF (DORGHR), which illustrates the use of this routine to compute the Schur factorization of a general matrix.

9.1  Program Text

Program Text (f08pefe.f90)

9.2  Program Data

Program Data (f08pefe.d)

9.3  Program Results

Program Results (f08pefe.r)

F08PEF (DHSEQR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012