NAG Library Routine Document
f08psf (zhseqr)
1
Purpose
f08psf (zhseqr) computes all the eigenvalues and, optionally, the Schur factorization of a complex Hessenberg matrix or a complex general matrix which has been reduced to Hessenberg form.
2
Specification
Fortran Interface
Subroutine f08psf ( 
job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info) 
Integer, Intent (In)  ::  n, ilo, ihi, ldh, ldz, lwork  Integer, Intent (Out)  ::  info  Complex (Kind=nag_wp), Intent (Inout)  ::  h(ldh,*), w(*), z(ldz,*)  Complex (Kind=nag_wp), Intent (Out)  ::  work(max(1,lwork))  Character (1), Intent (In)  ::  job, compz 

C Header Interface
#include <nagmk26.h>
void 
f08psf_ (const char *job, const char *compz, const Integer *n, const Integer *ilo, const Integer *ihi, Complex h[], const Integer *ldh, Complex w[], Complex z[], const Integer *ldz, Complex work[], const Integer *lwork, Integer *info, const Charlen length_job, const Charlen length_compz) 

The routine may be called by its
LAPACK
name zhseqr.
3
Description
f08psf (zhseqr) computes all the eigenvalues and, optionally, the Schur factorization of a complex upper Hessenberg matrix
$H$:
where
$T$ is an upper triangular matrix (the Schur form of
$H$), and
$Z$ is the unitary matrix whose columns are the Schur vectors
${z}_{i}$. The diagonal elements of
$T$ are the eigenvalues of
$H$.
The routine may also be used to compute the Schur factorization of a complex general matrix
$A$ which has been reduced to upper Hessenberg form
$H$:
In this case, after
f08nsf (zgehrd) has been called to reduce
$A$ to Hessenberg form,
f08ntf (zunghr) must be called to form
$Q$ explicitly;
$Q$ is then passed to
f08psf (zhseqr), which must be called with
${\mathbf{compz}}=\text{'V'}$.
The routine can also take advantage of a previous call to
f08nvf (zgebal) which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix
$H$ has the structure:
where
${H}_{11}$ and
${H}_{33}$ are upper triangular. If so, only the central diagonal block
${H}_{22}$ (in rows and columns
${i}_{\mathrm{lo}}$ to
${i}_{\mathrm{hi}}$) needs to be further reduced to Schur form (the blocks
${H}_{12}$ and
${H}_{23}$ are also affected). Therefore the values of
${i}_{\mathrm{lo}}$ and
${i}_{\mathrm{hi}}$ can be supplied to
f08psf (zhseqr) directly. Also,
f08nwf (zgebak) must be called after this routine to permute the Schur vectors of the balanced matrix to those of the original matrix. If
f08nvf (zgebal) has not been called however, then
${i}_{\mathrm{lo}}$ must be set to
$1$ and
${i}_{\mathrm{hi}}$ to
$n$. Note that if the Schur factorization of
$A$ is required,
f08nvf (zgebal) must
not be called with
${\mathbf{job}}=\text{'S'}$ or
$\text{'B'}$, because the balancing transformation is not unitary.
f08psf (zhseqr) uses a multishift form of the upper Hessenberg
$QR$ algorithm, due to
Bai and Demmel (1989). The Schur vectors are normalized so that
${\Vert {z}_{i}\Vert}_{2}=1$, but are determined only to within a complex factor of absolute value
$1$.
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
Arguments
 1: $\mathbf{job}$ – Character(1)Input

On entry: indicates whether eigenvalues only or the Schur form
$T$ is required.
 ${\mathbf{job}}=\text{'E'}$
 Eigenvalues only are required.
 ${\mathbf{job}}=\text{'S'}$
 The Schur form $T$ is required.
Constraint:
${\mathbf{job}}=\text{'E'}$ or $\text{'S'}$.
 2: $\mathbf{compz}$ – Character(1)Input

On entry: indicates whether the Schur vectors are to be computed.
 ${\mathbf{compz}}=\text{'N'}$
 No Schur vectors are computed (and the array z is not referenced).
 ${\mathbf{compz}}=\text{'V'}$
 The Schur vectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
 ${\mathbf{compz}}=\text{'I'}$
 The Schur vectors of $H$ are computed (and the array z is initialized by the routine).
Constraint:
${\mathbf{compz}}=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.
 3: $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $H$.
Constraint:
${\mathbf{n}}\ge 0$.
 4: $\mathbf{ilo}$ – IntegerInput
 5: $\mathbf{ihi}$ – IntegerInput

On entry: if the matrix
$A$ has been balanced by
f08nvf (zgebal),
ilo and
ihi must contain the values returned by that routine. Otherwise,
ilo must be set to
$1$ and
ihi to
n.
Constraint:
${\mathbf{ilo}}\ge 1$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ilo}},{\mathbf{n}}\right)\le {\mathbf{ihi}}\le {\mathbf{n}}$.
 6: $\mathbf{h}\left({\mathbf{ldh}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array
h
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the
$n$ by
$n$ upper Hessenberg matrix
$H$, as returned by
f08nsf (zgehrd).
On exit: if
${\mathbf{job}}=\text{'E'}$, the array contains no useful information.
If
${\mathbf{job}}=\text{'S'}$,
h is overwritten by the upper triangular matrix
$T$ from the Schur decomposition (the Schur form) unless
${\mathbf{info}}>{\mathbf{0}}$.
 7: $\mathbf{ldh}$ – IntegerInput

On entry: the first dimension of the array
h as declared in the (sub)program from which
f08psf (zhseqr) is called.
Constraint:
${\mathbf{ldh}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
 8: $\mathbf{w}\left(*\right)$ – Complex (Kind=nag_wp) arrayOutput

Note: the dimension of the array
w
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the computed eigenvalues, unless
${\mathbf{info}}>{\mathbf{0}}$ (in which case see
Section 6). The eigenvalues are stored in the same order as on the diagonal of the Schur form
$T$ (if computed).
 9: $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array
z
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if
${\mathbf{compz}}=\text{'V'}$ or
$\text{'I'}$ and at least
$1$ if
${\mathbf{compz}}=\text{'N'}$.
On entry: if
${\mathbf{compz}}=\text{'V'}$,
z must contain the unitary matrix
$Q$ from the reduction to Hessenberg form.
If
${\mathbf{compz}}=\text{'I'}$,
z need not be set.
On exit: if
${\mathbf{compz}}=\text{'V'}$ or
$\text{'I'}$,
z contains the unitary matrix of the required Schur vectors, unless
${\mathbf{info}}>{\mathbf{0}}$.
If
${\mathbf{compz}}=\text{'N'}$,
z is not referenced.
 10: $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array
z as declared in the (sub)program from which
f08psf (zhseqr) is called.
Constraints:
 if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
 if ${\mathbf{compz}}=\text{'N'}$, ${\mathbf{ldz}}\ge 1$.
 11: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace

On exit: if
${\mathbf{info}}={\mathbf{0}}$, the real part of
${\mathbf{work}}\left(1\right)$ contains the minimum value of
lwork required for optimal performance.
 12: $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array
work as declared in the (sub)program from which
f08psf (zhseqr) is called, unless
${\mathbf{lwork}}=1$, in which case a workspace query is assumed and the routine only calculates the minimum dimension of
work.
Constraint:
${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ or ${\mathbf{lwork}}=1$.
 13: $\mathbf{info}$ – IntegerOutput
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
 $999<{\mathbf{info}}<0$
If ${\mathbf{info}}=i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
 ${\mathbf{info}}=999$

Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
An explanatory message is output, and execution of the program is terminated.
 ${\mathbf{info}}>0$

The algorithm has failed to find all the eigenvalues after a total of $30\left({\mathbf{ihi}}{\mathbf{ilo}}+1\right)$ iterations.
7
Accuracy
The computed Schur factorization is the exact factorization of a nearby matrix
$\left(H+E\right)$, where
and
$\epsilon $ is the
machine precision.
If
${\lambda}_{i}$ is an exact eigenvalue, and
${\stackrel{~}{\lambda}}_{i}$ is the corresponding computed value, then
where
$c\left(n\right)$ is a modestly increasing function of
$n$, and
${s}_{i}$ is the reciprocal condition number of
${\lambda}_{i}$. The condition numbers
${s}_{i}$ may be computed by calling
f08qyf (ztrsna).
8
Parallelism and Performance
f08psf (zhseqr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08psf (zhseqr) 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 implementationspecific information.
The total number of real floatingpoint operations depends on how rapidly the algorithm converges, but is typically about:
 $25{n}^{3}$ if only eigenvalues are computed;
 $35{n}^{3}$ if the Schur form is computed;
 $70{n}^{3}$ if the full Schur factorization is computed.
The real analogue of this routine is
f08pef (dhseqr).
10
Example
This example computes all the eigenvalues and the Schur factorization of the upper Hessenberg matrix
$H$, where
See also
Section 10 in
f08ntf (zunghr), which illustrates the use of this routine to compute the Schur factorization of a general matrix.
10.1
Program Text
Program Text (f08psfe.f90)
10.2
Program Data
Program Data (f08psfe.d)
10.3
Program Results
Program Results (f08psfe.r)