f08psf 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.
The routine may be called by the names f08psf, nagf_lapackeig_zhseqr or its LAPACK name zhseqr.
3Description
f08psf computes all the eigenvalues and, optionally, the Schur factorization of a complex upper Hessenberg matrix $H$:
$$H=ZT{Z}^{\mathrm{H}}\text{,}$$
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 $H$:
$$\begin{array}{lll}A& =& QH{Q}^{\mathrm{H}}\text{, where}Q\text{ is unitary,}\\ & =& \left(QZ\right)T{\left(QZ\right)}^{\mathrm{H}}\text{.}\end{array}$$
In this case, after f08nsf has been called to reduce $A$ to Hessenberg form, f08ntf must be called to form $Q$ explicitly; $Q$ is then passed to f08psf, which must be called with ${\mathbf{compz}}=\text{'V'}$.
The routine can also take advantage of a previous call to f08nvf 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 directly. Also, f08nwf must be called after this routine to permute the Schur vectors of the balanced matrix to those of the original matrix. If f08nvf 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 must not be called with ${\mathbf{job}}=\text{'S'}$ or $\text{'B'}$, because the balancing transformation is not unitary.
f08psf 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 $1$.
4References
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
5Arguments
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, 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 $\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{ilo}},{\mathbf{n}})\le {\mathbf{ihi}}\le {\mathbf{n}}$.
Note: the second dimension of the array h
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: the $n\times n$ upper Hessenberg matrix $H$, as returned by f08nsf.
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 is called.
Note: the dimension of the array w
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
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).
Note: the second dimension of the array z
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$ 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 is called.
Constraints:
if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$;
if ${\mathbf{compz}}=\text{'N'}$, ${\mathbf{ldz}}\ge 1$.
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 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:
${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$ or ${\mathbf{lwork}}=-1$.
13: $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).
6Error 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 9 in the Introduction to the NAG Library FL Interface 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 $30({\mathbf{ihi}}-{\mathbf{ilo}}+1)$ iterations.
7Accuracy
The computed Schur factorization is the exact factorization of a nearby matrix $(H+E)$, where
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.
8Parallelism and Performance
f08psf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08psf 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
The total number of real floating-point 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.