f08pef 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.
The routine may be called by the names f08pef, nagf_lapackeig_dhseqr or its LAPACK name dhseqr.
3Description
f08pef computes all the eigenvalues and, optionally, the Schur factorization of a real upper Hessenberg matrix $H$:
$$H=ZT{Z}^{\mathrm{T}}\text{,}$$
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 ${z}_{i}$. See Section 9 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 $H$:
$$\begin{array}{lll}A& =& QH{Q}^{\mathrm{T}}\text{, where}Q\text{ is orthogonal,}\\ & =& \left(QZ\right)T{\left(QZ\right)}^{\mathrm{T}}\text{.}\end{array}$$
In this case, after f08nef has been called to reduce $A$ to Hessenberg form, f08nff must be called to form $Q$ explicitly; $Q$ is then passed to f08pef, which must be called with ${\mathbf{compz}}=\text{'V'}$.
The routine can also take advantage of a previous call to f08nhf 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 f08pef directly. Also, f08njf must be called after this routine to permute the Schur vectors of the balanced matrix to those of the original matrix. If f08nhf 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, f08nhf must not be called with ${\mathbf{job}}=\text{'S'}$ or $\text{'B'}$, because the balancing transformation is not orthogonal.
f08pef 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 factor $\pm 1$.
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 f08nhf, 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 $\mathrm{min}\phantom{\rule{0.125em}{0ex}}({\mathbf{ilo}},{\mathbf{n}})\le {\mathbf{ihi}}\le {\mathbf{n}}$.
6: $\mathbf{h}({\mathbf{ldh}},*)$ – Real (Kind=nag_wp) arrayInput/Output
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 f08nef.
On exit: if ${\mathbf{job}}=\text{'E'}$, the array contains no useful information.
If ${\mathbf{job}}=\text{'S'}$, h is overwritten by the upper quasi-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 f08pef is called.
8: $\mathbf{wr}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
9: $\mathbf{wi}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the arrays wr and wi
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On exit: the real and imaginary parts, respectively, of the computed eigenvalues, unless ${\mathbf{info}}>{\mathbf{0}}$ (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 9 for details.
10: $\mathbf{z}({\mathbf{ldz}},*)$ – Real (Kind=nag_wp) arrayInput/Output
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 orthogonal 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 orthogonal matrix of the required Schur vectors, unless ${\mathbf{info}}>{\mathbf{0}}$.
If ${\mathbf{compz}}=\text{'N'}$, z is not referenced.
11: $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08pef 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 ${\mathbf{compz}}=\text{'N'}$, ${\mathbf{ldz}}\ge 1$.
12: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{lwork}})\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
13: $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08pef 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 ${\mathbf{lwork}}=-1$.
14: $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The algorithm has failed to find all the eigenvalues after a total of $30({\mathbf{ihi}}-{\mathbf{ilo}}+1)$ iterations.
If ${\mathbf{info}}=i$, elements $1,2,\dots ,{\mathbf{ilo}}-1$ and $i+1,i+2,\dots ,n$ of wr and wi contain the real and imaginary parts of contain the eigenvalues which have been found.
If ${\mathbf{job}}=\text{'E'}$, then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix
$\hat{H}$, formed from ${\mathbf{h}}({\mathbf{ilo}}:{\mathbf{info}},{\mathbf{ilo}}:{\mathbf{info}})$, i.e., the ilo through info rows and columns of the final output matrix $H$.
for some matrix $U$, where ${H}_{i}$ is the input upper Hessenberg matrix and $\stackrel{~}{H}$ is an upper Hessenberg matrix formed from ${\mathbf{h}}({\mathbf{info}}+1:{\mathbf{ihi}},{\mathbf{info}}+1:{\mathbf{ihi}})$.
If ${\mathbf{compz}}=\text{'V'}$, then on exit
$${Z}_{\mathrm{out}}={Z}_{\mathrm{in}}U$$
where $U$ is defined in $(*)$ (regardless of the value of job).
If ${\mathbf{compz}}=\text{'I'}$, then on exit
$${Z}_{\mathrm{out}}=U$$
where $U$ is defined in $(*)$ (regardless of the value of job).
If ${\mathbf{info}}>{\mathbf{0}}$ and ${\mathbf{compz}}=\text{'N'}$, z is not accessed.
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 f08qlf.
8Parallelism and Performance
f08pef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pef 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 floating-point operations depends on how rapidly the algorithm converges, but is typically about:
$7{n}^{3}$ if only eigenvalues are computed;
$10{n}^{3}$ if the Schur form is computed;
$20{n}^{3}$ if the full Schur factorization is computed.
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; ${\mathbf{wr}}\left(\mathit{i}\right)={t}_{\mathit{i}\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, and ${\mathbf{wi}}\left(i\right)=0.0$.
If some of the computed eigenvalues form complex conjugate pairs, then $T$ has $2\times 2$ diagonal blocks. Each diagonal block has the form