NAG Library Routine Document

1Purpose

f08paf (dgees) computes the eigenvalues, the real Schur form $T$, and, optionally, the matrix of Schur vectors $Z$ for an $n$ by $n$ real nonsymmetric matrix $A$.

2Specification

Fortran Interface
 Subroutine f08paf ( sort, n, a, lda, sdim, wr, wi, vs, ldvs, work, info)
 Integer, Intent (In) :: n, lda, ldvs, lwork Integer, Intent (Out) :: sdim, info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), wr(*), wi(*), vs(ldvs,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Logical, External :: select Logical, Intent (Inout) :: bwork(*) Character (1), Intent (In) :: jobvs, sort
#include nagmk26.h
 void f08paf_ (const char *jobvs, const char *sort, logical (NAG_CALL *sel)(const double *wr, const double *wi),const Integer *n, double a[], const Integer *lda, Integer *sdim, double wr[], double wi[], double vs[], const Integer *ldvs, double work[], const Integer *lwork, logical bwork[], Integer *info, const Charlen length_jobvs, const Charlen length_sort)
The routine may be called by its LAPACK name dgees.

3Description

The real Schur factorization of $A$ is given by
 $A = Z T ZT ,$
where $Z$, the matrix of Schur vectors, is orthogonal and $T$ is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with $1$ by $1$ and $2$ by $2$ blocks. $2$ by $2$ blocks will be standardized in the form
 $a b c a$
where $bc<0$. The eigenvalues of such a block are $a±\sqrt{bc}$.
Optionally, f08paf (dgees) also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left. The leading columns of $Z$ form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.

4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5Arguments

1:     $\mathbf{jobvs}$ – Character(1)Input
On entry: if ${\mathbf{jobvs}}=\text{'N'}$, Schur vectors are not computed.
If ${\mathbf{jobvs}}=\text{'V'}$, Schur vectors are computed.
Constraint: ${\mathbf{jobvs}}=\text{'N'}$ or $\text{'V'}$.
2:     $\mathbf{sort}$ – Character(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
${\mathbf{sort}}=\text{'N'}$
Eigenvalues are not ordered.
${\mathbf{sort}}=\text{'S'}$
Eigenvalues are ordered (see select).
Constraint: ${\mathbf{sort}}=\text{'N'}$ or $\text{'S'}$.
3:     $\mathbf{select}$ – Logical Function, supplied by the user.External Procedure
If ${\mathbf{sort}}=\text{'S'}$, select is used to select eigenvalues to sort to the top left of the Schur form.
If ${\mathbf{sort}}=\text{'N'}$, select is not referenced and f08paf (dgees) may be called with the dummy function f08paz.
An eigenvalue ${\mathbf{wr}}\left(j\right)+\sqrt{-1}×{\mathbf{wi}}\left(j\right)$ is selected if ${\mathbf{select}}\left({\mathbf{wr}}\left(j\right),{\mathbf{wi}}\left(j\right)\right)$ is .TRUE.. If either one of a complex conjugate pair of eigenvalues is selected, then both are. Note that a selected complex eigenvalue may no longer satisfy ${\mathbf{select}}\left({\mathbf{wr}}\left(j\right),{\mathbf{wi}}\left(j\right)\right)=\mathrm{.TRUE.}$ after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case info is set to ${\mathbf{n}}+2$ (see info below).
The specification of select is:
Fortran Interface
 Function select ( wr, wi)
 Logical :: select Real (Kind=nag_wp), Intent (In) :: wr, wi
#include nagmk26.h
 Nag_Boolean select (const double *wr, const double *wi)
1:     $\mathbf{wr}$ – Real (Kind=nag_wp)Input
2:     $\mathbf{wi}$ – Real (Kind=nag_wp)Input
On entry: the real and imaginary parts of the eigenvalue.
select must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f08paf (dgees) is called. Arguments denoted as Input must not be changed by this procedure.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ matrix $A$.
On exit: a is overwritten by its real Schur form $T$.
6:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08paf (dgees) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{sdim}$ – IntegerOutput
On exit: if ${\mathbf{sort}}=\text{'N'}$, ${\mathbf{sdim}}=0$.
If ${\mathbf{sort}}=\text{'S'}$, ${\mathbf{sdim}}=\text{}$ number of eigenvalues (after sorting) for which select is .TRUE.. (Complex conjugate pairs for which select is .TRUE. for either eigenvalue count as $2$.)
8:     $\mathbf{wr}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array wr must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: see the description of wi.
9:     $\mathbf{wi}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array wi must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form $T$. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.
10:   $\mathbf{vs}\left({\mathbf{ldvs}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vs must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvs}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobvs}}=\text{'V'}$, vs contains the orthogonal matrix $Z$ of Schur vectors.
If ${\mathbf{jobvs}}=\text{'N'}$, vs is not referenced.
11:   $\mathbf{ldvs}$ – IntegerInput
On entry: the first dimension of the array vs as declared in the (sub)program from which f08paf (dgees) is called.
Constraints:
• if ${\mathbf{jobvs}}=\text{'V'}$, ${\mathbf{ldvs}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvs}}\ge 1$.
12:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – 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 f08paf (dgees) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lwork must generally be larger than the minimum, say $3×{\mathbf{n}}+\mathit{nb}×{\mathbf{n}}$, where $\mathit{nb}$ is the optimal block size for f08nef (dgehrd)
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,3×{\mathbf{n}}\right)$.
14:   $\mathbf{bwork}\left(*\right)$ – Logical arrayWorkspace
Note: the dimension of the array bwork must be at least $1$ if ${\mathbf{sort}}=\text{'N'}$, and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ otherwise.
If ${\mathbf{sort}}=\text{'N'}$, bwork is not referenced.
15:   $\mathbf{info}$ – 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.
${\mathbf{info}}=1 \text{to} {\mathbf{n}}$
If ${\mathbf{info}}=i$ and $i\le {\mathbf{n}}$, the $QR$ algorithm failed to compute all the eigenvalues.
${\mathbf{info}}={\mathbf{n}}+1$
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
${\mathbf{info}}={\mathbf{n}}+2$
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy ${\mathbf{select}}=\mathrm{.TRUE.}$. This could also be caused by underflow due to scaling.

7Accuracy

The computed Schur factorization satisfies
 $A+E=ZT ZT ,$
where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

8Parallelism and Performance

f08paf (dgees) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08paf (dgees) 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.

The total number of floating-point operations is proportional to ${n}^{3}$.
The complex analogue of this routine is f08pnf (zgees).

10Example

This example finds the Schur factorization of the matrix
 $A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 ,$
such that the real positive eigenvalues of $A$ are the top left diagonal elements of the Schur form, $T$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

10.1Program Text

Program Text (f08pafe.f90)

10.2Program Data

Program Data (f08pafe.d)

10.3Program Results

Program Results (f08pafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017