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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgeesx (f08pb)

## Purpose

nag_lapack_dgeesx (f08pb) computes the eigenvalues, the real Schur form $T$, and, optionally, the matrix of Schur vectors $Z$ for an $n$ by $n$ real nonsymmetric matrix $A$.

## Syntax

[a, sdim, wr, wi, vs, rconde, rcondv, info] = f08pb(jobvs, sort, select, sense, a, 'n', n)
[a, sdim, wr, wi, vs, rconde, rcondv, info] = nag_lapack_dgeesx(jobvs, sort, select, sense, a, 'n', n)

## Description

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, nag_lapack_dgeesx (f08pb) also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (rcondv). The leading columns of $Z$ form an orthonormal basis for this invariant subspace.
For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.8 of Anderson et al. (1999) (where these quantities are called $s$ and $\mathrm{sep}$ respectively).

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{jobvs}$ – string (length ≥ 1)
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:     $\mathrm{sort}$ – string (length ≥ 1)
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:     $\mathrm{select}$ – function handle or string containing name of m-file
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 nag_lapack_dgeesx (f08pb) may be called with the string '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 ${\mathbf{select}}\left({\mathbf{wr}}\left(j\right),{\mathbf{wi}}\left(j\right)\right)=\mathit{true}$ 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).
[result] = select(wr, wi)

Input Parameters

1:     $\mathrm{wr}$ – double scalar
2:     $\mathrm{wi}$ – double scalar
The real and imaginary parts of the eigenvalue.

Output Parameters

1:     $\mathrm{result}$ – logical scalar
$\mathbf{result}=\mathit{true}$ for selected eigenvalues.
4:     $\mathrm{sense}$ – string (length ≥ 1)
Determines which reciprocal condition numbers are computed.
${\mathbf{sense}}=\text{'N'}$
None are computed.
${\mathbf{sense}}=\text{'E'}$
Computed for average of selected eigenvalues only.
${\mathbf{sense}}=\text{'V'}$
Computed for selected right invariant subspace only.
${\mathbf{sense}}=\text{'B'}$
Computed for both.
If ${\mathbf{sense}}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$, ${\mathbf{sort}}=\text{'S'}$.
Constraint: ${\mathbf{sense}}=\text{'N'}$, $\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.
5:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ matrix $A$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores its real Schur form $T$.
2:     $\mathrm{sdim}$int64int32nag_int scalar
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 $2$.)
3:     $\mathrm{wr}\left(:\right)$ – double array
The dimension of the array wr will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
See the description of wi.
4:     $\mathrm{wi}\left(:\right)$ – double array
The dimension of the array wi will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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.
5:     $\mathrm{vs}\left(\mathit{ldvs},:\right)$ – double array
The first dimension, $\mathit{ldvs}$, of the array vs will be
• if ${\mathbf{jobvs}}=\text{'V'}$, $\mathit{ldvs}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise $\mathit{ldvs}=1$.
The second dimension of the array vs will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvs}}=\text{'V'}$ and $1$ otherwise.
If ${\mathbf{jobvs}}=\text{'V'}$, vs contains the orthogonal matrix $Z$ of Schur vectors.
If ${\mathbf{jobvs}}=\text{'N'}$, vs is not referenced.
6:     $\mathrm{rconde}$ – double scalar
If ${\mathbf{sense}}=\text{'E'}$ or $\text{'B'}$, contains the reciprocal condition number for the average of the selected eigenvalues.
If ${\mathbf{sense}}=\text{'N'}$ or $\text{'V'}$, rconde is not referenced.
7:     $\mathrm{rcondv}$ – double scalar
If ${\mathbf{sense}}=\text{'V'}$ or $\text{'B'}$, rcondv contains the reciprocal condition number for the selected right invariant subspace.
If ${\mathbf{sense}}=\text{'N'}$ or $\text{'E'}$, rcondv is not referenced.
8:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: jobvs, 2: sort, 3: select, 4: sense, 5: n, 6: a, 7: lda, 8: sdim, 9: wr, 10: wi, 11: vs, 12: ldvs, 13: rconde, 14: rcondv, 15: work, 16: lwork, 17: iwork, 18: liwork, 19: bwork, 20: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
${\mathbf{info}}=1 \text{to} {\mathbf{n}}$
If ${\mathbf{info}}=i$ and $i\le {\mathbf{n}}$, the $QR$ algorithm failed to compute all the eigenvalues.
W  ${\mathbf{info}}={\mathbf{n}}+1$
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
W  ${\mathbf{info}}={\mathbf{n}}+2$
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy ${\mathbf{select}}=\mathit{true}$. This could also be caused by underflow due to scaling.

## Accuracy

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

The total number of floating-point operations is proportional to ${n}^{3}$.
The complex analogue of this function is nag_lapack_zgeesx (f08pp).

## Example

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, $T$. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding invariant subspace are also returned.
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.
```function f08pb_example

fprintf('f08pb example results\n\n');

% Matrix A
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];

% Schur vectors of A, selecting real positive eigenvalues
jobvs = 'Vectors (Schur)';
sortp = 'Sort';
select = @(wr, wi) (wr > 0 && wi == 0);
sense = 'Both reciprocal condition numbers';

[~, sdim, wr, wi, vs, rconde, rcondv, info] = ...
f08pb( ...
jobvs, sortp, select, sense, a);

fprintf('Number of eigenvalues for which SELECT is true = %3d\n',sdim);
fprintf(' (dimension of invariant subspace)\n\n');
disp('Selected eigenvalues');
disp(wr(1:sdim) + i*wi(1:sdim));
fprintf('%s\n%61s = %9.1e\n\n', ...
'Reciprocal of projection norm onto the invariant subspace', ...
'rconde', rconde);
fprintf('%s\n%61s = %9.1e\n\n', ...
'Reciprocal condition number for the invariant subspace', ...
'rcondv', rcondv);

anorm = norm(a);
erbde = x02aj*anorm/rconde;
erbdv = x02aj*anorm/rcondv;

fprintf('%-61s = %9.1e\n', ...
'Approximate asymptotic error bound for selected eigenvalues', erbde);
fprintf('%-61s = %9.1e\n', ...
'Approximate asymptotic error bound for the invariant subspace', ...
erbdv);

```
```f08pb example results

Number of eigenvalues for which SELECT is true =   1
(dimension of invariant subspace)

Selected eigenvalues
0.7995

Reciprocal of projection norm onto the invariant subspace
rconde =   9.9e-01

Reciprocal condition number for the invariant subspace
rcondv =   8.2e-01

Approximate asymptotic error bound for selected eigenvalues   =   9.3e-17
Approximate asymptotic error bound for the invariant subspace =   1.1e-16
```