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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgeesx (f08pb)

## Purpose

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

## 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$A$ is given by
 A = Z T ZT , $A = Z T ZT ,$
where Z$Z$, the matrix of Schur vectors, is orthogonal and T$T$ is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with 1$1$ by 1$1$ and 2$2$ by 2$2$ blocks. 2$2$ by 2$2$ blocks will be standardized in the form
 [ a b c a ]
$[ a b c a ]$
where bc < 0$bc<0$. The eigenvalues of such a block are a ± sqrt(bc)$a±\sqrt{bc}$.
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$Z$ 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$s$ and sep$\mathrm{sep}$ 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:     jobvs – string (length ≥ 1)
If jobvs = 'N'${\mathbf{jobvs}}=\text{'N'}$, Schur vectors are not computed.
If jobvs = 'V'${\mathbf{jobvs}}=\text{'V'}$, Schur vectors are computed.
Constraint: jobvs = 'N'${\mathbf{jobvs}}=\text{'N'}$ or 'V'$\text{'V'}$.
2:     sort – string (length ≥ 1)
Specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
sort = 'N'${\mathbf{sort}}=\text{'N'}$
Eigenvalues are not ordered.
sort = 'S'${\mathbf{sort}}=\text{'S'}$
Eigenvalues are ordered (see select).
Constraint: sort = 'N'${\mathbf{sort}}=\text{'N'}$ or 'S'$\text{'S'}$.
3:     select – function handle or string containing name of m-file
If sort = 'S'${\mathbf{sort}}=\text{'S'}$, select is used to select eigenvalues to sort to the top left of the Schur form.
If sort = 'N'${\mathbf{sort}}=\text{'N'}$, select is not referenced and nag_lapack_dgeesx (f08pb) may be called with the string 'f08paz'.
An eigenvalue wr(j) + sqrt(1) × wi(j)${\mathbf{wr}}\left(j\right)+\sqrt{-1}×{\mathbf{wi}}\left(j\right)$ is selected if select(wr(j),wi(j))${\mathbf{select}}\left({\mathbf{wr}}\left(j\right),{\mathbf{wi}}\left(j\right)\right)$ 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 select(wr(j),wi(j)) = true${\mathbf{select}}\left({\mathbf{wr}}\left(j\right),{\mathbf{wi}}\left(j\right)\right)=\mathbf{true}$ 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 n + 2${\mathbf{n}}+2$ (see info below).
[result] = select(wr, wi)

Input Parameters

1:     wr – double scalar
2:     wi – double scalar
The real and imaginary parts of the eigenvalue.

Output Parameters

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

### Optional Input Parameters

1:     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$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldvs work lwork iwork liwork bwork

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores its real Schur form T$T$.
2:     sdim – int64int32nag_int scalar
If sort = 'N'${\mathbf{sort}}=\text{'N'}$, sdim = 0${\mathbf{sdim}}=0$.
If sort = 'S'${\mathbf{sort}}=\text{'S'}$, sdim = ${\mathbf{sdim}}=\text{}$ number of eigenvalues (after sorting) for which select is true. (Complex conjugate pairs for which select is true for either eigenvalue count as 2$2$.)
3:     wr( : $:$) – double array
Note: the dimension of the array wr must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
See the description of wi.
4:     wi( : $:$) – double array
Note: the dimension of the array wi must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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$T$. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.
5:     vs(ldvs, : $:$) – double array
The first dimension, ldvs, of the array vs will be
• if jobvs = 'V'${\mathbf{jobvs}}=\text{'V'}$, ldvs max (1,n) $\mathit{ldvs}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldvs1$\mathit{ldvs}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if jobvs = 'V'${\mathbf{jobvs}}=\text{'V'}$, and at least 1$1$ otherwise
If jobvs = 'V'${\mathbf{jobvs}}=\text{'V'}$, vs contains the orthogonal matrix Z$Z$ of Schur vectors.
If jobvs = 'N'${\mathbf{jobvs}}=\text{'N'}$, vs is not referenced.
6:     rconde – double scalar
If sense = 'E'${\mathbf{sense}}=\text{'E'}$ or 'B'$\text{'B'}$, contains the reciprocal condition number for the average of the selected eigenvalues.
If sense = 'N'${\mathbf{sense}}=\text{'N'}$ or 'V'$\text{'V'}$, rconde is not referenced.
7:     rcondv – double scalar
If sense = 'V'${\mathbf{sense}}=\text{'V'}$ or 'B'$\text{'B'}$, rcondv contains the reciprocal condition number for the selected right invariant subspace.
If sense = 'N'${\mathbf{sense}}=\text{'N'}$ or 'E'$\text{'E'}$, rcondv is not referenced.
8:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [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.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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.
INFO = 1ton${\mathbf{INFO}}=1 \text{to} {\mathbf{n}}$
If info = i${\mathbf{info}}=i$ and in$i\le {\mathbf{n}}$, the QR$QR$ algorithm failed to compute all the eigenvalues.
W INFO = N + 1${\mathbf{INFO}}={\mathbf{N}}+1$
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
W INFO = N + 2${\mathbf{INFO}}={\mathbf{N}}+2$
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select = true${\mathbf{select}}=\mathbf{true}$. This could also be caused by underflow due to scaling.

## Accuracy

The computed Schur factorization satisfies
 A + E = ZTZT , $A+E = ZTZT ,$
where
 ‖E‖2 = O(ε) ‖A‖2 , $‖E‖2 = O(ε) ‖A‖2 ,$
and ε$\epsilon$ 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 n3${n}^{3}$.
The complex analogue of this function is nag_lapack_zgeesx (f08pp).

## Example

```function nag_lapack_dgeesx_example
jobvs = 'Vectors (Schur)';
sortp = 'Sort';
select = @(wr, wi) (wi == 0);
sense = 'Both reciprocal condition numbers';
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];
[aOut, sdim, wr, wi, vs, rconde, rcondv, info] = ...
nag_lapack_dgeesx(jobvs, sortp, select, sense, a)
```
```

aOut =

0.7995   -0.0059   -0.0751   -0.0927
0   -0.1007    0.3937    0.3569
0         0   -0.0994   -0.5128
0         0    0.3132   -0.0994

sdim =

2

wr =

0.7995
-0.1007
-0.0994
-0.0994

wi =

0
0
0.4008
-0.4008

vs =

-0.6551   -0.1210   -0.5032    0.5504
-0.5236   -0.3286    0.7857    0.0229
0.5362   -0.5974    0.0904    0.5894
-0.0956   -0.7215   -0.3482   -0.5908

rconde =

0.5699

rcondv =

0.3102

info =

0

```
```function f08pb_example
jobvs = 'Vectors (Schur)';
sortp = 'Sort';
select = @(wr, wi) (wi == 0);
sense = 'Both reciprocal condition numbers';
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];
[aOut, sdim, wr, wi, vs, rconde, rcondv, info] = ...
f08pb(jobvs, sortp, select, sense, a)
```
```

aOut =

0.7995   -0.0059   -0.0751   -0.0927
0   -0.1007    0.3937    0.3569
0         0   -0.0994   -0.5128
0         0    0.3132   -0.0994

sdim =

2

wr =

0.7995
-0.1007
-0.0994
-0.0994

wi =

0
0
0.4008
-0.4008

vs =

-0.6551   -0.1210   -0.5032    0.5504
-0.5236   -0.3286    0.7857    0.0229
0.5362   -0.5974    0.0904    0.5894
-0.0956   -0.7215   -0.3482   -0.5908

rconde =

0.5699

rcondv =

0.3102

info =

0

```