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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgebal (f08nv)

Purpose

nag_lapack_zgebal (f08nv) balances a complex general matrix in order to improve the accuracy of computed eigenvalues and/or eigenvectors.

Syntax

[a, ilo, ihi, scale, info] = f08nv(job, a, 'n', n)
[a, ilo, ihi, scale, info] = nag_lapack_zgebal(job, a, 'n', n)

Description

nag_lapack_zgebal (f08nv) balances a complex general matrix A$A$. The term ‘balancing’ covers two steps, each of which involves a similarity transformation of A$A$. The function can perform either or both of these steps.
1. The function first attempts to permute A$A$ to block upper triangular form by a similarity transformation:
PAPT = A =
 A11 ′ A12 ′ A13 ′ 0 A22 ′ A23 ′ 0 0 A33 ′
$PAPT = A′ = A11′ A12′ A13′ 0 A22′ A23′ 0 0 A33′$
where P$P$ is a permutation matrix, and A11${A}_{11}^{\prime }$ and A33${A}_{33}^{\prime }$ are upper triangular. Then the diagonal elements of A11${A}_{11}^{\prime }$ and A33${A}_{33}^{\prime }$ are eigenvalues of A$A$. The rest of the eigenvalues of A$A$ are the eigenvalues of the central diagonal block A22${A}_{22}^{\prime }$, in rows and columns ilo${i}_{\mathrm{lo}}$ to ihi${i}_{\mathrm{hi}}$. Subsequent operations to compute the eigenvalues of A$A$ (or its Schur factorization) need only be applied to these rows and columns; this can save a significant amount of work if ilo > 1${i}_{\mathrm{lo}}>1$ and ihi < n${i}_{\mathrm{hi}}. If no suitable permutation exists (as is often the case), the function sets ilo = 1${i}_{\mathrm{lo}}=1$ and ihi = n${i}_{\mathrm{hi}}=n$, and A22${A}_{22}^{\prime }$ is the whole of A$A$.
2. The function applies a diagonal similarity transformation to A${A}^{\prime }$, to make the rows and columns of A22${A}_{22}^{\prime }$ as close in norm as possible:
A ′ ′ = DAD − 1 =
 I 0 0 0 D22 0 0 0 I
 A11 ′ A12 ′ A13 ′ 0 A22 ′ A23 ′ 0 0 A33 ′
 I 0 0 0 D22 − 1 0 0 0 I
.
$A′′ = DA′D-1 = I 0 0 0 D22 0 0 0 I A11′ A12′ A13′ 0 A22′ A23′ 0 0 A33′ I 0 0 0 D22-1 0 0 0 I .$
This scaling can reduce the norm of the matrix (i.e., A22 < A22$‖{A}_{22}^{\prime \prime }‖<‖{A}_{22}^{\prime }‖$) and hence reduce the effect of rounding errors on the accuracy of computed eigenvalues and eigenvectors.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     job – string (length ≥ 1)
Indicates whether A$A$ is to be permuted and/or scaled (or neither).
job = 'N'${\mathbf{job}}=\text{'N'}$
A$A$ is neither permuted nor scaled (but values are assigned to ilo, ihi and scale).
job = 'P'${\mathbf{job}}=\text{'P'}$
A$A$ is permuted but not scaled.
job = 'S'${\mathbf{job}}=\text{'S'}$
A$A$ is scaled but not permuted.
job = 'B'${\mathbf{job}}=\text{'B'}$
A$A$ is both permuted and scaled.
Constraint: job = 'N'${\mathbf{job}}=\text{'N'}$, 'P'$\text{'P'}$, 'S'$\text{'S'}$ or 'B'$\text{'B'}$.
2:     a(lda, : $:$) – complex 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 The second dimension of the array a.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda

Output Parameters

1:     a(lda, : $:$) – complex 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 the balanced matrix. If job = 'N'${\mathbf{job}}=\text{'N'}$, a is not referenced.
2:     ilo – int64int32nag_int scalar
3:     ihi – int64int32nag_int scalar
The values ilo${i}_{\mathrm{lo}}$ and ihi${i}_{\mathrm{hi}}$ such that on exit a(i,j)${\mathbf{a}}\left(i,j\right)$ is zero if i > j$i>j$ and 1j < ilo$1\le j<{i}_{\mathrm{lo}}$ or ihi < in${i}_{\mathrm{hi}}.
If job = 'N'${\mathbf{job}}=\text{'N'}$ or 'S'$\text{'S'}$, ilo = 1${i}_{\mathrm{lo}}=1$ and ihi = n${i}_{\mathrm{hi}}=n$.
4:     scale(n) – double array
Details of the permutations and scaling factors applied to A$A$. More precisely, if pj${p}_{j}$ is the index of the row and column interchanged with row and column j$j$ and dj${d}_{j}$ is the scaling factor used to balance row and column j$j$ then
scale(j) =
 { pj, j = 1,2, … ,ilo − 1 dj, j = ilo,ilo + 1, … ,ihi  and pj, j = ihi + 1,ihi + 2, … ,n.
$scalej = { pj, j=1,2,…,ilo-1 dj, j=ilo,ilo+1,…,ihi and pj, j=ihi+1,ihi+2,…,n.$
The order in which the interchanges are made is n$n$ to ihi + 1${i}_{\mathrm{hi}}+1$ then 1$1$ to ilo1${i}_{\mathrm{lo}}-1$.
5:     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

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: job, 2: n, 3: a, 4: lda, 5: ilo, 6: ihi, 7: scale, 8: 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.

Accuracy

The errors are negligible, compared with those in subsequent computations.

If the matrix A$A$ is balanced by nag_lapack_zgebal (f08nv), then any eigenvectors computed subsequently are eigenvectors of the matrix A${A}^{\prime \prime }$ (see Section [Description]) and hence nag_lapack_zgebak (f08nw) must then be called to transform them back to eigenvectors of A$A$.
If the Schur vectors of A$A$ are required, then this function must not be called with job = 'S'${\mathbf{job}}=\text{'S'}$ or 'B'$\text{'B'}$, because then the balancing transformation is not unitary. If this function is called with job = 'P'${\mathbf{job}}=\text{'P'}$, then any Schur vectors computed subsequently are Schur vectors of the matrix A${A}^{\prime \prime }$, and nag_lapack_zgebak (f08nw) must be called (with side = 'R'${\mathbf{side}}=\text{'R'}$) to transform them back to Schur vectors of A$A$.
The total number of real floating point operations is approximately proportional to n2${n}^{2}$.
The real analogue of this function is nag_lapack_dgebal (f08nh).

Example

```function nag_lapack_zgebal_example
job = 'Both';
a = [ 1.5 - 2.75i,  0 + 0i,  0 + 0i,  0 + 0i;
-8.06 - 1.24i,  -2.5 - 0.5i,  0 + 0i,  -0.75 + 0.5i;
-2.09 + 7.56i,  1.39 + 3.97i,  -1.25 + 0.75i,  -4.82 - 5.67i;
6.18 + 9.79i,  -0.92 - 0.62i,  0 + 0i,  -2.5 - 0.5i];
[aOut, ilo, ihi, scale, info] = nag_lapack_zgebal(job, a)
```
```

aOut =

-1.2500 + 0.7500i   1.3900 + 3.9700i  -4.8200 - 5.6700i  -2.0900 + 7.5600i
0.0000 + 0.0000i  -2.5000 - 0.5000i  -0.7500 + 0.5000i  -8.0600 - 1.2400i
0.0000 + 0.0000i  -0.9200 - 0.6200i  -2.5000 - 0.5000i   6.1800 + 9.7900i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   1.5000 - 2.7500i

ilo =

2

ihi =

3

scale =

3
1
1
1

info =

0

```
```function f08nv_example
job = 'Both';
a = [ 1.5 - 2.75i,  0 + 0i,  0 + 0i,  0 + 0i;
-8.06 - 1.24i,  -2.5 - 0.5i,  0 + 0i,  -0.75 + 0.5i;
-2.09 + 7.56i,  1.39 + 3.97i,  -1.25 + 0.75i,  -4.82 - 5.67i;
6.18 + 9.79i,  -0.92 - 0.62i,  0 + 0i,  -2.5 - 0.5i];
[aOut, ilo, ihi, scale, info] = f08nv(job, a)
```
```

aOut =

-1.2500 + 0.7500i   1.3900 + 3.9700i  -4.8200 - 5.6700i  -2.0900 + 7.5600i
0.0000 + 0.0000i  -2.5000 - 0.5000i  -0.7500 + 0.5000i  -8.0600 - 1.2400i
0.0000 + 0.0000i  -0.9200 - 0.6200i  -2.5000 - 0.5000i   6.1800 + 9.7900i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   1.5000 - 2.7500i

ilo =

2

ihi =

3

scale =

3
1
1
1

info =

0

```