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NAG Toolbox: nag_lapack_ddisna (f08fl)

Purpose

nag_lapack_ddisna (f08fl) computes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian m$m$ by m$m$ matrix A$A$, or for the left or right singular vectors of a general m$m$ by n$n$ matrix A$A$.

Syntax

[sep, info] = f08fl(job, m, n, d)
[sep, info] = nag_lapack_ddisna(job, m, n, d)

Description

The bound on the error, measured by the angle in radians, for the i$i$th computed vector is given by ε A2 / sepi $\epsilon {‖A‖}_{2}/{\mathrm{sep}}_{i}$, where ε$\epsilon$ is the machine precision and sepi ${\mathrm{sep}}_{i}$ is the reciprocal condition number for the vectors, returned in the array element sep(i) ${\mathbf{sep}}\left(i\right)$. sep(i) ${\mathbf{sep}}\left(i\right)$ is restricted to be at least ε A2 $\epsilon {‖A‖}_{2}$ in order to limit the size of the error bound.

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)
Specifies for which problem the reciprocal condition number should be computed.
job = 'E'${\mathbf{job}}=\text{'E'}$
The eigenvectors of a symmetric or Hermitian matrix.
job = 'L'${\mathbf{job}}=\text{'L'}$
The left singular vectors of a general matrix.
job = 'R'${\mathbf{job}}=\text{'R'}$
The right singular vectors of a general matrix.
Constraint: job = 'E'${\mathbf{job}}=\text{'E'}$, 'L'$\text{'L'}$ or 'R'$\text{'R'}$.
2:     m – int64int32nag_int scalar
m$m$, the number of rows of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
3:     n – int64int32nag_int scalar
n$n$, the number of columns of the matrix when job = 'L'${\mathbf{job}}=\text{'L'}$ or 'R'$\text{'R'}$.
If job = 'E'${\mathbf{job}}=\text{'E'}$, n is not referenced.
Constraint: if job = 'L'${\mathbf{job}}=\text{'L'}$ or 'R'$\text{'R'}$, n0${\mathbf{n}}\ge 0$.
4:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if job = 'E'${\mathbf{job}}=\text{'E'}$ and at least max (1,min (m,n))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ if job = 'L'${\mathbf{job}}=\text{'L'}$ or 'R'$\text{'R'}$.
The eigenvalues if job = 'E'${\mathbf{job}}=\text{'E'}$, or singular values if job = 'L'${\mathbf{job}}=\text{'L'}$ or 'R'$\text{'R'}$ of the matrix A$A$.
Constraints:
• the elements of the array d must be in either increasing or decreasing order;
• if job = 'L'${\mathbf{job}}=\text{'L'}$ or 'R'$\text{'R'}$ the elements of d must be non-negative.

None.

None.

Output Parameters

1:     sep( : $:$) – double array
Note: the dimension of the array sep must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if job = 'E'${\mathbf{job}}=\text{'E'}$ and at least max (1,min (m,n))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ if job = 'L'${\mathbf{job}}=\text{'L'}$ or 'R'$\text{'R'}$.
The reciprocal condition numbers of the vectors.
2:     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: m, 3: n, 4: d, 5: sep, 6: info.

Accuracy

The reciprocal condition numbers are computed to machine precision relative to the size of the eigenvalues, or singular values.

nag_lapack_ddisna (f08fl) may also be used towards computing error bounds for the eigenvectors of the generalized symmetric or Hermitian definite eigenproblem. See Golub and Van Loan (1996) for further details on the error bounds.

Example

```function nag_lapack_ddisna_example
job = 'Eigenvectors';
m = int64(4);
n = int64(4);
d = [-2.053115763536997;
-0.5146427793906143;
-0.2943264517738021;
12.86208499470141];
[sep, info] = nag_lapack_ddisna(job, m, n, d)
```
```

sep =

1.5385
0.2203
0.2203
13.1564

info =

0

```
```function f08fl_example
job = 'Eigenvectors';
m = int64(4);
n = int64(4);
d = [-2.053115763536997;
-0.5146427793906143;
-0.2943264517738021;
12.86208499470141];
[sep, info] = f08fl(job, m, n, d)
```
```

sep =

1.5385
0.2203
0.2203
13.1564

info =

0

```

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Chapter Introduction
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