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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgbcon (f07bu)

## Purpose

nag_lapack_zgbcon (f07bu) estimates the condition number of a complex band matrix A$A$, where A$A$ has been factorized by nag_lapack_zgbtrf (f07br).

## Syntax

[rcond, info] = f07bu(norm_p, kl, ku, ab, ipiv, anorm, 'n', n)
[rcond, info] = nag_lapack_zgbcon(norm_p, kl, ku, ab, ipiv, anorm, 'n', n)

## Description

nag_lapack_zgbcon (f07bu) estimates the condition number of a complex band matrix A$A$, in either the 1$1$-norm or the $\infty$-norm:
 κ1(A) = ‖A‖1‖A − 1‖1   or   κ∞(A) = ‖A‖∞‖A − 1‖∞ . $κ1(A)=‖A‖1‖A-1‖1 or κ∞(A)=‖A‖∞‖A-1‖∞ .$
Note that κ(A) = κ1(AH)${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{H}}\right)$.
Because the condition number is infinite if A$A$ is singular, the function actually returns an estimate of the reciprocal of the condition number.
The function should be preceded by a call to to compute A1${‖A‖}_{1}$ or A${‖A‖}_{\infty }$, and a call to nag_lapack_zgbtrf (f07br) to compute the LU$LU$ factorization of A$A$. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A11${‖{A}^{-1}‖}_{1}$ or A1${‖{A}^{-1}‖}_{\infty }$.

## References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

## Parameters

### Compulsory Input Parameters

1:     norm_p – string (length ≥ 1)
Indicates whether κ1(A)${\kappa }_{1}\left(A\right)$ or κ(A)${\kappa }_{\infty }\left(A\right)$ is estimated.
norm = '1'${\mathbf{norm}}=\text{'1'}$ or 'O'$\text{'O'}$
κ1(A)${\kappa }_{1}\left(A\right)$ is estimated.
norm = 'I'${\mathbf{norm}}=\text{'I'}$
κ(A)${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: norm = '1'${\mathbf{norm}}=\text{'1'}$, 'O'$\text{'O'}$ or 'I'$\text{'I'}$.
2:     kl – int64int32nag_int scalar
kl${k}_{l}$, the number of subdiagonals within the band of the matrix A$A$.
Constraint: kl0${\mathbf{kl}}\ge 0$.
3:     ku – int64int32nag_int scalar
ku${k}_{u}$, the number of superdiagonals within the band of the matrix A$A$.
Constraint: ku0${\mathbf{ku}}\ge 0$.
4:     ab(ldab, : $:$) – complex array
The first dimension of the array ab must be at least 2 × kl + ku + 1$2×{\mathbf{kl}}+{\mathbf{ku}}+1$
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 LU$LU$ factorization of A$A$, as returned by nag_lapack_zgbtrf (f07br).
5:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The pivot indices, as returned by nag_lapack_zgbtrf (f07br).
6:     anorm – double scalar
If norm = '1'${\mathbf{norm}}=\text{'1'}$ or 'O'$\text{'O'}$, the 1$1$-norm of the original matrix A$A$.
If norm = 'I'${\mathbf{norm}}=\text{'I'}$, the $\infty$-norm of the original matrix A$A$.
anorm may be computed by calling nag_blas_zlangb (f06ub) with the same value for the parameter norm_p.
anorm must be computed either before calling nag_lapack_zgbtrf (f07br) or else from a copy of the original matrix A$A$ (see Section [Example]).
Constraint: anorm0.0${\mathbf{anorm}}\ge 0.0$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the array ab.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

ldab work rwork

### Output Parameters

1:     rcond – double scalar
An estimate of the reciprocal of the condition number of A$A$. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, A$A$ is singular to working precision.
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: norm_p, 2: n, 3: kl, 4: ku, 5: ab, 6: ldab, 7: ipiv, 8: anorm, 9: rcond, 10: work, 11: rwork, 12: 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 computed estimate rcond is never less than the true value ρ$\rho$, and in practice is nearly always less than 10ρ$10\rho$, although examples can be constructed where rcond is much larger.

A call to nag_lapack_zgbcon (f07bu) involves solving a number of systems of linear equations of the form Ax = b$Ax=b$ or AHx = b${A}^{\mathrm{H}}x=b$; the number is usually 5$5$ and never more than 11$11$. Each solution involves approximately 8n(2kl + ku)$8n\left(2{k}_{l}+{k}_{u}\right)$ real floating point operations (assuming nkl$n\gg {k}_{l}$ and nku$n\gg {k}_{u}$) but takes considerably longer than a call to nag_lapack_zgbtrs (f07bs) with one right-hand side, because extra care is taken to avoid overflow when A$A$ is approximately singular.
The real analogue of this function is nag_lapack_dgbcon (f07bg).

## Example

```function nag_lapack_zgbcon_example
norm_p = '1';
m = int64(4);
kl = int64(1);
ku = int64(2);
ab = [complex(0),  0 + 0i,  0 + 0i,  0 + 0i;
0 + 0i,  0 + 0i,  0.97 - 2.84i,  0.59 - 0.48i;
0 + 0i,  -2.05 - 0.85i,  -3.99 + 4.01i,  3.33 - 1.04i;
-1.65 + 2.26i,  -1.48 - 1.75i,  -1.06 + 1.94i,  -0.46 - 1.72i;
0 + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0 + 0i];
anorm = 15.47935040206279;
[ab, ipiv, info] = nag_lapack_zgbtrf(m, kl, ku, ab);
[rcond, info] = nag_lapack_zgbcon(norm_p, kl, ku, ab, ipiv, anorm)
```
```

rcond =

0.0096

info =

0

```
```function f07bu_example
norm_p = '1';
m = int64(4);
kl = int64(1);
ku = int64(2);
ab = [complex(0),  0 + 0i,  0 + 0i,  0 + 0i;
0 + 0i,  0 + 0i,  0.97 - 2.84i,  0.59 - 0.48i;
0 + 0i,  -2.05 - 0.85i,  -3.99 + 4.01i,  3.33 - 1.04i;
-1.65 + 2.26i,  -1.48 - 1.75i,  -1.06 + 1.94i,  -0.46 - 1.72i;
0 + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0 + 0i];
anorm = 15.47935040206279;
[ab, ipiv, info] = f07br(m, kl, ku, ab);
[rcond, info] = f07bu(norm_p, kl, ku, ab, ipiv, anorm)
```
```

rcond =

0.0096

info =

0

```