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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpbcon (f07hg)

## Purpose

nag_lapack_dpbcon (f07hg) estimates the condition number of a real symmetric positive definite band matrix A$A$, where A$A$ has been factorized by nag_lapack_dpbtrf (f07hd).

## Syntax

[rcond, info] = f07hg(uplo, kd, ab, anorm, 'n', n)
[rcond, info] = nag_lapack_dpbcon(uplo, kd, ab, anorm, 'n', n)

## Description

nag_lapack_dpbcon (f07hg) estimates the condition number (in the 1$1$-norm) of a real symmetric positive definite band matrix A$A$:
 κ1(A) = ‖A‖1‖A − 1‖1 . $κ1(A)=‖A‖1‖A-1‖1 .$
Since A$A$ is symmetric, κ1(A) = κ(A) = AA1${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
Because κ1(A)${\kappa }_{1}\left(A\right)$ is infinite if A$A$ is singular, the function actually returns an estimate of the reciprocal of κ1(A)${\kappa }_{1}\left(A\right)$.

## 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:     uplo – string (length ≥ 1)
Specifies how A$A$ has been factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = UTU$A={U}^{\mathrm{T}}U$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = LLT$A=L{L}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix A$A$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
3:     ab(ldab, : $:$) – double array
The first dimension of the array ab must be at least kd + 1${\mathbf{kd}}+1$
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 Cholesky factor of A$A$, as returned by nag_lapack_dpbtrf (f07hd).
4:     anorm – double scalar
The 1$1$-norm of the original matrix A$A$, which may be computed by calling nag_blas_dlansb (f06re) with its parameter norm = '1'${\mathbf{norm}}=\text{'1'}$. anorm must be computed either before calling nag_lapack_dpbtrf (f07hd) or else from a copy of the original matrix A$A$.
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 iwork

### 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: uplo, 2: n, 3: kd, 4: ab, 5: ldab, 6: anorm, 7: rcond, 8: work, 9: iwork, 10: 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_dpbcon (f07hg) involves solving a number of systems of linear equations of the form Ax = b$Ax=b$; the number is usually 4$4$ or 5$5$ and never more than 11$11$. Each solution involves approximately 4nk$4nk$ floating point operations (assuming nk$n\gg k$) but takes considerably longer than a call to nag_lapack_dpbtrs (f07he) with one right-hand side, because extra care is taken to avoid overflow when A$A$ is approximately singular.
The complex analogue of this function is nag_lapack_zpbcon (f07hu).

## Example

```function nag_lapack_dpbcon_example
uplo = 'L';
kd = int64(1);
ab = [2.343074902771996, 2.078877201506509, 1.130612248337004, 1.146524711734229;
1.143796127400537, -1.149659055507477, -1.963537900164584, 5.495816452771857e+222];
anorm = 10.7;
[rcond, info] = nag_lapack_dpbcon(uplo, kd, ab, anorm)
```
```

rcond =

0.0135

info =

0

```
```function f07hg_example
uplo = 'L';
kd = int64(1);
ab = [2.343074902771996, 2.078877201506509, 1.130612248337004, 1.146524711734229;
1.143796127400537, -1.149659055507477, -1.963537900164584, 5.495816452771857e+222];
anorm = 10.7;
[rcond, info] = f07hg(uplo, kd, ab, anorm)
```
```

rcond =

0.0135

info =

0

```