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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dtbcon (f07vg)

## Purpose

nag_lapack_dtbcon (f07vg) estimates the condition number of a real triangular band matrix.

## Syntax

[rcond, info] = f07vg(norm_p, uplo, diag, kd, ab, 'n', n)
[rcond, info] = nag_lapack_dtbcon(norm_p, uplo, diag, kd, ab, 'n', n)

## Description

nag_lapack_dtbcon (f07vg) estimates the condition number of a real triangular 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(AT)${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.
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 computes A1${‖A‖}_{1}$ or A${‖A‖}_{\infty }$ exactly, and 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:     uplo – string (length ≥ 1)
Specifies whether A$A$ is upper or lower triangular.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A$A$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A$A$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
3:     diag – string (length ≥ 1)
Indicates whether A$A$ is a nonunit or unit triangular matrix.
diag = 'N'${\mathbf{diag}}=\text{'N'}$
A$A$ is a nonunit triangular matrix.
diag = 'U'${\mathbf{diag}}=\text{'U'}$
A$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1$1$.
Constraint: diag = 'N'${\mathbf{diag}}=\text{'N'}$ or 'U'$\text{'U'}$.
4:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals of the matrix A$A$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
5:     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 n$n$ by n$n$ triangular band matrix A$A$.
The matrix is stored in rows 1$1$ to kd + 1${k}_{d}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ij${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(1 + ij,j)​ for ​jimin (n,j + kd).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If diag = 'U'${\mathbf{diag}}=\text{'U'}$, the diagonal elements of A$A$ are assumed to be 1$1$, and are not referenced.

### 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: norm_p, 2: uplo, 3: diag, 4: n, 5: kd, 6: ab, 7: ldab, 8: rcond, 9: work, 10: iwork, 11: 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_dtbcon (f07vg) involves solving a number of systems of linear equations of the form Ax = b$Ax=b$ or ATx = b${A}^{\mathrm{T}}x=b$; the number is usually 4$4$ or 5$5$ and never more than 11$11$. Each solution involves approximately 2nk$2nk$ floating point operations (assuming nk$n\gg k$) but takes considerably longer than a call to nag_lapack_dtbtrs (f07ve) 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_ztbcon (f07vu).

## Example

```function nag_lapack_dtbcon_example
norm_p = '1';
uplo = 'L';
diag = 'N';
kd = int64(1);
ab = [-4.16, 4.78, 6.32, 0.16;
-2.25, 5.86, -4.82, 0];
[rcond, info] = nag_lapack_dtbcon(norm_p, uplo, diag, kd, ab)
```
```

rcond =

0.0144

info =

0

```
```function f07vg_example
norm_p = '1';
uplo = 'L';
diag = 'N';
kd = int64(1);
ab = [-4.16, 4.78, 6.32, 0.16;
-2.25, 5.86, -4.82, 0];
[rcond, info] = f07vg(norm_p, uplo, diag, kd, ab)
```
```

rcond =

0.0144

info =

0

```