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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztrcon (f07tu)

## Purpose

nag_lapack_ztrcon (f07tu) estimates the condition number of a complex triangular matrix.

## Syntax

[rcond, info] = f07tu(norm_p, uplo, diag, a, 'n', n)
[rcond, info] = nag_lapack_ztrcon(norm_p, uplo, diag, a, 'n', n)

## Description

nag_lapack_ztrcon (f07tu) estimates the condition number of a complex triangular 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:     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$n$ triangular matrix A$A$.
• If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, a$a$ is upper triangular and the elements of the array below the diagonal are not referenced.
• If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, a$a$ is lower triangular and the elements of the array above the diagonal are not referenced.
• 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 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 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: uplo, 3: diag, 4: n, 5: a, 6: lda, 7: rcond, 8: work, 9: rwork, 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_ztrcon (f07tu) 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 4n2$4{n}^{2}$ real floating point operations but takes considerably longer than a call to nag_lapack_ztrtrs (f07ts) 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_dtrcon (f07tg).

## Example

```function nag_lapack_ztrcon_example
norm_p = '1';
uplo = 'L';
diag = 'N';
a = [ 4.78 + 4.56i,  0 + 0i,  0 + 0i,  0 + 0i;
2 - 0.3i,  -4.11 + 1.25i,  0 + 0i,  0 + 0i;
2.89 - 1.34i,  2.36 - 4.25i,  4.15 + 0.8i,  0 + 0i;
-1.89 + 1.15i,  0.04 - 3.69i,  -0.02 + 0.46i,  0.33 - 0.26i];
[rcond, info] = nag_lapack_ztrcon(norm_p, uplo, diag, a)
```
```

rcond =

0.0268

info =

0

```
```function f07tu_example
norm_p = '1';
uplo = 'L';
diag = 'N';
a = [ 4.78 + 4.56i,  0 + 0i,  0 + 0i,  0 + 0i;
2 - 0.3i,  -4.11 + 1.25i,  0 + 0i,  0 + 0i;
2.89 - 1.34i,  2.36 - 4.25i,  4.15 + 0.8i,  0 + 0i;
-1.89 + 1.15i,  0.04 - 3.69i,  -0.02 + 0.46i,  0.33 - 0.26i];
[rcond, info] = f07tu(norm_p, uplo, diag, a)
```
```

rcond =

0.0268

info =

0

```