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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_ztpcon (f07uu)

Purpose

nag_lapack_ztpcon (f07uu) estimates the condition number of a complex triangular matrix, using packed storage.

Syntax

[rcond, info] = f07uu(norm_p, uplo, diag, n, ap)
[rcond, info] = nag_lapack_ztpcon(norm_p, uplo, diag, n, ap)

Description

nag_lapack_ztpcon (f07uu) estimates the condition number of a complex triangular matrix A$A$, in either the 1$1$-norm or the $\infty$-norm, using packed storage:
 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:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
5:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The n$n$ by n$n$ triangular matrix A$A$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.
If diag = 'U'${\mathbf{diag}}=\text{'U'}$, the diagonal elements of A$A$ are assumed to be 1$1$, and are not referenced; the same storage scheme is used whether diag = 'N'${\mathbf{diag}}=\text{'N'}$ or ‘U’.

None.

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: ap, 6: rcond, 7: work, 8: rwork, 9: 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_ztpcon (f07uu) 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_ztptrs (f07us) 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_dtpcon (f07ug).

Example

```function nag_lapack_ztpcon_example
norm_p = '1';
uplo = 'L';
diag = 'N';
n = int64(4);
ap = [ 4.78 + 4.56i;
2 - 0.3i;
2.89 - 1.34i;
-1.89 + 1.15i;
-4.11 + 1.25i;
2.36 - 4.25i;
0.04 - 3.69i;
4.15 + 0.8i;
-0.02 + 0.46i;
0.33 - 0.26i];
[rcond, info] = nag_lapack_ztpcon(norm_p, uplo, diag, n, ap)
```
```

rcond =

0.0268

info =

0

```
```function f07uu_example
norm_p = '1';
uplo = 'L';
diag = 'N';
n = int64(4);
ap = [ 4.78 + 4.56i;
2 - 0.3i;
2.89 - 1.34i;
-1.89 + 1.15i;
-4.11 + 1.25i;
2.36 - 4.25i;
0.04 - 3.69i;
4.15 + 0.8i;
-0.02 + 0.46i;
0.33 - 0.26i];
[rcond, info] = f07uu(norm_p, uplo, diag, n, ap)
```
```

rcond =

0.0268

info =

0

```