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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zhecon (f07mu)

Purpose

nag_lapack_zhecon (f07mu) estimates the condition number of a complex Hermitian indefinite matrix A$A$, where A$A$ has been factorized by nag_lapack_zhetrf (f07mr).

Syntax

[rcond, info] = f07mu(uplo, a, ipiv, anorm, 'n', n)
[rcond, info] = nag_lapack_zhecon(uplo, a, ipiv, anorm, 'n', n)

Description

nag_lapack_zhecon (f07mu) estimates the condition number (in the 1$1$-norm) of a complex Hermitian indefinite matrix A$A$:
 κ1(A) = ‖A‖1‖A − 1‖1 . $κ1(A)=‖A‖1‖A-1‖1 .$
Since A$A$ is Hermitian, 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 = PUDUHPT$A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = PLDLHPT$A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the factorization of A$A$, as returned by nag_lapack_zhetrf (f07mr).
3:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the interchanges and the block structure of D$D$, as returned by nag_lapack_zhetrf (f07mr).
4:     anorm – double scalar
The 1$1$-norm of the original matrix A$A$, which may be computed by calling nag_blas_zlanhe (f06uc) with its parameter norm = '1'${\mathbf{norm}}=\text{'1'}$. anorm must be computed either before calling nag_lapack_zhetrf (f07mr) 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 first dimension of the array a The second dimension of the arrays a, ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work

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: a, 4: lda, 5: ipiv, 6: anorm, 7: rcond, 8: work, 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_zhecon (f07mu) involves solving a number of systems of linear equations of the form Ax = b$Ax=b$; the number is usually 5$5$ and never more than 11$11$. Each solution involves approximately 8n2$8{n}^{2}$ real floating point operations but takes considerably longer than a call to nag_lapack_zhetrs (f07ms) 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_dsycon (f07mg).

Example

```function nag_lapack_zhecon_example
uplo = 'L';
a = [complex(-1.36),  0 + 0i,  0 + 0i,  0 + 0i;
1.58 - 0.9i,  -8.87 + 0i,  0 + 0i,  0 + 0i;
2.21 + 0.21i,  -1.84 + 0.03i,  -4.63 + 0i,  0 + 0i;
3.91 - 1.5i,  -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];
anorm = 14.6641984095488;
[a, ipiv, info] = nag_lapack_zhetrf(uplo, a);
[rcond, info] = nag_lapack_zhecon(uplo, a, ipiv, anorm)
```
```

rcond =

0.1497

info =

0

```
```function f07mu_example
uplo = 'L';
a = [complex(-1.36),  0 + 0i,  0 + 0i,  0 + 0i;
1.58 - 0.9i,  -8.87 + 0i,  0 + 0i,  0 + 0i;
2.21 + 0.21i,  -1.84 + 0.03i,  -4.63 + 0i,  0 + 0i;
3.91 - 1.5i,  -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];
anorm = 14.6641984095488;
[a, ipiv, info] = f07mr(uplo, a);
[rcond, info] = f07mu(uplo, a, ipiv, anorm)
```
```

rcond =

0.1497

info =

0

```