naginterfaces.library.lapacklin.zgtcon

naginterfaces.library.lapacklin.zgtcon(norm, dl, d, du, du2, ipiv, anorm)

zgtcon estimates the reciprocal condition number of a complex $n\times n$ tridiagonal matrix $A$, using the $LU$ factorization returned by zgttrf().

For full information please refer to the NAG Library document for f07cu

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07cuf.html

Parameters

normstr, length 1

Specifies the norm to be used to estimate $\kappa \left(A\right)$.

$norm=\text{'1'}$ or $\text{'O'}$

Estimate ${\kappa }_1\left(A\right)$.

$norm=\text{'I'}$

Estimate ${\kappa }_{\infty }\left(A\right)$.

dlcomplex, array-like, shape $(n-1)$

Must contain the $(n-1)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.

dcomplex, array-like, shape $\left(n\right)$

Must contain the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

ducomplex, array-like, shape $(n-1)$

Must contain the $(n-1)$ elements of the first superdiagonal of $U$.

du2complex, array-like, shape $(n-2)$

Must contain the $(n-2)$ elements of the second superdiagonal of $U$.

ipivint, array-like, shape $\left(n\right)$

Must contain the $n$ pivot indices that define the permutation matrix $P$. At the $i$th step, row $i$ of the matrix was interchanged with row $ipiv[i-1]$, and $ipiv[i-1]$ must always be either $i$ or $(i+1)$, $ipiv[i-1]=i$ indicating that a row interchange was not performed.

anormfloat

If $norm=\text{'1'}$ or $\text{'O'}$, the $1$-norm of the original matrix $A$.

If $norm=\text{'I'}$, the $\infty$-norm of the original matrix $A$.

$anorm$ may be computed by calling blas.zlangt with the same value for the argument $norm$.

$anorm$ must be computed either before calling zgttrf() or else from a copy of the original matrix $A$.

Returns

rcondfloat

Contains an estimate of the reciprocal condition number.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $norm$.

Constraint: $norm=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$.

(errno $-2$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-8$)

On entry, error in parameter $anorm$.

Constraint: $anorm\ge 0.0$.

Notes

zgtcon should be preceded by a call to zgttrf(), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as

$$A=PLU\text{,}$$

where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals. zgtcon then utilizes the factorization to estimate either ${\Vert A^{-1}\Vert }_1$ or ${\Vert A^{-1}\Vert }_{\infty }$, from which the estimate of the reciprocal of the condition number of $A$, $1/\kappa \left(A\right)$ is computed as either

$$1/{\kappa }_1\left(A\right)=1/\left({\parallel A\parallel }_1{\Vert A^{-1}\Vert }_1\right)$$

or

$$1/{\kappa }_{\infty }\left(A\right)=1/\left({\parallel A\parallel }_{\infty }{\Vert A^{-1}\Vert }_{\infty }\right)\text{.}$$

$1/\kappa \left(A\right)$ is returned, rather than $\kappa \left(A\right)$, since when $A$ is singular $\kappa \left(A\right)$ is infinite.

Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_1\left(A^T\right)$.

References

Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia