naginterfaces.library.lapacklin.dgbcon

naginterfaces.library.lapacklin.dgbcon(norm, kl, ku, ab, ipiv, anorm)

dgbcon estimates the condition number of a real band matrix $A$, where $A$ has been factorized by dgbtrf().

For full information please refer to the NAG Library document for f07bg

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f07/f07bgf.html

Parameters

normstr, length 1

Indicates whether ${\kappa }_1\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.

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

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

$norm=\text{'I'}$

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

klint

$k_l$, the number of subdiagonals within the band of the matrix $A$.

kuint

$k_u$, the number of superdiagonals within the band of the matrix $A$.

abfloat, array-like, shape $(2\times kl+ku+1,n)$

The $LU$ factorization of $A$, as returned by dgbtrf().

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

The pivot indices, as returned by dgbtrf().

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.dlangb with the same value for the argument $norm$.

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

Returns

rcondfloat

An estimate of the reciprocal of the condition number of $A$. $rcond$ is set to zero if exact singularity is detected or the estimate underflows. If $rcond$ is less than machine precision, $A$ is singular to working precision.

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 $-3$)

On entry, error in parameter $kl$.

Constraint: $kl\ge 0$.

(errno $-4$)

On entry, error in parameter $ku$.

Constraint: $ku\ge 0$.

(errno $-8$)

On entry, error in parameter $anorm$.

Constraint: $anorm\ge 0.0$.

Notes

dgbcon estimates the condition number of a real band matrix $A$, in either the $1$-norm or the $\infty$-norm:

$${\kappa }_1\left(A\right)={\parallel A\parallel }_1{\Vert A^{-1}\Vert }_1\text{or}{\kappa }_{\infty }\left(A\right)={\parallel A\parallel }_{\infty }{\Vert A^{-1}\Vert }_{\infty }\text{.}$$

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

Because the condition number is infinite if $A$ is singular, the function actually returns an estimate of the reciprocal of the condition number.

The function should be preceded by a call to blas.dlangb to compute ${\parallel A\parallel }_1$ or ${\parallel A\parallel }_{\infty }$, and a call to dgbtrf() to compute the $LU$ factorization of $A$. The function then uses Higham’s implementation of Hager’s method (see Higham (1988)) to estimate ${\Vert A^{-1}\Vert }_1$ or ${\Vert A^{-1}\Vert }_{\infty }$.

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