naginterfaces.library.lapacklin.dpbcon

naginterfaces.library.lapacklin.dpbcon(uplo, kd, ab, anorm)

dpbcon estimates the condition number of a real symmetric positive definite band matrix $A$, where $A$ has been factorized by dpbtrf().

For full information please refer to the NAG Library document for f07hg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07hgf.html

Parameters

uplostr, length 1

Specifies how $A$ has been factorized.

$uplo=\text{'U'}$

$A=U^{T}U$, where $U$ is upper triangular.

$uplo=\text{'L'}$

$A=L{L}^T$, where $L$ is lower triangular.

kdint

$k_d$, the number of superdiagonals or subdiagonals of the matrix $A$.

abfloat, array-like, shape $(kd+1,n)$

The Cholesky factor of $A$, as returned by dpbtrf().

anormfloat

The $1$-norm of the original matrix $A$, which may be computed by calling blas.dlansb with its argument $\text{norm}=\text{'1'}$. $anorm$ must be computed either before calling dpbtrf() 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 $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kd$.

Constraint: $kd\ge 0$.

(errno $-6$)

On entry, error in parameter $anorm$.

Constraint: $anorm\ge 0.0$.

Notes

dpbcon estimates the condition number (in the $1$-norm) of a real symmetric positive definite band matrix $A$:

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

Since $A$ is symmetric, ${\kappa }_1\left(A\right)={\kappa }_{\infty }\left(A\right)={\parallel A\parallel }_{\infty }{\Vert A^{-1}\Vert }_{\infty }$.

Because ${\kappa }_1\left(A\right)$ is infinite if $A$ is singular, the function actually returns an estimate of the reciprocal of ${\kappa }_1\left(A\right)$.

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

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