naginterfaces.library.lapacklin.zpbcon

naginterfaces.library.lapacklin.zpbcon(uplo, kd, ab, anorm)[source]

zpbcon estimates the condition number of a complex Hermitian positive definite band matrix , where has been factorized by zpbtrf().

For full information please refer to the NAG Library document for f07hu

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07huf.html

Parameters
uplostr, length 1

Specifies how has been factorized.

, where is upper triangular.

, where is lower triangular.

kdint

, the number of superdiagonals or subdiagonals of the matrix .

abcomplex, array-like, shape

The Cholesky factor of , as returned by zpbtrf().

anormfloat

The -norm of the original matrix , which may be computed by calling blas.zlanhb with its argument . must be computed either before calling zpbtrf() or else from a copy of the original matrix .

Returns
rcondfloat

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

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

zpbcon estimates the condition number (in the -norm) of a complex Hermitian positive definite band matrix :

Since is Hermitian, .

Because is infinite if is singular, the function actually returns an estimate of the reciprocal of .

The function should be preceded by a call to blas.zlanhb to compute and a call to zpbtrf() to compute the Cholesky factorization of . The function then uses Higham’s implementation of Hager’s method (see Higham (1988)) to estimate .

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