?pbtrf

Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band matrix.

Syntax

FORTRAN 77:

call spbtrf( uplo, n, kd, ab, ldab, info )

call dpbtrf( uplo, n, kd, ab, ldab, info )

call cpbtrf( uplo, n, kd, ab, ldab, info )

call zpbtrf( uplo, n, kd, ab, ldab, info )

FORTRAN 95:

call pbtrf( ab [, uplo] [,info] )

lapack_int LAPACKE_<?>pbtrf( int matrix_order, char uplo, lapack_int n, lapack_int kd, <datatype>* ab, lapack_int ldab );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.h

Description

The routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian positive-definite band matrix A:

`A = U^TU` for real data, `A = U^HU` for complex data	if `uplo='U'`
`A = LL^T` for real data, `A = LL^H` for complex data	if `uplo='L'`

where L is a lower triangular matrix and U is upper triangular.

Note

This routine supports the Progress Routine feature. See Progress Function for details.

Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

uplo	CHARACTER*1. Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of `A` is stored in the array `ab`, and how `A` is factored: If `uplo = 'U'`, the upper triangle of `A` is stored. If `uplo = 'L'`, the lower triangle of `A` is stored.
n	INTEGER. The order of matrix `A`; `n` ≥ 0.
kd	INTEGER. The number of superdiagonals or subdiagonals in the matrix `A`; `kd` ≥ 0.
ab	REAL for spbtrf DOUBLE PRECISION for dpbtrf COMPLEX for cpbtrf DOUBLE COMPLEX for zpbtrf. Array, DIMENSION `(,*)`. The array `ab` contains either the upper or the lower triangular part of the matrix `A` (as specified by `uplo`) in band storage (see Matrix Storage Schemes). The second dimension of `ab` must be at least `max(1, n)`.
ldab	INTEGER. The leading dimension of the array `ab`. `(ldab ≥ kd + 1)`

Output Parameters

The upper or lower triangular part of A (in band storage) is overwritten by the Cholesky factor U or L, as specified by uplo.

info

INTEGER. If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the factorization could not be completed. This may indicate an error in forming the matrix A.

Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine pbtrf interface are as follows:

ab	Holds the array `A` of size `(kd+1,n)`.
uplo	Must be 'U' or 'L'. The default value is 'U'.

Application Notes

If uplo = 'U', the computed factor U is the exact factor of a perturbed matrix A + E, where

c(n) is a modest linear function of n, and ε is the machine precision.

A similar estimate holds for uplo = 'L'.

The total number of floating-point operations for real flavors is approximately n(kd+1)². The number of operations for complex flavors is 4 times greater. All these estimates assume that kd is much less than n.

After calling this routine, you can call the following routines:

?pbtrs	to solve `A*X = B`
?pbcon	to estimate the condition number of `A`.