NAG FL Interface
f07hrf (zpbtrf)

Settings help

FL Name Style:

FL Specification Language:

1 Purpose

f07hrf computes the Cholesky factorization of a complex Hermitian positive definite band matrix.

2 Specification

Fortran Interface
Subroutine f07hrf ( uplo, n, kd, ab, ldab, info)
Integer, Intent (In) :: n, kd, ldab
Integer, Intent (Out) :: info
Complex (Kind=nag_wp), Intent (Inout) :: ab(ldab,*)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07hrf_ (const char *uplo, const Integer *n, const Integer *kd, Complex ab[], const Integer *ldab, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07hrf, nagf_lapacklin_zpbtrf or its LAPACK name zpbtrf.

3 Description

f07hrf forms the Cholesky factorization of a complex Hermitian positive definite band matrix A either as A=UHU if uplo='U' or A=LLH if uplo='L', where U (or L) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as A.

4 References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: uplo Character(1) Input
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
The upper triangular part of A is stored and A is factorized as UHU, where U is upper triangular.
The lower triangular part of A is stored and A is factorized as LLH, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: kd Integer Input
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
4: ab(ldab,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the n×n Hermitian positive definite band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in ab(kd+1+i-j,j)​ for ​max(1,j-kd)ij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab(1+i-j,j)​ for ​jimin(n,j+kd).
On exit: the upper or lower triangle of A is overwritten by the Cholesky factor U or L as specified by uplo, using the same storage format as described above.
5: ldab Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hrf is called.
Constraint: ldabkd+1.
6: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value.
If info=−999, dynamic memory allocation failed. See Section 9 in the Introduction to the NAG Library FL Interface for further information. An explanatory message is output, and execution of the program is terminated.
The leading minor of order value is not positive definite and the factorization could not be completed. Hence A itself is not positive definite. This may indicate an error in forming the matrix A. There is no routine specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling f07brf or as a full Hermitian matrix, by calling f07mrf.

7 Accuracy

If uplo='U', the computed factor U is the exact factor of a perturbed matrix A+E, where
|E|c(k+1)ε|UH||U| ,  
c(k+1) is a modest linear function of k+1, and ε is the machine precision.
If uplo='L', a similar statement holds for the computed factor L. It follows that |eij|c(k+1)εaiiajj.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07hrf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of real floating-point operations is approximately 4n (k+1) 2, assuming nk.
A call to f07hrf may be followed by calls to the routines:
The real analogue of this routine is f07hdf.

10 Example

This example computes the Cholesky factorization of the matrix A, where
A= ( 9.39+0.00i 1.08-1.73i 0.00+0.00i 0.00+0.00i 1.08+1.73i 1.69+0.00i -0.04+0.29i 0.00+0.00i 0.00+0.00i -0.04-0.29i 2.65+0.00i -0.33+2.24i 0.00+0.00i 0.00+0.00i -0.33-2.24i 2.17+0.00i ) .  

10.1 Program Text

Program Text (f07hrfe.f90)

10.2 Program Data

Program Data (f07hrfe.d)

10.3 Program Results

Program Results (f07hrfe.r)