NAG Library Routine Document

f07hrf (zpbtrf)

1
Purpose

f07hrf (zpbtrf) 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 <nagmk26.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 its LAPACK name zpbtrf.

3
Description

f07hrf (zpbtrf) 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 http://www.netlib.org/lapack/lawnspdf/lawn14.pdf
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.
uplo='U'
The upper triangular part of A is stored and A is factorized as UHU, where U is upper triangular.
uplo='L'
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 – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     kd – IntegerInput
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
4:     abldab* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab must be at least max1,n.
On entry: the n by 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 abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,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 – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hrf (zpbtrf) is called.
Constraint: ldabkd+1.
6:     info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

6
Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value.
If info=-999, dynamic memory allocation failed. See Section 3.7 in How to Use the NAG Library and its Documentation for further information. An explanatory message is output, and execution of the program is terminated.
info>0
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 (zgbtrf) or as a full Hermitian matrix, by calling f07mrf (zhetrf).

7
Accuracy

If uplo='U', the computed factor U is the exact factor of a perturbed matrix A+E, where
Eck+1εUHU ,  
ck+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 eijck+1εaiiajj.

8
Parallelism and Performance

f07hrf (zpbtrf) 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 (zpbtrf) may be followed by calls to the routines:
The real analogue of this routine is f07hdf (dpbtrf).

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)