nag_zpbstf (f08utc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zpbstf (f08utc)

+ Contents

    1  Purpose
    7  Accuracy
    9  Example

1  Purpose

nag_zpbstf (f08utc) computes a split Cholesky factorization of a complex Hermitian positive definite band matrix.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zpbstf (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kb, Complex bb[], Integer pdbb, NagError *fail)

3  Description

nag_zpbstf (f08utc) computes a split Cholesky factorization of a complex Hermitian positive definite band matrix B. It is designed to be used in conjunction with nag_zhbgst (f08usc).
The factorization has the form B=SHS, where S is a band matrix of the same bandwidth as B and the following structure: S is upper triangular in the first n+k/2 rows, and transposed — hence, lower triangular — in the remaining rows. For example, if n=9 and k=2, then
S = s11 s12 s13 s22 s23 s24 s33 s34 s35 s44 s45 s55 s64 s65 s66 s75 s76 s77 s86 s87 s88 s97 s98 s99 .

4  References


5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of B is stored.
The upper triangular part of B is stored.
The lower triangular part of B is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix B.
Constraint: n0.
4:     kbIntegerInput
On entry: if uplo=Nag_Upper, the number of superdiagonals, kb, of the matrix B.
If uplo=Nag_Lower, the number of subdiagonals, kb, of the matrix B.
Constraint: kb0.
5:     bb[dim]ComplexInput/Output
Note: the dimension, dim, of the array bb must be at least max1,pdbb×n.
On entry: the n by n Hermitian positive definite band matrix B.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Bij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Bij is stored in bb[kb+i-j+j-1×pdbb], for j=1,,n and i=max1,j-kb,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Bij is stored in bb[i-j+j-1×pdbb], for j=1,,n and i=j,,minn,j+kb;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Bij is stored in bb[j-i+i-1×pdbb], for i=1,,n and j=i,,minn,i+kb;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Bij is stored in bb[kb+j-i+i-1×pdbb], for i=1,,n and j=max1,i-kb,,i.
On exit: B is overwritten by the elements of its split Cholesky factor S.
6:     pdbbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix B in the array bb.
Constraint: pdbbkb+1.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
On entry, kb=value.
Constraint: kb0.
On entry, n=value.
Constraint: n0.
On entry, pdbb=value.
Constraint: pdbb>0.
On entry, pdbb=value and kb=value.
Constraint: pdbbkb+1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
The factorization could not be completed, because the updated element bvalue,value would be the square root of a negative number. Hence B is not positive definite. This may indicate an error in forming the matrix B.

7  Accuracy

The computed factor S is the exact factor of a perturbed matrix B+E, where
ck+1 is a modest linear function of k+1, and ε is the machine precision. It follows that eijck+1εbiibjj.

8  Further Comments

The total number of floating point operations is approximately 4n k+1 2, assuming nk.
A call to nag_zpbstf (f08utc) may be followed by a call to nag_zhbgst (f08usc) to solve the generalized eigenproblem Az=λBz, where A and B are banded and B is positive definite.
The real analogue of this function is nag_dpbstf (f08ufc).

9  Example

See Section 9 in nag_zhbgst (f08usc).

nag_zpbstf (f08utc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012