nag_dpbtrf (f07hdc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dpbtrf (f07hdc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpbtrf (f07hdc) computes the Cholesky factorization of a real symmetric positive definite band matrix.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpbtrf (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, NagError *fail)

3  Description

nag_dpbtrf (f07hdc) forms the Cholesky factorization of a real symmetric positive definite band matrix A either as A=UTU if uplo=Nag_Upper or A=LLT if uplo=Nag_Lower, 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:     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 3.2.1.3 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: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo=Nag_Upper
The upper triangular part of A is stored and A is factorized as UTU, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     kdIntegerInput
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
5:     ab[dim]doubleInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the n by n symmetric band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
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.
6:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+1.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
NE_INTERNAL_ERROR
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.
NE_POS_DEF
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 function specifically designed to factorize a band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_dgbtrf (f07bdc) or as a full matrix, by calling nag_dsytrf (f07mdc).

7  Accuracy

If uplo=Nag_Upper, the computed factor U is the exact factor of a perturbed matrix A+E, where
Eck+1εUTU ,
ck+1 is a modest linear function of k+1, and ε is the machine precision.
If uplo=Nag_Lower, a similar statement holds for the computed factor L. It follows that eijck+1εaiiajj.

8  Further Comments

The total number of floating point operations is approximately n k+1 2, assuming nk.
A call to nag_dpbtrf (f07hdc) may be followed by calls to the functions:
The complex analogue of this function is nag_zpbtrf (f07hrc).

9  Example

This example computes the Cholesky factorization of the matrix A, where
A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 .

9.1  Program Text

Program Text (f07hdce.c)

9.2  Program Data

Program Data (f07hdce.d)

9.3  Program Results

Program Results (f07hdce.r)


nag_dpbtrf (f07hdc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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