nag_dsptrf (f07pdc) (PDF version)
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f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dsptrf (f07pdc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dsptrf (f07pdc) computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dsptrf (Nag_OrderType order, Nag_UploType uplo, Integer n, double ap[], Integer ipiv[], NagError *fail)

3  Description

nag_dsptrf (f07pdc) factorizes a real symmetric matrix A, using the Bunch–Kaufman diagonal pivoting method and packed storage. A is factorized as either A=PUDUTPT if uplo=Nag_Upper or A=PLDLTPT if uplo=Nag_Lower, where P is a permutation matrix, U (or L) is a unit upper (or lower) triangular matrix and D is a symmetric block diagonal matrix with 1 by 1 and 2 by 2 diagonal blocks; U (or L) has 2 by 2 unit diagonal blocks corresponding to the 2 by 2 blocks of D. Row and column interchanges are performed to ensure numerical stability while preserving symmetry.
This method is suitable for symmetric matrices which are not known to be positive definite. If A is in fact positive definite, no interchanges are performed and no 2 by 2 blocks occur in D.

4  References

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 PUDUTPT, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as PLDLTPT, 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:     ap[dim]doubleInput/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the n by n symmetric matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
On exit: A is overwritten by details of the block diagonal matrix D and the multipliers used to obtain the factor U or L as specified by uplo.
5:     ipiv[n]IntegerOutput
On exit: details of the interchanges and the block structure of D. More precisely,
  • if ipiv[i-1]=k>0, dii is a 1 by 1 pivot block and the ith row and column of A were interchanged with the kth row and column;
  • if uplo=Nag_Upper and ipiv[i-2]=ipiv[i-1]=-l<0, di-1,i-1d-i,i-1 d-i,i-1dii  is a 2 by 2 pivot block and the i-1th row and column of A were interchanged with the lth row and column;
  • if uplo=Nag_Lower and ipiv[i-1]=ipiv[i]=-m<0, diidi+1,idi+1,idi+1,i+1 is a 2 by 2 pivot block and the i+1th row and column of A were interchanged with the mth row and column.
6:     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, n=value.
Constraint: n0.
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_SINGULAR
Dvalue,value is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

7  Accuracy

If uplo=Nag_Upper, the computed factors U and D are the exact factors of a perturbed matrix A+E, where
EcnεPUDUTPT ,
cn is a modest linear function of n, and ε is the machine precision.
If uplo=Nag_Lower, a similar statement holds for the computed factors L and D.

8  Further Comments

The elements of D overwrite the corresponding elements of A; if D has 2 by 2 blocks, only the upper or lower triangle is stored, as specified by uplo.
The unit diagonal elements of U or L and the 2 by 2 unit diagonal blocks are not stored. The remaining elements of U or L overwrite elements in the corresponding columns of A, but additional row interchanges must be applied to recover U or L explicitly (this is seldom necessary). If ipiv[i-1]=i, for i=1,2,,n (as is the case when A is positive definite), then U or L are stored explicitly in packed form (except for their unit diagonal elements which are equal to 1).
The total number of floating point operations is approximately 13n3.
A call to nag_dsptrf (f07pdc) may be followed by calls to the functions:
The complex analogues of this function are nag_zhptrf (f07prc) for Hermitian matrices and nag_zsptrf (f07qrc) for symmetric matrices.

9  Example

This example computes the Bunch–Kaufman factorization of the matrix A, where
A= 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ,
using packed storage.

9.1  Program Text

Program Text (f07pdce.c)

9.2  Program Data

Program Data (f07pdce.d)

9.3  Program Results

Program Results (f07pdce.r)


nag_dsptrf (f07pdc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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