nag_zsptrf (f07qrc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zsptrf (f07qrc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zsptrf (f07qrc) computes the Bunch–Kaufman factorization of a complex symmetric matrix, using packed storage.

2  Specification

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

3  Description

nag_zsptrf (f07qrc) factorizes a complex 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.

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]ComplexInput/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, 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 real floating point operations is approximately 43n3.
A call to nag_zsptrf (f07qrc) may be followed by calls to the functions:
The real analogue of this function is nag_dsptrf (f07pdc).

9  Example

This example computes the Bunch–Kaufman factorization of the matrix A, where
A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i ,
using packed storage.

9.1  Program Text

Program Text (f07qrce.c)

9.2  Program Data

Program Data (f07qrce.d)

9.3  Program Results

Program Results (f07qrce.r)


nag_zsptrf (f07qrc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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