nag_zppsv (f07gnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zppsv (f07gnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zppsv (f07gnc) computes the solution to a complex system of linear equations
AX=B ,
where A is an n by n Hermitian positive definite matrix stored in packed format and X and B are n by r matrices.

2  Specification

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

3  Description

nag_zppsv (f07gnc) uses the Cholesky decomposition to factor A as A=UHU if uplo=Nag_Upper or A=LLH if uplo=Nag_Lower, where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations AX=B.

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
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: if uplo=Nag_Upper, the upper triangle of A is stored.
If uplo=Nag_Lower, the lower triangle of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
4:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5:     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 Hermitian 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: if fail.code= NE_NOERROR, the factor U or L from the Cholesky factorization A=UHU or A=LLT, in the same storage format as A.
6:     b[dim]ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the n by r right-hand side matrix B.
On exit: if fail.code= NE_NOERROR, the n by r solution matrix X.
7:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
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_MAT_NOT_POS_DEF
The leading minor of order value of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

7  Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,
where
E1 = Oε A1
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,
where κA = A-11 A1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
nag_zppsvx (f07gpc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_herm_posdef_packed_lin_solve (f04cec) solves Ax=b  and returns a forward error bound and condition estimate. nag_herm_posdef_packed_lin_solve (f04cec) calls nag_zppsv (f07gnc) to solve the equations.

8  Further Comments

The total number of floating point operations is approximately 43 n3 + 8n2r , where r  is the number of right-hand sides.
The real analogue of this function is nag_dppsv (f07gac).

9  Example

This example solves the equations
Ax=b ,
where A  is the Hermitian positive definite matrix
A = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00
and
b = 3.93-06.14i 6.17+09.42i -7.17-21.83i 1.99-14.38i .
Details of the Cholesky factorization of A  are also output.

9.1  Program Text

Program Text (f07gnce.c)

9.2  Program Data

Program Data (f07gnce.d)

9.3  Program Results

Program Results (f07gnce.r)


nag_zppsv (f07gnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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