nag_zhpsv (f07pnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zhpsv (f07pnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zhpsv (f07pnc) computes the solution to a complex system of linear equations
AX=B ,
where A is an n by n Hermitian matrix stored in packed format and X and B are n by r matrices.

2  Specification

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

3  Description

nag_zhpsv (f07pnc) uses the diagonal pivoting method to factor A as A=UDUH if uplo=Nag_Upper or A=LDLH if uplo=Nag_Lower, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is Hermitian and block diagonal with 1 by 1 and 2 by 2 diagonal blocks. 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

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: the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUH or A=LDLH as computed by nag_zhptrf (f07prc), stored as a packed triangular matrix in the same storage format as A.
6:     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.
7:     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.
8:     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.
9:     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_SINGULAR
Dvalue,value is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be 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) and Chapter 11 of Higham (2002) for further details.
nag_zhpsvx (f07ppc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_herm_packed_lin_solve (f04cjc) solves AX=B  and returns a forward error bound and condition estimate. nag_herm_packed_lin_solve (f04cjc) calls nag_zhpsv (f07pnc) to solve the equations.

8  Parallelism and Performance

nag_zhpsv (f07pnc) is not threaded by NAG in any implementation.
nag_zhpsv (f07pnc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  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_dspsv (f07pac). The complex symmetric analogue of this function is nag_zspsv (f07qnc).

10  Example

This example solves the equations
Ax=b ,
where A  is the Hermitian matrix
A = -1.84i+0.00 0.11-0.11i -1.78-1.18i 3.91-1.50i 0.11+0.11i -4.63i+0.00 -1.84+0.03i 2.21+0.21i -1.78+1.18i -1.84-0.03i -8.87i+0.00 1.58-0.90i 3.91+1.50i 2.21-0.21i 1.58+0.90i -1.36i+0.00
and
b = 2.98-10.18i -9.58+03.88i -0.77-16.05i 7.79+05.48i .
Details of the factorization of A  are also output.

10.1  Program Text

Program Text (f07pnce.c)

10.2  Program Data

Program Data (f07pnce.d)

10.3  Program Results

Program Results (f07pnce.r)


nag_zhpsv (f07pnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014