nag_zsysv (f07nnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zsysv (f07nnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zsysv (f07nnc) computes the solution to a complex system of linear equations
AX=B ,
where A is an n by n symmetric matrix and X and B are n by r matrices.

2  Specification

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

3  Description

nag_zsysv (f07nnc) uses the diagonal pivoting method to factor A as
order uplo A
Nag_ColMajor Nag_Upper U D UT  
Nag_ColMajor Nag_Lower L D LT  
Nag_RowMajor Nag_Upper UT D U  
Nag_RowMajor Nag_Lower LT D L  
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric 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:     a[dim]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n symmetric matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if fail.code= NE_NOERROR, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUT, A=LDLT, A=UTDU or A=LTDL as computed by nag_zsytrf (f07nrc).
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax1,n.
7:     ipiv[dim]IntegerOutput
Note: the dimension, dim, of the array ipiv must be at least max1,n.
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.
8:     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.
9:     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.
10:   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, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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_zsysvx (f07npc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_complex_sym_lin_solve (f04dhc) solves Ax=b  and returns a forward error bound and condition estimate. nag_complex_sym_lin_solve (f04dhc) calls nag_zsysv (f07nnc) 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_dsysv (f07mac). The complex Hermitian analogue of this function is nag_zhesv (f07mnc).

9  Example

This example solves the equations
Ax=b ,
where A  is the complex symmetric matrix
A = -0.56+0.12i -1.54-2.86i 5.32-1.59i 3.80+0.92i -1.54-2.86i -2.83-0.03i -3.52+0.58i -7.86-2.96i 5.32-1.59i -3.52+0.58i 8.86+1.81i 5.14-0.64i 3.80+0.92i -7.86-2.96i 5.14-0.64i -0.39-0.71i
and
b = -6.43+19.24i -0.49-01.47i -48.18+66.00i -55.64+41.22i .
Details of the factorization of A  are also output.

9.1  Program Text

Program Text (f07nnce.c)

9.2  Program Data

Program Data (f07nnce.d)

9.3  Program Results

Program Results (f07nnce.r)


nag_zsysv (f07nnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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