nag_dsysv (f07mac) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dsysv (f07mac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dsysv (f07mac) computes the solution to a real 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_dsysv (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, double a[], Integer pda, Integer ipiv[], double b[], Integer pdb, NagError *fail)

3  Description

nag_dsysv (f07mac) 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.
Note that, in general, different permutations (pivot sequences) and diagonal block structures are obtained for uplo=Nag_Upper or Nag_Lower

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
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 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]doubleInput/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_dsytrf (f07mdc).
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]doubleInput/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.
  • 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

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
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.
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.
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.
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 ,
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_dsysvx (f07mbc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_real_sym_lin_solve (f04bhc) solves Ax=b  and returns a forward error bound and condition estimate. nag_real_sym_lin_solve (f04bhc) calls nag_dsysv (f07mac) to solve the equations.

8  Further Comments

The total number of floating point operations is approximately 13 n3 + 2n2r , where r  is the number of right-hand sides.
The complex analogues of nag_dsysv (f07mac) are nag_zhesv (f07mnc) for Hermitian matrices, and nag_zsysv (f07nnc) for symmetric matrices.

9  Example

This example solves the equations
Ax=b ,
where A  is the symmetric matrix
A = -1.81 2.06 0.63 -1.15 2.06 1.15 1.87 4.20 0.63 1.87 -0.21 3.87 -1.15 4.20 3.87 2.07   and   b = 0.96 6.07 8.38 9.50 .
Details of the factorization of A  are also output.

9.1  Program Text

Program Text (f07mace.c)

9.2  Program Data

Program Data (f07mace.d)

9.3  Program Results

Program Results (f07mace.r)

nag_dsysv (f07mac) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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