hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dsysv (f07ma)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dsysv (f07ma) 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.

Syntax

[a, ipiv, b, info] = f07ma(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, ipiv, b, info] = nag_lapack_dsysv(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dsysv (f07ma) uses the diagonal pivoting method to factor A as A=UDUT if uplo='U' or A=LDLT if uplo='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='U' or 'L'

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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo='U', the upper triangle of A is stored.
If uplo='L', the lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by n symmetric matrix A.
  • If uplo='U', the upper triangular part of a must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo='L', the lower triangular part of a must be stored and the elements of the array above the diagonal are not referenced.
3:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The n by r right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays a, b and the second dimension of the array a.
n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
If info=0, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUT or A=LDLT as computed by nag_lapack_dsytrf (f07md).
2:     ipiv: int64int32nag_int array
The dimension of the array ipiv will be max1,n
Details of the interchanges and the block structure of D. More precisely,
  • if ipivi=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='U' and ipivi-1=ipivi=-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='L' and ipivi=ipivi+1=-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.
3:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
If info=0, the n by r solution matrix X.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Element _ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

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_lapack_dsysvx (f07mb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_symm_solve (f04bh) solves Ax=b  and returns a forward error bound and condition estimate. nag_linsys_real_symm_solve (f04bh) calls nag_lapack_dsysv (f07ma) to solve the equations.

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_lapack_dsysv (f07ma) are nag_lapack_zhesv (f07mn) for Hermitian matrices, and nag_lapack_zsysv (f07nn) for symmetric matrices.

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.
function f07ma_example


fprintf('f07ma example results\n\n');

% Indefinite matrix A
uplo = 'Upper';
a = [-1.81, 2.06,  0.63, -1.15;
      0,    1.15,  1.87,  4.20;
      0,    0,    -0.21,  3.87;
      0,    0,     0,     2.07];

% RHS
b = [0.96;
     6.07;
     8.38;
     9.50];

% Solve
[af, ipiv, x, info] = f07ma( ...
                             uplo, a, b);

disp('Solution');
disp(x');

[ifail] = x04ca( ...
                 uplo, 'Non-unit', af, 'Details of factorization');

fprintf('\nPivot indices\n   ');
fprintf('%11d', ipiv);
fprintf('\n');


f07ma example results

Solution
   -5.0000   -2.0000    1.0000    4.0000

 Details of factorization
             1          2          3          4
 1      0.4074     0.3031    -0.5960     0.6537
 2                -2.5907     0.8115     0.2230
 3                            1.1500     4.2000
 4                                       2.0700

Pivot indices
             1          2         -2         -2

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015