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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dspsvx (f07pb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dspsvx (f07pb) uses the diagonal pivoting factorization
A=UDUT   or   A=LDLT  
to compute the solution to a real system of linear equations
AX=B ,  
where A is an n by n symmetric matrix stored in packed format and X and B are n by r matrices. Error bounds on the solution and a condition estimate are also provided.

Syntax

[afp, ipiv, x, rcond, ferr, berr, info] = f07pb(fact, uplo, ap, afp, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[afp, ipiv, x, rcond, ferr, berr, info] = nag_lapack_dspsvx(fact, uplo, ap, afp, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dspsvx (f07pb) performs the following steps:
1. If fact='N', the diagonal pivoting method is used 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.
2. If some dii=0, so that D is exactly singular, then the function returns with info=i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, infon+1 is returned as a warning, but the function still goes on to solve for X and compute error bounds as described below.
3. The system of equations is solved for X using the factored form of A.
4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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

Parameters

Compulsory Input Parameters

1:     fact – string (length ≥ 1)
Specifies whether or not the factorized form of the matrix A has been supplied.
fact='F'
afp and ipiv contain the factorized form of the matrix A. afp and ipiv will not be modified.
fact='N'
The matrix A will be copied to afp and factorized.
Constraint: fact='F' or 'N'.
2:     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'.
3:     ap: – double array
The dimension of the array ap must be at least max1,n×n+1/2
The n by n symmetric matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.
4:     afp: – double array
The dimension of the array afp must be at least max1,n×n+1/2
If fact='F', afp contains 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_dsptrf (f07pd), stored as a packed triangular matrix in the same storage format as A.
5:     ipivn int64int32nag_int array
If fact='F', ipiv contains details of the interchanges and the block structure of D, as determined by nag_lapack_dsptrf (f07pd).
  • 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.
6:     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 dimension of the array ipiv.
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:     afp: – double array
The dimension of the array afp will be max1,n×n+1/2
If fact='N', afp contains 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_dsptrf (f07pd), stored as a packed triangular matrix in the same storage format as A.
2:     ipivn int64int32nag_int array
If fact='N', ipiv contains details of the interchanges and the block structure of D, as determined by nag_lapack_dsptrf (f07pd), as described above.
3:     xldx: – double array
The first dimension of the array x will be max1,n.
The second dimension of the array x will be max1,nrhs_p.
If info=0 or n+1, the n by r solution matrix X.
4:     rcond – double scalar
The estimate of the reciprocal condition number of the matrix A. If rcond=0.0, the matrix may be exactly singular. This condition is indicated by info>0andinfon. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by infon+1.
5:     ferrnrhs_p – double array
If info=0 or n+1, an estimate of the forward error bound for each computed solution vector, such that x^j-xj/xjferrj where x^j is the jth column of the computed solution returned in the array x and xj is the corresponding column of the exact solution X. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
6:     berrnrhs_p – double array
If info=0 or n+1, an estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
7:     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>0andinfon
Element _ of the diagonal is exactly zero. The factorization has been completed, but the factor D is exactly singular, so the solution and error bounds could not be computed. rcond=0.0 is returned.
W  info=n+1
D is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

Accuracy

For each right-hand side vector b, the computed solution x^ is the exact solution of a perturbed system of equations A+Ex^=b, where
E1 = Oε A1 ,  
where ε is the machine precision. See Chapter 11 of Higham (2002) for further details.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x^ wc condA,x^,b  
where condA,x^,b = A-1 A x^ + b / x^ condA = A-1 A κ A. If x^  is the j th column of X , then wc  is returned in berrj  and a bound on x - x^ / x^  is returned in ferrj . See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The factorization of A  requires approximately 13 n3  floating-point operations.
For each right-hand side, computation of the backward error involves a minimum of 4n2  floating-point operations. Each step of iterative refinement involves an additional 6n2  operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form Ax=b ; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n2  operations.
The complex analogues of this function are nag_lapack_zhpsvx (f07pp) for Hermitian matrices, and nag_lapack_zspsvx (f07qp) 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 3.93 6.07 19.25 8.38 9.90 9.50 27.85 .  
Error estimates for the solutions, and an estimate of the reciprocal of the condition number of the matrix A  are also output.
function f07pb_example


fprintf('f07pb example results\n\n');

% Symmetric matrix, upper triangle stored in packed format
uplo = 'U';
ap = [-1.81;    
       2.06;    1.15;
       0.63;    1.87;   -0.21;
      -1.15;    4.2;     3.87;    2.07];
% RHS
b = [0.96,  3.93;
     6.07, 19.25;
     8.38,  9.90;
     9.50, 27.85];

% Factorize and solve
fact = 'Not Factorized';
ipiv = zeros(4,1,'int64');
afp  = ap;

[afp, ipiv, x, rcond, ferr, berr, info] = ...
  f07pb(...
        fact, uplo, ap, afp, ipiv, b);

disp('Solution');
disp(x);
fprintf('Condition number      = %7.3f\n',1/rcond);
fprintf('Forward  error bounds = %10.3e  %10.3e\n',ferr); 
fprintf('Backward error bounds = %10.3e  %10.3e\n',berr); 


f07pb example results

Solution
   -5.0000    2.0000
   -2.0000    3.0000
    1.0000    4.0000
    4.0000    1.0000

Condition number      =  75.687
Forward  error bounds =  2.424e-14   3.379e-14
Backward error bounds =  9.514e-17   8.933e-17

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