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NAG Toolbox: nag_linsys_real_symm_packed_solve (f04bj)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_linsys_real_symm_packed_solve (f04bj) computes 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. An estimate of the condition number of A and an error bound for the computed solution are also returned.

Syntax

[ap, ipiv, b, rcond, errbnd, ifail] = f04bj(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[ap, ipiv, b, rcond, errbnd, ifail] = nag_linsys_real_symm_packed_solve(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

Description

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. The factored form of A is then used to solve the system of equations AX=B.

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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo='U', the upper triangle of the matrix A is stored.
If uplo='L', the lower triangle of the matrix A is stored.
Constraint: uplo='U' or 'L'.
2:     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 column-wise in a linear array. The jth column of the matrix A is stored in the array ap as follows:
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.
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 matrix of right-hand sides B.

Optional Input Parameters

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

Output Parameters

1:     ap: – double array
The dimension of the array ap will be max1,n×n+1/2
If ifail0, 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 ifail0, details of the interchanges and the block structure of D, as determined by nag_lapack_dsptrf (f07pd).
  • If ipivk>0, then rows and columns k and ipivk were interchanged, and dkk is a 1 by 1 diagonal block;
  • if uplo='U' and ipivk=ipivk-1<0, then rows and columns k-1 and -ipivk were interchanged and dk-1:k,k-1:k is a 2 by 2 diagonal block;
  • if uplo='L' and ipivk=ipivk+1<0, then rows and columns k+1 and -ipivk were interchanged and dk:k+1,k:k+1 is a 2 by 2 diagonal block.
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 ifail=0 or n+1, the n by r solution matrix X.
4:     rcond – double scalar
If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=1/A1A-11.
5:     errbnd – double scalar
If ifail=0 or n+1, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and x is the corresponding column of the exact solution X. If rcond is less than machine precision, then errbnd is returned as unity.
6:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

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

   ifail>0andifailn
Diagonal block _ of the block diagonal matrix is zero. The factorization has been completed, but the solution could not be computed.
W  ifail=n+1
A solution has been computed, but rcond is less than machine precision so that the matrix A is numerically singular.
   ifail=-1
On entry, uplo not one of 'U' or 'u' or 'L' or 'l'.
   ifail=-2
Constraint: n0.
   ifail=-3
Constraint: nrhs_p0.
   ifail=-7
Constraint: ldbmax1,n.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.
The integer allocatable memory required is n, and the double allocatable memory required is 2×n. Allocation failed before the solution could 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. nag_linsys_real_symm_packed_solve (f04bj) uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The packed storage scheme is illustrated by the following example when n=4 and uplo='U'. Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14 a22 a23 a24 a33 a34 a44 aij = aji  
Packed storage of the upper triangle of A:
ap= a11, a12, a22, a13, a23, a33, a14, a24, a34, a44  
The total number of floating-point operations required to solve the equations AX=B is proportional to 13n3+2n2r. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The complex analogues of nag_linsys_real_symm_packed_solve (f04bj) are nag_linsys_complex_herm_packed_solve (f04cj) for complex Hermitian matrices, and nag_linsys_complex_symm_packed_solve (f04dj) for complex symmetric matrices.

Example

This example solves the equations
AX=B,  
where A is the symmetric indefinite 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 .  
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.
function f04bj_example


fprintf('f04bj example results\n\n');

% Solve Ax = b for symmetric packed A with error bound and condition number
uplo = 'U';
ap = [-1.81              ...
       2.06  1.15        ...
       0.63  1.87 -0.21  ...
      -1.15  4.20  3.87  2.07];
b = [0.96,  3.93;
     6.07, 19.25;
     8.38,  9.90;
     9.50, 27.85];

[ap, ipiv, x, rcond, errbnd, ifail] = ...
  f04bj(uplo, ap, b);

disp('Solution');
disp(x);
disp('Estimate of condition number');
fprintf('%10.1f\n\n',1/rcond);
disp('Estimate of error bound for computed solutions');
fprintf('%10.1e\n\n',errbnd);


f04bj example results

Solution
   -5.0000    2.0000
   -2.0000    3.0000
    1.0000    4.0000
    4.0000    1.0000

Estimate of condition number
      75.7

Estimate of error bound for computed solutions
   8.4e-15


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