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NAG Toolbox: nag_linsys_real_posdef_band_solve (f04bf)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_linsys_real_posdef_band_solve (f04bf) computes the solution to a real system of linear equations AX=B, where A is an n by n symmetric positive definite band matrix of band width 2k+1, 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

[ab, b, rcond, errbnd, ifail] = f04bf(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, b, rcond, errbnd, ifail] = nag_linsys_real_posdef_band_solve(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

The Cholesky factorization is used to factor A as A=UTU, if uplo='U', or A=LLT, if uplo='L', where U is an upper triangular band matrix with k superdiagonals, and L is a lower triangular band matrix with k subdiagonals. 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:     kd int64int32nag_int scalar
The number of superdiagonals k (and the number of subdiagonals) of the band matrix A.
Constraint: kd0.
3:     abldab: – double array
The first dimension of the array ab must be at least kd+1.
The second dimension of the array ab must be at least max1,n.
The n by n symmetric band matrix A. The upper or lower triangular part of the symmetric matrix is stored in the first kd+1 rows of the array. The jth column of A is stored in the jth column of the array ab as follows:
The matrix is stored in rows 1 to k+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abk+1+i-jj​ for ​max1,j-kij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+k.
See Further Comments below for further details.
4:     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:     abldab: – double array
The first dimension of the array ab will be kd+1.
The second dimension of the array ab will be max1,n.
If ifail=0 or n+1, the factor U or L from the Cholesky factorization A=UTU or A=LLT, in the same storage format as A.
2:     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.
3:     rcond – double scalar
If ifail=0 or n+1, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=1/A1A-11.
4:     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.
5:     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
The principal minor of order _ of the matrix A is not positive definite. The factorization has not been completed and 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: kd0.
   ifail=-4
Constraint: nrhs_p0.
   ifail=-6
Constraint: ldabkd+1.
   ifail=-8
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 3×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-11A1, the condition number of A with respect to the solution of the linear equations. nag_linsys_real_posdef_band_solve (f04bf) uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The band storage scheme for the array ab is illustrated by the following example, when n=6, k=2, and uplo='U':
On entry:
* * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66  
On exit:
* * u13 u24 u35 u46 * u12 u23 u34 u45 u56 u11 u22 u33 u44 u55 u66  
Similarly, if uplo='L' the format of ab is as follows:
On entry:
a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * a31 a42 a53 a64 * *  
On exit:
l11 l22 l33 l44 l55 l66 l21 l32 l43 l54 l65 * l31 l42 l53 l64 * *  
Array elements marked * need not be set and are not referenced by the function.
Assuming that nk, the total number of floating-point operations required to solve the equations AX=B is approximately nk+12 for the factorization and 4nkr for the solution following the factorization. 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 analogue of nag_linsys_real_posdef_band_solve (f04bf) is nag_linsys_complex_posdef_band_solve (f04cf).

Example

This example solves the equations
AX=B,  
where A is the symmetric positive definite band matrix
A= 5.49 2.68 0 0 2.68 5.63 -2.39 0 0 -2.39 2.60 -2.22 0 0 -2.22 5.17   and   B= 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .  
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.
function f04bf_example


fprintf('f04bf example results\n\n');

% Solve Ax = b for symmetric banded A with error bound and condition number
uplo = 'U';
kd = int64(1);
ab = [ 0.00,  2.68, -2.39, -2.22;
       5.49,  5.63,  2.60,  5.17];
b = [ 22.09,  5.10;
       9.31, 30.81;
      -5.24,-25.82;
      11.83, 22.90];

[ab, x, rcond, errbnd, ifail] = ...
  f04bf(uplo, kd, ab, 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);


f04bf example results

Solution
    5.0000   -2.0000
   -2.0000    6.0000
   -3.0000   -1.0000
    1.0000    4.0000

Estimate of condition number
      74.2

Estimate of error bound for computed solutions
   8.2e-15


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