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NAG Toolbox: nag_linsys_real_band_solve (f04bb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_linsys_real_band_solve (f04bb) computes the solution to a real system of linear equations AX=B, where A is an n by n band matrix, with kl subdiagonals and ku superdiagonals, 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, ipiv, b, rcond, errbnd, ifail] = f04bb(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, ipiv, b, rcond, errbnd, ifail] = nag_linsys_real_band_solve(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

The LU decomposition with partial pivoting and row interchanges is used to factor A as A=PLU, where P is a permutation matrix, L is the product of permutation matrices and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. 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:     kl int64int32nag_int scalar
The number of subdiagonals kl, within the band of A.
Constraint: kl0.
2:     ku int64int32nag_int scalar
The number of superdiagonals ku, within the band of A.
Constraint: ku0.
3:     abldab: – double array
The first dimension of the array ab must be at least 2×kl+ku+1.
The second dimension of the array ab must be at least max1,n.
The n by n matrix A.
The matrix is stored in rows kl+1 to 2kl+ku+1; the first kl rows need not be set, more precisely, the element Aij must be stored in
abkl+ku+1+i-jj=Aij  for ​max1,j-kuiminn,j+kl. 
See Further Comments 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 and the second dimension of the array ab.
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 2×kl+ku+1.
The second dimension of the array ab will be max1,n.
If ifail0, ab stores details of the factorization.
The upper triangular band matrix U, with kl+ku superdiagonals, is stored in rows 1 to kl+ku+1 of the array, and the multipliers used to form the matrix L are stored in rows kl+ku+2 to 2kl+ku+1.
2:     ipivn int64int32nag_int array
If ifail0, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row ipivi. ipivi=i indicates a row interchange was not required.
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.

W  ifail>0andifailn
Diagonal element _ of the upper triangular factor 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
Constraint: n0.
   ifail=-2
Constraint: kl0.
   ifail=-3
Constraint: ku0.
   ifail=-4
Constraint: nrhs_p0.
   ifail=-6
Constraint: ldab2×kl+ku+1.
   ifail=-9
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. In this case the factorization and the solution X have been computed, but rcond and errbnd have not been 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_band_solve (f04bb) 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, kl=1, and ku=2. Storage of the band matrix A in the array ab:
* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *  
Array elements marked * need not be set and are not referenced by the function. Array elements marked + need not be set, but are defined on exit from the function and contain the elements u14, u25 and u36.
The total number of floating-point operations required to solve the equations AX=B depends upon the pivoting required, but if nkl+ku then it is approximately bounded by O n kl kl + ku  for the factorization and O n 2 kl + ku r  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_band_solve (f04bb) is nag_linsys_complex_band_solve (f04cb).

Example

This example solves the equations
AX=B,  
where A is the band matrix
A= -0.23 2.54 -3.66 0 -6.98 2.46 -2.73 -2.13 0 2.56 2.46 4.07 0 0 -4.78 -3.82   and   B= 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 .  
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.
function f04bb_example


fprintf('f04bb example results\n\n');

% Solve Ax = b for banded A with error bound and condition number
kl = int64(1);
ku = int64(2);
ab = [ 0,      0,      0,      0;
       0,      0,     -3.66,  -2.13;
       0,      2.54,  -2.73,   4.07;
      -0.23,   2.46,   2.46,  -3.82;
      -6.98,   2.56,  -4.78,   0];
b = [  4.42, -36.01;
      27.13, -31.67;
      -6.14,  -1.16;
      10.5,  -25.82];

[ab, ipiv, x, rcond, errbnd, ifail] = ...
  f04bb(kl, ku, 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);


f04bb example results

Solution
   -2.0000    1.0000
    3.0000   -4.0000
    1.0000    7.0000
   -4.0000   -2.0000

Estimate of condition number
      56.4

Estimate of error bound for computed solutions
   6.3e-15


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