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NAG Toolbox: nag_lapack_dtbtrs (f07ve)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dtbtrs (f07ve) solves a real triangular band system of linear equations with multiple right-hand sides, AX=B or ATX=B.

Syntax

[b, info] = f07ve(uplo, trans, diag, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dtbtrs(uplo, trans, diag, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dtbtrs (f07ve) solves a real triangular band system of linear equations AX=B or ATX=B.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
2:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans='N'
The equations are of the form AX=B.
trans='T' or 'C'
The equations are of the form ATX=B.
Constraint: trans='N', 'T' or 'C'.
3:     diag – string (length ≥ 1)
Indicates whether A is a nonunit or unit triangular matrix.
diag='N'
A is a nonunit triangular matrix.
diag='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
4:     kd int64int32nag_int scalar
kd, the number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'.
Constraint: kd0.
5:     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 triangular band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • 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+kd.
If diag='U', the diagonal elements of A are assumed to be 1, and are not referenced.
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 second dimension of the array ab.
n, 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.
Constraint: nrhs_p0.

Output Parameters

1:     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.
The n by r solution matrix X.
2:     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. A is singular and the solution has not been computed.

Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
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
EckεA ,  
ck is a modest linear function of k, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x ckcondA,xε ,   provided   ckcondA,xε<1 ,  
where condA,x=A-1Ax/x.
Note that condA,xcondA=A-1AκA; condA,x can be much smaller than condA and it is also possible for condAT to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling nag_lapack_dtbrfs (f07vh), and an estimate for κA can be obtained by calling nag_lapack_dtbcon (f07vg) with norm_p='I'.

Further Comments

The total number of floating-point operations is approximately 2nkr if kn.
The complex analogue of this function is nag_lapack_ztbtrs (f07vs).

Example

This example solves the system of equations AX=B, where
A= -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16   and   B= -16.64 -4.16 -13.78 -16.59 13.10 -4.94 -14.14 -9.96 .  
Here A is treated as a lower triangular band matrix with one subdiagonal.
function f07ve_example


fprintf('f07ve example results\n\n');

% Solve AX=B, where A is lower triangular banded
% and stored in triangular/symmetric banded format 
kd = int64(1);
ab = [-4.16, 4.78,  6.32, 0.16;
      -2.25, 5.86, -4.82, 0.00];
b = [-16.64,  -4.16;
     -13.78, -16.59;
      13.10,  -4.94;
     -14.14,  -9.96];

% Solve
uplo  = 'L';
trans = 'N';
diag  = 'N';
[x, info] = f07ve( ...
                   uplo, trans, diag, kd, ab, b);

% Display solution
[ifail] = x04ca( ...
                 'Gen', diag, x, 'Solution(s)');


f07ve example results

 Solution(s)
             1          2
 1      4.0000     1.0000
 2     -1.0000    -3.0000
 3      3.0000     2.0000
 4      2.0000    -2.0000

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