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NAG Toolbox: nag_lapack_dgbtrs (f07be)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgbtrs (f07be) solves a real band system of linear equations with multiple right-hand sides,
AX=B   or   ATX=B ,  
where A has been factorized by nag_lapack_dgbtrf (f07bd).

Syntax

[b, info] = f07be(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dgbtrs(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgbtrs (f07be) is used to solve a real band system of linear equations AX=B or ATX=B, the function must be preceded by a call to nag_lapack_dgbtrf (f07bd) which computes the LU factorization of A as A=PLU. The solution is computed by forward and backward substitution.
If trans='N', the solution is computed by solving PLY=B and then UX=Y.
If trans='T' or 'C', the solution is computed by solving UTY=B and then LTPTX=Y.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans='N'
AX=B is solved for X.
trans='T' or 'C'
ATX=B is solved for X.
Constraint: trans='N', 'T' or 'C'.
2:     kl int64int32nag_int scalar
kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
3:     ku int64int32nag_int scalar
ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
4:     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 LU factorization of A, as returned by nag_lapack_dgbtrf (f07bd).
5:     ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
The pivot indices, as returned by nag_lapack_dgbtrf (f07bd).
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

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

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
EckεPLU ,  
ck is a modest linear function of k=kl+ku+1, and ε is the machine precision. This assumes kn.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x ckcondA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA, and condAT can be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling nag_lapack_dgbrfs (f07bh), and an estimate for κA can be obtained by calling nag_lapack_dgbcon (f07bg) with norm_p='I'.

Further Comments

The total number of floating-point operations is approximately 2n2kl+kur, assuming nkl and nku.
This function may be followed by a call to nag_lapack_dgbrfs (f07bh) to refine the solution and return an error estimate.
The complex analogue of this function is nag_lapack_zgbtrs (f07bs).

Example

This example solves the system of equations AX=B, where
A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82   and   B= 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_dgbtrf (f07bd).
function f07be_example


fprintf('f07be example results\n\n');

m  = int64(4);
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;
      10.50, -25.82];
% Factorize A
[abf, ipiv, info] = f07bd( ...
                           m, kl, ku, ab);

% Compute Solution
trans = 'N';
[x, info] = f07be( ...
                      trans, kl, ku, abf, ipiv, b);

disp('Solution(s)');
disp(x);


f07be example results

Solution(s)
   -2.0000    1.0000
    3.0000   -4.0000
    1.0000    7.0000
   -4.0000   -2.0000


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