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NAG Toolbox: nag_lapack_dgtsv (f07ca)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgtsv (f07ca) computes the solution to a real system of linear equations
AX=B ,  
where A is an n by n tridiagonal matrix and X and B are n by r matrices.

Syntax

[dl, d, du, b, info] = f07ca(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)
[dl, d, du, b, info] = nag_lapack_dgtsv(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgtsv (f07ca) uses Gaussian elimination with partial pivoting and row interchanges to solve the equations AX=B . The matrix A  is factorized as A=PLU , where P  is a permutation matrix, L  is unit lower triangular with at most one nonzero subdiagonal element per column, and U  is an upper triangular band matrix, with two superdiagonals.
Note that the equations ATX=B may be solved by interchanging the order of the arguments du and dl.

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

Parameters

Compulsory Input Parameters

1:     dl: – double array
The dimension of the array dl must be at least max1,n-1
Must contain the n-1 subdiagonal elements of the matrix A.
2:     d: – double array
The dimension of the array d must be at least max1,n
Must contain the n diagonal elements of the matrix A.
3:     du: – double array
The dimension of the array du must be at least max1,n-1
Must contain the n-1 superdiagonal elements of the matrix A.
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.
Note: to solve the equations Ax=b, where b is a single right-hand side, b may be supplied as a one-dimensional array with length ldb=max1,n.
The n by r right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array b and the dimension of the array d.
n, the number of linear equations, i.e., 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, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     dl: – double array
The dimension of the array dl will be max1,n-1
If no constraints are violated, dl stores the (n-2) elements of the second superdiagonal of the upper triangular matrix U from the LU factorization of A, in dl1,dl2,,dln-2.
2:     d: – double array
The dimension of the array d will be max1,n
If no constraints are violated, d stores the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
3:     du: – double array
The dimension of the array du will be max1,n-1
If no constraints are violated, du stores the n-1 elements of the first superdiagonal of U.
4:     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.
Note: to solve the equations Ax=b, where b is a single right-hand side, b may be supplied as a one-dimensional array with length ldb=max1,n.
If info=0, the n by r solution matrix X.
5:     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, and the solution has not been computed. The factorization has not been completed unless n=_.

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^-x 1 x1 κA E1 A1 ,  
where κA = A-11 A1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Alternatives to nag_lapack_dgtsv (f07ca), which return condition and error estimates are nag_linsys_real_tridiag_solve (f04bc) and nag_lapack_dgtsvx (f07cb).

Further Comments

The total number of floating-point operations required to solve the equations AX=B  is proportional to nr .
The complex analogue of this function is nag_lapack_zgtsv (f07cn).

Example

This example solves the equations
Ax=b ,  
where A  is the tridiagonal matrix
A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1   and   b = 2.7 -0.5 2.6 0.6 2.7 .  
function f07ca_example


fprintf('f07ca example results\n\n');

% Tridiagonal matrix A stored as diagonals:
du = [        2.1    -1.0      1.9     8.0];
d  = [3.0     2.3    -5.0     -0.9     7.1];
dl = [3.4     3.6     7.0     -6.0        ];

% RHS B
b  = [2.7;   -0.5;    2.6;     0.6;    2.7];

% Solve Ax = B
[dlf, df, duf, x, info] = f07ca( ...
                                 dl, d, du, b);

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


f07ca example results

Solution
   -4.0000    7.0000    3.0000   -4.0000   -3.0000


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