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NAG Toolbox: nag_lapack_dgttrs (f07ce)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgttrs (f07ce) computes the solution to a real system of linear equations AX=B  or ATX=B , where A  is an n  by n  tridiagonal matrix and X  and B  are n  by r  matrices, using the LU  factorization returned by nag_lapack_dgttrf (f07cd).

Syntax

[b, info] = f07ce(trans, dl, d, du, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dgttrs(trans, dl, d, du, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgttrs (f07ce) should be preceded by a call to nag_lapack_dgttrf (f07cd), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A  as
A=PLU ,  
where P  is a permutation matrix, L  is unit lower triangular with at most one nonzero subdiagonal element in each column, and U  is an upper triangular band matrix, with two superdiagonals. nag_lapack_dgttrs (f07ce) then utilizes the factorization to solve the required equations.

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:     trans – string (length ≥ 1)
Specifies the equations to be solved as follows:
trans='N'
Solve AX=B for X.
trans='T' or 'C'
Solve ATX=B for X.
Constraint: trans='N', 'T' or 'C'.
2:     dl: – double array
The dimension of the array dl must be at least max1,n-1
Must contain the n-1 multipliers that define the matrix L of the LU factorization of A.
3:     d: – double array
The dimension of the array d must be at least max1,n
Must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
4:     du: – double array
The dimension of the array du must be at least max1,n-1
Must contain the n-1 elements of the first superdiagonal of U.
5:     du2: – double array
The dimension of the array du2 must be at least max1,n-2
Must contain the n-2 elements of the second superdiagonal of U.
6:     ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
Must contain the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row ipivi, and ipivi must always be either i or i+1, ipivi=i indicating that a row interchange was not performed.
7:     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 dimension of the arrays d, ipiv.
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, i.e., the number of columns of the matrix B.
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

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
where
E1 =OεA1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x 1 x 1 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.
Following the use of this function nag_lapack_dgtcon (f07cg) can be used to estimate the condition number of A  and nag_lapack_dgtrfs (f07ch) can be used to obtain approximate error bounds.

Further Comments

The total number of floating-point operations required to solve the equations AX=B  or ATX=B  is proportional to nr .
The complex analogue of this function is nag_lapack_zgttrs (f07cs).

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 6.6 -0.5 10.8 2.6 -3.2 0.6 -11.2 2.7 19.1 .  
function f07ce_example


fprintf('f07ce 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        ];
n  = numel(d);

% Factorize A.
[dlf, df, duf, du2f, ipiv, info] = ...
  f07cd(dl, d, du);

% Rhs B
b = [ 2.7,   6.6;
     -0.5,  10.8;
      2.6,  -3.2;
      0.6, -11.2;
      2.7,  19.1];

% Solve AX = B
trans = 'No transpose';
[x, info] = f07ce( ...
                   trans, dlf, df, duf, du2f, ipiv, b);

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


f07ce example results

Solution(s)
   -4.0000    5.0000
    7.0000   -4.0000
    3.0000   -3.0000
   -4.0000   -2.0000
   -3.0000    1.0000


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