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NAG Toolbox

NAG Toolbox: nag_lapack_dgtrfs (f07ch)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgtrfs (f07ch) computes error bounds and refines 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) and an initial solution returned by nag_lapack_dgttrs (f07ce). Iterative refinement is used to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07ch(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dgtrfs(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgtrfs (f07ch) should normally be preceded by calls to nag_lapack_dgttrf (f07cd) and nag_lapack_dgttrs (f07ce). nag_lapack_dgttrf (f07cd) 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 compute a solution, X^ , to the required equations. Letting x^  denote a column of X^ , nag_lapack_dgtrfs (f07ch) computes a component-wise backward error, β , the smallest relative perturbation in each element of A  and b  such that x^  is the exact solution of a perturbed system
A+E x^=b+f , with  eij β aij , and  fj β bj .  
The function also estimates a bound for the component-wise forward error in the computed solution defined by max xi - xi^ / max xi^ , where x  is the corresponding column of the exact solution, X .

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 subdiagonal elements of the matrix 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 matrix A.
4:     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.
5:     dlf: – double array
The dimension of the array dlf must be at least max1,n-1
Must contain the n-1 multipliers that define the matrix L of the LU factorization of A.
6:     df: – double array
The dimension of the array df must be at least max1,n
Must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
7:     duf: – double array
The dimension of the array duf must be at least max1,n-1
Must contain the n-1 elements of the first superdiagonal of U.
8:     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.
9:     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.
10:   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.
11:   xldx: – double array
The first dimension of the array x must be at least max1,n.
The second dimension of the array x must be at least max1,nrhs_p.
The n by r initial solution matrix X.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays b, x and the dimension of the arrays d, df, ipiv.
n, the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the arrays b, x.
r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     xldx: – double array
The first dimension of the array x will be max1,n.
The second dimension of the array x will be max1,nrhs_p.
The n by r refined solution matrix X.
2:     ferrnrhs_p – double array
Estimate of the forward error bound for each computed solution vector, such that x^j-xj/x^jferrj, where x^j is the jth column of the computed solution returned in the array x and xj is the corresponding column of the exact solution X. The estimate is almost always a slight overestimate of the true error.
3:     berrnrhs_p – double array
Estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
4:     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
E=OεA  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x x κA E A ,  
where κA=A-1 A , 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.
Function nag_lapack_dgtcon (f07cg) can be used to estimate the condition number of A .

Further Comments

The total number of floating-point operations required to solve the equations AX=B  or ATX=B  is proportional to nr . At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this function is nag_lapack_zgtrfs (f07cv).

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 .  
Estimates for the backward errors and forward errors are also output.
function f07ch_example


fprintf('f07ch 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 = [ 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);

% Refine solution
[x, ferr, berr, info] = ...
  f07ch( ...
         trans, dl, d, du, dlf, df, duf, du2f, ipiv, b, x);

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

fprintf('Forward  error bounds = %10.1e  %10.1e\n',ferr); 
fprintf('Backward error bounds = %10.1e  %10.1e\n',berr); 


f07ch example results

Solution:
   -4.0000    5.0000
    7.0000   -4.0000
    3.0000   -3.0000
   -4.0000   -2.0000
   -3.0000    1.0000

Forward  error bounds =    9.4e-15     1.4e-14
Backward error bounds =    7.2e-17     5.9e-17

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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