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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgtrfs (f07cv)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgtrfs (f07cv) computes error bounds and refines the solution to a complex system of linear equations AX=B  or ATX=B  or AHX=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_zgttrf (f07cr) and an initial solution returned by nag_lapack_zgttrs (f07cs). Iterative refinement is used to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07cv(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zgtrfs(trans, dl, d, du, dlf, df, duf, du2, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgtrfs (f07cv) should normally be preceded by calls to nag_lapack_zgttrf (f07cr) and nag_lapack_zgttrs (f07cs). nag_lapack_zgttrf (f07cr) 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_zgttrs (f07cs) then utilizes the factorization to compute a solution, X^ , to the required equations. Letting x^  denote a column of X^ , nag_lapack_zgtrfs (f07cv) 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'
Solve ATX=B for X.
trans='C'
Solve AHX=B for X.
Constraint: trans='N', 'T' or 'C'.
2:     dl: – complex 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: – complex 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: – complex 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: – complex 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: – complex 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: – complex 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: – complex 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: – complex 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: – complex 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: – complex 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_zgtcon (f07cu) 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  or AHX=B  is proportional to nr . At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The real analogue of this function is nag_lapack_dgtrfs (f07ch).

Example

This example solves the equations
AX=B ,  
where A  is the tridiagonal matrix
A = -1.3+1.3i 2.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i -1.3+1.3i 2.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -1.3+3.3i -1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i -0.3+4.3i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -3.3+1.3i  
and
B = 2.4-05.0i 2.7+06.9i 3.4+18.2i -6.9-05.3i -14.7+09.7i -6.0-00.6i 31.9-07.7i -3.9+09.3i -1.0+01.6i -3.0+12.2i .  
Estimates for the backward errors and forward errors are also output.
function f07cv_example


fprintf('f07cv example results\n\n');

% Tridiagonal matrix stored by diagonals
du = [              2   - 1i     2   + 1i    -1   + 1i     1   - 1i  ];
d  = [-1.3 + 1.3i  -1.3 + 1.3i  -1.3 + 3.3i  -0.3 + 4.3i  -3.3 + 1.3i];
dl = [ 1   - 2i     1   + 1i     2   - 3i     1   + 1i               ];

% Factorize
[dlf, df, duf, du2, ipiv, info] = ...
  f07cr(dl, d, du);

% RHS
b  = [ 2.4 -  5i,    2.7 +  6.9i;
       3.4 + 18.2i, -6.9 -  5.3i;
     -14.7 +  9.7i, -6   -  0.6i;
      31.9 -  7.7i, -3.9 +  9.3i;
      -1   +  1.6i, -3   + 12.2i];

% Solve for x
trans = 'No transpose';
[x, info] = f07cs( ...
                   trans, dlf, df, duf, du2, ipiv, b);

% Refine solution
[x, ferr, berr, info] = ...
  f07cv( ...
         trans, dl, d, du, dlf, df, duf, du2, 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); 


f07cv example results

Solution:
   1.0000 + 1.0000i   2.0000 - 1.0000i
   3.0000 - 1.0000i   1.0000 + 2.0000i
   4.0000 + 5.0000i  -1.0000 + 1.0000i
  -1.0000 - 2.0000i   2.0000 + 1.0000i
   1.0000 - 1.0000i   2.0000 - 2.0000i

Forward  error bounds =    5.5e-14     7.7e-14
Backward error bounds =    3.6e-17     1.0e-16

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Chapter Contents
Chapter Introduction
NAG Toolbox

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