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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgttrs (f07cs)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgttrs (f07cs) computes 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).

Syntax

[b, info] = f07cs(trans, dl, d, du, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zgttrs(trans, dl, d, du, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgttrs (f07cs) should be preceded by a call to nag_lapack_zgttrf (f07cr), 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_zgttrs (f07cs) 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
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)
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 multipliers that define the matrix L of the LU factorization of 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 upper triangular matrix U from the LU factorization of A.
4:     du: – complex 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: – 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.
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: – 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.

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: – complex 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_zgtcon (f07cu) can be used to estimate the condition number of A  and nag_lapack_zgtrfs (f07cv) 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  or AHX=B  is proportional to nr .
The real analogue of this function is nag_lapack_dgttrs (f07ce).

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 .  
function f07cs_example


fprintf('f07cs 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               ];

% Rhs B
b = [  2.4 -  5.0i   2.7 +  6.9i; 
       3.4 + 18.2i  -6.9 -  5.3i; 
     -14.7 +  9.7i  -6.0 -  0.6i; 
      31.9 -  7.7i  -3.9 +  9.3i; 
      -1   +  1.6i  -3.0 + 12.2i];

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

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

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


f07cs example results

Solution(s)
   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


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Chapter Contents
Chapter Introduction
NAG Toolbox

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