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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgtsvx (f07cp)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgtsvx (f07cp) uses the LU factorization to compute the solution to a complex system of linear equations
AX=B ,  ATX=B   or   AHX=B ,  
where A is a tridiagonal matrix of order n and X and B are n by r matrices. Error bounds on the solution and a condition estimate are also provided.

Syntax

[dlf, df, duf, du2, ipiv, x, rcond, ferr, berr, info] = f07cp(fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[dlf, df, duf, du2, ipiv, x, rcond, ferr, berr, info] = nag_lapack_zgtsvx(fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgtsvx (f07cp) performs the following steps:
1. If fact='N', the LU decomposition is used to factor the matrix A as A=LU, where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
2. If some uii=0, so that U is exactly singular, then the function returns with info=i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, infon+1 is returned as a warning, but the function still goes on to solve for X and compute error bounds as described below.
3. The system of equations is solved for X using the factored form of A.
4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     fact – string (length ≥ 1)
Specifies whether or not the factorized form of the matrix A has been supplied.
fact='F'
dlf, df, duf, du2 and ipiv contain the factorized form of the matrix A. dlf, df, duf, du2 and ipiv will not be modified.
fact='N'
The matrix A will be copied to dlf, df and duf and factorized.
Constraint: fact='F' or 'N'.
2:     trans – string (length ≥ 1)
Specifies the form of the system of equations.
trans='N'
AX=B (No transpose).
trans='T'
ATX=B (Transpose).
trans='C'
AHX=B (Conjugate transpose).
Constraint: trans='N', 'T' or 'C'.
3:     dl: – complex array
The dimension of the array dl must be at least max1,n-1
The n-1 subdiagonal elements of A.
4:     d: – complex array
The dimension of the array d must be at least max1,n
The n diagonal elements of A.
5:     du: – complex array
The dimension of the array du must be at least max1,n-1
The n-1 superdiagonal elements of A.
6:     dlf: – complex array
The dimension of the array dlf must be at least max1,n-1
If fact='F', dlf contains the n-1 multipliers that define the matrix L from the LU factorization of A.
7:     df: – complex array
The dimension of the array df must be at least max1,n
If fact='F', df contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
8:     duf: – complex array
The dimension of the array duf must be at least max1,n-1
If fact='F', duf contains the n-1 elements of the first superdiagonal of U.
9:     du2: – complex array
The dimension of the array du2 must be at least max1,n-2
If fact='F', du2 contains the (n-2) elements of the second superdiagonal of U.
10:   ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
If fact='F', ipiv contains the pivot indices from the LU factorization of A.
11:   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 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 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 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:     dlf: – complex array
The dimension of the array dlf will be max1,n-1
If fact='N', dlf contains the n-1 multipliers that define the matrix L from the LU factorization of A.
2:     df: – complex array
The dimension of the array df will be max1,n
If fact='N', df contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
3:     duf: – complex array
The dimension of the array duf will be max1,n-1
If fact='N', duf contains the n-1 elements of the first superdiagonal of U.
4:     du2: – complex array
The dimension of the array du2 will be max1,n-2
If fact='N', du2 contains the (n-2) elements of the second superdiagonal of U.
5:     ipiv: int64int32nag_int array
The dimension of the array ipiv will be max1,n
If fact='N', ipiv contains the pivot indices from the LU factorization of A; row i of the matrix was interchanged with row ipivi. ipivi will always be either i or i+1; ipivi=i indicates a row interchange was not required.
6:     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.
If info=0 or n+1, the n by r solution matrix X.
7:     rcond – double scalar
The estimate of the reciprocal condition number of the matrix A. If rcond=0.0, the matrix may be exactly singular. This condition is indicated by info>0andinfon. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by infon+1.
8:     ferrnrhs_p – double array
If info=0 or n+1, an estimate of the forward error bound for each computed solution vector, such that x^j-xj/xjferrj 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 as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
9:     berrnrhs_p – double array
If info=0 or n+1, an 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).
10:   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>0andinfo<n
Element _ of the diagonal is exactly zero. The factorization has not been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. rcond=0.0 is returned.
W  info>0andinfo=n
Element _ of the diagonal is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. rcond=0.0 is returned.
W  info=n+1
U is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

Accuracy

For each right-hand side vector b, the computed solution x^ is the exact solution of a perturbed system of equations A+Ex^=b, where
E c n ε L U ,  
cn is a modest linear function of n, and ε is the machine precision. See Section 9.3 of Higham (2002) for further details.
If x is the true solution, then the computed solution x^ satisfies a forward error bound of the form
x-x^ x^ wc condA,x^,b  
where condA,x^,b = A-1 A x^ + b / x^ condA = A-1 A κ A. If x^  is the j th column of X , then wc  is returned in berrj  and a bound on x - x^ / x^  is returned in ferrj . See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The total number of floating-point operations required to solve the equations AX=B  is proportional to nr .
The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of this function is nag_lapack_dgtsvx (f07cb).

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, forward errors and condition number are also output.
function f07cp_example


fprintf('f07cp 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               ];
n = numel(d);

% 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];

% Input parameters
fact = 'No factors';
trans = 'No transpose';
dlf = dl;
df  = d;
duf = du;
du2 = complex(zeros(n-2,1));
ipiv = zeros(n,1,'int64');

% Solve
[dlf, df, duf, du2, ipiv, x, rcond, ferr, berr, info] = ...
  f07cp( ...
         fact, trans, dl, d, du, dlf, df, duf, du2, ipiv, b);

disp('Solution(s)');
disp(x);
disp('Backward errors (machine-dependent)');
fprintf('%10.1e',berr);
fprintf('\n');
disp('Estimated forward error bounds (machine-dependent)');
fprintf('%10.1e',ferr);
fprintf('\n\n');
disp('Estimate of reciprocal condition number');
fprintf('%10.1e\n',rcond);


f07cp 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

Backward errors (machine-dependent)
   3.6e-17   1.0e-16
Estimated forward error bounds (machine-dependent)
   5.5e-14   7.7e-14

Estimate of reciprocal condition number
   5.4e-03

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