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NAG Toolbox: nag_lapack_zgttrs (f07cs)
Purpose
nag_lapack_zgttrs (f07cs) computes the solution to a complex system of linear equations
or
or
, where
is an
by
tridiagonal matrix and
and
are
by
matrices, using the
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
as
where
is a permutation matrix,
is unit lower triangular with at most one nonzero subdiagonal element in each column, and
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:
– string (length ≥ 1)
-
Specifies the equations to be solved as follows:
- Solve for .
- Solve for .
- Solve for .
Constraint:
, or .
- 2:
– complex array
-
The dimension of the array
dl
must be at least
Must contain the multipliers that define the matrix of the factorization of .
- 3:
– complex array
-
The dimension of the array
d
must be at least
Must contain the diagonal elements of the upper triangular matrix from the factorization of .
- 4:
– complex array
-
The dimension of the array
du
must be at least
Must contain the elements of the first superdiagonal of .
- 5:
– complex array
-
The dimension of the array
du2
must be at least
Must contain the elements of the second superdiagonal of .
- 6:
– int64int32nag_int array
-
The dimension of the array
ipiv
must be at least
Must contain the pivot indices that define the permutation matrix . At the th step, row of the matrix was interchanged with row , and must always be either or , indicating that a row interchange was not performed.
- 7:
– complex array
-
The first dimension of the array
b must be at least
.
The second dimension of the array
b must be at least
.
The by matrix of right-hand sides .
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the array
b and the dimension of the arrays
d,
ipiv.
, the order of the matrix .
Constraint:
.
- 2:
– int64int32nag_int scalar
-
Default:
the second dimension of the array
b.
, the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
Output Parameters
- 1:
– complex array
-
The first dimension of the array
b will be
.
The second dimension of the array
b will be
.
The by solution matrix .
- 2:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
-
If , argument 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,
, satisfies an equation of the form
where
and
is the
machine precision. An approximate error bound for the computed solution is given by
where
, the condition number of
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
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 or or is proportional to .
The real analogue of this function is
nag_lapack_dgttrs (f07ce).
Example
This example solves the equations
where
is the tridiagonal matrix
and
Open in the MATLAB editor:
f07cs_example
function f07cs_example
fprintf('f07cs example results\n\n');
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 ];
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];
[dlf, df, duf, du2, ipiv, info] = ...
f07cr(dl, d, du);
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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