f07csf computes the solution to a complex system of linear equations
or
or
, where
is an
tridiagonal matrix and
and
are
matrices, using the
factorization returned by
f07crf.
f07csf should be preceded by a call to
f07crf, 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.
f07csf then utilizes the factorization to solve the required equations.
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
https://www.netlib.org/lapack/lug
-
1:
– Character(1)
Input
-
On entry: specifies the equations to be solved as follows:
- Solve for .
- Solve for .
- Solve for .
Constraint:
, or .
-
2:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
3:
– Integer
Input
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
-
4:
– Complex (Kind=nag_wp) array
Input
-
Note: the dimension of the array
dl
must be at least
.
On entry: must contain the multipliers that define the matrix of the factorization of .
-
5:
– Complex (Kind=nag_wp) array
Input
-
Note: the dimension of the array
d
must be at least
.
On entry: must contain the diagonal elements of the upper triangular matrix from the factorization of .
-
6:
– Complex (Kind=nag_wp) array
Input
-
Note: the dimension of the array
du
must be at least
.
On entry: must contain the elements of the first superdiagonal of .
-
7:
– Complex (Kind=nag_wp) array
Input
-
Note: the dimension of the array
du2
must be at least
.
On entry: must contain the elements of the second superdiagonal of .
-
8:
– Integer array
Input
-
Note: the dimension of the array
ipiv
must be at least
.
On entry: 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.
-
9:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
b
must be at least
.
On entry: the matrix of right-hand sides .
On exit: the solution matrix .
-
10:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f07csf is called.
Constraint:
.
-
11:
– Integer
Output
-
On exit:
unless the routine detects an error (see
Section 6).
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 routine
f07cuf can be used to estimate the condition number of
and
f07cvf can be used to obtain approximate error bounds.
Background information to multithreading can be found in the
Multithreading documentation.
The real analogue of this routine is
f07cef.
This example solves the equations
where
is the tridiagonal matrix
and