nag_zgttrs (f07csc) 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_zgttrf (f07crc).
nag_zgttrs (f07csc) should be preceded by a call to
nag_zgttrf (f07crc), 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_zgttrs (f07csc) 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
http://www.netlib.org/lapack/lug
- 1:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_TransTypeInput
-
On entry: specifies the equations to be solved as follows:
- Solve for .
- Solve for .
- Solve for .
Constraint:
, or .
- 3:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
- 5:
– const ComplexInput
-
Note: the dimension,
dim, of the array
dl
must be at least
.
On entry: must contain the multipliers that define the matrix of the factorization of .
- 6:
– const ComplexInput
-
Note: the dimension,
dim, of the array
d
must be at least
.
On entry: must contain the diagonal elements of the upper triangular matrix from the factorization of .
- 7:
– const ComplexInput
-
Note: the dimension,
dim, of the array
du
must be at least
.
On entry: must contain the elements of the first superdiagonal of .
- 8:
– const ComplexInput
-
Note: the dimension,
dim, of the array
du2
must be at least
.
On entry: must contain the elements of the second superdiagonal of .
- 9:
– const IntegerInput
-
Note: the dimension,
dim, 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.
- 10:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix of right-hand sides .
On exit: the by solution matrix .
- 11:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 12:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
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_zgtcon (f07cuc) can be used to estimate the condition number of
and
nag_zgtrfs (f07cvc) can be used to obtain approximate error bounds.
The real analogue of this function is
nag_dgttrs (f07cec).
This example solves the equations
where
is the tridiagonal matrix
and