f07crf 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.
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:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
2:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
dl
must be at least
.
On entry: must contain the subdiagonal elements of the matrix .
On exit: is overwritten by the multipliers that define the matrix of the factorization of .
-
3:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
d
must be at least
.
On entry: must contain the diagonal elements of the matrix .
On exit: is overwritten by the diagonal elements of the upper triangular matrix from the factorization of .
-
4:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
du
must be at least
.
On entry: must contain the superdiagonal elements of the matrix .
On exit: is overwritten by the elements of the first superdiagonal of .
-
5:
– Complex (Kind=nag_wp) array
Output
-
On exit: contains the elements of the second superdiagonal of .
-
6:
– Integer array
Output
-
On exit: contains the pivot indices that define the permutation matrix . At the th step, row of the matrix was interchanged with row . will always be either or , indicating that a row interchange was not performed.
-
7:
– Integer
Output
-
On exit:
unless the routine detects an error (see
Section 6).
The computed factorization satisfies an equation of the form
where
and
is the
machine precision.
Following the use of this routine,
f07csf can be used to solve systems of equations
or
or
, and
f07cuf can be used to estimate the condition number of
.
Background information to multithreading can be found in the
Multithreading documentation.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The real analogue of this routine is
f07cdf.
This example factorizes the tridiagonal matrix
given by