f07cuf estimates the reciprocal condition number of a complex
tridiagonal matrix
, using the
factorization returned by
f07crf.
f07cuf 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.
f07cuf then utilizes the factorization to estimate either
or
, from which the estimate of the reciprocal of the condition number of
,
is computed as either
or
is returned, rather than
, since when
is singular
is infinite.
-
1:
– Character(1)
Input
-
On entry: specifies the norm to be used to estimate
.
- or
- Estimate .
- Estimate .
Constraint:
, or .
-
2:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
3:
– 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 .
-
4:
– 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 .
-
5:
– 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 .
-
6:
– 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 .
-
7:
– 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.
-
8:
– Real (Kind=nag_wp)
Input
-
On entry: if
or
, the
-norm of the
original matrix
.
If , the -norm of the original matrix .
anorm may be computed by calling
f06unf with the same value for the argument
norm.
anorm must be computed either
before calling
f07crf or else from a
copy of the original matrix
.
Constraint:
.
-
9:
– Real (Kind=nag_wp)
Output
-
On exit: contains an estimate of the reciprocal condition number.
-
10:
– Complex (Kind=nag_wp) array
Workspace
-
-
11:
– Integer
Output
-
On exit:
unless the routine detects an error (see
Section 6).
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.
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 condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The total number of floating-point operations required to perform a solve is proportional to .
The real analogue of this routine is
f07cgf.
This example estimates the condition number in the
-norm of the tridiagonal matrix
given by