f07cuc estimates the reciprocal condition number of a complex
tridiagonal matrix
, using the
factorization returned by
f07crc.
f07cuc should be preceded by a call to
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.
f07cuc 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:
– Nag_NormType
Input
-
On entry: specifies the norm to be used to estimate
.
- Estimate .
- Estimate .
Constraint:
or .
-
2:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
3:
– const Complex
Input
-
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 .
-
4:
– const Complex
Input
-
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 .
-
5:
– const Complex
Input
-
Note: the dimension,
dim, of the array
du
must be at least
.
On entry: must contain the elements of the first superdiagonal of .
-
6:
– const Complex
Input
-
Note: the dimension,
dim, of the array
du2
must be at least
.
On entry: must contain the elements of the second superdiagonal of .
-
7:
– const Integer
Input
-
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.
-
8:
– double
Input
-
On entry: if
, the
-norm of the
original matrix
.
If , the -norm of the original matrix .
anorm may be computed as demonstrated in
Section 10 for the
-norm. The
-norm may be similarly computed by swapping the
dl and
du arrays in the code for the
-norm.
anorm must be computed either
before calling
f07crc or else from a
copy of the original matrix
.
Constraint:
.
-
9:
– double *
Output
-
On exit: contains an estimate of the reciprocal condition number.
-
10:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
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 function. 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 function is
f07cgc.
This example estimates the condition number in the
-norm of the tridiagonal matrix
given by