F07CGF (DGTCON) estimates the reciprocal condition number of a real
n
by
n
tridiagonal matrix
A
, using the
LU
factorization returned by
F07CDF (DGTTRF).
F07CGF (DGTCON) should be preceded by a call to
F07CDF (DGTTRF), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix
A
as
where
P
is a permutation matrix,
L
is unit lower triangular with at most one nonzero subdiagonal element in each column, and
U
is an upper triangular band matrix, with two superdiagonals. F07CGF (DGTCON) then utilizes the factorization to estimate either
A-11
or
A-1∞
, from which the estimate of the reciprocal of the condition number of
A
,
1/κA
is computed as either
or
1/κA
is returned, rather than
κA
, since when
A
is singular
κA
is infinite.
Higham N J (2002)
Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
- 1: NORM – CHARACTER(1)Input
On entry: specifies the norm to be used to estimate
κA.
- NORM='1' or 'O'
- Estimate κ1A.
- NORM='I'
- Estimate κ∞A.
Constraint:
NORM='1', 'O' or 'I'.
- 2: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 3: DL(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DL
must be at least
max1,N-1.
On entry: must contain the n-1 multipliers that define the matrix L of the LU factorization of A.
- 4: D(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
- 5: DU(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DU
must be at least
max1,N-1.
On entry: must contain the n-1 elements of the first superdiagonal of U.
- 6: DU2(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DU2
must be at least
max1,N-2.
On entry: must contain the n-2 elements of the second superdiagonal of U.
- 7: IPIV(*) – INTEGER arrayInput
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On entry: must contain the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row IPIVi, and IPIVi must always be either i or i+1, IPIVi=i indicating that a row interchange was not performed.
- 8: ANORM – REAL (KIND=nag_wp)Input
On entry: if
NORM='1' or
'O', the
1-norm of the
original matrix
A.
If NORM='I', the ∞-norm of the original matrix A.
ANORM may be computed by calling
F06RNF with the same value for the parameter
NORM.
ANORM must be computed either
before calling
F07CDF (DGTTRF) or else from a
copy of the original matrix
A (see
Section 9).
Constraint:
ANORM≥0.0.
- 9: RCOND – REAL (KIND=nag_wp)Output
On exit: contains an estimate of the reciprocal condition number.
- 10: WORK(2×N) – REAL (KIND=nag_wp) arrayWorkspace
- 11: IWORK(N) – INTEGER arrayWorkspace
- 12: INFO – INTEGEROutput
On exit:
INFO=0 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.
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
n
.
The complex analogue of this routine is
F07CUF (ZGTCON).
This example estimates the condition number in the
1-norm of the tridiagonal matrix
A
given by