NAG Library Routine Document
F07HGF (DPBCON) estimates the condition number of a real symmetric positive definite band matrix
has been factorized by F07HDF (DPBTRF)
||N, KD, LDAB, IWORK(N), INFO
||AB(LDAB,*), ANORM, RCOND, WORK(3*N)
The routine may be called by its
F07HGF (DPBCON) estimates the condition number (in the
-norm) of a real symmetric positive definite band matrix
Because is infinite if is singular, the routine actually returns an estimate of the reciprocal of .
The routine should be preceded by a call to F06REF
and a call to F07HDF (DPBTRF)
to compute the Cholesky factorization of
. The routine then uses Higham's implementation of Hager's method (see Higham (1988)
) to estimate
Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396
- 1: UPLO – CHARACTER(1)Input
: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
- 2: N – INTEGERInput
On entry: , the order of the matrix .
- 3: KD – INTEGERInput
On entry: , the number of superdiagonals or subdiagonals of the matrix .
- 4: AB(LDAB,) – REAL (KIND=nag_wp) arrayInput
the second dimension of the array AB
must be at least
: the Cholesky factor of
, as returned by F07HDF (DPBTRF)
- 5: LDAB – INTEGERInput
: the first dimension of the array AB
as declared in the (sub)program from which F07HGF (DPBCON) is called.
- 6: ANORM – REAL (KIND=nag_wp)Input
-norm of the original
, which may be computed by calling F06REF
with its parameter
must be computed either before
calling F07HDF (DPBTRF)
or else from a copy
of the original matrix
- 7: RCOND – REAL (KIND=nag_wp)Output
: an estimate of the reciprocal of the condition number of
is set to zero if exact singularity is detected or the estimate underflows. If RCOND
is less than machine precision
is singular to working precision.
- 8: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 9: IWORK(N) – INTEGER arrayWorkspace
- 10: INFO – INTEGEROutput
unless the routine detects an error (see Section 6
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , the th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
The computed estimate RCOND
is never less than the true value
, and in practice is nearly always less than
, although examples can be constructed where RCOND
is much larger.
A call to F07HGF (DPBCON) involves solving a number of systems of linear equations of the form
; the number is usually
and never more than
. Each solution involves approximately
floating point operations (assuming
) but takes considerably longer than a call to F07HEF (DPBTRS)
with one right-hand side, because extra care is taken to avoid overflow when
is approximately singular.
The complex analogue of this routine is F07HUF (ZPBCON)
This example estimates the condition number in the
-norm) of the matrix
is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by F07HDF (DPBTRF)
. The true condition number in the
9.1 Program Text
Program Text (f07hgfe.f90)
9.2 Program Data
Program Data (f07hgfe.d)
9.3 Program Results
Program Results (f07hgfe.r)