NAG Library Routine Document
F07GGF (DPPCON) estimates the condition number of a real symmetric positive definite matrix
has been factorized by F07GDF (DPPTRF)
, using packed storage.
||N, IWORK(N), INFO
||AP(*), ANORM, RCOND, WORK(3*N)
The routine may be called by its
F07GGF (DPPCON) estimates the condition number (in the
-norm) of a real symmetric positive definite 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 F06RDF
and a call to F07GDF (DPPTRF)
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: AP() – REAL (KIND=nag_wp) arrayInput
the dimension of the array AP
must be at least
: the Cholesky factor of
stored in packed form, as returned by F07GDF (DPPTRF)
- 4: ANORM – REAL (KIND=nag_wp)Input
-norm of the original
, which may be computed by calling F06RDF
with its parameter
must be computed either before
calling F07GDF (DPPTRF)
or else from a copy
of the original matrix
- 5: 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.
- 6: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 7: IWORK(N) – INTEGER arrayWorkspace
- 8: 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 F07GGF (DPPCON) 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 but takes considerably longer than a call to F07GEF (DPPTRS)
with one right-hand side, because extra care is taken to avoid overflow when
is approximately singular.
The complex analogue of this routine is F07GUF (ZPPCON)
This example estimates the condition number in the
-norm) of the matrix
is symmetric positive definite, stored in packed form, and must first be factorized by F07GDF (DPPTRF)
. The true condition number in the
9.1 Program Text
Program Text (f07ggfe.f90)
9.2 Program Data
Program Data (f07ggfe.d)
9.3 Program Results
Program Results (f07ggfe.r)