NAG Library Routine Document
estimates the condition number of a real symmetric positive definite matrix
has been factorized by f07fdf (dpotrf)
|Integer, Intent (In)||:: ||n, lda|
|Integer, Intent (Out)||:: ||iwork(n), info|
|Real (Kind=nag_wp), Intent (In)||:: ||a(lda,*), anorm|
|Real (Kind=nag_wp), Intent (Out)||:: ||rcond, work(3*n)|
|Character (1), Intent (In)||:: ||uplo|
The routine may be called by its
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 f06rcf
and a call to f07fdf (dpotrf)
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: – Character(1)Input
: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
- 2: – IntegerInput
On entry: , the order of the matrix .
- 3: – Real (Kind=nag_wp) arrayInput
the second dimension of the array a
must be at least
: the Cholesky factor of
, as returned by f07fdf (dpotrf)
- 4: – IntegerInput
: the first dimension of the array a
as declared in the (sub)program from which f07fgf (dpocon)
- 5: – Real (Kind=nag_wp)Input
-norm of the original
, which may be computed by calling f06rcf
with its argument
must be computed either before
calling f07fdf (dpotrf)
or else from a copy
of the original matrix
- 6: – 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.
- 7: – Real (Kind=nag_wp) arrayWorkspace
- 8: – Integer arrayWorkspace
- 9: – IntegerOutput
unless the routine detects an error (see Section 6
Error Indicators and Warnings
If , argument 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.
Parallelism and Performance
f07fgf (dpocon) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
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.
A call to f07fgf (dpocon)
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 f07fef (dpotrs)
with one right-hand side, because extra care is taken to avoid overflow when
is approximately singular.
The complex analogue of this routine is f07fuf (zpocon)
This example estimates the condition number in the
-norm) of the matrix
is symmetric positive definite and must first be factorized by f07fdf (dpotrf)
. The true condition number in the
Program Text (f07fgfe.f90)
Program Data (f07fgfe.d)
Program Results (f07fgfe.r)