NAG Library Function Document
nag_dppcon (f07ggc) estimates the condition number of a real symmetric positive definite matrix
has been factorized by nag_dpptrf (f07gdc)
, using packed storage.
||nag_dppcon (Nag_OrderType order,
const double ap,
nag_dppcon (f07ggc) estimates the condition number (in the
-norm) of a real symmetric positive definite matrix
Because is infinite if is singular, the function actually returns an estimate of the reciprocal of .
The function should be preceded by a call to nag_dsp_norm (f16rdc)
and a call to nag_dpptrf (f07gdc)
to compute the Cholesky factorization of
. The function 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
order – Nag_OrderTypeInput
: the order
argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See Section 188.8.131.52
in the Essential Introduction for a more detailed explanation of the use of this argument.
uplo – Nag_UploTypeInput
: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
n – IntegerInput
On entry: , the order of the matrix .
ap – const doubleInput
the dimension, dim
, of the array ap
must be at least
: the Cholesky factor of
stored in packed form, as returned by nag_dpptrf (f07gdc)
anorm – doubleInput
-norm of the original
, which may be computed by calling nag_dsp_norm (f16rdc)
with its argument
must be computed either before
calling nag_dpptrf (f07gdc)
or else from a copy
of the original matrix
rcond – double *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.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
On entry, .
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.
8 Parallelism and Performance
nag_dppcon (f07ggc) is not threaded by NAG in any implementation.
nag_dppcon (f07ggc) 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 Users' Note
for your implementation for any additional implementation-specific information.
A call to nag_dppcon (f07ggc) 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 nag_dpptrs (f07gec)
with one right-hand side, because extra care is taken to avoid overflow when
is approximately singular.
The complex analogue of this function is nag_zppcon (f07guc)
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 nag_dpptrf (f07gdc)
. The true condition number in the
10.1 Program Text
Program Text (f07ggce.c)
10.2 Program Data
Program Data (f07ggce.d)
10.3 Program Results
Program Results (f07ggce.r)