NAG Library Function Document
nag_dpbcon (f07hgc) estimates the condition number of a real symmetric positive definite band matrix
has been factorized by nag_dpbtrf (f07hdc)
nag_dpbcon (f07hgc) estimates the condition number (in the
-norm) of a real symmetric positive definite band 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_dsb_norm (f16rec)
and a call to nag_dpbtrf (f07hdc)
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
, the order of the matrix .
kd – IntegerInput
On entry: , the number of superdiagonals or subdiagonals of the matrix .
ab – const doubleInput
the dimension, dim
, of the array ab
must be at least
: the Cholesky factor of
, as returned by
pdab – IntegerInput
: the stride separating row or column elements (depending on the value of order
) of the matrix in the array
anorm – doubleInput
-norm of the original
, which may be computed
by calling nag_dsb_norm (f16rec)
with its argument
must be computed either before
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, .
On entry, .
On entry, .
On entry, and .
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.
A call to nag_dpbcon (f07hgc) 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 nag_dpbtrs (f07hec)
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_zpbcon (f07huc)
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 nag_dpbtrf (f07hdc)
. The true condition number in the
9.1 Program Text
Program Text (f07hgce.c)
9.2 Program Data
Program Data (f07hgce.d)
9.3 Program Results
Program Results (f07hgce.r)