NAG Library Routine Document
f07hgf (dpbcon)
1
Purpose
f07hgf (dpbcon) estimates the condition number of a real symmetric positive definite band matrix
$A$, where
$A$ has been factorized by
f07hdf (dpbtrf).
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n, kd, ldab  Integer, Intent (Out)  ::  iwork(n), info  Real (Kind=nag_wp), Intent (In)  ::  ab(ldab,*), anorm  Real (Kind=nag_wp), Intent (Out)  ::  rcond, work(3*n)  Character (1), Intent (In)  ::  uplo 

C Header Interface
#include nagmk26.h
void 
f07hgf_ (const char *uplo, const Integer *n, const Integer *kd, const double ab[], const Integer *ldab, const double *anorm, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_uplo) 

The routine may be called by its
LAPACK
name dpbcon.
3
Description
f07hgf (dpbcon) estimates the condition number (in the
$1$norm) of a real symmetric positive definite band matrix
$A$:
Since
$A$ is symmetric,
${\kappa}_{1}\left(A\right)={\kappa}_{\infty}\left(A\right)={\Vert A\Vert}_{\infty}{\Vert {A}^{1}\Vert}_{\infty}$.
Because ${\kappa}_{1}\left(A\right)$ is infinite if $A$ is singular, the routine actually returns an estimate of the reciprocal of ${\kappa}_{1}\left(A\right)$.
The routine should be preceded by a call to
f06ref to compute
${\Vert A\Vert}_{1}$ and a call to
f07hdf (dpbtrf) to compute the Cholesky factorization of
$A$. The routine then uses Higham's implementation of Hager's method (see
Higham (1988)) to estimate
${\Vert {A}^{1}\Vert}_{1}$.
4
References
Higham N J (1988) FORTRAN codes for estimating the onenorm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396
5
Arguments
 1: $\mathbf{uplo}$ – Character(1)Input

On entry: specifies how
$A$ has been factorized.
 ${\mathbf{uplo}}=\text{'U'}$
 $A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
 ${\mathbf{uplo}}=\text{'L'}$
 $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
 2: $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
 3: $\mathbf{kd}$ – IntegerInput

On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint:
${\mathbf{kd}}\ge 0$.
 4: $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array
ab
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the Cholesky factor of
$A$, as returned by
f07hdf (dpbtrf).
 5: $\mathbf{ldab}$ – IntegerInput

On entry: the first dimension of the array
ab as declared in the (sub)program from which
f07hgf (dpbcon) is called.
Constraint:
${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
 6: $\mathbf{anorm}$ – Real (Kind=nag_wp)Input

On entry: the
$1$norm of the
original matrix
$A$, which may be computed by calling
f06ref with its argument
${\mathbf{norm}}=\text{'1'}$.
anorm must be computed either
before calling
f07hdf (dpbtrf) or else from a
copy of the original matrix
$A$.
Constraint:
${\mathbf{anorm}}\ge 0.0$.
 7: $\mathbf{rcond}$ – Real (Kind=nag_wp)Output

On exit: an estimate of the reciprocal of the condition number of
$A$.
rcond is set to zero if exact singularity is detected or the estimate underflows. If
rcond is less than
machine precision,
$A$ is singular to working precision.
 8: $\mathbf{work}\left(3\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace

 9: $\mathbf{iwork}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace

 10: $\mathbf{info}$ – IntegerOutput
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
 ${\mathbf{info}}<0$
If ${\mathbf{info}}=i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
7
Accuracy
The computed estimate
rcond is never less than the true value
$\rho $, and in practice is nearly always less than
$10\rho $, although examples can be constructed where
rcond is much larger.
8
Parallelism and Performance
f07hgf (dpbcon) 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 implementationspecific information.
A call to
f07hgf (dpbcon) involves solving a number of systems of linear equations of the form
$Ax=b$; the number is usually
$4$ or
$5$ and never more than
$11$. Each solution involves approximately
$4nk$ floatingpoint operations (assuming
$n\gg k$) but takes considerably longer than a call to
f07hef (dpbtrs) with one righthand side, because extra care is taken to avoid overflow when
$A$ is approximately singular.
The complex analogue of this routine is
f07huf (zpbcon).
10
Example
This example estimates the condition number in the
$1$norm (or
$\infty $norm) of the matrix
$A$, where
Here
$A$ 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
$1$norm is
$74.15$.
10.1
Program Text
Program Text (f07hgfe.f90)
10.2
Program Data
Program Data (f07hgfe.d)
10.3
Program Results
Program Results (f07hgfe.r)