NAG CL Interface
f07bgc (dgbcon)

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1 Purpose

f07bgc estimates the condition number of a real band matrix A, where A has been factorized by f07bdc.

2 Specification

#include <nag.h>
void  f07bgc (Nag_OrderType order, Nag_NormType norm, Integer n, Integer kl, Integer ku, const double ab[], Integer pdab, const Integer ipiv[], double anorm, double *rcond, NagError *fail)
The function may be called by the names: f07bgc, nag_lapacklin_dgbcon or nag_dgbcon.

3 Description

f07bgc estimates the condition number of a real band matrix A, in either the 1-norm or the -norm:
κ1(A)=A1A-11   or   κ(A)=AA-1 .  
Note that κ(A)=κ1(AT).
Because the condition number is infinite if A is singular, the function actually returns an estimate of the reciprocal of the condition number.
The function should be preceded by a call to f16rbc to compute A1 or A, and a call to f07bdc to compute the LU factorization of A. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11 or A-1.

4 References

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

5 Arguments

1: order Nag_OrderType Input
On entry: 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 order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: norm Nag_NormType Input
On entry: indicates whether κ1(A) or κ(A) is estimated.
norm=Nag_OneNorm
κ1(A) is estimated.
norm=Nag_InfNorm
κ(A) is estimated.
Constraint: norm=Nag_OneNorm or Nag_InfNorm.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: kl Integer Input
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5: ku Integer Input
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
6: ab[dim] const double Input
Note: the dimension, dim, of the array ab must be at least max(1,pdab×n).
On entry: the LU factorization of A, as returned by f07bdc.
7: pdab Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.
Constraint: pdab2×kl+ku+1.
8: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On entry: the pivot indices, as returned by f07bdc.
9: anorm double Input
On entry: if norm=Nag_OneNorm, the 1-norm of the original matrix A.
If norm=Nag_InfNorm, the -norm of the original matrix A.
anorm may be computed by calling f16rbc with the same value for the argument norm.
anorm must be computed either before calling f07bdc or else from a copy of the original matrix A.
Constraint: anorm0.0.
10: rcond double * 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.
11: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, kl=value.
Constraint: kl0.
On entry, ku=value.
Constraint: ku0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
NE_INT_3
On entry, pdab=value, kl=value and ku=value.
Constraint: pdab2×kl+ku+1.
NE_INTERNAL_ERROR
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 for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, anorm=value.
Constraint: anorm0.0.

7 Accuracy

The computed estimate rcond is never less than the true value ρ, and in practice is nearly always less than 10ρ, although examples can be constructed where rcond is much larger.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07bgc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

A call to f07bgc involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n(2kl+ku) floating-point operations (assuming nkl and nku) but takes considerably longer than a call to f07bec with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The complex analogue of this function is f07buc.

10 Example

This example estimates the condition number in the 1-norm of the matrix A, where
A= ( -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 ) .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by f07bdc. The true condition number in the 1-norm is 56.40.

10.1 Program Text

Program Text (f07bgce.c)

10.2 Program Data

Program Data (f07bgce.d)

10.3 Program Results

Program Results (f07bgce.r)