NAG CL Interface
f07huc (zpbcon)

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1 Purpose

f07huc estimates the condition number of a complex Hermitian positive definite band matrix A, where A has been factorized by f07hrc.

2 Specification

#include <nag.h>
void  f07huc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, const Complex ab[], Integer pdab, double anorm, double *rcond, NagError *fail)
The function may be called by the names: f07huc, nag_lapacklin_zpbcon or nag_zpbcon.

3 Description

f07huc estimates the condition number (in the 1-norm) of a complex Hermitian positive definite band matrix A:
κ1(A)=A1A-11 .  
Since A is Hermitian, κ1(A)=κ(A)=AA-1.
Because κ1(A) is infinite if A is singular, the function actually returns an estimate of the reciprocal of κ1(A).
The function should be preceded by a call to f16uec to compute A1 and a call to f07hrc to compute the Cholesky factorization of A. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11.

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: uplo Nag_UploType Input
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=UHU, where U is upper triangular.
uplo=Nag_Lower
A=LLH, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: kd Integer Input
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
5: ab[dim] const Complex Input
Note: the dimension, dim, of the array ab must be at least max(1,pdab×n).
On entry: the Cholesky factor of A, as returned by f07hrc.
6: 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: pdabkd+1.
7: anorm double Input
On entry: the 1-norm of the original matrix A, which may be computed by calling f16uec with its argument norm=Nag_OneNorm. anorm must be computed either before calling f07hrc or else from a copy of the original matrix A.
Constraint: anorm0.0.
8: 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.
9: 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, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+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.
f07huc 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 f07huc involves solving a number of systems of linear equations of the form Ax=b; the number is usually 5 and never more than 11. Each solution involves approximately 16nk real floating-point operations (assuming nk) but takes considerably longer than a call to f07hsc with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The real analogue of this function is f07hgc.

10 Example

This example estimates the condition number in the 1-norm (or -norm) of the matrix A, where
A= ( 9.39+0.00i 1.08-1.73i 0.00+0.00i 0.00+0.00i 1.08+1.73i 1.69+0.00i -0.04+0.29i 0.00+0.00i 0.00+0.00i -0.04-0.29i 2.65+0.00i -0.33+2.24i 0.00+0.00i 0.00+0.00i -0.33-2.24i 2.17+0.00i ) .  
Here A is Hermitian positive definite, and is treated as a band matrix, which must first be factorized by f07hrc. The true condition number in the 1-norm is 153.45.

10.1 Program Text

Program Text (f07huce.c)

10.2 Program Data

Program Data (f07huce.d)

10.3 Program Results

Program Results (f07huce.r)