NAG FL Interface
f07btf (zgbequ)

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1 Purpose

f07btf computes diagonal scaling matrices DR and DC intended to equilibrate a complex m × n band matrix A of band width (kl+ku+1) , and reduce its condition number.

2 Specification

Fortran Interface
Subroutine f07btf ( m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info)
Integer, Intent (In) :: m, n, kl, ku, ldab
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Out) :: r(m), c(n), rowcnd, colcnd, amax
Complex (Kind=nag_wp), Intent (In) :: ab(ldab,*)
C Header Interface
#include <nag.h>
void  f07btf_ (const Integer *m, const Integer *n, const Integer *kl, const Integer *ku, const Complex ab[], const Integer *ldab, double r[], double c[], double *rowcnd, double *colcnd, double *amax, Integer *info)
The routine may be called by the names f07btf, nagf_lapacklin_zgbequ or its LAPACK name zgbequ.

3 Description

f07btf computes the diagonal scaling matrices. The diagonal scaling matrices are chosen to try to make the elements of largest absolute value in each row and column of the matrix B given by
B = DR A DC  
have absolute value 1 . The diagonal elements of DR and DC are restricted to lie in the safe range (δ,1/δ) , where δ is the value returned by routine x02amf. Use of these scaling factors is not guaranteed to reduce the condition number of A but works well in practice.

4 References

None.

5 Arguments

1: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
2: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
3: kl Integer Input
On entry: kl, the number of subdiagonals of the matrix A.
Constraint: kl0.
4: ku Integer Input
On entry: ku, the number of superdiagonals of the matrix A.
Constraint: ku0.
5: ab(ldab,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the m×n band matrix A whose scaling factors are to be computed.
The matrix is stored in rows 1 to kl+ku+1, more precisely, the element Aij must be stored in
ab(ku+1+i-j,j)  for ​max(1,j-ku)imin(m,j+kl).  
See Section 9 in f07bnf for further details.
6: ldab Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07btf is called.
Constraint: ldabkl+ku+1.
7: r(m) Real (Kind=nag_wp) array Output
On exit: if info=0 or info>m, r contains the row scale factors, the diagonal elements of DR. The elements of r will be positive.
8: c(n) Real (Kind=nag_wp) array Output
On exit: if info=0, c contains the column scale factors, the diagonal elements of DC. The elements of c will be positive.
9: rowcnd Real (Kind=nag_wp) Output
On exit: if info=0 or info>m, rowcnd contains the ratio of the smallest value of r(i) to the largest value of r(i). If rowcnd0.1 and amax is neither too large nor too small, it is not worth scaling by DR.
10: colcnd Real (Kind=nag_wp) Output
On exit: if info=0, colcnd contains the ratio of the smallest value of c(i) to the largest value of c(i).
If colcnd0.1, it is not worth scaling by DC.
11: amax Real (Kind=nag_wp) Output
On exit: max|aij|. If amax is very close to overflow or underflow, the matrix A should be scaled.
12: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0andinfom
Row value of A is exactly zero.
info>m
Column value of A is exactly zero.

7 Accuracy

The computed scale factors will be close to the exact scale factors.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07btf 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 implementation-specific information.

9 Further Comments

The real analogue of this routine is f07bff.

10 Example

This example equilibrates the complex band matrix A given by
A = ( -1.65+2.26i (-2.05-0.85i)×10−10 -(0.97-2.84i ((0 -0.00+6.30i (-1.48-1.75i)×10−10 (-3.99+4.01i ((0.59-0.48i -0 (-0.77+2.83i (-1.06+1.94i)×1010 ((3.33-1.04i)×1010 -0 -(0 -(0.48-1.09i -0.46-1.72i ) .  
Details of the scaling factors, and the scaled matrix are output.

10.1 Program Text

Program Text (f07btfe.f90)

10.2 Program Data

Program Data (f07btfe.d)

10.3 Program Results

Program Results (f07btfe.r)