NAG Library Routine Document

f07btf  (zgbequ)


    1  Purpose
    7  Accuracy


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


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 nagmk26.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 its LAPACK name zgbequ.


f07btf (zgbequ) 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.




1:     m – IntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
2:     n – IntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
3:     kl – IntegerInput
On entry: kl, the number of subdiagonals of the matrix A.
Constraint: kl0.
4:     ku – IntegerInput
On entry: ku, the number of superdiagonals of the matrix A.
Constraint: ku0.
5:     abldab* – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the m by 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
abku+1+i-jj  for ​max1,j-kuiminm,j+kl. 
See Section 9 in f07bnf (zgbsv) for further details.
6:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07btf (zgbequ) is called.
Constraint: ldabkl+ku+1.
7:     rm – Real (Kind=nag_wp) arrayOutput
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:     cn – Real (Kind=nag_wp) arrayOutput
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 ri to the largest value of ri. 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 ci to the largest value of ci.
If colcnd0.1, it is not worth scaling by DC.
11:   amax – Real (Kind=nag_wp)Output
On exit: maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
12:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

Error Indicators and Warnings

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


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

Parallelism and Performance

f07btf (zgbequ) 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.

Further Comments

The real analogue of this routine is f07bff (dgbequ).


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.

Program Text

Program Text (f07btfe.f90)

Program Data

Program Data (f07btfe.d)

Program Results

Program Results (f07btfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017