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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zpbequ (f07ht)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zpbequ (f07ht) computes a diagonal scaling matrix S  intended to equilibrate a complex n  by n  Hermitian positive definite band matrix A , with bandwidth 2kd+1 , and reduce its condition number.


[s, scond, amax, info] = f07ht(uplo, kd, ab, 'n', n)
[s, scond, amax, info] = nag_lapack_zpbequ(uplo, kd, ab, 'n', n)


nag_lapack_zpbequ (f07ht) computes a diagonal scaling matrix S  chosen so that
sj=1 / ajj .  
This means that the matrix B  given by
B=SAS ,  
has diagonal elements equal to unity. This in turn means that the condition number of B , κ2B , is within a factor n  of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).


Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia


Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A is stored in the array ab, as follows:
The upper triangle of A is stored.
The lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2:     kd int64int32nag_int scalar
kd, the number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'.
Constraint: kd0.
3:     abldab: – complex array
The first dimension of the array ab must be at least kd+1.
The second dimension of the array ab must be at least max1,n.
The upper or lower triangle of the Hermitian positive definite band matrix A whose scaling factors are to be computed.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+kd.
Only the elements of the array ab corresponding to the diagonal elements of A are referenced. (Row kd+1 of ab when uplo='U', row 1 of ab when uplo='L'.)

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the second dimension of the array ab.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     sn – double array
If info=0, s contains the diagonal elements of the scaling matrix S.
2:     scond – double scalar
If info=0, scond contains the ratio of the smallest value of s to the largest value of s. If scond0.1 and amax is neither too large nor too small, it is not worth scaling by S.
3:     amax – double scalar
maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

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.
The _th diagonal element of A is not positive (and hence A cannot be positive definite).


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

Further Comments

The real analogue of this function is nag_lapack_dpbequ (f07hf).


This example equilibrates the Hermitian positive definite matrix A  given by
A = 9.39 -i1.08-1.73i -i0 -i0 1.08+1.73i -i1.69 -0.04+0.29i×1010 -i0 0 -0.04-0.29i×1010 2.65×1020 -0.33+2.24i×1010 0 -i0 -0.33-2.24i×1010 -i2.17 .  
Details of the scaling factors and the scaled matrix are output.
function f07ht_example

fprintf('f07ht example results\n\n');

% Symmetric banded A
uplo = 'U';
kd = int64(1);
n  = int64(4);
ab = [0,          1.08 - 1.73i,  -0.04e10 + 0.29e10i,  -0.33e10 + 2.24e10i;
      9.39 + 0i,  1.69 + 0i,      2.65e20 + 0i,         2.17    + 0i];

% Scale A
[s, scond, amax, info] = f07ht( ...
                                uplo, kd, ab);

fprintf('scond = %8.1e, amax = %8.1e\n\n', scond, amax);
disp('Diagonal scaling factors');

% Apply scalings
asp = ab*diag(s);
for i = 1:n
  for j = 0:min(kd,n-i)
    asp(kd+1-j,i+j) = s(i)*asp(kd+1-j,i+j);

kl = int64(0);
[ifail] = x04de( ...
                 n, n, kl, kd, asp, 'Scaled matrix');

f07ht example results

scond =  8.0e-11, amax =  2.6e+20

Diagonal scaling factors
   3.3e-01   7.7e-01   6.1e-11   6.8e-01

 Scaled matrix
             1          2          3          4
 1      1.0000     0.2711
        0.0000    -0.4343

 2                 1.0000    -0.0189
                   0.0000     0.1370

 3                            1.0000    -0.1376
                              0.0000     0.9341

 4                                       1.0000

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Chapter Introduction
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