.Modelica.Math.Matrices.balance

Information

Syntax

(D,B) = Matrices.balance(A);

Description

This function returns a vector D, such that B=inv(diagonal(D))*A*diagonal(D) has a better condition as matrix A, i.e., conditionNumber(B) ≤ conditionNumber(A). The elements of D are multiples of 2 which means that this function does not introduce round-off errors. Balancing attempts to make the norm of each row of B equal to the norm of the respective column.

Balancing is used to minimize roundoff errors induced through large matrix calculations like Taylor-series approximation or computation of eigenvalues.

Example

       - A = [1, 10,  1000; 0.01,  0,  10; 0.005,  0.01,  10]
       - Matrices.norm(A, 1);
         = 1020.0
       - (T,B)=Matrices.balance(A)
       - T
         = {256, 16, 0.5}
       - B
         =  [1,     0.625,   1.953125;
             0.16,  0,       0.3125;
             2.56,  0.32,   10.0]
       - Matrices.norm(B, 1);
         = 12.265625

The Algorithm is taken from

H. D. Joos, G. Grübel:

RASP'91 Regulator Analysis and Synthesis Programs
DLR - Control Systems Group 1991

which based on the balance function from EISPACK.

Interface

function balance
  extends Modelica.Icons.Function;
  input Real A[:, size(A, 1)];
  output Real D[size(A, 1)] "diagonal(D)=T is transformation matrix, such that
            B = inv(T)*A*T has smaller condition as A";
  output Real B[size(A, 1), size(A, 1)] "Balanced matrix (= inv(diagonal(D))*A*diagonal(D) )";
end balance;

Revisions

Release Notes:

• July 5, 2002 by H. D. Joos and Nico Walther
  Implemented.