This information is part of the Modelica Standard Library maintained by the Modelica Association.

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.

(D, B) = balance(A)

A	Type: Real[:,size(A, 1)]

D	Type: Real[size(A, 1)] Description: diagonal(D)=T is transformation matrix, such that B = inv(T)*A*T has smaller condition as A
B	Type: Real[size(A, 1),size(A, 1)] Description: Balanced matrix (= inv(diagonal(D))*A*diagonal(D) )