.Modelica_LinearSystems2.Math.Matrices.householderSimilarityTransformation

Information

Syntax

Matrices.householderSimilarityTransformation(A,u);

Description

This function computes the Housholder similarity transformation

As = S*A*S

with

S = I -2*u*u'/(u'*u).

This transformation is widely used for transforming non-symmetric matrices to a Hessenberg form.

Example

// First step of Hessenberg decomposition
Real A[4,4] = [1,2,3,4;
               3,4,5,6;
               9,8,7,6;
               1,2,0,0];
Real Ar[4,4];
Real u[4]={0,0,0,0};

u[2:4] = Modelica_LinearSystems2.Math.Vectors.householderVector(A[2:4,1],{1,0,0});
// u= = {0, 0.8107, 0.5819, 0.0647}

Ar = householderSimilarityTransformation(A,u);
//  Ar = [1.0,     -3.8787,    -1.2193,    3.531;
        -9.5394, 11.3407,      6.4336,   -5.9243;
         0.0,     3.1307,      0.7525,   -3.3670;
         0.0,     0.8021,     -1.1656,   -1.0932]

See also

householderReflexion

Interface

function householderSimilarityTransformation
  extends Modelica.Icons.Function;
  import Modelica.Math.Vectors.length;
  input Real A[:, size(A, 1)] "Square matrix A";
  input Real u[size(A, 1)] "Householder vector";
  output Real SAS[size(A, 1), size(A, 1)];
end householderSimilarityTransformation;