functionhouseholderReflection

Reflect each of the vectors a_i of matrix A=[a_1, a_2, ..., a_n] on a plane with orthogonal vector u

Extends from Modelica.Icons.Function (Icon for functions), Modelica.Icons.ObsoleteModel (Icon for classes that are obsolete and will be removed in later versions).

Information

Syntax

Matrices.householderReflection(A,u);

Description

This function computes the Householder reflection (transformation)

Ar = Q*A

with

Q = I -2*u*u'/(u'*u)

where u is Householder vector, i.e., the normal vector of the reflection plane.

Householder reflection is widely used in numerical linear algebra, e.g., to perform QR decompositions.

Example

// First step of QR decomposition
  import   ObsoleteModelica4.Math.Vectors.Utilities;

  Real A[3,3] = [1,2,3;
                 3,4,5;
                 2,1,4];
  Real Ar[3,3];
  Real u[:];

  u=Utilities.householderVector(A[:,1],{1,0,0});
  // u= {0.763, 0.646, 0}

  Ar=householderReflection(A,u);
 // Ar = [-6.0828,   -5.2608,   -4.4388;
 //        0.0,      -1.1508,   -2.3016;
 //        0.0,       2.0,       0.0]

Inputs

Type	Name	Default	Description
Real[:,:]	A		Rectangular matrix
Real[size(A, 1)]	u		Householder vector

Outputs

Type	Name	Default	Description
Real[size(A, 1),size(A, 2)]	RA		Reflection of A

Revisions

2010/04/30 by Marcus Baur, DLR-RM