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

Syntax

Vectors.Utilities.householderVector(a,b);

Description

The function call "householderVector(a, b)" returns the normalized Householder vector u for Householder reflection of input vector a onto vector b, i.e., Householder vector u is the normal vector of the reflection plane. Algebraically, the reflection is performed by transformation matrix Q

Q = I - 2*u*u',

i.e., vector a is mapped to

a -> Q*a=c*b

with scalar c, |c| = ||a|| / ||b||. Q*a is the reflection of a about the hyperplane orthogonal to u. Q is an orthogonal matrix, i.e.

Q = inv(Q) = Q'

Example

  a = {2, -4, -2, -1};
  b = {1, 0, 0, 0};

  u = householderVector(a,b);    // {0.837, -0.478, -0.239, -0.119}
                               // Computation (identity(4) - 2*matrix(u)*transpose(matrix(u)))*a results in
                               // {-5, 0, 0, 0} = -5*b

See also

Vectors.Utilities.householderReflection
Matrices.Utilities.householderReflection
Matrices.Utilities.householderSimilarityTransformation

u = householderVector(a, b)

a	Type: Real[:] Description: Real vector to be reflected
b	Type: Real[size(a, 1)] Description: Real vector b vector a is mapped onto

u	Type: Real[size(a, 1)] Description: Householder vector to map a onto b