Modelica.Math.Vectors.Utilities

Utility functions that should not be directly utilized by the user

Information

This package contains utility functions that are utilized by higher level vector and matrix functions. These functions are usually not useful for an end-user.

Extends from Modelica.Icons.UtilitiesPackage (Icon for utility packages).

Package Content

Name	Description
householderVector	Calculate a normalized householder vector to reflect vector a onto vector b
householderReflection	Reflect a vector a on a plane with orthogonal vector u
roots	Compute zeros of a polynomial where the highest coefficient is assumed as not to be zero

Modelica.Math.Vectors.Utilities.householderVector

Calculate a normalized householder vector to reflect vector a onto vector b

Information

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',

a -> Q*a=c*b

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

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica.Math.Vectors.Utilities.householderReflection

Reflect a vector a on a plane with orthogonal vector u

Information

Syntax

Vectors.Utilities.householderReflection(a,u);

Description

Function "householderReflection(a, u)" performs the reflection of vector a about a plane orthogonal to vector u (Householder vector). Algebraically the operation is defined by

b=Q*a

Q = I - 2*u*u',

Q = inv(Q) = Q'

Example

  a = {2, -4, -2, -1};
  u = {0.837, -0.478, -0.239, -0.119};

  householderReflection(a,u);    //  = {-5.0, -0.001, -0.0005, -0.0044}

See also

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica.Math.Vectors.Utilities.roots

Compute zeros of a polynomial where the highest coefficient is assumed as not to be zero

Information

Syntax

  r = Vectors.Utilities.roots(p);

Description

This function computes the roots of a polynomial P of x

  P = p[1]*x^n + p[2]*x^(n-1) + ... + p[n-1]*x + p[n+1];

with the coefficient vector p. It is assumed that the first element of p is not zero, i.e., that the polynomial is of order size(p,1)-1.

To compute the roots, the eigenvalues of the corresponding companion matrix C

         |-p[2]/p[1]  -p[3]/p[1]  ...  -p[n-2]/p[1]  -p[n-1]/p[1]  -p[n]/p[1] |
         |    1            0                0               0           0     |
         |    0            1      ...       0               0           0     |
  C =    |    .            .      ...       .               .           .     |
         |    .            .      ...       .               .           .     |
         |    0            0      ...       0               1           0     |

are calculated. These are the roots of the polynomial.
Since the companion matrix has already Hessenberg form, the transformation to Hessenberg form has not to be performed. Function eigenvaluesHessenberg
provides efficient eigenvalue computation for those matrices.

Example

  r = roots({1,2,3});
  // r = [-1.0,  1.41421356237309;
  //      -1.0, -1.41421356237309]
  // which corresponds to the roots: -1.0 +/- j*1.41421356237309

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs