Modelica.Math.Vectors.Utilities

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.

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

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

Inputs

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

Outputs

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

Modelica.Math.Vectors.Utilities.householderReflection

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

Inputs

Name	Description
a[:]	Real vector a to be reflected
u[size(a, 1)]	householder vector

Outputs

Name	Description
ra[size(u, 1)]	reflexion of a

Modelica.Math.Vectors.Utilities.roots

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

Inputs

Name	Description
p[:]	Vector with polynomial coefficients p[1]*x^n + p[2]*x^(n-1) + p[n]*x +p[n-1]

Outputs

Name	Description
roots[max(0, size(p, 1) - 1), 2]	roots[:,1] and roots[:,2] are the real and imaginary parts of the roots of polynomial p