.Modelica.Math.Matrices.singularValues

Information

Syntax

         sigma = Matrices.singularValues(A);
(sigma, U, VT) = Matrices.singularValues(A);

Description

This function computes the singular values and optionally the singular vectors of matrix A. Basically the singular value decomposition of A is computed, i.e.,

A = U Σ VT
  = U*Sigma*VT

where U and V are orthogonal matrices (UU^T=I, VV^T=I). Σ = [diagonal(σ_i), zeros(n,m-n)], if n=size(A,1) ≤ m=size(A,2)) or [diagonal(σ_i); zeros(n-m,m)], if n > m=size(A,2)). Σ has the same size as matrix A with nonnegative diagonal elements in decreasing order and with all other elements zero (σ₁ is the largest element). The function returns the singular values σ_i in vector sigma and the orthogonal matrices in matrices U and VT.

Example

A = [1, 2,  3,  4;
     3, 4,  5, -2;
    -1, 2, -3,  5];
(sigma, U, VT) = singularValues(A);
results in:
   sigma = {8.33, 6.94, 2.31};
i.e.
   Sigma = [8.33,    0,    0, 0;
               0, 6.94,    0, 0;
               0,    0, 2.31, 0]

See also

Interface

function singularValues
  extends Modelica.Icons.Function;
  input Real A[:, :] "Matrix";
  output Real sigma[min(size(A, 1), size(A, 2))] "Singular values";
  output Real U[size(A, 1), size(A, 1)] = identity(size(A, 1)) "Left orthogonal matrix";
  output Real VT[size(A, 2), size(A, 2)] = identity(size(A, 2)) "Transposed right orthogonal matrix";
end singularValues;