.Modelica_LinearSystems2.Math.Matrices.leastSquares

Information

Syntax

x = Matrices.leastSquares(A,b);

Description

A linear system of equations A*x = b has no solutions or infinitely many solutions if A is not square. Function "leastSquares" returns a solution in a least squarse sense:

  size(A,1) > size(A,2):  returns x such that |A*x - b|^2 is a minimum
  size(A,1) = size(A,2):  returns x such that A*x = b
  size(A,1) < size(A,2):  returns x such that |x|^2 is a minimum for all
                          vectors x that fulfill A*x = b

Note

The solution is computed with the LAPACK function "dgelsx", i.e., QR or LQ factorization of A with column pivoting. If A does not have full rank, the solution is not unique and from the infinitely many solutions the one is selected that minimizes both |x|^2 and |A*x - b|^2.

Interface

function leastSquares
  extends Modelica.Icons.Function;
  input Real A[:, :] "Matrix A";
  input Real b[size(A, 1)] "Vector b";
  output Real x[size(A, 2)] "Vector x such that min|A*x-b|^2 if size(A,1) >= size(A,2) or min|x|^2 and A*x=b, if size(A,1) < size(A,2)";
end leastSquares;