functionleastSquares

Solve overdetermined or underdetermined real system of linear equations A*x=b in a least squares sense (A may be rank deficient)

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

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.

Inputs

Type	Name	Default	Description
Real[:,:]	A		Matrix A
Real[size(A, 1)]	b		Vector b

Outputs

Type	Name	Default	Description
Real[size(A, 2)]	x		Vector x such that min\|Ax-b\|^2 if size(A,1) >= size(A,2) or min\|x\|^2 and Ax=b, if size(A,1) < size(A,2)