com.github.bruneli.scalaopt.core.leastsquares
Minimize an objective function acting on a vector of real values.
Minimize an objective function acting on a vector of real values.
real-valued objective function
initial Variables
algorithm configuration parameters
Variables at a local minimum or failure
Solve a constrained linear system with the help of QR factorization.
Solve a constrained linear system with the help of QR factorization.
Given an m by n matrix a, an n by n diagonal matrix d, and an m-vector b, the problem is to determine an x which solves the system
a*x = b and d*x = 0
in the least squares sense.
This subroutine completes the solution of the problem if it is provided with the necessary information from the QR factorization, with column pivoting, of a. That is, if a*p = q*r, where p is a permutation matrix, q has orthogonal columns, and r is an upper triangular matrix with diagonal elements of non-increasing magnitude, then qrSolv expects the full upper triangle of r, the permutation matrix p, and the first n components of (q transpose)*b. The system a*x = b, d*x = 0, is then equivalent to
r*z = qT*b, pT*d*p*z = 0,
where x = p*z. If this system does not have full rank, then a least squares solution is obtained.
Least squares solution of a non-linear problem with the Levenberg-Marquardt method.
The Levenberg-Marquardt method is equivalent to the Gauss-Newton method but with a trust region strategy meaning that beyond some distance delta, a gradient descent method is used instead of a Gauss-Newton algorithm. The Gauss-Newton method is a modified Newton's method with line search that simplifies via an approximation the computation of the Hessian in case of quadratic Loss function (Least squares problem). Example: find the best exponential curve passing through points
Although modified, the code implemented below has been inspired from the minpack package licensed under:
Minpack Copyright Notice (1999) University of Chicago. All rights reserved
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