Next Article in Journal
Design of an Optimized Fuzzy Classifier for the Diagnosis of Blood Pressure with a New Computational Method for Expert Rule Optimization
Next Article in Special Issue
On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich’s Technique
Previous Article in Journal
New Methodology to Approximate Type-Reduction Based on a Continuous Root-Finding Karnik Mendel Algorithm
Previous Article in Special Issue
Expanding the Applicability of Some High Order Househölder-Like Methods
Article Menu

Export Article

Open AccessArticle
Algorithms 2017, 10(3), 78; https://doi.org/10.3390/a10030078

An Efficient Algorithm for the Separable Nonlinear Least Squares Problem

Department of Mathematics, Western Washington University, Bellingham, WA 98225-9063, USA
*
Authors to whom correspondence should be addressed.
Received: 7 June 2017 / Revised: 23 June 2017 / Accepted: 1 July 2017 / Published: 10 July 2017
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems 2017)
Full-Text   |   PDF [403 KB, uploaded 13 July 2017]

Abstract

The nonlinear least squares problem m i n y , z A ( y ) z + b ( y ) , where A ( y ) is a full-rank ( N + ) × N matrix, y R n , z R N and b ( y ) R N + with n , can be solved by first solving a reduced problem m i n y f ( y ) to find the optimal value y * of y, and then solving the resulting linear least squares problem m i n z A ( y * ) z + b ( y * ) to find the optimal value z * of z. We have previously justified the use of the reduced function f ( y ) = C T ( y ) b ( y ) , where C ( y ) is a matrix whose columns form an orthonormal basis for the nullspace of A T ( y ) , and presented a quadratically convergent Gauss–Newton type method for solving m i n y C T ( y ) b ( y ) based on the use of QR factorization. In this note, we show how LU factorization can replace the QR factorization in those computations, halving the associated computational cost while also providing opportunities to exploit sparsity and thus further enhance computational efficiency. View Full-Text
Keywords: separable equations; nonlinear least squares; full-rank matrices; QR factorization; over-determined systems; Gauss–Newton method; least squares solutions; LU factorization; quadratic convergence separable equations; nonlinear least squares; full-rank matrices; QR factorization; over-determined systems; Gauss–Newton method; least squares solutions; LU factorization; quadratic convergence
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Shen, Y.; Ypma, T.J. An Efficient Algorithm for the Separable Nonlinear Least Squares Problem. Algorithms 2017, 10, 78.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Algorithms EISSN 1999-4893 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top