Blog Posts on numerical methods

Python: Marquardt algorithm for nonlinear regression now at version 0.0.4! by digital explorations on Feb 5, 2012We have updated our old Python implementation of Marquardt algorithm for nonlinear least squares. The new version now offers three numerical derivative approximation formulas: Forward difference: Backward difference: Central difference: We have also...

Python:Marquardt's algorithm for nonlinear least squares problems by digital explorations on Dec 4, 2011WARNING: The codes have not been thoroughly tested. If you know Python and is familiar with the Marquardt's algorithm, please help in cleaning up the code. It will be appreciated. When you need a routine to perform nonlinear least squares,most of the...

Python: Runge Kutta Nystrom for second order differential equations. by digital explorations on Oct 31, 2011Here is a Python module to perform Runge-Kutta-Nystrom for solving differential equations of the second order. I have forgotten the source but it is probably from Scheid's Numerical Analysis book. ?View Code PYTHON1 2 3 4 5 6 7 8 9 10 11 12 13 14 15...

Python : Graeffe Root Squaring Method Part 2: estimating roots by digital explorations on Oct 31, 2011Now that we have the grsm table,(see previous post /?=12572) it is time to determine estimated roots. By estimated roots, we mean that the absolute values of the roots. The signs can be tested by evaluating the positive and negative values on the ori...

Python : Graeffe Root Squaring Method Part 1: creating the grsm table. by digital explorations on Oct 30, 2011I was introduced into this method by one of my U.P. professors, Prof. Emeritus E. Quiwa, at the College of Engineering of the University of the Philippines in 1979, in the time of mainframes!. The Graeffe's method for finding all roots of polynomial...

Roots of polynomials with Ralston-Rabinowitz implementation of Laguerre's method, revised to 0.0.2! by digital explorations on Oct 29, 2011We have already presented a muller method,in /?p=297 for computing roots or zeroes of general analytic functions which include polynomials. Another method, more specialized for polynomials, and which is sure to converge to a root when the coefficient...

Derivation of formula for sum of squares of the first n integers by Newton Interpolation formula (with Python codes) by digital explorations on Jun 24, 2011What is the value of ? A search in mathematical reference books gave the following formula: with the usual proof using recursion. But the proof does not really show how the final formula was arrived at. In this blog article we will demonstrate a meth...

Python: Newton interpolating polynomial for approximations by digital explorations on Jun 24, 2011As promised in the last recent post, we now show Python codes to compute the coefficients, which are divided differences of the Newton interpolating polynomial for approximation applications in tables. The centers (or the x-values) do not have to be...

Adams Predictor-Corrector Method for Solving ODEs of the first order. by digital explorations on Feb 27, 2010We have already discussed RKF45, Dormand Prince Python codes for solving ODEs of the first order. Here is an implementaion of Adams predictor-corrector method. The method is not self-starting and we have to make a call to initialize four values for t...

Heun’s method for solving Ordinary Differential Equations of the First Order. by digital explorations on Feb 26, 2010Odeheun.py is an implementation of Heun’s method and is a simple modification of the basic Euler method. It is the simplest predictor-corrector method for solving ODEs. ?Download odeheun.py1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21...