1_data_fitting

Harvard Applied Math 205: Code Examples

Unit 1: Data Fitting

vander_cond.txt

This is a transcript of an interactive Python session using NumPy to evaluate the condition number of several different Vandermonde matrices.

v_inter.py

This Python program does Vandermonde polynomial interpolation, outputting the results and error to text, which can be reading by plotting programs such as Gnuplot. If the program is run with

python v_inter.py >out

so that the data is sent to a file called out, then within Gnuplot the results can be shown with the command:

plot 'out' u 1:3 t 'Reference curve', 'out' u 1:2 t 'Polynomial fit'

The differences between the two can be shown with the command:

plot 'out' u 1:4 t 'Interpolation differences'

v_inter2.py

This is a second version of v_inter.py that uses Matplotlib to plot the results directly within Python.

ch_inter.py

This Python program performs Lagrange interpolation using Chebyshev or linearly-spaced points.

lsum.py

This program calculates the sum of absolute values of the Lagrange polynomial, needed to evaluate the Lebesgue Constant.

spline.py & spline2.py

spline.py calculates a cubic spline using the library functions in SciPy. It also calculates the differences between splines using slightly different control points. spline2.py is an example of defining a parametric curve using two cubic splines x(t) and y(t).

lfit.py

This program performs least-squares fitting of a polynomial to samples of cos(4*x). It uses two different methods: a direct approach based on the normal equations (which is generally ill-conditioned) and a built-in Python routine, lstsq, that uses a different algorithm with better conditioning. The methods used by lstsq will be discussed later in the course.

np_lfit.py

This program performs linear least squares fitting of a sum of exponentials to a given function. It demonstrates that the linear least squares does not just work for polynomials.

under_lfit.py

This program demonstrates underconstrained least squares, whereby a high-degree polynomial is fitted to a small number of data points by imposing additional constraints.

nonlinlsq.py

This program solves a nonlinear least squares problem based on finding the position of a radio transmitter based on signal received at nearby beacons.