Suppose \(f(t,x) = x_0 e^{x_1 t}\)

with data

t: 0     1    2     3
y: 2.0   0.7  0.3   0.1

Use Gauss Newton to fit the data.

In [4]:

import numpy as np
import scipy as sp
import matplotlib.pyplot as pt
import scipy.linalg as la

/usr/lib/python2.7/dist-packages/numpy/oldnumeric/__init__.py:11: ModuleDeprecationWarning: The oldnumeric module will be dropped in Numpy 1.9
  warnings.warn(_msg, ModuleDeprecationWarning)

In [5]:

def residual(x):
    return y - x[0] * np.exp(x[1] * t)

def jacobian(x):
    return np.array([
        ...,
        ...
        ]).T

(Edit this cell for solution)

In [6]:

# data
t = np.array([0.0, 1.0, 2.0, 3.0])
y = np.array([2.0, 0.7, 0.3, 0.1])

In [16]:

# initial guess
x = np.array([1, 0])
#x = np.array([0.4, 2])

def plot_iterate(x):
    pt.plot(t, y, 'ro', markersize=20, clip_on=False)
    T = np.linspace(t.min(), t.max(), 100)
    Y = x[0] * np.exp(x[1] * T)
    pt.plot(T, Y, 'b-')

plot_iterate(x)

In [25]:

# evaluate this cell in-place many times (Ctrl-Enter)

J = jacobian(x)
r = residual(x)
s = la.lstsq(...)[0]
x = x + s
print la.norm(r)

plot_iterate(x)

0.0446775329872

(Edit this cell for solution.)