1 \documentclass[12pt]{article}

     2 \title{Matrices and Solution of Equations}

     3 \author{FOSSEE}

     4 \begin{document}

     5 \date{}

     6 \vspace{-1in}

     7 \begin{center}

     8 \LARGE{Interpolation, Differentiation and Integration}\\

     9 \large{FOSSEE}

    10 \end{center}

    11 \section{}

    12 Loading a data file

    13 \begin{verbatim}

    14 In [2]: x, y = loadtxt('points.txt', unpack = True)

    15 #load data file directly into Arrays.

    16 \end{verbatim}

    17 \section{}

    18 Interpolate

    19 \begin{verbatim}

    20 In []: from scipy.interpolate import splrep

    21 In []: tck = splrep(x,y) #get spline curve representation for x,y.

    22 In []: from scipy.interpolate import splev

    23 #To evaluate spline and it's derivatives.

    24 In []: Xnew = arange(0.01,3,0.02)

    25 #missing set of points

    26 In []: Ynew = splev(Xnew, tck)

    27 #Value of function at Xnew

    28 In []: plot(Xnew, Ynew)

    29 \end{verbatim}

    30 

    31 \section{Differentiation}

    32 Taylor series - finite difference approximations

    33 $f(x+h)=f(x)+hf^{'}(x)$ Forward

    34 \begin{verbatim}

    35 In []: x = linspace(0, 2*pi, 100)

    36 In []: y = sin(x)

    37 In []: deltax = x[1] - x[0]

    38 In []: fD = (y[1:] - y[:-1]) / deltax 

    39 #fD is the required forward difference

    40 \end{verbatim}

    41 

    42 \section{Quadrature}

    43 $\int_0^1(sin(x) + x^2)$ 

    44 In []: def f(x):

    45            return sin(x)+x**2

    46 In []: from scipy.integrate import quad

    47 In []: quad(f, 0, 1)

    48 \end{document}

    49