#!/usr/bin/env python
# coding: utf-8

# # Numerical Methods 1
# ### [Gerard Gorman](http://www.imperial.ac.uk/people/g.gorman), [Matthew Piggott](http://www.imperial.ac.uk/people/m.d.piggott), [Nicolas Barral](http://www.imperial.ac.uk/people/n.barral)

# # Revision exercises

# ## Question 1
# 
# Below is a extract of shot data from a seismic survey.
# 
# |  Time (ms)  |  Shot  |
# |-----|-----|
# | 0 | -0.021373 |
# | 4 | -0.024578 |
# | 8 | -0.023914 |
# | 12 | -0.018227 |
# | 16 | -0.00781 |
# | 20 | 0.005602 |
# | 24 | 0.019264 |
# | 28 | 0.030235 |
# | 32 | 0.036059 |
# | 36 | 0.035334 |
# 
#  1. Calculate the Lagrange polynomial for these points. Plot both the Lagrange polynomial and the raw data points.
#  2. The full shot is available in the file [shot.txt](data/shot.txt) (in the data folder) - where the sample interval is 4ms as above. Note that the file only contains one column as you can calculate the time column yourself. Use cubic-polynomial splines to re-interpolate the data for a sample interval of 7.07ms. Plot both the original shot data and the interpolated time series.

# ## Question 2
# 
# 1. Calculate the upper triangular form for the matrix:
# \begin{align*}
# A = \begin{bmatrix}
# −5 &  3 &  4\\
# 10 & −8 & −9\\
# 15 &  1 &  2
# \end{bmatrix}
# \end{align*}
# 
# 2. Consider the matrix:
# \begin{align*}
# C = \begin{bmatrix}
# −5 &  3 &  4\\
# 10 & −8 & −9\\
# 15 &  1 &  2
# \end{bmatrix}
# \end{align*}
# Does matrix $C$ have an inverse? If not, then why not? 

# ## Question 3
# 
# Consider the function
# $$f(x) = \dfrac{1}{(x − 0.3)^2 + 0.01} - \dfrac{1}{(x − 0.8)^2 + 0.04}$$.
# 
# 
#  1. Write a function that computes the second derivative of $f(x)$ using central differencing. The interface to your function should look like *central_diff2(f, x, h)* where $f$ is the function to be differentiated, $x$ is the position at which the derivative should be estimated, and $h (= \Delta x)$ is the step size.
#  2. Use this function to compute the derivative at $x = 0.5$ for decreasing values of $\Delta x$. Start with $\Delta x=1.2$ and keep halving until the relative difference between solutions falls below $1.0^{-6}$ Plot the convergence of the method, i.e. plot $\Delta x$ against the absolute difference between the analytical value and the finite difference approximation.

# ## Question 4
# 
# Consider the integral:
# $$\int_0^{2\pi} x^2\cos(x) dx.$$
# 
# Show how the absolute error between the exact solution and the numerical solution varies with the size of the integration step for the four numerical integration methods: trapezoid rule; Simpson's rule; composite Simpson's rule; and Weddle's rule. Show your result by plotting error against integration step, $dx$.