Lab 10 (11/10/2017)

Problems and Even More Problems!

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1. Fill the following table using Bisection and Secant methods of root finding. Set error limit as 1e-10 and maximum number of iterations as 1000. Analyse your results.

   Function	Interval	Bisection Method		Secant Method
   		Root	Iterations	Root	Iterations
   $x^2 - 9$	$[0,5]$
   $x^5 - x - 1$	$[1,2]$
   $xe^{-x}$	$[-1,2]$
   $2\cos(3x)-e^{x}$	$[0,6]$
   $(x-1)^5$	$[0,3]$
   $xe^{-x}$	$[-1.5,1]$

   In the above table, you can see that the secant method can be either faster or slower than bisection. You may also observe convergence failures: either convergence to a value that is not near a root or convergence to a value substantially less accurate than expected. Regarding the bisection roots as accurate, are there any examples of convergence to a value that is not near a root in the table? If so, which? Are there any examples of inaccurate roots? If so, which?

2. Approximate the integral of the function $ f(x)=\sqrt{\vert x-0.5\vert}$ over the interval [-1,1] to a tolerance of 1.e-10. The exact value of this integral is $ \sqrt{2}/6+\sqrt{6}/2$. What are the estimated and true errors using all the different methods we learnt in class?
3. In this problem, we analyse solving systems of equations. Generate coefficient matrix A and the RHS vector b for a given N (the number of variables) using np.random.uniform() function of numpy package. You can also take a look at Lab4 for use of np.random.uniform() function. Then fill the following table and plot N vs. Time. You can get time using the code snippet given after the table.

   Size (N)	Time (ms)
   	Gauss Elimination	Gauss Seidel
   100
   200
   300
   400
   500
   1000
   5000
   10000
   20000

import time
import numpy as np

st_time = time.time()
# Put your code here. For example, multiplying two matrices!
A = np.random.uniform(-10., 10., (5000,1000))
B = np.random.uniform(-20., 20., (1000, 700))
C = A.dot(B)
print 'Time taken to multiply A(5000,1000) and B(1000, 700): ', \
    time.time() - st_time, ' s.' 
# When you do your own code, replace the above lines with your code

Time taken to multiply A(5000,1000) and B(1000, 700):  0.34871506691  s.