Numerical Analysis

Section 9 Assignments

#Assignment set 1
#Numerical Analysis
#September 5, 2019

##############################################
#Exercise 1
##############################################
#Useful commands:
#inline(), plot(), abs(), max()
##############################################
#A) Plot the functions
#f(x) = sin(x)
#T7(x) = x - x^3/3! + x^5/5! - x^7/7!
#on the same set of axes.
#
#B) Then, given the domain you've chosen, calculate the maximum error.
#
#C)Finally, find the largest interval you can so that the maximum absolute true
#error is less than 0.1.






##############################################
#Exercise 2
##############################################
#Useful commands:
#sym, syms, diff, factorial, function_handle, plot
##############################################
#A) Write a loop that constructs the degree 7 Taylor expansion of
#f(x) = sin(x)
#about the point a = 0.
#
#B) Rewrite your loop so that it can find the Taylor series of degree n about
#any center a for f(x) = sin(x).
#
#C) Rewrite your loop so that it can take an arbitrary function, produce the
#degree n Taylor polynomial, and plot the function and the Taylor polynomial
#near a.




################################################
#Exercise 3 (optional)
################################################
#Modify your answer to Exercise 2 so that given an error tolerance,
#the program produces the largest domain near a for which the maximum error is
#less than that tolerance. The output should be both the Taylor polynomial
#and a domain where the approximation is good.

#Assignment set 2
#Numerical Analysis
#September 13, 2019

##############################################
#Exercise 1
##############################################
##############################################
#useful commands:
#for/endfor, while/endwhile, break, @, if/elseif/else/endif
##############################################
#A) Plot the function f(x) = x^6 - x - 1. Use your plot
#to construct reasonable brackets for the roots of f.
#
#B) Use the bisection method to compute the roots of f. Set
#your tolerance so that the approximation is good to six
#significant figures.






##############################################
#Exercise 2
##############################################
#Useful commands:
#function, return, for, while, break
##############################################
#Write a function that takes as input a function,
#a bracket [a,b], a tolerance, and a maximum number
#of iterations and produces an approximation to the
#root in the interval [a,b]. You should consider
#adding a logic check to make sure that a root really
#is inside the bracket before executing.




################################################
#Exercise 3 (optional)
################################################
#Rewrite the function in Exercise 2 so that the user specifies
#a desired number of significant figures instead of a tolerance.

################################################
#Exercise 1
################################################
#Find an approximation for the square root of 2 using
#the method of false position.


################################################
#Exercise 2
################################################
#Find an approximation for the square root of 2 using
#the Newton-Raphson method. Compare the number of steps and
#the running time with Exercise 1.

################################################
#Exercise 3
################################################
#Find a function for which your bisection solver runs faster than
#your false position solver. Explain what you think is happening.

################################################
#Exercise 4
################################################
#Find a function that gives you extremely slow convergence
#in Newton's method. Explain what you think is happening.

################################################
#Exercise 1
################################################
#Use Newtons' method to find the roots of
#f(x) = x^6 - 9x^3 + 2x - 10


################################################
#Exercise 2
################################################
#Repeat exercise 1 but using the secant method.

################################################
#Exercise 3
################################################
#Turn your secant and Newton's method routines into funtions.
#You should build in failure checks (that is, the x's get huge
#or the answers don't converge).


################################################
#Exercise 4
################################################
#Write a hybrid method that first tries the secant method to find a root
#and then uses the bisection method if the secant method fails.

################################################
#Exercise 1 (challenge)
################################################
#Implement synthetic division. Input should be an array of
#numbers representing the coefficients and one number representing
#the point to be evaluated. Output should be the quotient and
#the function value.


################################################
#Exercise 2 (challenge)
################################################
#Use the synthetic division method and Newton's method to factor the polynomial
#f(x) = x^3 - 2x^2 - 5x + 6 completely.

################################################
#Exercise 3
################################################
#Find the Lagrange interpolating polynomial for the points
#(2,8), (4,2), (8,0.125).
#Plot the polynomial and the function f(x) = 2^(5-x),
#which generates the points, on the same graph.
#Is the Lagrange polynomial a good interpolant?
#Is it a good extrapolant?


################################################
#Exercise 4
################################################
#Find the Lagrange interpolating function for the points
#(-3, 4/10), (-2, 4/5), (-1, 2), (0,4), (1, 2), (2, 4/5), (3, 4/10)
#These points come from the function f(x) = 1/(x^2 + 1).
#What happens? Is the Lagrange polynomial a good approximation
#for f? Why or why not?

################################################
#Exercise 1
################################################
#Write a script that produces the coefficients to the Newton
#polynomial corresponding to the points (1,1), (2,3), (5,10), (6, -2)
#and then produces the polynomial. Verify that the polynomial is
#correct by plotting the points and the polynomial on the same axes.


################################################
#Exercise 2
################################################
#Write a function that takes a matrix containing n
#interpolation points and produces the corresponding Newton
#polynomial.

################################################3
#Exercise 1
################################################
#Compute the cubic spline function that interpolates the points
#(-1, 3), (0,1), (1, 3). Plot your function and the points.


####################################################
#Exercise 2
###################################################
#Extend your previous result by appending the point (2,0). Pay attention to the organization
#of the matrix that you use to find the spline coefficients.


#########################################################
#Exercise 3
##########################################################
#Use array appending to build a script that will compute and plot the spline interpolation for
#an arbitrary number of points.

############################################################
#Exercise 4
###########################################################
#Sample the function f(x) = 1/(x^2 + 16) with an odd number of equally spaced points.
#Compare the graph of the original function, the graph of the unique polynomial interpolating function through
#the sample points, and the spline interpolating function through the sample points.

#############################################################
#Exercise 1
################################################################
#Write a function that implements the n segment trapezoid rule
#for inputs consisting of an interval (a,b), a number of segments n,
#and an anonymous function f. Then wrap that function into a driver
#that takes the above inputs plus a tolerance and increases the
#number of segments until a given tolerance is met (in absolute
#relative error). Compute the area under the function e^(-x^2) from
# x=0 to x=2.

###############################################################
#Exercise 2
#################################################################
#Write a function that implements Richardson's first order
#extrapolation of the trapezoid rule. Compare the speed of
#convergence with the trapezoid rule for the function x^2 sin(2x)
#on the interval [0,10].

################################################################
#Exercise 3
##################################################################
#Write a function that implements Simpson's 1/3 rule. Then create a
#driver function that takes the same inputs plus a tolerance and
#increases the number of segments until the tolerance is met (in
#absolute relative error).

##############################################################
#Exercise 1
##############################################################
#Compute the Fourier coefficients of the function y = x on the
#interval [-pi, pi] (this is an example of a sawtooth wave).
#It will be easiest to use the trapz function to compute
#the trapezoid rule approximations of the sample point vectors.

##############################################################
#Exercise 2
##############################################################
#Construct a finite Fourier series approximation for the previous
#exercise, and plot it against your function. If you increase the
#number of terms, what happens to the approximation near the
#discontinuous points?