hw_interpolation

Homework for Garcia 1.2-1.4, functions/interpolation/etc

Contents

• Tutorials
• Problem 1
• Problem 2
• Problem 3
• Problem 4

Tutorial 2 Exercises

For your in-class work, read tutorial 2 and complete the items marked as "Exercise". You'll be working with interpolation.m

• For loops
• While loops
• If/else statements
• Lagrange polynomial function
• Writing a new function

The problems below are homework -- continue editing interpolation.m as you solve these problems.

Problem 1

A simple population growth model for some species can be approximated with the logistic equation

x_(n+1) = ρ x_n (1 - x_n)

where x_n is the current population density and x_(n+1) is the density some time later. The population density has a maximum value of 1.0 and a minimum value of 0.0 (upon extinction). ρ is a parameter that influences the population growth.

As an example, consider the case in which the initial population density is 0.5 and n is in years. Write a program to:

• Consider population growth parameters of ρ = 0.7, 1.4, 2.8, 3.1, 3.5, 3.7.
• Generate graphs of x_n versus n for each growth parameter with n = 0, 1, … , 50.
• Use the subplot command to graph each case in a separate subplot.
• Use num2str along with legend to produce a legend in each subplot identifying the growth parameter value.

Problem 2

Consider the following compounding interest formula')

I = V[(R/100)/k]

where V is the current value of the investment, R is the annual interest rate, k is the number of compounding periods within a year and I is the interest earned over the most recent compounding period (that is added to the investment).

Write a program to calculate the period-by-period investment value for an interest rate of 10% and for different compounding periods.

• Consider investments that compound annually (k=1), monthly (k=12), weekly (k=52) and daily (k=365). Assume that there are no leap years.
• In each case, use a while loop to calculate the increasing investments until the initial investment quadruples.
• Plot the growth in the investment versus time for each case on a single plot. Be careful that the time scales are consistent.

Problem 3

Numerical derivative of f = x*cos(x).

• Calculate f for x = 0:pi/20:3*pi.
• Use a for loop along with if-elseif-else statements to numerically calculate the derivative df/dx at each point. Calculate the derivative for the first point (i.e., x=0) based on the first two values. Calculate the derivative for the last point (i.e., x=3*π) based on the last two values. Calculate the derivative for interior points based on the surrounding two values.
• Calculate the exact value for df/dx.
• Plot the numerical (solid) and exact (dashed) values for df/dx.
• In a separate figure, plot the difference between the numerical and exact values for the derivative.

Problem 4

Modify the interpolation program (interp.m and intrpf.m) to evaluate a Lagrange polynomial for any number of (x,y) pairs. Prompt the user to enter the number of points (num) and the (x,y) pairs.

Demonstrate that your program works for num=4, with the following (x,y) pairs: (0,1), (1,0), (3,4), (5,3).

Published with MATLAB® R2014b