104 ECS Engineering Computational Tools CIE Sections 002 & 007
Exercise 1: Plotting functions (to be done in lab as a learning exercise) The purpose of this exercise is to plot the function
f ( x) = x 3 + 5 x 2 x 6
so that its zeroes can be approximately found. (Note that this is a minor modification of the polynomial we have been using in lecture.) As usual, start off by putting your name and an identifier (ECS104, Lab5, Exercise 1) in cells A1 and A2 of Sheet1. Now, after skipping a line, make a table of x and f(x), with x between -2 and +2 (in steps of 0.2). Make sure to format the table properly (with column headers in bold font and a reasonable number of decimal places for the values). Now generate a presentation quality plot of f vs. x. Make a note in your worksheet as to the approximate values of the two zeroes. Save the spreadsheet. Exercise 2: Newtons Method (to be done in lab as a learning exercise) The purpose of this exercise is to use Newtons method to find the zeroes of the polynomial
f ( x) = x 3 + 5 x 2 x 6
On Sheet2, put your name and identifying information in cells A1 and A2. As we have been doing in lecture, we always start out by writing out the steps you will take (in pseudo-code). So, in the space below, write down a description as to how you will find the zeroes using Newtons method. Recall that Newtons method generates new values of x by the formula
x j +1 = x j
f (x j ) f ( x j )
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STOP HERE AND SHOW YOUR INSTRUCTOR YOUR WORK!!!!! The next step is to convert your written description into a flowchart. Do that in this space.
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STOP HERE AND SHOW YOUR INSTRUCTOR YOUR WORK!!!!! Finally, we are ready to use the computer. After skipping a line, start a table with the column headings j, xj, f(xj), f(xj), and change (bold font please). Start with a initial guess for x=1.5. [By looking at the plot in Exercise 1, which zero do you think Newtons method will find?]. Put formulae in the first rows of the table for f(xj) [see above], f(xj) [use calculus], and change. So, in the second row of the table, insert the above formula to get the new x; copy down the formulae for f(x), f(x), and change. Now you can continue the iteration by simply copying the last row of the matrix down until the zero is found. [How far did you have to copy it down?] Box your final answer (that is, for what value of x does f(x)=0 ?) If you wanted to get the other zero, you could adjust your initial guess for x (say to 2) and you should converge to the other zero. Save the spreadsheet again. STOP HERE AND SHOW YOUR INSTRUCTOR YOUR WORK!!!!!
Exercise 3: Bisection Method (to be done in lab as a learning exercise) The purpose of this exercise is to use the bisection method to find the zeroes of the polynomial
f ( x) = x 3 + 5 x 2 x 6
Start with pseudo-code for how you will solve the problem. Recall that the bisection method is similar to the old game that goes like this: I think of a number between 1 and 100 and you have to guess it. You guess the mid-point (50) and I say higher. Now you know the number is between 50 and 100. You repeat the process until you guess the number. Write your pseudo-code here:
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The flowchart goes here:
Now to the computer. Start on Sheet3 with your name and identifier in cells A1 and A2. After skipping a line or two, create column headings xleft, f(xleft), xmidl, f(xmidl), xrite, f(xrite), and change. In the following row, put you initial guess for xleft (say 0.0) and xrite (say a 2.0), formula to calculate xmidl, and formulae to compute f(xleft), f(xmidl), and f(xrite). The value for change can be computed using xrite-xleft. [Recall that we need to start with xleft and xrite such that their respective function values have opposite signs.]. The real trick in the following rows is to determine which values to use for xleft and xrite. If f(xleft) and f(xmidl) are different signs, then we know the root is between them and the new xleft should be the same as the old xleft and the new xrite should be the same as the old xmidl. On the other hand, if f(xleft) and f(xmidl) are the same sign, then we know the zero is between xmidl and xrite, so the new xleft should be the old xmidl and the new xrite should be the same as the old xrite. [Draw a picture to follow this!!] We can implement this in Excel via the if() worksheet function [see the help page for more information.] Specifically, in the location for the new xleft we should put the formula =if(f(xleft)*f(xmidl)<0,xleft,xmidl). [Note that the terms in italics are not to be inserted directly as I have written; instead, use their cell addresses!] What this if() does is check the product of f(xleft) and x(xmidl). If they have opposite signs, then the product will be negative and we should use the old xleft; otherwise we should use the old xmidl. Put the similar formula in the position for the new xrite. The other formulae on the first line can now be dragged down to the line containing the new xleft and new
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xrite. Finally, you can continue to drag the formulae down until you have converged to the zero. [How will you know this?] Box your final answer. Save the spreadsheet. STOP HERE AND SHOW YOUR INSTRUCTOR YOUR WORK!!!!!
Homework: 1. Plotting functions (case 1) Plot the functions
f1 ( x) = sin( x) x2 f 2 ( x) = 2 2
from x=0 to x=3. Make sure your plot is presentation quality. Determine (approximately) the value of x for which f1(x)=f2(x). Note this (in a box) on your worksheet. 2. Newtons method Use Newtons method for finding the value of x for which f1(x)=f2(x). Make sure that you include ps...