Math 472/572 - Assignment 1 - University of Michigan
Math 472/572 - Assignment 2
Due: Tuesday, September 26. Nothing accepted after Thursday, September 28. 10% off for being late. Please work by yourself. See me if you need help.
1. (6 points) Suppose one measures x and computes y = sin( 10x ). Here sinu means sin of u radians. Suppose the following three measurements of values of x are made
i. x1a = 0.101 ( 0.0005
ii. x2a = 1.01 ( 0.005
iii. x3a = 10.1 ( 0.05
and one calculates the corresponding value of yja = sin( 10xja ). Use the function rule for error propagation to get an approximate upper bound for the relative error in the corresponding computed value of yja = sin( 10xja ). For each j determine by how many times the relative error in xj might be magnified to get the relative error in yj. You may assume your computer or calculator calculates the value of sin( 10xja ) exactly.
2. A certain application requires one to calculate y = sin(x + h) – sin(x) for various values of x and h where h is considerably smaller than x. Here sinu means sin of u radians. In particular, consider the case where x = 1 and h = 1/300.
a. (1 point) Find the value of y using your calculator or favorite piece of mathematical software.
b. (2 points) Find the value of y when you round the result of each operation to five significant decimal digits. (This means it is rounded to five digits starting with the first non-zero digit.) Use your calculator or computer to evaluate the sin function, but round the result to five significant digits also. What is the relative error assuming the value in a is the true value?
c. (2 points) Show that an equivalent formula for y is y = 2 sin( ) cos( x + ).
d. (2 points) Find the value of y using the equivalent formula in c. Again round the result of each operation to five significant decimal digits and use your calculator or computer to evaluate the sin function. What is the relative error assuming the value in a is the true value?
e. (1 point) Why is the value in d more accurate than the one in b?
3. (6 points) Suppose y = x2(3-x) – . Suppose we are going to evaluate y for various values of x between 0 and 1. Let ya be the value of y that we get doing the computation on a computer using floating point arithmetic with machine ( equal to (, i.e. the result of each arithmetic operation is rounded and the relative error between the mathematical result of the operation and the rounded value is no more than (. Estimate the maximum possible absolute error | y – ya | for x in the range 0 ( x ( 1. For simplicity you may assume that the starting value of x is exact, i.e. there is no rounding in the value of x. Your answer should be of the form C(, where C is a constant you are to find. Explain your work. Here are some things to keep on mind on this problem.
i. Work your way up from small to large. For example, when x2 is rounded, the result has a relative error of (. Suppose integers like 2, 3, 4 and are represented exactly with no error. Then x2/4 has a relative error of about ( before rounding. This is because relative errors approximately add when one multiplies or divides. Rounding the result introduces another relative error of e. These relative errors approximately add, so x2/4 has a relative error of about 2( after rounding. Continue in this fashion calculating the error in increasingly larger pieces.
ii. Keep in mind that when you add or subtract the absolute errors add. This is before rounding. Rounding makes another relative error of (. When you multiply or divide the relative errors approximately add, again not taking into account the rounding of the result.
iii. When you do the subtraction, you need to convert the error from relative form to absolute form. This is done by multiplying by the value.
iv. After you find the absolute error in x2(3 - x) - the expression will be of the form g(x) (. Make a graph of g(x) for 0 ( x ( 1. Use the graph to find the maximum value of g(x). This is C.
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