Linear Algebra/Topic: Accuracy of Computations/Solutions

SolutionsEdit

Problem 1

Using two decimal places, add   and  .

Answer

Sceintific notation is convienent to express the two-place restriction. We have  . The   has no apparent effect.

Problem 2

This intersect-the-lines problem contrasts with the example discussed above.

                    

Illustrate that in this system some small change in the numbers will produce only a small change in the solution by changing the constant in the bottom equation to   and solving. Compare it to the solution of the unchanged system.

Answer

The reduction

 

gives a solution of  .

Problem 3

Solve this system by hand (Rice 1993).

 
  1. Solve it accurately, by hand.
  2. Solve it by rounding at each step to four significant digits.
Answer
  1. The fully accurate solution is that   and  .
  2. The four-digit conclusion is quite different.
     
Problem 4

Rounding inside the computer often has an effect on the result. Assume that your machine has eight significant digits.

  1. Show that the machine will compute   as unequal to  . Thus, computer arithmetic is not associative.
  2. Compare the computer's version of   and  . Is twice the first equation the same as the second?
Answer
  1. For the first one, first,   is   and so  . For the other one, first   and so  .
  2. The first equation is   while the second is  .
Problem 5

Ill-conditioning is not only dependent on the matrix of coefficients. This example (Hamming 1971) shows that it can arise from an interaction between the left and right sides of the system. Let   be a small real.

 
  1. Solve the system by hand. Notice that the  's divide out only because there is an exact cancelation of the integer parts on the right side as well as on the left.
  2. Solve the system by hand, rounding to two decimal places, and with  .
Answer
  1. This calculation
     
    gives a third equation of  . Substituting into the second equation gives   so   and thus  . With those, the first equation says that  .
  2. The solution with two digits kept
     
     
    comes out to be  ,  , and  .

ReferencesEdit

  • Hamming, Richard W. (1971), Introduction to Applied Numerical Analysis, Hemisphere Publishing .
  • Rice, John R. (1993), Numerical Methods, Software, and Analysis, Academic Press .