Ch 1.3.4 Chapter Summary

Comments (Page 1 of 5)

  • We have established some basic information and introduced numerical analysis.

  • We have distinguished what we do in numerical analysis from symbolic computing, the realm of both computer algebra systems and pen and paper.

  • We have also introduced the basic data types and structure that provide R with the power and elegance necessary to do numerical analysis.

  • With a basic toolkit, we finally took a look at a number of algorithms to solve common problems in mathematics.

Comments (Page 2 of 5)

  • Using these tools, we are going to explore a number of problems that arise in applied mathematics across fields.

  • Before we get to more algorithms, we need to talk about error and what it means for the computer to be wrong.

  • The computer can be wrong for a lot of reasons, and some of them we can limit.

  • In other cases, there is little we can do other than recognize the problem and brace ourselves.

Comments (Page 3 of 5)

  • The algorithms for division and summation range from straightforward to very complex, and for good reason.
  • Each has a place and an application.
  • Sometimes an algorithm for a numerical problem will be very fast, but only apply in a subset of cases.
  • Or it may produce a result that is not as accurate.
  • Numerical analysis then moves from mathematics to an art, as we select the algorithms and tools necessary to meet our needs for the least amount of computing effort.
  • Sometimes meeting this requirement, however, requires more human and programming effort.

Comments (Page 4 of 5)

  • These sorts of analysis are the core of numerical analysis.
  • We want to find the best possible algorithm to suit the task at hand, but the task goes beyond the problem as stated.
  • It includes how fast a solution must be calculated, how accurate it must be, and how much time we have to implement a solution.
  • All of these come together as our constraints when solving numerical problems and each has a different aspect.
  • Project managers often say you can have a solution fast, cheap, or correct, provided you pick only two.
  • The same sort of constraints play out here.

Comments (Page 5 of 5)

  • The rest of this text will develop an understanding of error analysis, and the meaning of both precision and accuracy.
  • Then we will develop a set of tools for performing tasks in linear algebra, followed by a suite of tools for interpolation, of various types.
  • Then we will move into calculus and discuss algorithms for both differentiation and integration.