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• Chapter 1.1:

  • Significant Digits of Precision, Absolute and Relative Errors

    Know what these terms mean, and the contexts within which each type of error measure is most useful.

    If I give you a number and an approximation to that number, you should be able to tell me what the absolute and relative errors are in the approximation.

  • Rounding and Chopping

    If I give you a number, you should be able to round it to n digits or chop it to n digits. Be sure to remember the best practice rule for rounding numbers when the discarded portion has exactly one significant digit and that digit is 5.

  • Nested Multiplication

    If I give you a function, you should be able to rewrite that function in nested form.

    You should be able to use synthetic division to deflate a polynomial, given one of its roots.

• Chapter 1.2:

  • Review of Taylor Series

    Understand basic concepts such as what a Taylor series represents, and the significance of the point of expansion. Be able to derive the Taylor series for various functions at different points of expansion. Note that this will require that you know how to compute derivatives of simple functions, that you know how to apply things like the chain rule, etc. Be able to determine the range of values over which a Taylor Series can (or cannot) be used to approximate a function.

  • Taylor's Theorem

    Know what it represents. Be able to use it to determine the range of x over which a particular Taylor series can be used to represent a particular function f(x) (an example is given in our textbook on pages 25-26).

  • Mean Value Theorem

    Be familiar with the Mean Value Theorem and understand its implications.

  • Alternating Series Theorem

    Understand it and its implications. Know when it applies. Be able to use it to determine how many terms will be required in a particular Taylor Series expansion in order to approximate a function f(x) at a given x within a specified error tolerance.

• Chapter 2.1:

  • Normalized Floating Point Representation

    If I give you a number, you should be able to express it in a normalized floating point representation.

  • Machine Numbers

    Understand the concept that there are many real numbers that cannot be represented exactly on a digital computer. Know what numbers it will and won't be possible to store exactly on a computer, given the specification of how it stores numbers. Know about the hole at zero. Know the types of errors in number representation that can occur on the computer (i.e. overflow, underflow, etc.). Be able to quantify the error in a machine number approximation to a given real number.

  • IEEE Single-Precision Floating Point Number Representation

    If I give you a decimal number, you should be able to tell me its IEEE floating point machine representation, in hex format, and vice versa. Be sure that you are familiar with the IEEE floating point machine representations for quantities such as positive and negative infinity, NaN, and positive and negative zero. Know what a subnormal number is.

  • Computer Errors in Representing Numbers

    Given the specification of a floating point machine representation, be able to determine the machine epsilon, the smallest and largest positive numbers that can be represented in normalized form, and the smallest and largest numbers of any kind that can be stored using that representation.

    Understand how errors can accumulate through sequences of mathematical operations on a computer.

• Chapter 2.2

  • Loss of Significance

    Know how loss of significance can occur. Be familiar with the "loss of precision theorem" and how it can be used to determine a bound on the number of significant bits that will be lost as a result of various operations Be able to use various strategies to rewrite expressions to avoid loss of significance in the evaluation on a computer of certain functions near certain points. Understand what "range reduction" is and how/when it occurs.

• Chapter 3:

  • Locating Roots of Equations

    Know the five root-finding algorithms discussed in this chapter: the bisection method, the method of false position, the modified method of false position, Newton's method, and the secant method, and be able to use each of them to find an approximation to a root of a given function within a specified accuracy range. Know the strength/weaknesses of each approach. Be able to determine for a given function when each method may be applicable and when it won't be. Be able to describe the possibilities for convergence or lack of convergence when each method is used. Know how quickly each of these methods can converge when it does converge.

• Chapter 4.1:

  • Polynomial Interpolation

    • Know how to determine the Lagrange form of the interpolating polynomial for a given table of values (x, f(x)). Understand the characteristics of the basis functions used in the Lagrange representation (the Lagrange cardinal polynomials).
    • Know how to determine the Newton form of the interpolating polynomial for a given table of values (x, f(x)).
    • Know how to compute a divided differences table and use it to determine the coefficients of the Newton polynomials used as the basis functions in the Newton form of the interpolating polynomial for a given table of values (x, f(x)).
    • Be familiar with other possible basis functions that can be used to represent an interpolating polynomial, such as the monomials, and the Chebyshev polynomials. Be sure to remember that although a particular polynomial may be represented in a variety of different ways, i.e. as a linear combination of different kinds of basis functions, the identity of the polynomial is not affected by the choice of basis functions.
    • Be able to determine the degree of the lowest degree polynomial that interpolates a given table of numbers.
    • Understand the existence and uniqueness theorems for interpolating polynomials. This will enable you to answer questions such as these:
      • Given a set of n tabulated points (x,y) under what conditions is it guaranteed that there exists a polynomial that passes through all of these points? Under these conditions, what is the highest possible degree of the unique polynomial that interpolates the n given points?
      • Is it possible that there exists a polynomial of degree 2 that interpolates 5 distinct tabulated points (x,y)?
      • Is it possible that there exists a polynomial of degree 6 that interpolates 3 distinct tabulated points (x,y)?
      • Is it possible that there exist two different polynomials of degree 2 that interpolate the same set of 5 distinct tabulated points (x,y)?
      • Is it possible that there exist two different polynomials of degree 6 that interpolate the same set of 3 distinct tabulated points (x,y)?

    Be familiar with the concept of inverse interpolation, and the circumstances under which it can be used to estimate the root of a function.

Topics that are mentioned in the book but which will not be queried in the exam:
• Achieving good programming practices
• Matlab syntax
• Using Newton's method with systems of non-linear equations
• Fractal basins of attraction