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• Chapter 4.2:

  • Errors in Polynomial Interpolation

    • The error in a polynomial approximation p(x) of a function f(x), where p(x) is constrained to be equal to f(x) at a given set of nodes x_i, will decrease as the number of nodes increases:
      1. always
      2. some of the time
      3. never
      Explain.
    • What is a reliable way to reduce the expected error in a polynomial approximation of a given function f(x)?
      1. use more nodes
      2. use a nonuniform spacing for the nodes
      3. both
      4. neither
      Explain.
    • Be able to compute the Chebyshev nodes for a given interval, and be aware that the Runge function f(x) = 1/(1 + x²) will be better approximated by the polynomial that interpolates a set of Chebyshev nodes than by the polynomial that interpolates a set of evenly spaced nodes.
    • Know how to determine an upper bound on the error in a polynomial approximation p(x) of a given function f(x) where p(x) interpolates f(x) at n distinct equally spaced nodes over a specified interval [a,b], given the length of the interval and the number of nodes.
    • Know how to determine an upper bound on the error in a linear interpolation of a simple trigonometric function from a table of values, given the accuracy of the table entries, and the distance between the nodes.
    • Know how to use the divided differences corollary to determine the degree of the lowest degree polynomial that interpolates all of the values in a given table.

• Chapter 4.3:

  • Estimating Derivatives and Richardson Extrapolation

    • Know how to combine two (or more) Taylor Series to achieve an estimate of the first derivative f'(x) of a function f(x).
    • Know how to combine two (or more) Taylor series to achieve an estimate of the second derivative f''(x) of a function f(x).
    • Be able to determine the error term for a formula for f'(x) that was achieved using a combination of multiple Taylor series. (see problems 4.3.1, etc.)
    • Be able to explain the basic process used in Richardson extrapolation.
    • Know how to use Richardson extrapolation to estimate the derivative of a given function f(x) at a point x₀, given a specific step size h and total number of iterations n.

• Chapter 5.1:

  • Numerical Integration using the Method of Upper and/or Lower Sums

    • Know how to approximate the definite integral of a function f(x) using the method of lower sums over a given partition P = {a = x₀<x₁<...<x_n = b} of the closed interval [a,b].
    • Know how to approximate the definite integral of a function f(x) using the method of upper sums over a given partition P = {a = x₀<x₁<...<x_n = b} of the closed interval [a,b].
    • Know how to approximate the definite integral of a function f(x) using the average of the methods of lower and upper sums over a given partition P = {a = x₀<x₁<...<x_n = b} of the closed interval [a,b].
    • Know how to determine a lower bound on the number of subintervals that would be needed in order to approximate the definite integral of a given function f(x) over a specific interval [a,b] to within a specified error bound using a uniform partition.
    • Likewise, be able to determine the bound on the error that would result from estimating the definite integral of a given function f(x) over a specified interval using a partition defined by a given number of evenly spaced points.

• Chapter 5.2:

  • Numerical Integration using the Trapezoid Rule

    • Know how to approximate the definite integral of a function f(x) over a given interval [a,b] using the basic trapezoid rule.
    • Know how to approximate the definite integral of a function f(x) over a given interval [a,b] using the composite trapezoid rule with n equal sized subintervals.
    • Know how to obtain an upper bound on the error in the approximation of the definite integral of a function f(x) over a given interval [a,b] obtained using the composite trapezoid rule with uniform spacing.
    • Know how to determine the minimum number of equally spaced points n that will be required to estimate the definite integral of a given function f(x) over a specified interval [a,b] to within a specified error bound, using the composite trapezoid rule with uniform spacing.
    • Know how to use the recursive trapezoid formula to efficiently compute a series of approximations to a definite integral.

• Chapter 5.3:

  • Estimating Definite Integrals using the Romberg Algorithm

    • Be able to use the Romberg algorithm to obtain an array of estimates of a specified definite integral.
    • Know how to determine whether the function to be integrated satisfies the smoothness criterion required for the use of the Romberg algorithm (i.e. know how to check to see if the Romberg algorithm is working).

• Chapter 6.1:

  • Numerical Integration using Simpson's Rule

    • Be able to use the basic Simpson's rule to estimate the definite integral of a function f(x) over an interval [a,b], and be able to determine an upper bound on the error in the approximation thus obtained.
    • Be able to use the composite Simpson's rule to estimate the definite integral of a function f(x) over an interval [a,b], and be able to determine an upper bound on the error in the approximation thus obtained.
    • Be able to use the adaptive Simpson's rule to estimate the definite integral of a function f(x) over an interval [a,b], and be able to determine when the desired level of accuracy has been obtained.
    • Be familiar with the basic Newton-Cotes Quadrature Rules, including the Midpoint Rule, the Trapezoid Rule, Simpson's 1/3 Rule, and Simpson's 3/8 Rule. Be able to explain how each of these the Newton-Cotes Rules works to approximate the value of the definite integral of a given function over a given interval. Be able to explain why the higher-order Newton-Cotes rules are generally not as widely used as the lower-order Newton-Cotes rules.
    • Know the difference between open rules and closed rules, including the circumstances under which one type might be more preferable to use than the other.

• Chapter 6.2:

  • Gaussian Quadrature

    • Be able to derive a quadrature rule that will give exact values for the definite integral of every polynomial of degree <= n, given a specified set of nodes over a specified interval. (see example 1 on page 230)
    • Be able to use the Gaussian Quadrature Theorem to determine the Gaussian quadrature formula that will give exact values for the definite integral of every polynomial of degree <= 2n+1 over a specified interval, given a specified number of nodes.
    • Know how to define the Legendre polynomials, and how to use them to obtain the nodes for Gaussian quadrature over the interval [-1,1].
    • Given a Gaussian quadrature formula over a standard interval (such as [-1,1]), know how to use a change of intervals to use this rule over a different interval.

• Chapter 7.1:

  • Naïve Gaussian Elimination

    • Be able to solve a system of linear equations using naïve Gaussian elimination
    • Be able to characterize the situations in which the naïve Gaussian elimination algorithm will fail; in cases where a solution exists, but the naïve Gaussian elimination algorithm cannot be used to find it, be able to determine whether (and how) the problem can be recast so that a solution can be obtained
    • Be able to describe the naïve Gaussian elimination algorithm in your own words (specifically, you should know that it consists of two phases - a forward elimination phase followed by a back-substitution phase and you should be able to say what is accomplished in each phase)
    • Understand what a residual vector is and how it is used. Be able to compute the residual vector r for a given approximate solution x to a given system of linear equations Ax=b. Be able to discuss the implications of a residual vector being relatively large or relatively small.

• Chapter 7.2:

  • Gaussian Elimination with Scaled Partial Pivoting

    • Be familiar with the concepts of complete pivoting, partial pivoting, and scaled partial pivoting, and be able to discuss the relative advantages and disadvantages of using each of these types of pivoting in conjunction with Gaussian elimination.
    • Know how to solve a system of linear equations using Gaussian elimination with scaled partial pivoting

• Chapter 7.3:

  • Tridiagonal and Banded Systems

    • Be able to identify a tridiagonal or banded system
    • Be able to explain why it is desirable to solve tridiagonal or banded systems with special-case methods
    • For a tridiagonal system of n equations with n unknowns, how much more efficient is it to use the special case algorithm (called Tri in our textbook) to solve the system rather than to use the general purpose algorithm (Naive_Gauss in our textbook)? [to answer this question you should count the number of arithmetic operations required in each case]
    • Be able to describe the conditions under which the special case solution Tri will be guaranteed to not fail
    • Be able to determine whether a matrix is diagonally dominant

• Chapter 8.1:

  • Matrix Factorizations

    • Be able to determine the LU factorization of a given matrix A
    • Know how the LU factorization of a coefficient matrix A can used to find the solution to the system of linear equations Ax = b. Be able to explain the advantages of using LU factorization in the solution of systems of linear equations.
    • Know the conditions under which an LDL^T factorization of a matrix can be found.
    • Be able to determine the LDL^T factorization of a matrix.
    • Know the conditions under which the Cholesky factorization of a matrix can be computed.
    • Be able to determine the Cholesky factorization of a matrix.
    • Be able to explain how to efficiently solve a series of systems of linear equations Ax_i = b_i that share the same coefficient matrix but have different right hand sides.
    • Be able to use Gaussian elimination to compute the inverse of a matrix A.