Midterms and Final Review

CS/Math 166 -- D. C. Smolarski, S.J.
Santa Clara University, Department of Mathematics and Computer Science

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Sample Topics and Questions

This is not an exclusive list. It is merely meant to highly some of the major topics covered since the beginning of the course. In general, to prepare for the midterms and the final, check over homework problems and examples in the book and in the notes.

Depending on when midterms are scheduled, some topics referenced below may not yet have been covered in the lectures.

Page number cross-references to the class notes are indicated in brackets after each topic.

Sample questions for some of the following topics may be found at sampleexamquests.html. The questions listed all refer to examples in the class notes, and the full solutions may be found in the class notes.

  1. Convert from decimal numbers (including fractions) to binary and vice versa. [2-1], [2-2]
  2. Perform arithmetic using limited precision, e.g., 3 significant chopping or rounding. [2-3, 2-4]
  3. Avoid "Catastrophic cancellation" in certain expressions. [2-4], [2-4A]
  4. Perform "constrained" arithmetic, i.e., arithmetic using n digits and "chopping" or "rounding." [2-4], [2-4A], BF sec 1.2 HW
  5. Find the absolute, relative, and percent errors, given the approximate answer and the exact answer. [2-5]
  6. Use "interval arithmetic" to determine a new interval from given ones. Notes following [2-6]
  7. Detect and correct sources of errors. [2-4], [2-8]
  8. Re-write a polynomial via Horner's Rule. [2-9]
  9. Find the Taylor (Maclaurin) series expansion for a specified function. [2-10]
  10. Compute the order of convergence of a particular iterative method. [2-14]
  11. Be able to set up the key iteration step in various methods of finding roots of non-linear equations. [3-1]-[3-9]
  12. Find the next n intervals using the bisection method (or False Position method or Newton-Raphson method, or secant method) for a function. [3-5], [3-8], [3-9].
  13. Discuss the pros and cons of the various methods for finding roots of non-linear equations. [3-10]
  14. Perform matrix/vector arithmetic. [4-4], [4-5]
  15. Determine whether a matrix is singular. [4-12]
  16. Find the inverse of a 2 by 2 matrix. [4-13]
  17. Find the eigenvalues & eigenvectors of a 2 by 2 (3 by 3) matrix. [4-13]
  18. Find the characteristic polynomial of a matrix. [4-14]
  19. Determine whether a matrix is orthogonal. [4-15]
  20. Discuss (and find) the condition number of a matrix. [4-16A]
  21. Find the norm of a vector and a matrix. [4-16R]
  22. Discuss how many digits accuracy are possible given the condition number of a matrix and the arithmetic limitations of a computer. [4-16R]
  23. Find the normal equations for a linear system. [4-17]
  24. Solve a linear system using Gaussian Elimination. [4-18] - [4-24]
  25. Decompose a matrix into an LU-decomposition. [4-23]
  26. Given the time it takes to solve a certain sized system via Gaussian elimination, how long does it take to solve a different sized system. [4-24]
  27. Solve a linear system using Gaussian Elimination with Partial Pivoting. [4-28]
  28. Create the Jacobi/Gauss-Seidel equations for a linear system and perform several iterations. [4-37]
  29. Use the Gauss-Seidel iteration method to find several iterations for a linear system. [4-37]
  30. Find the interval of the eigenvalues of a matrix via Gerschgorin's Disk Theorem. [4-42]
  31. Use the Power Method to determine the largest eigenvalue for a matrix. [4-44]
  32. Find the linear (least squares) regression line for a set of data points. [5-4]-[5-5]
  33. Find the Lagrange interpolating polynomial for a set of data points. [5-13]
  34. Topic: finite difference operator conversions and equalities. [5-14]-[5-16]
  35. Compute forward (divided) difference tables. [5-18]
  36. Correct errors in data given the forward difference table. [5-26]
  37. Use the Newton Forward Difference formula to interpolate values. [5-31]
  38. What is a cubic spline curve, and why use one? [5-34]
  39. Prove/disprove orthogonality of a set of vectors/functions. [5-45]
  40. Topic: approximation of a known curve by Chebyshev polynomials. [5-47]
  41. Use the trapezoidal rule to approximate an integral. [6-4]
  42. Use Simpson's 1/3 rule to approximate an integral. [6-7]
  43. How many panels are needed to get a certain degree of accuracy via Simpson's rule? cf. [6-7], BF sec 4.2, Ex 2.
  44. Use Richardson's extrapolation to get a better approximation. [6-10]
  45. Use Romberg Integration to approximate the integral of a function, given a set of discrete values. [6-12]
  46. Create a new quadrature formula. [6-13]
  47. Use Gaussian quadrature to evaluate an integral. [6-14]
  48. Change an n-th order ODE into a coupled system of first order ODEs. [6-18]
  49. "Solve" an ODE-BVP by discretization techniques, i.e., set up the appropriate linear system in matrix form. [6-38A] and following.

This page is maintained by Dennis C. Smolarski, S.J. Email: dsmolarski "at" scu.edu
© Copyright 2017 Dennis C. Smolarski, SJ, All rights reserved.
Last changed: 24 March 2017.