Sample Exam 1

Sample Exam 1 Solutions

You will be given a formula sheet containing some useful information.

Disclaimer: The exam covers all material presented in class even if omitted from the outline.

Chapter 0

• 0.1 - Horner's method for evaluating a polynomial
• 0.3 - floating point nubmers and floating point arith.
• 0.4 - computer errors - truncation and roundoff error; machine epsilon; loss of significant digits;

Chapter 1 - Nonlinear Equations

• 1.1 - bisection - error estimate
• 1.2 - fixed point interation - cobweb, convergence condition
• 1.3 - forward, backward error, condition number
• 1.4 - Newton's method - convergence, complex roots
• 1.5 - Secant method, idea of Muller's and Brent's Method
• multiple roots - change in convergence of Newton's method;

Chapter 2 - Numerical Linear Algebra

• - matrix notation and review - must be able to do calculations
• 2.1 and 2.2 - Gaussian elimination and LU decomposition using GE; computation count
• 2.3 - errors, norms, condition numbers
• 2.4 - reducing error in GE, partial pivoting, virtual scaling
• 2.5 - indirect methods - Jacobi and Gauss-Seidel itereation; conditions for convergence;

• Final Exam Thursday, July 19, 316 DH - Time: 10:30-12:30

  Final Exam Review Questions

  Formula Sheet Final Exam Version (to be updated)

  Tentative Topics (subject to change)
  • Nonlinear Equations - 10%
    • Newton's method and fixed point iteration
  • Linear Alegbra - 15%
    • Gaussian elimination
    • Power method and inverse power method
  • Interpolation - 15%
    • Lagrange form
    • Newton's form and error - nested multiplication
    • Cubic spline - what are they and why important
  • Differentiation and Integration - 40%
    • rules for first and second derivatives; errors
    • Richardson extrapolation
    • Newton-Cotes Rules - rectangular, trapezodal, Simpson's 1/3 and error estimates
    • Rhomberg integration
    • Gaussian quadrature
  • Differential equations - 20%
    • Euler's method, derivation and algorithm; error
    • Modified Euler
    • Systems, converting 2nd order equation, all methods for scalar equations
    • Runge-Kutta schemes
    • Error analysis
  • Random number generators and Monte Carlo simulation - ??%

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