COSC 250 Introduction to Numerical Methods

  • Prerequisite:  COSC 110, MATH 121, or MATH 125

    Algorithmic methods for function evaluation, roots of equations, solutions to systems of linear equations, function interpolation, numerical differentiation and integration, and use spline functions for curve fitting.  Focus on managing and measuring errors in computation.   Also offered as MATH 250; either COSC 250 or MATH may be substituted for the other and may be used interchangeably for D or F repeats but may not be counted for duplicate credit.

    Course Outcomes

    Upon completion of this course, students will be able to

    1. Explain the role of and the limitations of the computer in solving mathematical and engineering problems.
    2. Implement mathematical algorithms to
      1. evaluate functions
      2. find approximate roots of equations
      3. solve systems of linear equations
      4. perform numerical differentiation and integration
      5. fit a curve to a set of data.
    3. Discuss selected numerical algorithms for solving a variety of commonly encountered mathematical problems.
    4. Analyze a computation for error and discuss the types and sources of errors involved.
    5. Explain how error accumulates and discuss the errors inherent in using standard floating point numbers.

    Detailed Course Outline
        
    A.  Errors in Computation — 0.5 week
    1.  Representational error
    2.  Computational error – relative and absolute
    3.  Computer rounding approaches

    B.  Taylor Series representation of a function — 1.5 week
    1.  Error term in the representation
    2.  Properties of alternating series
    3.  Appropriate and inappropriate applications

    C.  Representation of Numbers — 1.0 week
    1.  Conversion of integers: binary – decimal – hex
    2.  Conversion of floating point numbers: binary – decimal – hex
    3.  Properties of the IEEE standard floating point representation
    a.  Hole at zero
    b.  Implied leading bit
    c.  +/- infinity and Not-a-Number
    d.  Machine epsilon
    e.  Calculating roundoff error
    f.  Propagated error
    4.  Loss of significance
    5.  Loss of precision theorem
    6.  Techniques for avoiding loss of significance
    a.  Rationalization
    b.  Use of the Taylor series
    c.  Use of identities

    D.  Finding Roots of Equations — 2 weeks
    1.  Techniques
    a.  Bisection method
    b.  Newton's method
    c.  Secant method
    d.  Variation and hybrid methods
    2.  Analysis of convergence for each technique
    3.  Conditions under which convergence is a problem

    E.  Interpolation — 2 weeks
    1.  Lagrange's form for the interpolating polynomial
    2.  Newton's form for the interpolating polynomial
    3.  Evaluation
    4.  Divided differences algorithm
    5.  Inverse interpolation
    6.  Errors in interpolation
    7.  Theorems regarding error
    8.  Derivatives and divided differences
    9.  Richardson extrapolation
    10.  First and second derivatives using interpolation

    F.  Numerical Integration — 2.5 weeks
    1.  Using upper and lower sums
    2.  Trapezoidal rule
    3.  Error analysis
    4.  Recursive trapezoid formula
    5.  Romberg algorithm
    6.  Simpson's rule
    7.  Gaussian quadrature
    8.  Method of undetermined coefficients
    9.  Legendre polynomials

    G.  Systems of equations — 2.0 weeks
    1.  Gaussian elimination
    2.  Poorly conditioned matrices
    3.  Scaled partial pivoting
    4.  Tridiagonal systems
    5.  LU factorization
    6.  Iterative solution of linear equations
    a.  Gauss-Seidel method
    b.  Jacobi method

    H.  Splines for curve fitting — 1.5 weeks
    1.  First degree splines
    2.  Interpolation
    3.  Modulus of continuity
    4.  Second degree splines
    5.  Cubic splines

    In-class examinations — 1 week

    Total: 14 weeks

    Final Exam: During Final Exam week

    Evaluation Methods

    The final grade for the course will be determined as follows:

    65% Examinations — Three exams during the term and the final consisting of mathematical problems.

    35% Projects and homework — At least four computer projects will be assigned in addition to one or two sets of mathematical problems.  The projects and problems will be similar to those provided in the text being used.

    Example Grading Scale
    90%–100% — A
    80%–89% — B
    70%–79% — C
    60%–69% — D
    Below 60% — F

    Undergraduate Attendance Policy

    Class attendance is regarded as being very important.  Individual faculty members may establish penalties for excessive numbers of unexcused absences.  Excused absences will be allowed for illness, family emergencies, and involvement in university activities, such as sports.  The penalties specified will meet university guidelines and be distributed to students with the course syllabus on the first day of class.

    Required Textbooks, Supplemental Books and Readings

    Cheney & Kincaid, Numerical Mathematics and Computing, Fifth Edition, Brooks/Cole, 2004.

    Special Resource Requirements

    None