AMS 527, Numerical Analysis II 
Numerical methods based upon functional approximation: polynomial interpolation and approximation; and numerical differentiation and integration. Solution methods for ordinary differential equations. AMS 527 may be taken whether or not the student has completed AMS 526. 
3 credits, ABCF grading 

Required Text:
"Scientific Computing: An Introductory Survey" by Michael T. Heath, Revised 2nd Edition, SIAM, 2018. ISBN#: 978-1611975574

AMS 527 Instructor page

 
Learning Outcomes:

1) Build understanding of fundamentals of numerical approximations:
      * Classification of sources of errors;
      * Effect of floating-point arithmetic;
      * Accuracy and stability.

2) Master concepts and numerical methods for solving nonlinear equations:
      * Methods for nonlinear equations in 1-D: interval bisection method, fixed-point iteration, Newton’s method, secant method;
      * Methods for nonlinear equations in n-D: Newton’s method, Newton-like method;
      * Sensitivity, convergence rates, and stopping criteria.

3) Build fundamental understanding of concepts and numerical methods for optimization:
      * Unconstrained vs. constrained optimization, global vs. local minimum, convexity, optimality conditions;
      * Algorithms for unconstrained optimization in 1-D and n-D: golden section search, Newton’s method, Quasi-Newton methods, steepest descent, and conjugate radient;
      * Algorithms for constrained optimization: Lagrange multiplier.

4) Build fundamental understanding of interoperation and approximation:
      * Interpolation versus approximation, basis functions, convergence, Taylor polynomial;
      * Polynomial interpolation, piecewise polynomial interpolation, orthogonal polynomial interpolation, lease squares approximations;
      * Trigonometric interoperation.

5) Master concepts and numerical methods for numerical integration and differentiation:
      * Newton-Cotes rules, Gaussian quadrature rules, change of interval;
      * Derivation with method of undetermined coefficients and orthogonal polynomials;
      * Finite difference approximation, forward difference, backward difference, and centered difference.

6) Master basic numerical methods for initial-value and boundary-value problems:
     * Stability of solutions of ODEs; global error vs. local error; stiffness; explicit vs. implicit methods; analysis of stability;
     * Basic algorithms/schemes and their derivations: Euler’s methods (forward and backward); trapezoid method; Heun’s method; fourth-order Runge-Kutta method;
     * Finite-difference methods and finite element methods.

7) Demonstrate programming skills for numerical methods.