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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
"Scientific Computing: An Introductory Survey" by Michael T. Heath, 2nd Edition, McGraw-Hill, 2002
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.