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5.24. Numerical Methods (Mandatory)

5.24.1. Justification ↑ Back to top

Numerical methods are a fundamental tool shared by computer science and engineering disciplines for approximating solutions to mathematical problems that cannot be solved analytically. This course provides an introduction to the most common numerical methods, including error analysis, solution of nonlinear equations, interpolation, numerical integration, and solution of differential equations, in scientific and engineering contexts.

5.24.2. Generales Goals ↑ Back to top

1. Understand the importance of numerical methods in solving scientific and engineering problems.
2. Apply different numerical methods to approximate solutions to mathematical problems.
3. Analyze the accuracy and efficiency of the numerical methods used.

5.24.3. Contribution to Outcomes ↑ Back to top

AG-C08) Problem Analysis: Identifies, formulates, and analyzes complex computing problems. (Usage)

AG-C11) Use of Tools: Applies modern computing tools in problem solving. (Usage)

5.24.4. Content ↑ Back to top

5.24.4.1. Error Analysis and Floating-Point Arithmetic (6 hours) [Skills AG-C08,AG-C11] ↑ Back to top

Bibliography: (Chapra and Canale, 2015; Burden and Faires, 2010)

Topics

1. IEEE 754 floating-point representation: machine epsilon, overflow, underflow, and rounding modes
2. Rounding errors, catastrophic cancellation, and loss of significance
3. Forward and backward error analysis; condition number of a problem
4. Numerical stability: stable vs. unstable algorithms; Wilkinson's backward error analysis
5. Error propagation in arithmetic operations and function evaluation

Learning Outcomes

1. Explain the IEEE 754 floating-point standard and identify the sources of rounding and cancellation errors [Familiarity]
2. Estimate the condition number of a problem and predict the accuracy of a numerical result [Usage]
3. Analyze an algorithm for numerical stability using forward and backward error bounds [Assessment]

5.24.4.2. Root-Finding for Nonlinear Equations (10 hours) [Skills AG-C08,AG-C11] ↑ Back to top

Bibliography: (Chapra and Canale, 2015; Burden and Faires, 2010)

Topics

1. Bisection method: interval bracketing, stopping criteria, and error bound
2. Newton-Raphson method: derivation, quadratic convergence, and failure cases
3. Secant method as an alternative to Newton's method without an explicit derivative
4. Order of convergence and efficiency comparison among iterative methods
5. Extension of Newton's method to systems of nonlinear equations via the Jacobian matrix, the basis of solvers used in machine learning

Learning Outcomes

1. Compare the convergence guarantees of the bisection, Newton-Raphson, and secant methods [Familiarity]
2. Implement the Newton-Raphson method to approximate roots of a nonlinear function [Usage]
3. Analyze the order of convergence of an iterative method and justify its choice based on computational cost [Assessment]

5.24.4.3. Approximation Theory and Interpolation (10 hours) [Skills AG-C08,AG-C11] ↑ Back to top

Bibliography: (Chapra and Canale, 2015; Burden and Faires, 2010)

Topics

1. Polynomial interpolation: Lagrange, Newton divided differences, and Runge's phenomenon
2. Spline interpolation: cubic splines, B-splines, and piecewise polynomial methods
3. Best approximation in normed spaces: Chebyshev (minimax) and least squares
4. Chebyshev polynomials: properties, orthogonality, and spectral convergence
5. Trigonometric approximation and the fast Fourier transform (FFT)

Learning Outcomes

1. Explain Runge's phenomenon and justify the choice of Chebyshev nodes to mitigate it [Familiarity]
2. Construct cubic spline and Chebyshev interpolants for given data and estimate the interpolation error [Usage]
3. Apply the FFT to efficiently compute trigonometric approximations of a sampled function [Assessment]

5.24.4.4. Numerical Integration and Quadrature (10 hours) [Skills AG-C08,AG-C11] ↑ Back to top

Bibliography: (Chapra and Canale, 2015; Burden and Faires, 2010)

Topics

1. Newton-Cotes rules: trapezoidal, Simpson's, and composite rules; error analysis
2. Adaptive quadrature and automatic error control
3. Gaussian quadrature: optimal nodes and weights, orthogonal polynomial connection
4. Numerical treatment of improper and singular integrals
5. Monte Carlo integration and quasi-Monte Carlo methods for high-dimensional integrals

Learning Outcomes

1. Derive the error formula for composite Simpson's rule and identify its order of accuracy [Familiarity]
2. Select and apply an appropriate quadrature rule (Gaussian, adaptive) based on integrand regularity [Usage]
3. Apply Monte Carlo integration to estimate high-dimensional integrals and quantify the statistical error [Assessment]

5.24.4.5. Numerical Methods for Differential Equations (12 hours) [Skills AG-C08,AG-C11] ↑ Back to top

Bibliography: (Chapra and Canale, 2015; Burden and Faires, 2010)

Topics

1. Runge-Kutta methods: Euler, RK4, and embedded methods for error control
2. Linear multistep methods: Adams-Bashforth, Adams-Moulton, and BDF methods
3. Stability analysis: zero-stability, absolute stability regions, and stiff ODEs
4. Finite difference methods for parabolic and elliptic PDEs: stability and convergence

Learning Outcomes

1. Compare explicit and implicit ODE solvers and explain their relative stability for stiff problems [Familiarity]
2. Implement a Runge-Kutta method with adaptive step control and apply it to a system of ODEs [Usage]
3. Analyze the stability and convergence of a finite difference scheme for a parabolic PDE [Assessment]

5.24.5. Bibliography ↑ Back to top

Chapra, S. C. and Canale, R. P. (2015). Numerical Methods for Engineers. McGraw-Hill Education.

Burden, R. L. and Faires, J. D. (2010). Numerical Analysis. Cengage Learning.