SyDe312 - Numerical Methods
Lecture Topics and Notes
Lecture Topics and Notes
Stephen
Birkett
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SyDe312 - Numerical Methods Lecture Topics and Notes |
Stephen
Birkett
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Research
Teaching
Linear Algebra
Numerical Methods Topics and Lectures Assignments Tests Computational Mathematics Musical Instruments Music
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The outline below will be expanded continuously as we go through the course, giving details of specific topics covered and references to the text sections required in addition to the material in the lecture notes. Links to pdf files of the lecture slides will become active as these are available for downloading. | ||||
TEXT REFERENCES | APPROXIMATE # LECTURES | DETAILED TOPICS | |||
2.1-2.4 1.1-1.3 Gilat intro FPV applet Cleve's corner | 3 | COURSE INTRODUCTION | |||
| UNIT 0 - NUMERICAL COMPUTATION | |||||
| Digital representation of numbers. Integers. Floating point arithmetic [including IEEE 754 standard]. Implications for routine calculations. Sources of errors. Introduction to Matlab. [Read 2.3-2.4 for background.] | |||||
6.1-6.6 7.2 supplement | 7 | UNIT I - LINEAR ALGEBRA | |||
| General introduction (6.1-6.2). Direct solution methods: Gaussian and Gauss-Jordan elimination with pivoting (6.3), matrix factorizations - LU (6.4), Cholesky (slides) & QR (unit III). Quantifying inaccuracy, conditioning (6.5). Iterative solution methods: Jacobi & Gauss-Siedel (6.6). Iterative improvement (slides). Over-determined systems: singular value decomposition SVD (slides). Finding eigenvalues (7.2 - not 7.2.4). Some of this material is not covered in the text - see lecture slides and supplementary problems. | |||||
3.1-3.5 7.3 | 3 | UNIT II - ROOT FINDING AND NON-LINEAR SYSTEMS | |||
| Basic strategy and tactics required for finding roots effectively. Simple methods: fixed-point iteration (3.4 including corollary 3.4.3 - not Aitken error est., not contraction mapping theorem), bracketing and bisection (3.1). Interpolation methods: secant (3.3- not error analysis) and regula falsi (not in text), Newton’s method (3.2 - not error analysis, not error estimation). Special tactics for polynomials (3.5+lecture slides). Laguerre's method (not in text). Nonlinear systems: fixed point iteration (not in text, see also 3.4), Newton-Rhapson method (7.3) | |||||
7.1, 4.1 & 4.3; much extra material in lecture slides | 4 | UNIT III - CURVE FITTING AND INTERPOLATION | |||
| Curve-fitting: linear least-squares problem (7.1+lecture slides extra), linearizing transformations and arbitrary basis functions (lecture slides), three LS solution methods - normal equations (7.1), QR decomposition and SVD/pseudo-inverse (lecture slides). Interpolation: polynomial interpolation (4.1) with different basis functions - mononomials, Lagrange polynomials, Newton polynomials & divided differences (all 4.1); polynomial wiggle at high order (4.2.2, NOT 4.2 in general), piecewise polynomial interpolation (4.3), cubic splines including different endpoint conditions. NOT 4.4-4.6. NOT Bezier curves or B-splines. | |||||
| 5.1, 5.3, 4.7.1, 5.4; much extra material in lecture slides | 4 | UNIT IV - INTEGRATION AND DIFFERENTIATION OF FUNCTIONS | |||
| Quadrature: classical newton-Cotes formulas (5.1+lecture notes - only important formulas indicated in the lecture need be recalled), generalization in terms of weights and nodes, techniques for improper integrals, variable node spacing, Gaussian quadrature (5.3+lecture notes), orthogonal polynomials (4.7.1+lecture notes). Legendre polynomials. Gauss-Legendre quadrature, including with general limits. Gauss-Laguerre qaudrature. Matlab quadrature techniques. Numerical differentiation: (5.4) Finite differences (forward, backward, central). Generalzied nodes and weights. Smoothing methods (Lagrange and newton interpolation; cubic splines). NOT 5.2 | |||||
| (8.1), 8.2, 8.4.2, 8.5 lecture
slides | 3 | UNIT V - ORDINARY DIFFERENTIAL EQUATIONS | |||
| General background (8.1). Initial value problems. Explicit one-step methods: Euler (8.2), midpoint (lecture slides). Difference between LTE and GE errors. General knowledge of what an implicit method is, e.g. trapezoidal (8.4.2). General knowledge about Taylor methods (order). Runge-Kutta methods, and especially RK-4 (8.5). Implementation in matlab. (NOT any of: 8.1.x, 8.3, 8.4-except 8.4.2, 8.5.2, 8.6) | |||||
