SyDe312  Numerical Methods Lecture Topics and Notes 
Stephen
Birkett

Research
Teaching
Linear Algebra
Numerical Methods Topics and Lectures Assignments Tests Computational Mathematics Musical Instruments Music

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  
2.12.4 1.11.3 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 2.32.4 for background.]  
6.16.6 7.2 supplement  7  UNIT I  LINEAR ALGEBRA  
General introduction (6.16.2). Direct solution methods: Gaussian and GaussJordan elimination with pivoting (6.3), matrix factorizations  LU (6.4), Cholesky (slides) & QR (unit III). Quantifying inaccuracy, conditioning (6.5). Iterative solution methods: Jacobi & GaussSiedel (6.6). Iterative improvement (slides). Overdetermined 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.  
3.13.5 7.3  3  UNIT II  ROOT FINDING AND NONLINEAR SYSTEMS  
Basic strategy and tactics required for finding roots effectively. Simple methods: fixedpoint 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), NewtonRhapson method (7.3)  
7.1, 4.1 & 4.3; much extra material in lecture slides  4  UNIT III  CURVE FITTING AND INTERPOLATION  
Curvefitting: linear leastsquares 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/pseudoinverse (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 4.44.6. NOT Bezier curves or Bsplines.  
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 newtonCotes 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. GaussLegendre quadrature, including with general limits. GaussLaguerre 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 onestep 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). RungeKutta methods, and especially RK4 (8.5). Implementation in matlab. (NOT any of: 8.1.x, 8.3, 8.4except 8.4.2, 8.5.2, 8.6)  