1 27 2012 Computational Methods CMSC AMSC 460 Ramani Duraiswami Dept of Computer Science Syllabus Introduction Computer Arithmetic and Errors Chapter 1 approx 4 lectures course survey introduction to Matlab machine arithmetic and error analysis stability and conditioning Solving Linear Systems of Equations Chapter 2 approx 6 lectures Gaussian elimination well conditioning vs ill conditioning matrix and vector norms Notions of algorithm complexity sparse systems direct and iterative methods 1 1 27 2012 Syllabus Interpolation Chapters 3 approx 6 lectures polynomial interpolation Other basis functions and polynomials piecewise polynomial interpolation spline interpolation Zeros and Roots Chapter 4 approx 4 lectures Linear and Nonlinear systems of equations Bisection Secant and Newton method Introduction to optimization Solving Linear Least Squares Problems Chapter 5 approx 4 lectures data fitting and least squares QR factorization Syllabus Integration Quadrature Chapter 6 6 lectures elementary integration formulas midpoint trapezoid etc compound and adaptive integration formulas Gaussian quadrature Fourier Analysis Chapter 8 3 lectures Ordinary Differential Equations Chapter 9 approx 6 lectures ordinary differential equations and Euler s method adaptive methods for ordinary differential equations methods for stiff systems 2 1 27 2012 MATLAB Overview History of MATLAB Strengths of MATLAB Weaknesses of MATLAB What is MATLAB MATLAB MATrix LABoratory Interactive system Programming language Extendable 3 1 27 2012 What is MATLAB 2 Considering MATLAB at home Standard edition Available for roughly 2 thousand dollars Student edition Available for roughly 1 hundred dollars Some limitations Shorter license period On campus Site license History of MATLAB Ancestral software to MATLAB Fortran subroutines for solving linear LINPACK and eigenvalue EISPACK problems 4 1 27 2012 History of MATLAB con t 2 One of the developers of these packages Cleve Moler wanted his students to be able to use LINPACK and EISPACK without requiring knowledge of Fortran MATLAB developed as an interactive system to access LINPACK and EISPACK History of MATLAB con t 3 MATLAB gained popularity primarily through word of mouth because it was not officially distributed In the 1980 s MATLAB was rewritten in C with more functionality such as plotting routines Commercialized by a company The Mathworks In many fields it is the software for quantitative analysis Finance biology defence image processing audio etc Some competing packages Octave an open source alternative Mathematica IDL 5 1 27 2012 Strengths of MATLAB MATLAB is relatively easy to learn MATLAB code is optimized to be relatively quick when performing matrix operations MATLAB may behave like a calculator or as a programming language MATLAB is interpreted errors are easier to fix Although primarily procedural MATLAB does have some object oriented elements Weaknesses of MATLAB MATLAB is NOT a general purpose programming language MATLAB is usually used as an interpreted language making it for the most part slower than a compiled language such as C MATLAB is designed for scientific computation and is less suitable for some things such as parsing text Many of these weakness have been addressed somewhat 6 1 27 2012 Matlab Windows o Command line Interface Main Window o Editor Window o Present Directory o Directory Contents and Workspace variables o Command line o Command History Matrices in Matlab o Entering a Matrix A 0 0 8 0 6 0 8 0 36 0 48 0 6 0 48 0 64 A A 1 ans 0 0 8000 0 6000 0 0 8000 0 6000 0 8000 0 3600 0 4800 0 6000 0 4800 0 6400 o Matrix referencing A 1 2 ans 0 8000 A 2 ans 0 8000 0 3600 0 4800 o Matrix Operations A A A A 3 A A A ans 1 0000 0 0 0 0 2800 0 9600 0 0 9600 0 2800 7 1 27 2012 Built in functions o Inverting a Matrix inv A o ans Determinant det A ans 0 0000 0 8000 0 6000 0 8000 0 3600 0 4800 0 6000 0 4800 0 6400 1 000 o o Transpose of a Matrix A Rank rank A ans ans 0 0 8000 0 6000 0 8000 0 3600 0 4800 0 6000 0 4800 0 6400 3 Solving Linear System o Linear system of algebraic equations x1 x2 2x3 2 3x1 x2 x3 6 x1 3x2 4x3 4 Ax b A 1 1 2 3 1 1 1 3 4 A 1 1 3 1 1 3 2 1 4 b 2 6 4 b 2 6 4 rank A ans 3 x A b also could do inv A b but not recommended x 1 0000 1 0000 2 0000 8 1 27 2012 Plotting a function y1 7x 0 6 x y2 5x 0 08 x x 0 0 01 5 y1 7 x 0 6 x y2 5 x 0 08 x plot x y1 x y2 legend y1 y2 Introduction to MATLAB Vectors Matrices Syntax Vector operations including the dot commands length size linspace logspace size rand randn randperm Special vectors and matrices zeros ones eye magic Scripts and functions Diary Graphing plot special fonts plot3 semilogx semilogy title xlabel ylabel axis grid legend subplot Formatted output Sprintf disp input Programming for if while General misc commands ginput set size max sum close figure hist any all floor fix round Graphical programming and callbacks 9