Computational Methods CMSC AMSC MAPL 460 Ramani Duraiswami Dept of Computer Science Course Goals Introduction to the use of scientific computing techniques to solve problems in various domains Understand principles behind algorithms Intelligent choice and use of available software Understand how to Convert a model into a discrete system on the computer How to deal with data Perform simulations for applications Display and evaluate simulation results Appreciate which computations are feasible Understand how erroneous the computations could be New Paradigm Scientific Discovery through Computing Paradigm A set of assumptions concepts values and practices that constitutes a way of viewing reality for the community that shares them especially in an intellectual discipline Engineering aeronautics fluid dynamics circuit design radar antennas signal processing Physics stellar dynamics materials Economics Sociology modeling and analyzing data computational statistics stock picking Biology biostatistics computational biology genomics and proteomics Chemistry molecular dynamics drug design activity recognition Computer Science modeling systems network performance information retrieval Your field Another paradigm Data driven science Grab data and process it Audio video text MRI X Ray weather strain gage flow gene chip seismograph Moore s law drives both processing power memory sensor cost and capability Moore s law Processor speed doubles every 18 months More generally Technology X capability will double in Y months Need algorithms to process larger and larger data sets and extract information from them Fit data Extract model parameters Learn relationships In general compute with the data The Course Two lectures a week Homework every week or other week 40 homework 25 exam 1 35 final Attendance participation will be a factor Class web site http www umiacs umd edu ramani cmsc460 index html Required Book Numerical Computing with MATLAB by Cleve Moler The good news The complete book is online Book is also not as expensive as some others 40 Software Matlab Matlab is essential Two choices Choice 1 Use university computers that already have Matlab on them Choice 2 Buy the student edition of Matlab for your personal use Can give pointers but cannot support the installation or access to the software Book comes with Matlab software that is downloadable from the book web site Some homework will use that software Homework Homework will involve programming in MATLAB mainly problems from the text Style Clarity Cleanliness of output will count Work Results must be easily understood to be interpreted Visualization graphs Commented code Remember the TA is a busy student doing his own courses research Syllabus Introduction Computer Arithmetic and Errors Chapter 1 approx 3 lectures course survey introduction to Matlab machine arithmetic and error analysis stability and conditioning Solving Linear Systems of Equations Chapter 2 approx 4 lectures Gaussian elimination well conditioning vs ill conditioning matrix and vector norms Notions of algorithm complexity sparse systems direct and iterative methods Syllabus Interpolation Chapters 3 approx 4 lectures polynomial interpolation Other basis functions and polynomials piecewise polynomial interpolation spline interpolation Zeros and Roots Chapter 4 approx 3 lectures Linear and Nonlinear systems of equations Bisection Secant and Newton method Introduction to optimization Solving Linear Least Squares Problems Chapter 5 approx 3 lectures data fitting and least squares QR factorization Syllabus Integration Quadrature Chapter 6 elementary integration formulas midpoint trapezoid etc compound and adaptive integration formulas Gaussian quadrature Fourier Analysis Chapter 8 Ordinary Differential Equations Chapter 9 approx 4 lectures ordinary differential equations and Euler s method adaptive methods for ordinary differential equations methods for stiff systems MATLAB Overview History of MATLAB Strengths of MATLAB Weaknesses of MATLAB What is MATLAB MATLAB MATrix LABoratory Interactive system Programming language Extendable 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 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 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 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 o Matrix Operations A A A A 3 A A A ans A 2 ans 0 8000 0 3600 0 4800 1 0000 0 0 0 0 2800 0 9600 0 0 9600 0 2800 Built in functions o Inverting a Matrix inv A ans o Determinant det A ans 1 000 o Rank rank A ans 3 0 0000 0 8000 0 6000 0 8000 0 3600 0 4800 0 6000 0 4800 0 6400 o Transpose of a Matrix A ans 0 0 8000 0 6000 0 8000 0 3600 0 4800 0 6000 0 4800 0 6400