Syllabus of Numerical Analysis of Different Universities Introduction to Numerical Analysis

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1 Syllabus of Numerical Analysis of Different Universities In this appendix we give the syllabus for the courses of numerical analysis held in different universities of USA, UK, Saudi Arabia, and others. ROWAN UNIVERSITY, Department of Mathematics Syllabus Introduction to Numerical Analysis CATALOG DESCRIPTION: Numerical Analysis 3 s.h. (Prerequisites: Linear Algebra, Ordinary Differential Equations (or concurrently) and prior computer programming experience) This course includes: elements of error analysis, real roots of an equation, polynomial approximation by finite difference and least square methods, interpolation, quadrature, numerical solution of ordinary differential equations, and numerical solutions of systems of linear equations. The student should expect to program a computer in addition to using a graphing calculator. OBJECTIVES: The purpose of numerical analysis is two-fold: (1) to find acceptable approximate solutions when exact solutions are either impossible or so arduous and time-consuming as to be impractical, and (2) to devise alternate methods of solution better suited to the capabilities of computers. While this course will involve the student in considerable computation in order to apply techniques and obtain acceptable answers, the main emphasis will be on the underlying theory. It will be necessary to draw upon a good bit of calculus, linear algebra, computer science and other branches of mathematics during the course. CONTENT: 1. Errors in Computation 1

2 2. Finding Roots of Equations by Approximation 2.1 Graphical and other rough methods 2.2 Methods of refinement, false position, iteration 2.3 Newton-Raphson method 3. Finite Differences and Polynomial Approximations 3.1 Finite differences, definition and theorems 3.2 Approximating polynomials, Gregory-Newton formula 3.3 Interpolation and extrapolation of tables 3.4 Error Analysis 4. Finite Integration 4.1 Finite integrals, definition and theorems 4.2 Summation of series 4.3 Quadrature formulas, Trapezoidal, Simpson, Weddle rules. 4.4 Richardson Extrapolation and Romberg Integration 5. Solutions of Systems of Equations 5.1 Scaled Gaussian Elimination 5.2 The Gauss-Seidel and Jacobi Iterative Methods Additional topics may be selected, as time permits, from: Approximation by Least Square Method Numerical Solution of Differential Equations Fractal and Chaos TEXTS: The following might be possible texts for this course: 1.K. Atkinson and W. Han, ELEMENTARY NUMERICAL ANALYSIS, John Wiley, 3rd edition 2. Burden, R.L. and Faires, D.F., Numerical Analysis, 5th ed. PWS-Kent, Boston, MA. 3. Cheney, Ward and Kincaid, David, Numerical Analysis and Computing, 2nd ed., Brooks/Cole, Pacific Grove, CA. 4. Marion, M.J., Numerical Analysis, A Practical Approach, Macmillian, New York, NY. 2

3 Math-254 (NUMERICAL ANALYSIS) TEXTBOOK: Lecture Note of Numerical Analysis Using MATLAB AUTHOR: Rizwan Butt Note: The Contents of the course will be covered by the following sections: CHAPTER 2: 2.1,2.2,2.4,2.5,2.6,2.7,2.8,2.9. CHAPTER 3: 3.1,3.2,3.3,3.4,3.6,3.7. CHAPTER 4: 4.1,4.2,4.3. CHAPTER 5: 5.1,5.2,5.3,5.5,5.6. CHAPTER 6: 6.1,6.2,6.3. Theorems with Proofs: Theorem 2.2,3.26,5.2,5.3,5.4. Note: The proofs of Linear Lagrange formula+unique Lagrange polynomial+trapezoidal rule for two-points + Simpson s rule for three-point + differentiation [two-point + threepoint (Forward+Central+Backward)formulas] can be ask. Theorems and Lemmas without Proofs: Theorems 2.1,3.1,3.2,3.7,3.8,3.14,3.20,3.21,3.22,4.1,4.2,4.3,4.4,5.1,5.5,6.1. Lemmas 2.1,2.2,2.3,2.4. Note: Before we start Chapter 2, we must discuss Computer Representation of Number, Error(in details), the Taylor s polynomial. These topic can be found in the first and fourth chapters of the recommended book(sections:1.2,1.3,1.4,and 4.1). Note: About the 10 marks we do the following: 2 Quizzes + computer assignments + Attendance ( ) marks Note: No tutorial classes for the course. Note: Please advise your students to use the recommended book. The students should study the following topics: Chapter 2: Solution of Nonlinear Equations The bisection method: How to apply it and to compute an error bound for the approximate solutions derived by the method. The Newton s method: How to apply it and the analysis of its error. The secant method: How to apply it. 3

4 The fixed point iterative method: How to formulate the function g(x) which will satisfies the conditions of the Theorem 2.2, then apply the iterative scheme and the analysis of the error. The rate of convergence of the iterative methods including the Newton s method. The multiple roots: How to define it and discuss the conditions under which the root is said to be simple or multiple. Here some attention should be given for the rate of the convergence of Newton s method for both the simple and the multiple roots. Also, the following modified Newton s methods should be discussed: x n+1 = x n mf(x n) f (x n ), f (x n ) 0, for n = 0, 1, 2,... where m is the multiplicity of the multiple roots, and the other one is x n+1 = x n f(x n )f (x n ), for n = 0, 1, 2,... [f (x n )] 2 f(x n )f (x n ) The Newton s method for the nonlinear systems (only for two nonlinear equations). Chapter 3 Systems of Linear Equations How to apply the Gaussian elimination method (without pivoting)(algorithmic approach) and also, discuss the partial pivoting. Give examples showing that the system has infinite number of solutions or no solution at all (singular matrix). How to apply LU factorization [only l ii = 1 (in lecture class), and u ii = 1 (in tutorial class)]. How to apply the iterative methods (Jacobi and Gauss-Seidel) to solve a linear system. The analysis of the error related to these methods (condition for convergence, diagonally dominant matrix...). Also, how to compute an error bound for both methods. Error in solving linear systems. Residual vector, condition number of a given matrix... etc. How to compute an upper bound for both absolute and relative errors. Here we must give the definitions of the vector and matrix norms (l norm only). Chapter 4: Approximating Functions How to construct the Lagrange polynomial which approximate a function f(x) at an (n + 1) distinct numbers(not the uniqueness). How to apply the divided differences to construct the Newton s polynomial (without discussing the forward or backward cases). Error in polynomial interpolation: How to compute an error bound for any x value in the interval [a, b] and a given x = x [a, b]. Interpolation using spline functions(linear Spline Only). Chapter 5 Differentiation and Integration How to derive the first and second order finite difference formulas for approximating the first and second derivatives of the function f(x) at a point x 0 using Lagrange and Taylor 4

5 polynomials (note that for the first derivative we study (Forward+Central+Backward) and for the second derivative we study only the central difference formula). How to apply these formulas including the estimation of an error bound (also, discuss the effect of error in function values). How to derive the Trapezoidal and the Simpson rules, how to apply them and to compute the error bounds (for both single and composite formulas). Chapter 6: Ordinary Differential Equations How to use the Euler s method, the Taylor method of order N and the Runge Kutta method of order two (only modified Euler s method) and order four for solving first order initial-value problems in ordinary differential equations. Also, discuss the local truncation errors for Euler and Taylor methods. 5

6 Course Syllabus- MAD6405 Numerical Analysis Spring 2007 Course Prefix/Number: MAD Course title: Numerical Analysis Course Credit Hours: 3 Website: Prerequisites or Co-Requisites: MAD4401 and MAS5107or MAS3105 Text: Matrix Computation, 3rd ed., G. GOLUB and C. VAN LOAN Course Description: Theoretical treatment of numerical methods of linear algebra supplemented with use of computers; polynomial approximations, uniform approximations, least square approximations; error analysis for numerical solutions of linear equations, algebraic eigenvalue problems. Student Learning Outcomes: After successfully completing this course, with the help of computers and software, the student will be able to 1. solve linear systems of equations; 2. solve some eigenvalue problems; 3. solve least square problems; 4. find polynomial approximations of functions. Topics Covered: Part I: Numerical Linear Algebra. General Linear Systems 1. Triangular System. 2. LU factorization. 3. Error analysis. Special Linear Systems 1. LDM and LD LT factorizations. 2. Positive definite system. 3. Banded system. 6

7 The Symmetric Eigenvalue Problems 1. Properties and decompositions. 2. Perturbations. 3. The symmetric QR algorithm. 4. Bisection method, R.Q.I, subspace iterations, D&C method and homotopy method. The unsymmetric Eigenvalue Problems 1. Properties and decompositions. 2. Perturbations. 3. Power iterations Part II: Approximation Theories. Uniform Approximation 1. Weierstrass approximation theorem, 2. Bernstein polynomials. 3. Approximation by interpolations. Least Square Approximation 1. Inner product space. 2. orthonormal system. 3. Full rank least square problems. 4. Rank deficient least square problems. Grading / Evaluation: Test1 20% Quizzes/homework 50% Final 30% There will be absolutely NO early or make-up test given. A missed test will be counted as 0, unless a valid reason is presented to the instructor. In this case, the weight will be added to final. Special Technology Utilized by Students: N/A Expectations for Academic Conduct/Plagiarism Policy: As members of the University of West Florida, we commit ourselves to honesty. As we strive for excellence in performance, integritypersonal and institutionalis our most precious asset. Honesty in our academic work is vital, and we will not knowingly act in ways which erode that integrity. Accordingly, we pledge not to cheat, nor to tolerate cheating, nor to plagiarize the work of others. We pledge to share community resources in ways that are responsible and that comply with established policies of fairness. Cooperation and competition are means to high achievement and are encouraged. Indeed, cooperation is expected unless our directive is to individual performance. We will com- 7

8 pete constructively and professionally for the purpose of stimulating high performance standards. Finally, we accept adherence to this set of expectations for academic conduct as a condition of membership in the UWF academic community. ASSISTANCE: Students with special needs who require specific examination-related or other courserelated accommodations should contact Barbara Fitzpatrick, Director of Disabled Student Services (DSS), dss@uwf.edu, (850) DSS will provide the student with a letter for the instructor that will specify any recommended accommodations. 8

9 Student Syllabus for Numerical Analysis I- Spring 2007 United Arab Emirates University Faculty of Science Department of Mathematics and Computer Science General Information:- Teacher s name: Dr. Fathi M. Allan f.allan@uaeu.ac.ae Course Title : Numerical Analysis I (3 Credit.O) Text Book : Numerical Analysis, by R. L. Burden and J. D. Faires. Course Description: Error analysis, solving nonlinear equations in one variable using : Fixed point method, Bisection method and Newton s method. Solving linear systems of equations using : Gaussian elimination method and Cholesky s method. Lagrange Interpolation, Hermite Interpolation. Eigenvalue problems. Course Objectives: Study numerical analysis techniques and methods. Learn how to use Mathematica software to solve numerical analysis problems. To explore topics related to nonlinear equations in one variable. To illustrate the applications the various topics covered by the course such as: interpolation, and linear systems to problems in science, engineering and other related areas. Student Learning Outcomes: 1. Demonstrate ability to think critically. 2. Compute the absolute and realtive errors. 3. Analyze algorithms and study their convergence. 4. Approximate the solution of nonlinear equations of one variable using various methods such as bisection method, fixed point method and Newton s methods. 5. Approximate zeros of polynomials using Muller s method. 6. Use Lagrange and Newton s interpolation techniques to approximate functions. 7. Use interpolation techniques to solve different mathematical problems. 8. Solve linear systems using different strategies such as partial pivoting. 9. Solve linear systems using different iterative methods such as Jacobi, Gauss-Seidel, SOR methods. 10. Analyze the Jacobi, Gauss-Seidel, SOR methods. 11. Compute different decomposition of matrices such as LU and Choleski s decompositions and use them to solve linear systems. 12. Use numerical differentiation and integration. 9

10 13. Use Mathematica to solve numerically mathematical problems. 14. work effectively with others. 10

11 Syllabus for the Numerical Analysis Prelim based on APPM effective August 2001 Texts: K. Atkinson, Introduction to Numerical Analysis (except Chapter 1). G. Golub and C. Van Loan, Matrix Computations, Chapters 2-5, 7, 10. K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Chapters 2.2, 2.4, , 3.1, 3.2, 4.2, Recommended Supplemental Text: J. Stoer and R. Bulirsch, Introduction to Numerical Analysis. The following topics are covered in APPM The prelim does NOT cover any additional APPM 6610 topics. Interpolation Theory polynomial interpolation theory Newton divided differences finite differences and table-oriented interpolation formulae errors in data and forward differences further results on interpolation error Hermite interpolation piecewise polynomial interpolation, B-splines Fourier series, DFT and FFT trigonometric interpolation Approximation of Functions the Weierstrass Theorem and Taylor s Theorem the minimax approximation problem the least squares approximation problem orthogonal polynomials minimax approximation near-minimax approximations Rootfinding for Nonlinear Equations the bisection method Newton s method the secant method Muller s method a general theory for one-point iteration methods Aitken extrapolation for linearly convergent sequences the numerical evaluation of multiple roots Brent s rootfinding algorithm roots of polynomials systems of nonlinear equations Newton s method for nonlinear systems unconstrained optimization Numerical Integration the trapezoidal rule and Simpson s rule Newton-Cotes integration formulae Gaussian quadrature asymptotic error formulae and their applications 11

12 automatic numerical integration singular integrals numerical differentiation Linear Algebra vector spaces, matrices, and linear systems eigenvalues and canonical forms for matrices vector and matrix norms, condition numbers convergence and perturbation theorems Sherman-Morrison formula Numerical Solution of Systems of Linear Equations, Direct methods Gaussian elimination pivoting and scaling in Gaussian elimination variants of Gaussian elimination error analysis the residual correction method Numerical Solution of Systems of Linear Equations Iterative Methods Jacobi, Gauss-Seidel error prediction and acceleration the numerical solution of Poisson s equation the conjugate gradient method The Matrix Eigenvalue Problem eigenvalue location, error, and stability results the power method orthogonal transformations using Householder matrices, the eigenvalues of a symmetric tridiagonal matrix the QR Method, the calculation of eigenvectors and inverse iteration least squares solution of linear systems Numerical Methods for Ordinary Differential Equations, existence, uniqueness, and stability theory Euler s method, multi step methods, the midpoint method, the trapezoidal method a low-order predictor-corrector algorithm derivation of higher order multistep methods convergence and stability theory for multistep methods stiff differential equations and the method of lines single-step and Runge-Kutta methods boundary value problems Introduction to Linear Parabolic and Hyperbolic PDEs parabolic problems in one space dimension an explicit scheme, convergence, Fourier analysis an implicit scheme, parabolic problems in two space dimensions an explicit scheme ADI hyperbolic problems in one space dimension the CFL condition finite differences, stability, accuracy, and consistency, the Lax Equivalence Theorem 12

13 Numerical Analysis Syllabus Dr. Abdul Hassen Phone (856) ext Prerequisite: Linear Algebra, Ordinary Differential Equations (or concurrently) and prior computer programming experience. Text: Atkinson and Han, Elementary Numerical Analysis, 3rd Edition, Published by Wiley. Catalog Description: This course includes: elements of error analysis, real roots of an equation, polynomial approximation by finite difference and least square methods, interpolation, quadrature, numerical solution of ordinary differential equations, and numerical solutions of systems of linear equations. The student should expect to program a computer in addition to using a graphing calculator. Objective: The purpose of numerical analysis is two-fold: (1) to find acceptable approximate solutions when exact solutions are either impossible or so arduous and timeconsuming as to be impractical, and (2) to devise alternate methods of solution better suited to the capabilities of computers. While this course will involve the student in considerable computation in order to apply techniques and obtain acceptable answers, the main emphasis will be on the underlying theory. It will be necessary to draw upon a good bit of calculus, linear algebra, computer science and other branches of mathematics during the course. 13

14 School of Engineering, Tohoku University 2004 Syllabus Mano, Akira ( Professor ) / Disaster Control Research Center Course Object and Description Numerical analysis is a fundamental tool for most graduate students in science and engineering, and frequently constructs the core of their studies. The purpose of this course is to furnish the auditors with elementary skills of the analysis together with applicability to step up the higher level. Along the purpose, the course lectures on the basic theories and techniques, and requires the programming to solve the assignment. The exercise on numerical modeling for physical problems is also an important step to promote the applicability. Course Plan 1st Numerical errors. Nonlinear algebra 2nd Interporation and numerical integration 3rd Direct methods for simultaneous linear equations 4th Iterative methods for simultaneous linear equations 5th Eigenvalues and eigenvectors 6th Ordinal differential equations (1) 7th Ordinal differential equations (2) 8th Miscellaneousness 9th Partial differential equations 10th Finite difference method 11th Numerical analysis of evolution equation 12th Stability and convergence 13th Application to practical problems (1) 14th Application to practical problems (2) 15th Application to practical problems (3) Required Text and Recommended References Computational Techniques for Fluid Dynamics, Vol.1, (C.A.J. Fletcher, Springer-Verlag Lecture note. A First Course in Numerical Analysis ( Anthony Ralston and Philip Rabinowitz, McGraw-Hill, Inc.) 14

15 Undergraduate courses in numerical analysis 451. (CSE) NUMERICAL COMPUTATIONS (3) Algorithms for interpolation, numerical integration, numerical solution of nonlinear equations, linear systems and ordinary differential equations emphasizing computational properties and implementation. Students may take only one course for credit from MATH 451 and 455. Prerequisites: CMPSC 201C, 201F, or CSE 103; MATH 230 or 231. MATH/CSE 451. Numerical Computations This is an undergraduate course introducing most of the basic and classical numerical algorithms. The course is more focused on the study and implementation of these basic algorithms. The students will have to complete several computing projects in addition to other homeworks (CSE) INTRODUCTION TO NUMERICAL ANALYSIS I (3) Floating point computation, numerical root finding, interpolation, numerical quadrature, direct methods for linear systems. Students may take only one course for credit from MATH 451 and 455. Prerequisites: CMPSC 201C, 201F, or CSE 103; MATH 220; MATH 230 or (CSE) INTRODUCTION TO NUMERICAL ANALYSIS II (3) Polynomial and piecewise polynomial approximation, matrix least squares problems, numerical solution of eigenvalue problems, numerical solution of ordinary differential equations. Prerequisite: MATH 455. Graduate courses in numerical analysis 523. NUMERICAL ANALYSIS I. Approximation and Interpolation; Numerical Quadrature; Direct Methods of Numerical Linear Algebra; Numerical solution of nonlinear systems and optimization NUMERICAL ANALYSIS II. Numerical Solution of Ordinary Differential Equations; Numerical Solution of Partial Differential Equations; Some Iterative Methods of Numerical Linear Algebra. 15

16 552. (CSE 552) Numerical Solution of Partial Differential Equations (3) Finite difference methods for elliptic, parabolic, and hyperbolic differential equations. Solutions techniques for discretized systems. Finite element methods for elliptic problems. Prerequisite: MATH 402 or 404; MATH (CSE) 451 or (CSE 556) Finite Element Methods (3) Sobolev spaces, variational formulations of boundary value problems; piecewise polynomial approximation theory, convergence and stability, special methods and applications. Prerequisite: Math 412, MATH 501 and 502, or consent from instructor. MATH 523 and MATH 524. Numerical Analysis These courses form a two-semester graduate level introduction to numerical analysis. They will be mainly focused on the design and the analysis of classical as well as recently developed numerical algorithms and techniques, for the solution of variety of problems in mathematical analysis and algebra. In short, this course will provide an introduction to the basics of the mathematical theory behind scientific and engineering computing. The students who take this course should have a very good and stable knowledge of single and multivariable calculus, linear algebra and be familiar with basic facts from functional, real and complex analysis and the theory of partial differential equations. MATH 523 Numerical analysis I Approximation and Interpolation; Numerical Quadrature; Direct Methods of Numerical Linear Algebra; Numerical solution of nonlinear systems and optimization Polynomial Approximation Lagrange Interpolation Least Squares Polynomial Approximation Piecewise polynomial approximation and interpolation The Fast Fourier Transform Multipole method for special dense matrix vector product Numerical Quadrature Basic quadrature The Peano Kernel Theorem Richardson Extrapolation Asymptotic error expansions Romberg Integration Gaussian Quadrature Adaptive quadrature Monte Carlo methods for higher dimensional integrals. Direct Methods of Numerical Linear Algebra Triangular systems Gaussian elimination and LU decomposition Pivoting Backward error analysis Conditioning and roundoff errors. Numerical solution of nonlinear systems and optimization 16

17 One-point iteration Newton s method Unconstrained minimization Newton s method Line search methods Conjugate gradients MATH 524 Numerical analysis II Numerical Solution of Ordinary Differential Equations Euler s Method Linear multistep methods One step methods Stiffness Numerical Solution of Partial Differential Equations BVPs for 2nd order elliptic PDEs The five-point discretization of the Laplacian Finite element methods Difference methods for the heat equation Difference methods for hyperbolic equations Hyperbolic conservation laws Some Iterative Methods of Numerical Linear Algebra Classical iterations Multigrid methods Reference Analysis of Numerical Methods, by Eugene Isaacson and Herbert Bishop Keller; Dover Publications Introduction to Numerical Analysis, by J. Stoer and R. Bulirsch; Springer-Verlag ISBN MATH/CSE 552. Numerical Solution of Partial Differential Equations This is an introductory graduate course on numerical methods for partial differential equations. It is designed mainly for graduate students in department of mathematics. The course should also be appropriate for non-math graduate students who are good at advanced calculus and linear algebra. The students are required to do computing projects in addition to theoretical homework problems. All the graduate students in computational and applied math program are required to take this course. MATH/CSE 552. Numerical Solution of Partial Differential Equations A review of basic methods for the Poisson Equations on regular domain (1 week) The finite difference method The finite element method The finite volume method Second order elliptic boundary value equations (5 weeks) A review of qualitative properties (2 hour) Maximal principle 17

18 Existence and uniqueness (existence of classical and weak solutions) Regularity (H2 regularity of the solutions for smooth or convex domain) The finite difference method (4 hours) Basic finite difference schemes Discrete maximal principle and M-matrices Error estimates Boundary treatments The finite element method (5 hours) Linear finite element methods Error estimates: estimates in H1 norm and L2 norm Construction of more general finite element methods The finite volume method (3 hours) Basic finite volume schemes Conservation properties Relation with finite element method Error estimates High order finite volume method Direct and iterative methods for solving the discrete systems (2 weeks) Direct methods (1) Sparse matrix data structure (1) Basic iterative methods (2) Basic iterative methods Jacobi and Gauss-Seidel methods The method of subspace corrections and its convergence properties (2) Conjugate gradient methods and preconditioning (2) The multigrid method (2 weeks) Introduction of the algorthm using one dimensional problem (2 hours) Algorithmic details for /-cycle, cycle, (2 hours) V-cycle and W-cycle algorithms Convergence analysis using the method of subspace corrections (2 hours) Parabolic and hyperbolic problems (4 weeks) Model problems and stability estimates (2 hours) Examples of the methods of lines (2 hours) The Lax-Richtmyer equivalence theorem (1 hour) Stability analysis (2 hours) Discrete Fourier series (.5) von Neumann stability analysis (.5) The Kreiss matrix theory (1) Consistency, convergence and error estimates (1) Convection dominated problems (1 week) The failure of standard discretization Monotone schemes and Godunov theorem Higher order methods Nonlinear problems Textbooks and references There are many text books available, but there is no single one that would fit the aforementioned syllabus. 18

19 Major references Hackbusch, W., Elliptic differential equations : theory and numerical treatment Berlin ; New York : Springer-Verlag, c1992. [good reference for elliptic problems] Strikwerda, John C., Finite difference schemes and partial differential equations / John C. Strikwerda. Pacific Grove, Calif. : Wadsworth & Brooks/Cole Advanced Books & Software, c1989. [good reference for linear parabolic and hyperbolic problems] Johnson, Claes, Numerical solution of partial differential equations by the finite element method / Claes Johnson. Cambridge [Cambridgeshire] ; New York: Cambridge University Press, c1987. [Overall good reference for finite element method for this course; but not enough materials for theoretical analysis] Susanne Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, New York : Springer-Verlag, c1994. [Chapter 3 is a good reference on the construction of a finite element space; other parts of the book are too theoretical/technical for this course] Jinchao Xu, Lecture notes for MATH/CSE 552 [covers materials that can not be found in the above three books and other major text books, especially good materials for iterative and multigrid methods, finite volume methods] 19

20 HOME PEOPLE UNDERGRADUATE GRADUATE RESEARCH DEPART- MENT HIGH SCHOOL Home > Undergraduate > Courses > Syllabi > AMSC/CMSC 466 (Introduction to Numerical Analysis I) DESCRIPTION Floating point computations, direct methods for linear systems, interpolation, solution of nonlinear equations, numerical differentiation and integration. PREREQUISITES Math 240 and 241, CMSC 105 or CMSC 106 or CMSC 114 or ENEE 114 TOPICS Floating point computation (1 week) Properties of machine arithmetic Direct methods for linear systems (4 weeks) Gaussian elimination Pivoting strategies Cholesky factorization Vector and matrix norms Conditioning and the effect of rounding error Interpolation (2 weeks) Polynomial approximation Newton and Lagrange forms Error formula (derived) Solution of nonlinear equations (3 weeks) Bisection, Secant, and Newton s method Fixed point methods Newton s method for systems Numerical differentiation and integration (3 weeks) Numerical differentiation Numerical integration TEXT: Text(s) typically used in this course. 20

21 Numerical Analysis + Lab MA : 5 Credits Prerequsites: None Root finding for nonlinear equations- Newton-Raphson, Secant, Regula-Falsi methods and their convergence, Newton?s method for system of nonlinear equations. Interpolation - Newton?s formulae, Lagrange, Hermite, Spline interpolation with error analysis. Numerical differentiation. Numerical integration- Newton-Cotes formulae - open and closed type-trapezoidal, Simpson and Weddle rules, Gaussian quadrature formulae- Gauss-Laguerre, Gauss-Hermite integration, composite integration methods, double integration. System of linear algebraic equations- Gauss elimination, Jacobi, Gauss-Seidel, relaxation methods and their convergence. Numerical methods for determining eigenvalues. 21

22 Syllabus: MATH 5338 NUMERICAL ANALYSIS (I) Web: 1. COURSE PREREQUISITES: Knowledge of calculus, linear algebra, and programming or consent of the instructor. 2. COURSE GOALS: Solution of one variable equations Interpolation and polynomial approximation Numerical differentiation and integration Direct methods for solving linear systems Iterative techniques in matrix algebra 3. TEXTBOOK: Numerical Analysis by R.L.Burden and J.D.Faires (7th Edition) 22

23 University of Houston Department of Mathematics MATH 4364 Numerical Analysis Prerequisites: MATH 2431 (Linear Algebra), MATH 3331 (Differential Equations), COSC 1301 or 2101 or equivalent experience with one programming language (computer assignments). Textbook: Numerical Analysis (8th edition), by R.L. Burden and J.D. Faires, Brooks- Cole Publishers Brief Description of MATH 4364 : This course introduces students to classical numerical methods for approximating the solutions of common mathematical problems. It allows students to deal with numerical methods both at a theoretical level and for programming purposes. This is an introductory course and will be a mix of mathematics and computing. The mathematical content of the course for this semester covers interpolation, numerical integration, direct and iterative methods for solving linear systems of algebraic equations and numerical methods for solving nonlinear algebraic equations. N.B. This is the first semester of a two semester course. The emphasis the second semester (MATH 4365) will be in particular on numerical methods for ordinary differential equations and partial differential equations. Course structure: This a ex-cathedra course, that is a mix of mathematics of computing. Student evaluation will be based on in-class exams and one final exam, homework extracted from the text and computer projects. Tentative Syllabus of the Fall semester (MATH 4364) Chapter 2 : Resolution of nonlinear equations in one variable Chapter 3 : Interpolation Chapter 4 : Numerical differentiation and numerical integration Chapter 6 : Resolution of linear systems (direct methods) Chapter 7 : Resolution of linear systems (indirect methods) Chapter 10 : Resolution of nonlinear systems (as time permits) Tentative Syllabus of the Spring semester (MATH 4365) Chapter 5 : Numerical solution of ODEs Chapter 8 : Selected topics of numerical approximation Chapter 9 : Numerical methods for computation of eigenvalues Chapter 11 : Boundary-value problem for ODEs Chapter 12 : Finite differences methods for PDE s, Introduction to finite elements for elliptic problems 23