MAT128A: Numerical Analysis Lecture One: Course Logistics and What is Numerical Analysis?

Transcription

1 MAT128A: Numerical Analysis Lecture One: Course Logistics and What is Numerical Analysis? September 26, 2018 Lecture 1 September 26, / 19

2 Course Logistics My contact information: James Bremer Office: MSB 2230 Office hours: MW 2:10-3:10 PM, F 12:00-1:00 PM The address for our course website is: The official syllabus is available there. I will post homework and programming assignments there as well. Lecture slides (including these) will also be posted to the course website. Canvas will be used to submit programming assignments and to post grades. Lecture 1 September 26, / 19

3 Course Logistics Grading scheme: Programming projects: 30% Midterm: 30% Final: 40% Homework will be assigned each week, but not collected. Instead, solutions will be posted on the course website. The first midterm will on November 5 during class. The final is Wednesday December 12 from 8:00 AM - 10:00 AM. Only university approved excuses for missing the exams will be accepted. Please arrange any necessary accommodations through the Student Disability Center at the beginning of the course. Lecture 1 September 26, / 19

4 Course Logistics Programming projects: Five programming projects will be assigned during the quarter. Each will be worth 6in Canvas. Late projects will not be accepted. In each project, you will be asked to implement one or more functions in the C programming language. Careful documentation of the proper behavior of the functions will be provided to you. You will be given a C program which, when compiled against your code, tests that the functions are implemented correctly. Half of your grade for each project will depend on the results of running that publicly available test code. There will be a second test code, which we will not make publicly available until after the project is due. The other half of your grade for the project will depend on that code functioning properly. Your programs will be compiled with the version of GNU C available on the math department s servers and they will be tested on those servers. They must compile without errors and run on the math department s servers, using the versions of the GNU C present on those machines. You will have access to these servers through your class account. Lecture 1 September 26, / 19

5 Course Logistics Textbook: Timothy Sauer, Numerical Analysis Second Edition, Pearson, ISBN: I will deviate substantially from the textbook, but I will post my lectures slides for each class on the course website. The same textbook will be used in MAT128B and MAT128C this year, however, so if you are planning to take those course, it would probably behoove you to buy a copy. It might also be useful to you to see a different presentation of some of the material I cover. I might assign homework from the textbook, in which case I will reference the second edition, but otherwise any edition of the book will suffice. Lecture 1 September 26, / 19

6 What is Numerical Analysis? Roughly speaking, numerical analysis is the field of mathematics concerned with the solution of mathematics problems (usually, but not exclusively, those from the area of mathematics known as analysis) using finite precision arithmetic on computers. For the most part, the term numerical analysis does not encompass symbolic methods, such as those used by computer algebra systems like Mathematica. The term does encompass some problems which are more algebraic than analytic in nature such as solving systems of linear and nonlinear algebraic equations. Typical problems addressed by numerical analysts include: Solution of ordinary and partial differential equations Stochastic (random) processes Optimization problems Lecture 1 September 26, / 19

7 Why finite precision arithmetic? In a finite precision arithmetic system, real numbers are represented using string of bits of a fixed length. On most computers in use today, 32-bit (single precision) or 64-bit (double precision) strings are used. The advantage of doing this is that arithmetic operations can be performed on these fixed length strings very very quickly. A single processor core on a typical laptop computer can, when operating at peak efficiency, perform something on the order of 6 billion such operations per second!!! There are alternatives to doing this. Computer algebra systems, for instance, typically represent real numbers using symbolic expressions. Operations on such expressions are generally very expensive perhaps 100s or 1000s of such operations can be performed per second. Variable precision arithmetic, in which real numbers are represented as strings of bits of variable lengths, is sometimes used. Again, this is many orders of magnitude slower than using finite precision arithmetic. Lecture 1 September 26, / 19

8 What is novel about numerical analysis? Most of the objects we study in mathematics contain an infinite amount of information. Examples include real numbers such as π = and functions such as f (x) = cos(x). Obviously, if we wish to store these objects on a computer, we must find a way to approximate them using a finite amount of information. In fact, for the reasons we just discussed, we usually want to find a way to accurately store them using finite precision numbers. Moreover, we wish to store them in such a way that whatever operations we perform on them can be done quickly and accurately. Lecture 1 September 26, / 19

9 What is novel about numerical analysis? In other words, the main difference between numerical analysis and most fields of pure mathematics is a focus on: Methods for the efficient representation of mathematical objects Analyses of the efficiency of performing mathematical operations on these objects We usually measure the former by the number of bits required to store an object and the later by the number of arithmetic operations required to perform the operation. During this quarter, we will mostly study the problem of representing smooth or piecewise smooth functions accuracy and efficiently. Lecture 1 September 26, / 19

10 A simple example The first few terms of the Taylor expansion of cosine around x = 0 are cos(x) x x x x x 10. This is a reasonably good way to represent cos(x) near 0. For instance, when x = 1 2, the above approximation gives us cos(x) , which differs from the correct answer by a quantity on the magnitude of Moreover, this expansion only involves 6 coefficients which need to be stored and a straightforward approach to evaluating it requires 16 operations. Lecture 1 September 26, / 19

11 Complications introduced by finite precision arithmetic If we evaluate the formula p(x) = x x x x x x x x x x x x x x x x x x x x at x = 0.9 using double precision arithmetic, we get p(0.9) But if we use the alternate expression p(x) = 20 i=1 (x i), for the polynomial p(x), then we get p(0.9) Lecture 1 September 26, / 19

12 Which one is right? Using a computer algebra system, we can determine that p(0.9) is exactly equal to So the approximation obtained via the formula is excellent, while the approximation p(0.9) p(x) = 20 i=1 (x i) p(0.9) obtained via the monomial expansion of p(x) is way off (the second digit is already wrong!!!) Lecture 1 September 26, / 19

13 What happened? The formula p(x) = x x x x x x x x x x x x x x x x x x x x is ostensibly an efficient way to represent p it only involves 21 coefficients, all of which can be exactly represented using double precision arithmetic. However, although the coefficients in the formula can be stored easily enough, in order to obtain reasonable accuracy when using the formula, we would need to perform arithmetic operations using very high precision precision arithmetic. Lecture 1 September 26, / 19

14 What this course is about This course principally concerns methods for the representation of piecewise smooth functions such as this: We will also discuss the numerical integration and differentiation of such functions. Our focus will be on highly accurate, rapidly convergent methods. Our previous discussion should convince you that developing such methods might be more difficult than one might initially suspect. Lecture 1 September 26, / 19

15 How is relates to the other courses in the MAT128 sequences MAT128B focuses on the numerical solution of nonlinear algebraic equations and numerical linear algebra. MAT128C is principally concerned with the numerical solution of differential equations. When solving differential or partial differential equations (the topic of MAT128), an underlying scheme for representing the solutions (such as those we will discuss in MAT128A) is necessary. Usually, that underlying scheme gives rise to a linear system or a sequence of linear systems which must be solved or otherwise manipulated (using the techniques from MAT128B). Lecture 1 September 26, / 19

16 A rough outline of topics Finite Precision Arithmetic Basics of finite precision arithmetic and the IEEE format Typical problems which occur: underflow, overflow, cancellation The condition number of evaluation of a function; examples: f (x) = x 2 ; f (x) = cos(λx) especially a comparison of f (x) = 1/x with f (x) = 1/(1 x) and a discussion of the consequences of the logarithmic distribution of finite precision numbers and the desirability of placing singularities at 0 Forward and backward error estimates Lecture 1 September 26, / 19

17 A rough outline of topics Representing Piecewise Smooth Functions Why we don t use Taylor Series Basics of Fourier series Expansions of Smooth, Periodic Functions in Exponentials Chebyshev Expansions of Smooth Functions Legendre Expansions of Smooth Functions Expansions of Piecewise Smooth Functions Lecture 1 September 26, / 19

18 A rough outline of topics Polynomial Interpolation Lagrange polynomials and the Lagrange interpolation formula The unsuitability of equispaced nodes, even in the case of well-behaved functions The suitability of Chebyshev and Legendre nodes and the barycentric form of the Lagrange formula Minimax polynomials and their relation to Chebyshev expansions Lecture 1 September 26, / 19

19 A rough outline of topics Numerical Integration and Differentiation Review of the trapezoidal rule, Gauss-Legendre and Gauss-Chebyshev quadrature rules Curtis-Clenshaw quadrature rules Adaptive integration Spectral integration and differentiation Finite differences approximations of derivatives Lecture 1 September 26, / 19