Numerical Analysis. Yutian LI. 2018/19 Term 1 CUHKSZ. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 1 / 41

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1 Numerical Analysis Yutian LI CUHKSZ 2018/19 Term 1 Yutian LI (CUHKSZ) Numerical Analysis 2018/19 1 / 41

2 Reference Books BF R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition, Thomsom Brooks/Cole, At K. E. Atkinson, An Introduction to Numerical Analysis, 2nd edition, John Wiley, DMN D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice-Hall, St G. W. Stewart, Afternotes on Numerical Analysis, SIAM, Software Matlab Python Julia Yutian LI (CUHKSZ) Numerical Analysis 2018/19 2 / 41

3 Contents 1 Introduction 2 Nonlinear Equations 3 Interpolation 4 Numerical Integration 5 Numerical Differentiation 6 Linear System of Equations Yutian LI (CUHKSZ) Numerical Analysis 2018/19 3 / 41

4 Chapter 1: Introduction 1 Overview & Motivation 2 Computational Errors 3 Computer Arithmetic 4 Convergence and Stability 5 Some basic algorithms and Matlab codes Yutian LI (CUHKSZ) Numerical Analysis 2018/19 4 / 41

5 Overview & Motivation 1. Solve Nonlinear Equations (Chapter 2) The LambertW function is the inverse function of fpwq we w : that is z : wpzqe wpzq. One needs to solve a nonlinear equation to find the value of wpzq. e.g., to fnid wp1q, we need solve we w 1. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 5 / 41

6 2. Interpolation (Chapter 3) (a) piecewise linear function (b) polynomials Yutian LI (CUHKSZ) Numerical Analysis 2018/19 6 / 41

7 3. Numerical Integration (Chapter 4) The c.d.f. (cumulative distribution function) of the standard normal distribution is given by Npxq? 1» x e t2 1 2 dt 2π 8 2» 1 x? e t2 2 dt 2π 0 To find the (approximate) values of Npxq, we need perform a numerical method to evaluate the above integral. This is the quadrature rules. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 7 / 41

8 4. Solve linear system (Chapter 5) In solving the application problems, the last step is often to find the solution of a linear system of equations AX b. (e.g., when we use finite difference or finite element methods to solve a differential equation) When A is large, it is necessary to use some numerical method and to solve it on a computer. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 8 / 41

9 5. Least squares Figure: The result of fitting a set of data points with a quadratic function Yutian LI (CUHKSZ) Numerical Analysis 2018/19 9 / 41

10 Computational Errors Truncation error the error made by numerical algorithms that arises from taking finite number of steps in computation. Round-off error the error produced when a computer is used to perform real number calculations. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 10 / 41

11 Truncation error: Consider Taylor s theorem fpxq P n pxq R n pxq, where P n pxq f px 0 q f 1 px 0 qpx x 0 q R n pxq f pn 1q pξq pn 1q! px x 0q n 1. f 2 px 0 q px x 0 q 2 2! f pnq px 0 q px x 0 q n, n! If we use P n pxq to approximate fpxq, the error R n pxq is called the truncation error. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 11 / 41

12 Round-off error: In the computational world, each representable number has only a fixed and finite number of digits. Only a small subset of the real number can be represented in a typical computer So for example, 1{3, π,? 2, these numbers cannot be represented exactly in the computation by a computer. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 12 / 41

13 Round-Off Error Can Be Fatal. On February 25, 1991 an Iraqi Scud missle hit an American Army barracks and killed 28 soldiers. The Patriot Missle System that was supposed to intercept and destroy the Scud failed. The General Accounting Office report, GAO/IMTEC-92-26, Patriot Missile Defense: Software Problem Led to System Failure at Dhahran, Saudi Arabia gave the reason as round-off error. See When we do computations using computers, the round-off errors are inevitable. But we can try to minimize the negative impact of these errors by rewriting the formula you wish to compute. This must be done on a case-by-case basis. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 13 / 41

14 Computer Arithmetic Binary Floating Point Number. In 1985, the IEEE (Institute for Electrical and Electronic Engineers) published a report called Binary Floating Point Arithmetic Standard An updated version was published in 2008 as IEEE A 64-bit (binary digit) representation is used for a real number. The first bit is a sign indicator, denoted s. This is followed by an 11-bit exponent, c, called the characteristic, and a 52-bit binary fraction, f, called the mantissa. The base for the exponent is 2. p 1q s 2 c 1023 p1 fq Yutian LI (CUHKSZ) Numerical Analysis 2018/19 14 / 41

15 e.g., s 0, c , f and the machine number precisely represents the decimal number p 1q s 2 c 1023 p1 fq Numbers occurring in calculations that have a magnitude less than p1 0q result in underflow and are generally set to zero. Numbers greater than p q result in overflow and typically cause the computations to stop. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 15 / 41

16 Decimal floating-point number 0.d 1 d 2 d k 10 n, 1 d 1 9, 0 d 2, d 3 9. Any positive real number within the numerical range of the machine can be normalized to the form y 0.d 1 d 2 d k d k 1 d k 2 10 n. The floating-point form of y, denoted f lpyq, is obtained by terminating the mantissa of y at k decimal digits. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 16 / 41

17 There are two common ways of performing this termination Chopping: simply chops off the digits d k 1 d k 2 This produces the floating-point form flpyq 0.d 1 d 2 d k 10 n. Rounding: adds 5 10 n pk 1q to y and then chops the result to obtain a number of the form flpyq 0.δ 1 δ 2 δ k 10 n. For rounding, when d k 1 5, we add 1 to d k to obtain flpyq; that is, we round up. When d k 1 5, we simply chop off all but the first k digits; that is, round down. The error that results from replacing a number with its floating-point form is called round-off error regardless of whether the rounding or the chopping method is used. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 17 / 41

18 Example Determine the five-digit (a) chopping and (b) rounding values of π. The number π has an infinite decimal expansion of the form π Written in normalized decimal form, we have π The floating-point form of π using five-digit chopping is flpπq Since the sixth digit of the decimal expansion of π is a 9, the floating-point of π using five-digit rounding is flpπq p q Yutian LI (CUHKSZ) Numerical Analysis 2018/19 18 / 41

19 Definition Suppose that p is an approximation to p. The actual error is p p, the absolute error is p p, and the relative error is p p p, provided that p 0. Definition The number p is said to approximate p to t significant digits (or figures) if t is the largest nonnegative integer for which p p p 5 10 t. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 19 / 41

20 Example Determine the actual, absolute, and relative errors when approximating p by p when (a) p and p ; (b) p and p ; (c) p and p As a measure of accuracy, the absolute error can be misleading and the relative error more meaningful because the relative error takes into consideration the size of the value. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 20 / 41

21 Inaccurate representation of numbers. We see that the floating-point representation flpyq for the number y has the relative error y flpyq y. If k decimal digits and chopping are used for the machine representation of y 0.d 1 d 2 d k d k 1 10 n, then y flpyq y 0.d k 1 d k 2 10 n k 0.d 1 d 2 10 n k 10 k 1. In a similar manner, a bound for the relative error when using k-digit rounding arithmetic is k 1. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 21 / 41

22 Inaccurate calculation. Assume that the floating-point representations f lpxq and f lpyq are given for the real numbers x and y and that the symbols `, a, b, c represent machine addition, substraction, multiplication and division operations, respectively. We will assume a finite-digit arithmetic given by x ` y flpflpxq flpyqq, x a y flpflpxq flpyqq, x b y flpflpxq flpyqq, x c y flpflpxq flpyqq. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 22 / 41

23 Example Suppose x 5 7 and y 1 3 x and y. we want to do arithmetic calculations involving x , y Five-digit chopping x ` y flpflpxq flpyqq flp q flp q Five-digit rounding x ` y flpflpxq flpyqq flp q flp q Yutian LI (CUHKSZ) Numerical Analysis 2018/19 23 / 41

24 Absolute Error , Relative Error , Yutian LI (CUHKSZ) Numerical Analysis 2018/19 24 / 41

25 How to avoid loss of accuracy due to round-off error? Reformulate the problem. One of the most common error producing calculations involves the cancelation of significant digits due to subtraction of nearly equal numbers. So we should reformulate the problem to avoid subtracting nearly equal numbers. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 25 / 41

26 Example The two roots of ax 2 bx c 0 are given by x 1 b? b 2 4ac 2a, x 2 b? b 2 4ac, 2a Consider the equation x x 1 0. The two roots are x , x In this equation, b 2 is much larger than 4ac, so the numerator of x 1 is a subtraction of two nearly equal numbers. Use 4-digit rounding we get flpx 1 q Yutian LI (CUHKSZ) Numerical Analysis 2018/19 26 / 41

27 which is a poor approximation to x 1 with the large relative error To improve the accuracy of the calculation, we change the formula by rationalizing the numerator, x 1 b Using this we get? b 2 4ac b? b 2 4ac 2a b? b 2 4ac 2c? b b 2 4ac. flpx 1 q , which has the small relative error Yutian LI (CUHKSZ) Numerical Analysis 2018/19 27 / 41

28 Rewrite polynomials into nested form. Example Evaluate fpxq x 3 6.1x 2 3.2x 1.5 for x Using 3-digit arithmetic we have, x x 2 x 3 6.1x 2 3.2x Result Relative error Exact Chopping Rounding Yutian LI (CUHKSZ) Numerical Analysis 2018/19 28 / 41

29 In both cases, large relative errors occur. To improve the calculation, we change the polynomial to the following nested form, fpxq x 3 6.1x 2 3.2x 1.5 ppx 6.1qx 3.2qx 1.5 Then using chopping we have f p4.71q pp q q The relative error is Yutian LI (CUHKSZ) Numerical Analysis 2018/19 29 / 41

30 Using rounding we have f p4.71q pp q q The relative error is Polynomials should always be expressed in nested form before performing an evaluation, because this form minimized the number of arithmetic calculations. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 30 / 41

31 Avoid large numbers eat small numbers. We should add those numbers with small magnitude first when we do the summation Example fl ņ i1 flpx i q x and x i 0.01, i 2,, 101, using 4-digit arithmetic, we have flpx 1 q and flpx i q , i 2,, 101. So after doing the summation s 101 flp i1 flpx iqq, we have s If we define x i 0.01, i 1,, 100 and x , then we have s which is the true value of the summation.. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 31 / 41

32 Convergence and Stability Convergence Definition Suppose tβ n u 8 n1 is a sequence known to converge to zero, and tα nu 8 n1 converges to a number α. If a positive constant K exists with α n α K β n for large n, then we say that tα n u 8 n1 converges to α with the rate (or, order) of convergence Opβ n q (big oh of β n ), and written as α n α Opβ n q. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 32 / 41

33 In most cases, β n 1{n p, (p 0), so we have α n α Op1{n p q Similarly, if a function F phq satisfies F phq L Kh p for small positive h, we write F phq L Oph p q. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 33 / 41

34 Stability. If small changes in the initial data produce correspondingly small changes in the final results, then we say that the algorithm is stable. Otherwise, the algorithm is said unstable. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 34 / 41

35 Example The solution of the recursive equation is p n 10 3 p n 1 p n 2, n 2, 3, p n c n c 2 3 n. If p 0 1 and p 1 1{3, we have c 1 1 and c 2 0, then p n p1{3q n. If we use 5 digit rounding to compute p n, then the computed sequence ˆp , ˆp , which corresponding to ĉ and ĉ , and 1 n ˆp n n. 3 The round-off error is p n ˆp n p3 n q This error grows exponentially, so the algorithm is unstable. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 35 / 41

36 Example The solution of the recursive equation p n 3 4 p n p n 2, n 2, 3, is p n c n 1 n c 2. 4 If the computed values for c 1 and c 2 are ĉ 1 and ĉ 2, due to round-off errors, then we have the round-off error in computing p n is p n ˆp n 1 n 1 n pc 1 ĉ 1 q pc 2 ĉ 2 q c 1 ĉ 1 1 c 2 ĉ Small changes in the initial errors c 1 ĉ 1 and c 2 ĉ 2 will result in a small change in p n ˆp n, the algorithm is stable. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 36 / 41

37 Relation between convergence and stability. The concept of convergence is about the theoretical errors (truncation errors) between the exact value and the true value of an algorithm. The concept of stability is about the computational errors (round-off error) between the true value of an algorithm and the computed value from the algorithm. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 37 / 41

38 Consider the algorithm p n 10 3 p n 1 p n 2, n 2, 3, When p 0 1 and p 1 1{3, we have p n p1{3q n. Clearly, However, for the computed value lim p n 0. nñ8 lim ˆp n 8. nñ8 Yutian LI (CUHKSZ) Numerical Analysis 2018/19 38 / 41

39 Some basic algorithms and Matlab codes Sum n i1 x i s = 0; % initialize for i = 1 : n s = s + x(i); % A common mistake is forget the s here! end Or we can use Matlab command sum(x). Yutian LI (CUHKSZ) Numerical Analysis 2018/19 39 / 41

40 Product ± n i1 x i s = 0; % initialize for i = 1 : n s = s * x(i); % A common mistake is forget the s here! end Or we can use Matlab command prod(x). Yutian LI (CUHKSZ) Numerical Analysis 2018/19 40 / 41

41 Matrix-vector multiplication y A x for i = 1 : n y(i) = 0; % initialize for j = 1 : n y(i) = y(i)+a(i, j) x(j); end end % Use ; to compress the outputs. Or we can use Matlab command y = A*x. Yutian LI (CUHKSZ) Numerical Analysis 2018/19 41 / 41