Intro to Numerical Methods

Transcription

1 COLUMBIA UNIVERSITY Intro to Numerical Methods APAM E4300 (1) MIDTERM IN-CLASS SAMPLE EXAM MARCH 11, 013 INSTRUCTOR: SANDRO FUSCO FAMILY NAME: GIVEN NAME: UNI: INSTRUCTIONS: 1. Attempt all seven (7) questions.. Your work must justify the answer you give. 3. Point values are as shown. Work is required for full credit, and may earn partial credit. 4. Be sure to indicate your answer VERY clearly. Put your answer in a circle or box when applicable. 5. No lecture notes and/or books are permitted. 6. This is the first of nine (9) pages. Question Points Marks Extra Credit 10 Total 100 APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE 1 OF 9

2 Problem 1: (10 Points) a) [ points] For a general root finding problem, list the following three algorithms in order of increasing speed (where by faster we mean takes less steps to converge to an answer): Secant method, Newton s method, Bisection method b) [ points] Write the equation for the tangent line to y = f(x) at x = p. c) [ points] Solve for the x-intercept of the line in point (b). What formula have you derived, with what roles for p and x? d) [ points] Write the equation of the line that intersects the curve y = f(x) at x = p and x = q. e) [ points] Solve for the x-intercept of the line in point (d). What formula have you derived, with what roles for p, q, and x? a) [ points]: Bisection method, Secant method, Newton s method b) [ points]: Write the equation for the tangent line to y = f(x) at x = p. y fp f px p c) [ points] Solve for the x-intercept of the line in point (b). What formula have you derived, with what roles for p and x? When y 0, x p. This is the step of Newton s method with the current iterate x k = p and the new iterate x k+1 = x. d) [ points] Write the equation of the line that intersects the curve y = f(x) at x = p and x = q. y fp fq fp x p q p e) [ points] Solve for the x-intercept of the line in point (d). What formula have you derived, with what roles for p, q, and x? When y 0, x p fp. This is the step of Secant method with x k = p, x k-1 = q and the new iterate x k+1 = x. APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE OF 9

3 Problem : (15 Points) a) [5 points] Find the n-th Taylor Polynomial approximation to expanded around x=0. (Maclaurin Series expansion) b) [5 points] Starting with 1, add terms one at the time in order to estimate e 0.5. After each new term is added, compute the true ε t and approximate ε a percent relative errors (use the true value e 0.5 = ) c) [5 points] Add new terms until the absolute value of the approximate error ε a falls below a pre-specified error criterion ε s conforming to three (3) significant figures. a) Recall that the Taylor series expansion of a function f(x) about a point x 0 is From this we can define a sequence of polynomials of increasing degree that approximate the function near the point x 0. Since all derivatives of f(x) = e x are again just e x, the Taylor polynomials from a series expansion about the point x 0 = 0 are: T 1, T 1 1!, T 1 1!!, T 1 1!! 3!, T 1 1!! 3!!. b) The first estimate is simply equal to 1. The second estimate is then given by Hence we have:!.. 100% 9.0% and.. 100%. 33.3%, and so on. c) The error criterion that ensures a result that is correct to at least three significant figures is given by the formula % 0.05%. Thus, we will add terms to the series until falls below this level. Term Result Thus, after six terms are included the approximate error falls below 0.05% and the computation is terminated. APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE 3 OF 9

4 Problem 3: (15 Points) Let n and m be positive integers and consider the binary floating point system where our machine stores numbers of the following form: 1. where 0, 1,,, and 0,1 for all k. Remember: In a binary system each a k gets multiplied by -k and all these get added up to produce the number being represented. a) [5 points] In terms of n and m, how many numbers are in this system? Don t forget the ± b) [5 points] Let m= and n=5. What is machine precision if we approximate all numbers by rounding to the closest number in the floating point system? c) [5 points] Let m=. Find the smallest value of n that will ensure that some number in the system is bigger than 1.8. a) There are x k x (m+1) = (m+1) x k+1 numbers in this system. The first is for the ±. The other terms are obvious. b) Machine precision is the smallest number ε such that 1+ ε > 1 in the system. First we have to find the smallest floating point number larger than 1. When n=5, it is = = 1+1/3. A number gets rounded up to if it is closer to it; i.e., if it is larger than the midpoint 1+1/64. Hence the machine epsilon is ε = 1/64. c) The largest number in the system is = n. When n = 1, the largest number is = 1.5 < 1.8 When n =, the largest number is = 1.75 < 1.8 When n = 3, the largest number is = + -3 = = > 1.8 Hence n=3 is the smallest value of n that will ensure that some number in the system is bigger than 1.8. APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE 4 OF 9

5 Problem 4: (15 Points) Suppose that you are given a continuous function f and numbers a < b such that f(a) and f(b) have different signs. You may ignore floating point error for the purpose of this problem, a) [10 points] Write a short MATLAB function utilizing the bisection method to find the root of f in [a,b]. The output should be [c iter] where c is within ½ x of a root of f and iter is the number of iterations (i.e., bisections) that were performed. The first line can be written as follows: % The program starts here. function [c iter] = bisect(f,a,b) b) [5 points] Suppose that b - a = 64 and note that log (10) 3.3. What is the number of iterations that the function described in part (a) should require to finish? a) See code on the course website. where delta = ½ x b) We know that: b a k 64 = 10 k Hence we can stop after k = 40 iterations. k 6 k k 10log ( 10) APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE 5 OF 9

6 Problem 5: (15 Points) a) [10 points] In class we have seen one way to approximate the derivative of a function f: for some small number h (forward finite difference). Assuming that f C, use Taylor s Theorem to determine the accuracy of this approximation. b) [5 points] Show that, with this formula, we can approximate a derivative to only about the square root of the machine precision. a) To determine the accuracy of this approximation, we use Taylor s Theorem, assuming that f C : This shows that the approximation is first-order accurate. "ξ, ξx,xh ". b) The roundoff also plays a role in the evaluation of the forward finite difference. For example, if h is so small that x+h is rounded to x, then the computed finite difference is zero. More generally, even if the only error made is in rounding the values f(x+h) and f(x), then the computed difference quotient will be: 1 1 Since each is less than the machine precision ε, this implies that the rounding error is less than or equal to Since the truncation error is proportional to h and the rounding error is proportional to 1/h, the best accuracy is achieved when the two quantities are approximately equal. Ignoring the constants, this means that Then, with the forward finite difference, we can approximate a derivative to only about the square root of the machine precision. APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE 6 OF 9

7 Problem 6: (15 Points) Below is the MATLAB m-file for a mystery function. function output = mystery( input,n,n) for j=1:n t=1:*n; t=t'/(*n); y=0; for p=1:n y=y+^(p)*rand*t.^(p-1); X(:,p)=t.^(p-1); end beta=x\y; L=@(z) z.^(0:n-1)*beta; Lout(j)=L(input); end output=mean(lout); end Answer the following questions regarding the behavior of this function. Remember: The MATLAB command rand will produce a random number that is uniformly distributed between 0 and 1, hence with mean ½. a) [10 points] What will be the value of mystery(1/5, N, n) as both N and n get large? In other words, find lim lim 1,, 5 b) [5 points] What will be the value of mystery(½, N, n) as N gets large and n stays fixed? In other words, find lim 1,, a) lim,,. Hence: lim lim 1 5,, b) lim,,. APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE 7 OF 9

8 Problem 7: (15 Points) In class we saw that Newton s method is quadratically convergent, under suitable hypotheses. Theorem: If f C, if x 0 is sufficiently close to a root x* of f, and if f (x*) 0, then Newton s Method converges to x* and ultimately the convergence rate is quadratic; that is there exists a constant C* = f (x*)/f (x*) such that: Write the proof of this Theorem. lim k xk + 1 x* = C * x x* k See page 86 of the textbook. APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE 8 OF 9

9 Extra Credit Problem: (10 Points) a) [5 points] State Horner s Method for Polynomial Evaluation. b) [5 points] Use Horner s method to find P(3) for the polynomial P(x) = x 5 6x 4 + 8x 3 +4x 40. a) See page 10 of b) See page 11 of APAM E4300 (1) MIDTERM SAMPLE EXAM PAGE 9 OF 9