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1 Homework and Computer Problems for Math*2130 (W17). MARCUS R. GARVIE 1 December 21, Department of Mathematics & Statistics, University of Guelph

2 NOTES: These questions are a bare minimum. You should also: 1. Work through all the class examples. 2. Work through past exam questions. 3. Look in any introductory Numerical Analysis/Methods book for similar problems to do (see the course outline for some recommended texts). Furthermore, during the first week you should work through one of the MATLAB tutorials listed on the course webpage. 1

3 Chapter 1 Basic Tools 1.4 Taylor s Theorem 1. What is the third-order Taylor polynomial for f(x) = x + 1, about x0 = 0? 2. Given that R(x) = x 6 6! eξ for x [ 1, 1], where ξ is between x and 0, find and upper bound for R, valid for all x [ 1, 1], that is independent of x and ξ. 3. Given that for x R(x) = x 4 4! [ 1 2, 1 2 ( ) ξ ], where ξ is between x and 0, find 2

4 and upper bound for R, value for all x that is independent of x and ξ. 4. What is the fourth-order Taylor polynomial for about x 0 = 0? [ 1 2, 1 ], 2 1 x Find the Taylor polynomial of third-order for sin(x) using x 0 = π For the function below construct the third-order Taylor polynomial approximation, using x 0 = 0, and then estimate the error by computing an upper bound on the remainder, [ over the given interval: f(x) = ln(1 + x), x 1 2, 1 ] Find the Taylor Series expansion of f(x) = 3 x of degree 2 about x 0 = 8. Using the remainder term estimate the maximum absolute error on the interval [7, 9]. 8. Construct a Taylor polynomial approximation that is accurate to within 10 3, over the indicated interval, for the following function, using x 0 = 0: f(x) = e x, x [0, 1]. 3

5 9. For each function below, use the Mean Value Theorem to find a value M such that f(x 1 ) f(x 2 ) M x 1 x 2 that is valid for all x 1, x 2 in the stated interval: (a) f(x) = ln(1 + x) on the interval [ 1, 1], (b) f(x) = sin(x) on the interval [0, π]. 10. A function is called monotone on an interval if its derivative is strictly positive or strictly negative on the interval. Suppose f is continuous and monotone on the interval [a, b], and f(a)f(b) < 0; prove that there is exactly one value α [a, b] such that f(α) = (MATLAB) Use the program GRAPH_PRES to estimate the (true) maximum absolute error in using the Taylor polynomial of Question 7 to approximate 3 x on [7, 9]. 12. (MATLAB) Use the program GRAPH_MANY_PRES to graphically investigate how well the Taylor series T n (x) (for n = 1, 2,... ) approximates the functions f(x) near x = x 0 (see Questions 1, 4, 5, 6, 7, 8). 4

6 1.5 Error and Asymptotic Error 1. Use Taylor s Theorem to show that 1 + x = x + O(x2 ), for x sufficiently small. 2. Use Taylor s Theorem to show that for x sufficiently small. e x = 1 + x + O(x 2 ), 3. Use Taylor s Theorem to show that 1 cos(x) = 1 x 2 x + O(x3 ), for x sufficiently small. 4. Show that sin(x) = x + O(x 3 ). 5. Recall the summation formula n 1+r+r 2 +r r n = r k = 1 rn+1 1 r, r 1. k=0 Use this to prove that for r < 1 n r k = 1 1 r + O(rn+1 ). k=0 Hint: What is the definition of O notation? 5

7 6. Use the above result to show that 9 terms are all that is needed to compute S = e k k=0 to within 10 4 absolute accuracy. 7. Recall the summation formula n n(n + 1) S = k =. 2 k=1 Use this to show that n k = 1 2 n2 + O(n). k=0 8. Use the definition of O to show that if y = y h + O(h p ), then hy = hy h + O(h p+1 ). 9. Show that if a n = O(n p ) and b n = O(n q ) then a n b n = O(n p+q ). 10. (MATLAB) Recall that the infinite (classic) geometric series is S = 1 + r + r 2 + r 3 + = 1, r < 1. 1 r 6

8 In fact we saw in Question 5 that S = 1 + r + r 2 + r r n + O(r n+1 ), i.e., the error in using only n + 1 terms in the approximation of S is O(r n+1 ). The MATLAB program GEOMETRIC_PRES sums this finite series for a n given r until the absolute error S r k < tol, k=0 where tol is a tolerance supplied by the user. Use this program with tol = and (a) r = and (b) r = Explain the difference in results for these two cases. 1.6 Computer Arithmetic 1. Perform the indicated computations in each of three ways: (i) Exactly; (ii) Using 3 digit decimal arithmetic, with chopping; (iii) Using 3 digit decimal arithmetic, with rounding. For both approximations, compute the absolute error and the relative error. (a) (b) ( ) For f(x) = ex 1, how many terms in the Taylor x expansion are needed to get single precision accuracy 7

9 [ (7 decimal digits) for all x 0, 1 ]? How many 2 terms are needed for double precision accuracy (14 decimal digits) over this same range? 3. Using 3 digit decimal arithmetic, find three values a, b, and c, such that (a + b) + c a + (b + c). 4. Assume we are using 3 digit decimal arithmetic. For ɛ = and a 1 = 5, compute ( ) 1 a 2 = a 0 + a 1 ɛ for a 0 equal to each of 1, 2, and 3. Comment. 5. What is the machine epsilon for a computer that uses binary arithmetic, 24 bits for the fraction, and rounds? What if it chops? 6. (MATLAB) Perform the following calculation in MAT- LAB: >> What does MATLAB give as the answer? What is the exact answer? Explain the discrepancy. 8

10 7. (MATLAB) Perform the following calculation in MAT- LAB: >> format long >> What does MATLAB give as the answer? What is the exact answer? Explain the discrepancy. 8. (MATLAB) Open the MATLAB program EPS_CALC_PRES by typing: >> open eps_calc_pres Explain what the program does. If necessary type >> help while Also run the program and compare the answer you get with the value of machine epsilon in MATLAB, obtained by simply entering the command eps 9

11 9. (MATLAB) Recall the well-known expression for the number e: ( e = lim x = x x) The program EXP_CALC_PRES uses this result with integer x values to approximate e. Run this program and explain the pattern in the errors. What is the optimal x value? 10

12 Chapter 2 A Survey of Simple Methods and Tools 2.1 Horner s Rule 1. Write the following polynomial in nested form: x 3 + 3x Write the following polynomial in nested form, but this time take advantage of the fact that they involve only even powers of x to minimize the computations: x x4. 3. Write the following polynomial in nested form: 1 x x4 1 6 x6. 11

13 4. Consider the polynomial p(x) = 1+(x 1)+ 1 6 (x 1)(x 2)+1 7 (x 1)(x 2)(x 4). This can be written in nested-like form by factoring out each binomial term as far as it will go, thus: ( ( 1 p(x) = 1 + (x 1) 1 + (x 2) )) (x 4). 7 Write the following polynomial in this kind of nested form: p(x) = ( x 1 ) 5 ( x 1 ) (x 4) + ( x 1 ) (x 4)(x 2) Difference Approximations 1. Compute, by hand, approximations to f (1) for each of the following functions, with h = 1/16 using the Forward and Centered Difference Approximations: (a) f(x) = arctan(x), (b) f(x) = e x. 2. Use the Forward Difference Approximation to approximate f (1), the Centered Difference Approximation to approximate f (1.1), and the Backward Difference Approximation to approximate f (1.2), using the data in the table below: 12

14 x f(x) Use the Centered Difference Approximation to the first derivative to prove that this approximation will be exact for any quadratic polynomial. 4. Find coefficients A, B, and C so that (a) f (x) = Af(x) + Bf(x + h) + Cf(x + 2h) + O(h 2 ) (hard!). Hint: Use Taylor s theorem. 5. Use Taylor s Theorem to show that the approximation f (x) 8f(x + h) 8f(x h) f(x + 2h) + f(x 2h) 12h is O(h 4 ) (hard!). Again, use Taylor s theorem. 6. Use the derivative approximation from the previous problem with h = 0.1 to evaluate f (1.2), using the data in the following table: 13

15 x f(x) Use Taylor expansions for f(x±h) to derive an O(h 2 ) accurate approximation to f (x) using f(x) and f(x± h). Provide all the details of the error estimate. Hint: Go out as far as the fourth derivative term, and then add the two expansions. 8. (MATLAB) Use the programs FDA_PRES, BDA_PRES and CDA_PRES to investigate how the errors in difference approximations change with a decrease in the step sizes h. Explain what you see. (Note: the above programs compute approximations to the derivative of a function y = f(x) at some point x 0 using the Forward, Backward and Centered Difference Approximations respectively.) 2.3 Euler s Method 1. Use Euler s method with h = 0.25 to compute approximate solution values for the initial value prob- 14

16 lem y = sin(t + y), y(0) = 1. You should eventually get y 4 = (be sure that your calculator is set to radians). 2. Use Euler s method with h = 0.25 to compute approximate solution values for y = e t y, y(0) = 1. What approximate value do you get for y(1) = ? 3. Repeat the above with h = What value do you now get for y 8 y(1)? 4. (MATLAB) Use the program EULER_PRES to compute approximations to each of the following initial value 1 problems on [0, 1], using M = 2, 4, 8, 16 steps. (i) f(t, y) = y + sin(t), y(0) = 1, where y(t) = (3/2) exp( t) + (sin(t) cos(t))/2, (ii) f(t, y) = t y, y(0) = 2, where y(t) = 3 exp( t) + t 1, (iii) f(t, y) = exp(t y), y(0) = 1, where y(t) = log(exp(t) 1 + exp( 1)), (iv) f(t, y) = y log(y), y(0) = 3, 1 In MATLAB log means natural log. where y(t) = exp(log(3) exp( t)). 15

17 Where is the maximum error on [0, 1] for each value of h = 1/M? How does the approximation change as h is reduced? 2.4 Linear Interpolation 1. Use linear interpolation to find an approximation to erf(0.34), where f(x) = erf(x) is the error function, using the data in the following table: x f(x) Also, give an upper bound on the error in the approximation. 2. The gamma function, denoted by Γ(x), occurs in a number of applications, most notably probability theory and the solution of certain differential equations. It is basically the generalization of the factorial function to non-integer values, in that Γ(n + 1) = n!. The table below gives values of Γ(x) for x between 1 and 2. Use linear interpolation to approximate Γ(1.005) =

18 x Γ(x) Construct a linear interpolating polynomial to the function f(x) = x 1 using x 0 = 1/2 and x 1 = 1 as nodes. What is the upper bound on the error over the interval [1/2, 1], according to the error estimate? 4. Repeat the above for f(x) = x 1 3, using the interval [1/8, 1]. 5. (MATLAB) Apply the program INTEROL_PRES to Question 3. 17

19 2.5 Trapezoid Rule 1. Apply the trapezoid rule with h = 1, to approximate 8 the integral I = x 4 dx = How small does h have to be to get an error less that 10 3? 10 6? 2. Use the trapezoid rule with h = π, to approximate 4 the integral I = π 2 0 sin(x) dx = 1. How small does h have to be to get an error less that 10 3? 10 6? 3. Apply the trapezoid rule with h = 1, to approximate 8 the integral I = 1 0 x(1 x 2 ) dx = 1 4. How small does h have to be to get an error less than 10 3? 10 6? 18

20 4. Apply the trapezoid rule with h = 1 8, to approximate the integral I = 1 0 ln(1 + x) dx = 2 ln(2) 1. How small does h have to be to get an error less than 10 3? 10 6? 5. (MATLAB) Apply the program TRAPEZOID_PRES to numerically verify any of the above results. For example, what is the actual error incurred for Question 3 when using h = 1/23? 2.6 Solution of Tridiagonal Linear Systems 1. Use the tridiagonal algorithm to solve the following system of equations: x x x 3 = x 4 π π 9 2. Use the tridiagonal algorithm to solve the following 19

21 system of equations: x 1 x 2 x 3 x 4 = Is the following system diagonally dominant? Use the triadiagonal algorithm to find the solution x x x 3 = x (MATLAB) Use MATLAB to verify any of the solutions you have found by using the backslash command. That is, with the system Ax = b to find x just type >> A\b. See any tutorial to see how to enter vectors and matrices in MATLAB. 5. (MATLAB) We can use some simple commands in MATLAB to implement the individual row operations in the tridiagonal algorithm. The commands applied to a hypothetical augmented matrix B are as follows (r i refers to the ith row of B and k is a 20

22 nonzero scalar): To do kr i r i (multiply row i by k) type >> B(i,:) = k*b(i,:) To do r i kr j r i (subtract k times row j from row i) type >> B(i,:) = B(i,:) - k*b(j,:) To do r i r j (swap rows i and j) type >> B([i,j],:) = B([j,i],:) 2.7 Boundary Value Problems 1. Solve, by hand, the following two-point BVP: u + u = f(x), x [0, 1], u(0) = u(1) = 0, where f(x) = x, using h = 1/4. Write out the linear system explicitly prior to a solution. You should get the following 3 3 system: U 1 U 2 U 3 =

23 2. Repeat the above problem, but this time using h = 1/5. What does the new system become? 3. Approximate the solution of the two-point BVP u + u + u = 1, x [0, 1], u(0) = u(1) = 0. Assuming h < 2 determine if the resulting coefficient matrix is diagonally dominant? Hint: For the approximation of the 2nd derivative see your lecture notes (MATLAB) Use the program BVP_SIMPLE_PRES to numerically solve the BVP u + u = f(x), x [0, 1] u(0) = u(1) = 0, where f and the exact solutions u are given by (a) f(x) = (π 2 + 1) sin(πx), u(x) = sin(πx); 1 Note: the problem of determining diagonal dominance is harder than I would give you in an exam, but shows you the sort of calculations one might have to do in practice. However, the initial part is fair game! 22

24 (b) f(x) = 4 exp( x) 4x exp( x), u(x) = x(1 x) exp( x); (c) f(x) = π(π sin(πx) + 2 cos(πx)) exp( x), u(x) = exp( x) sin(πx); (d) f(x) = 3 1/x (x 2 x 2) ln(x), u(x) = x(1 x) ln(x). Using h = 1/2 n where n = 2, 3, 4,... numerically investigate the accuracy of the approximations. Recall that with n equal subintervals we have the approximations u k u(x k ), k = 1, 2,..., n 1 where the nodes are x k = hk, h = 1/n. As a measure of the accuracy choose the error to be E(h) := max k u(x k ) u k, k = 1, 2,..., n 1. 23

25 Chapter 3 Root - Finding 3.1 Bisection Method 1. Do three iterations of the bisection method, applied to f(x) = x 3 2 using a = 0 and b = (a) How many iterations of the bisection method are needed to find the root of f(x) = x e x2 on [0, 1] to an accuracy of 0.1? (b) Apply the method to find the root to the specified accuracy. (c) If the actual root is verify that your root does indeed have the required accuracy. 3. (MATLAB) Use the program BISECT_PRES to apply the bisection method to the following test problems: (a) f(x) = 2 exp(x) on [0, 1], 24

26 (b) f(x) = x 3 2 on [0, 2], (c) f(x) = 5 1/x on [0.1, 0.25], (d) f(x) = x 2 sin(x) on [0.5, 1]. Can you check the exact roots of the problems with the answers given by the program? Use an appropriate error bound to check a priori the minimum number of steps needed for the roots to be accurate within Newton s Method 1. (a) Write down Newton s method applied to the function f(x) = x 3 2. Simplify the computation as much as possible. (b) What has been accomplished if we find the root of this function? 2. (a) Write down Newton s method applied to the function f(x) = a x 1. Simplify the resulting computation as much as possible. (b) What has been accomplished if we find the root of this function? 25

27 3. (a) Do three iterations of Newton s method for f(x) = 3 e x, using x 0 = 1. Repeat, with x 0 = 2, 4, 8, 16, but using as many iterations as needed in order to see a pattern. (b) Comment on your results. 4. Figure 3.1 shows the geometry of a planetary orbit around the sun. The position of the sun is given by S, the position of the planet is given by P. Let x denote the angle defined by P 0 OA, measured in radians. The dotted line is a circle concentric to the ellipse and having a radius equal to the major axis of the ellipse. Let T be the total orbital period of the planet, and let t be the time required for the planet to go from A to P. Then Kepler s equation from orbital mechanics, relating x and t, is x ɛ sin(x) = 2πt T. Here ɛ is the eccentricity of the elliptical orbit (the extent to which it deviates from a circle). For an orbit of eccentricity ɛ = 0.01 (roughly equivalent to that of the Earth), what is the value of x corresponding to t = T? What is the value of x corresponding to 4 t = T? Use Newton s method to solve the required 8 equation. 26

28 Figure 3.1: Orbital geometry 27

29 5. (MATLAB) Use the program NEWTON_RAPH_PRES to apply Newton s method to the following test problems: (a) f(x) = x 2 4 on [1, 6.5], x 0 = 6, tol = 1e 06; (b) f(x) = (x 2) 2 on [1.5, 3.2], x 0 = 3, tol = 1e 03; (c) f(x) = 6(x 2) 5 on [2, 3], x 0 = 3, tol = 1e 03; (d) f(x) = (4/3) exp((2 x/2)(1 + ln(x)/x)) on [0, 20], x 0 = 2.5, f (x) = (4/3) exp((2 x/2)(1 + ln(x)/x)) ((2 x/2)(1 ln(x))/x 2 (1 + ln(x)/2)/2); (e) f(x) = x 3 2x + 2 on [ 0.5, 1.5], x 0 = 0; (f) f(x) = 1 x 2 on [ 1, 1], x 0 = 0. The number tol refers to the tolerance that the program uses in it s stopping criterion for Newton s method (default is 10 6 ). In each case: (i) explain the behaviour of Newton s method that you see, and (ii) for those cases where we have convergence, explain how the rate of convergence is related to the multiplicity of the roots. 28

30 3.3 How to Stop Newton s Method 1. Approximate the root of f(x) = e x + x using Newton s Method. Stop the method using an appropriate Stopping Criterion with a tolerance of (see your lecture notes). (Hint: to start, show using the IMVT that there is a root in [ 1, 0].) 2. (MATLAB) Use the program NEWTON_RAPH_PRES to check your written answer in Problem Secant Method 1. Do three steps of the secant method for f(x) = x 3 2, using x 0 = 0, x 1 = Repeat the above using x 0 = 1, x 1 = 0. Comment on how the algorithm is performing compared to the last example. 3. (MATLAB) Use the program SECANT_PRES to apply the Secant method to the following test problems: (a) f(x) = x 2 4 on [2, 6], x 0 = 6, x 1 = 5.5, tol = 1e 03; 29

31 (b) f(x) = (x 2) 2 on [2, 3], x 0 = 3, x 1 = 2.7, tol = 1e 06; (c) f(x) = 6(x 2) 5 on [2, 3], x 0 = 3, x 1 = 2.95, tol = 1e 03; (d) f(x) = x 2 4 on [ 3, 3], x 0 = 1, x 1 = 1; (e) f(x) = (4/3) exp((2 x/2)(1 + ln(x)/x)) on [2, 20], x 0 = 2.5, x 1 = 3.5. The number tol refers to the tolerance that the program uses in it s stopping criterion for the Secant method (default is 10 6 ). In each case: (i) explain the behaviour of the Secant method that you see, and (ii) for those cases where we have convergence, explain how the rate of convergence is related to the multiplicity of the roots. 3.5 Fixed Point Iteration 1. Do three steps of the following fixed point iteration problem: x n+1 = ln(1 + x n ), x 0 = Let y 0 and h = 1/8 be fixed numbers. Do three steps 30

32 of the following fixed point iteration y n+1 = y h( y 0 ln(y 0 ) y n ln(y n )), y 0 = 1/2. 3. Consider the fixed point iteration x n+1 = 1 + e x n. Show that this iteration converges for any x 0 [1, 2]. How many iterations does the theory predict it will take to achieve 10 5 accuracy? 4. For each function listed below, find an interval [a, b] such that g([a, b]) [a, b]. Draw a graph of y = g(x) and y = x over this interval, and confirm that a fixed point exists there. Estimate (by eye) the value of the fixed point, and use this as a starting value for a fixed point iteration. Does the iteration converge? Explain: (a) g(x) = 1 ( x + 2 ), (b) g(x) = 1+e x. 2 x 5. Let h(x) = 1 x 2 /4. Show that this function has a root at x = α = 2. Using x 0 = 1/2, apply fixed point iteration with x n+1 = h(x n ) to approximate the fixed point of h. Comment on your results. 31

33 6. (MATLAB) Use the program FIXED_POINT_ITER_PRES to apply Fixed Point iteration to the following iteration functions: (a) g(x) = sin(x) on [0, 2.5], x 0 = 2, tol = 1e 03; (b) g(x) = 1 + 2/x on [0, 3.5], x 0 = 1, tol = 1e 03; (c) g(x) = x 2 /4 + x/2 on [2, 20], x 0 = 2.3. In each case (i) work out by hand (if possible) the fixed points, (ii) explain what you see, and state whether the iterative process converges or diverges. 32

34 Chapter 4 Interpolation and Approximation 4.1 Lagrange Interpolation 1. Find the polynomial of degree 2 that interpolates y = x 3 at the nodes x 0 = 0, x 1 = 1 and x 2 = 2. Plot y = x 3 and the interpolating polynomial over the interval [0, 2]. 2. Construct the quadratic polynomial that interpolates y = x at the nodes x 0 = 1/4, x 1 = 9/16, and x 2 = Find the polynomial of degree 3 that interpolates y = x 3 at the nodes x 0 = 0, x 1 = 1, x 2 = 2, and x 3 = 3. (Simplify your interpolating polynomial as 33

35 much as possible.) Hint: This is easy if you think about the implications of the uniqueness of the interpolating polynomial. 4. What is the error in quadratic interpolation to f(x) = x, using equally spaced nodes on the interval [1/4, 1]? 5. Repeat the above for f(x) = x 1 on [ ] 1 2, (MATLAB) Use the program LAGRANGE_PRES to construct and plot the Lagrange interpolant P n (x) for the following functions f(x) with n nodes on the specified intervals [a, b]: (a) f(x) = exp(x) on [ 2, 2] with n = 2, 3, 4; (b) f(x) = x 3 on [0, 2] with n = 2, 3, 4; (c) f(x) = tan(x) on [ 1.5, 1.5] with n = 4, 5, 6, 7, 8; (d) f(x) = 1/(1+25x 2 ) on [ 1, 1] with n = 4, 8, 16, 32. We can measure the fit between f(x) and the interpolating polynomial P n (x) using the maximum norm f P n := max x [a,b] f(x) P n(x), 34

36 which is just the maximum absolute error between f and P n for all x [a, b]. We say that the sequence of Lagrange interpolation polynomials P n converges to f if f P n 0 as n. For each problem above decide in which cases the Lagrange interpolation polynomials are convergent to the given functions. If the Lagrange polynomials are divergent explain why based on the plots. 4.2 Hermite Interpolation 1. Construct the cubic Hermite interpolant to f(x) = 1 + x using the nodes a = 0 and b = 1. Plot the interpolant and the function together. Where is the approximation error the greatest? 2. Construct the cubic Hermite interpolant to f(x) = sin(x) using the nodes a = 0 and b = π. Plot the error between the interpolant and the function. Where is the error greatest? 3. Construct the cubic [ Hermite ] interpolatant to f(x) = x on the interval 8, 1. What is the maximum er- 35

37 ror as predicted by theory? 4. (MATLAB) Use the program HERMITE_PRES to construct and plot the Hermite interpolant H(x) for the following functions f(x) with n nodes on the specified intervals [a, b]: (a) f(x) = exp(x) on [ 1, 1] with n = 2, 3 (b) f(x) = 1/x on [1/2, 4] with n = 2, 3, 4 (c) f(x) = x on [ 1, 1] with n = 2, 3, 4, 5; (d) f(x) = 1/(1+25x 2 ) on [ 1, 1] with n = 2, 3, 4, 5, 6; We can measure the fit between f(x) and the interpolating polynomial H(x) using the maximum norm f H := max f(x) H(x), x [a,b] which is just the maximum absolute error between f and H for all x [a, b]. We say that the sequence of Hermite interpolation polynomials H converges to f if f H 0 as n. For each problem above decide in which cases the Hermite interpolation polynomials are convergent to the given functions. If the Hermite polynomials are divergent explain why based on the plots. How do the graphs of the Hermite interpolating polynomials differ from the graphs of the Lagrange interpolating polynomials? 36

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