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All Textbook Solutions for Numerical Analysis

Show that the following equations have at least one solution in the given intervals. a. x cos x 2x2 + 3x 1 = 0, [0.2, 0.3] and [1.2, 1.3] b. (x 2)2 ln x = 0, [1, 2] and [e, 4] c. 2x cos(2x) (x 2)2 = 0, [2, 3] and [3, 4] d. x (ln x)x = 0, [4, 5]Show that the following equations have at least one solution in the given intervals. a. x cos x = 0, [0, 1] b. ex x2 + 3x 2 = 0, [0, 1] c. 3 tan(2x) + x = 0, [0, 1] d. ln x x2 + 52x 1 = 0, [12,1]Find intervals containing solutions to the following equations. a. x 2x = 0 b. 2x cos(2x) (x + 1)2 = 0 c. 3x ex = 0 d. x + 1 2 sin(x) = 0Find intervals containing solutions to the following equations. a. x 3x = 0 b. 4x2 ex = 0 c. x3 2x2 4x + 2 = 0 d. x3 + 4.001x2 + 4.002x + 1.101 = 0Find maxaxb |f(x)| for the following functions and intervals. a. f(x) = (2 ex + 2x)/3, [0, 1] b. f(x) = (4x 3)/(x2 2x), [0.5, 1] c. f(x) = 2x cos(2x) (x 2)2, [2, 4] d. f(x) = 1 + ecos(x1), [1, 2]Find maxaxb | f(x)| for the following functions and intervals. a. f(x) = 2x/(x2 + 1), [0, 1] b. f(x) = x2(4x),, [0, 4] c. f(x) = x3 4x + 2, [1, 2] d. f(x) = x(3x2),, [0, 1]Show that f(x) is 0 at least once in the given intervals. a. f(x) = 1 ex + (e 1) sin((/2)x), [0, 1] b. f(x) = (x 1) tan x + x sin x, [0, 1] c. f(x) = x sin x (x 2) ln x, [1, 2] d. f(x) = (x 2) sin x ln(x + 2), [1, 3]Suppose f C[a, b] and f (x) exists on (a, b). Show that if f (x) 0 for all x in (a, b), then there can exist at most one number p in [a, b] with f(p) = 0.Let f(x) = x3. a. Find the second Taylor polynomial P2(x) about x0 = 0. b. Find R2(0.5) and the actual error in using P2(0.5) to approximate f(0.5). c. Repeat part (a) using x0 = 1. d. Repeat part (b) using the polynomial from part (c).Find the third Taylor polynomial P3(x) for the function f(x)=x+1 about x0 = 0. Approximate 0.5,0.75,1.25 and 1.5 using P3(x) and find the actual errors.Find the second Taylor polynomial P2(x) for the function f(x) = ex cos x about x0 = 0. a. Use P2(0.5) to approximate f(0.5). Find an upper bound for error |f(0.5) P2(0.5)| using the error formula and compare it to the actual error. b. Find a bound for the error |f(x) P2(x)| in using P2(x) to approximate f(x) on the interval [0, 1]. c. Approximate 01f(x)dx using 01P2(x)dx. d. Find an upper bound for the error in (c) using 01|R2(x)dx| and compare the bound to the actual error.Repeat Exercise 11 using x0 = /6. 11. Find the second Taylor polynomial P2(x) for the function f(x) = ex cos x about x0 = 0. a. Use P2(0.5) to approximate f(0.5). Find an upper bound for error |f(0.5) P2(0.5)| using the error formula and compare it to the actual error. b. Find a bound for the error |f(x) P2(x)| in using P2(x) to approximate f(x) on the interval [0, 1]. c. Approximate 01f(x)dx using 01P2(x)dx. d. Find an upper bound for the error in (c) using 01|R2(x)dx| and compare the bound to the actual error.13ES14ES15ESUse the error term of a Taylor polynomial to estimate the error involved in using sin x x to approximate sin 1.Use a Taylor polynomial about /4 to approximate cos 42 to an accuracy of 106.Let f(x) = (1 x)1 and x0 = 0. Find the nth Taylor polynomial Pn(x) for f(x) about x0. Find a value of n necessary for Pn(x) to approximate f(x) to within 106 on [0, 0.5].Let f(x) = ex and x0 = 0. Find the nth Taylor polynomial Pn(x) for f(x) about x0. Find a value of n necessary for Pn(x) to approximate f(x) to within 106 on [0, 0.5].20ESThe polynomial P2(x)=112x2 is to be used to approximate f(x) = cos x in [12,12]. Find a bound for the maximum error.Use the Intermediate Value Theorem 1.11 and Rolles Theorem 1.7 to show that the graph of f(x) = x3 + 2x + k crosses the x-axis exactly once, regardless of the value of the constant k.23ES24ES25ES26ES27ES28ES29ES30ESIn your own words, describe the Lipschitz condition. Give several examples of functions that satisfy this condition or give examples of functions that do not satisfy this condition.Compute the absolute error and relative error in approximations of p by p. a. p = , p = 22/7 b. p = , p = 3.1416 c. p = e, p = 2.718 d. p = 2, p = 1.414Compute the absolute error and relative error in approximations of p by p. a. p = e10, p = 22000 b. p = 10, p = 1400 c. p = 8!, p = 39900 d. p = 9!, p = 18(9/e)93ESFind the largest interval in which p must lie to approximate p with relative error at most 104 for each value of p. a. b. e c. 2 d. 73Perform the following computations (i) exactly, (ii) using three-digit chopping arithmetic, and (iii) using three-digit rounding arithmetic. (iv) Compute the relative errors in parts (ii) and (iii). a. 45+13 b. 4513 c. (13311)+320 d. (13+311)320Use three-digit rounding arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits. a. 133 + 0.921 b. 133 0.499 c. (121 0.327) 119 d. (121 119) 0.327Use three-digit rounding arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits. a. 1314672e5.4 b. 10+6e362 c. (29)(97) d. 13+111311Repeat Exercise 7 using four-digit rounding arithmetic. 7. Use three-digit rounding arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits. a. 1314672e5.4 b. 10+6e362 c. (29)(97) d. 13+111311Repeat Exercise 7 using three-digit chopping arithmetic. 7. Use three-digit rounding arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits. a. 1314672e5.4 b. 10+6e362 c. (29)(97) d. 13+11131110ES11ES12ESLet f(x)=xcosxsinxxsinx. a. Find limx0 f(x). b. Use four-digit rounding arithmetic to evaluate f(0.1). c. Replace each trigonometric function with its third Maclaurin polynomial and repeat part (b). d. The actual value is f(0.1) = 1.99899998. Find the relative error for the values obtained in parts (b) and (c).Let f(x)=exexx. a. Find limx0(ex ex )/x. b. Use three-digit rounding arithmetic to evaluate f(0.1). c. Replace each exponential function with its third Maclaurin polynomial and repeat part (b). d. The actual value is f(0.1) = 2.003335000. Find the relative error for the values obtained in parts (b) and (c).Use four-digit rounding arithmetic and the formulas (1.1), (1.2), and (1.3) to find the most accurate approximations to the roots of the following quadratic equations. Compute the absolute errors and relative errors. a. 13x21234x+16=0 b. 13x2+1234x16=0 c. 1.002x2 11.01x + 0.01265 = 0 d. 1.002x2 + 11.01x + 0.01265 = 016ES17ESRepeat Exercise 16 using four-digit chopping arithmetic. 16. Use four-digit rounding arithmetic and the formulas (1.1), (1.2), and (1.3) to find the most accurate approximations to the roots of the following quadratic equations. Compute the absolute errors and relative errors. a. 13x21234x+16=0 b. 13x2+1234x16=0 c. 1.002x2 11.01x + 0.01265 = 0 d. 1.002x2 + 11.01x + 0.01265 = 0Use the 64-bit-long real format to find the decimal equivalent of the following floating-point machine numbers. a. 0 10000001010 1001001100000000000000000000000000000000000000000000 b. 1 10000001010 1001001100000000000000000000000000000000000000000000 c. 0 01111111111 0101001100000000000000000000000000000000000000000000 d. 0 01111111111 010100110000000000000000000000000000000000000000000120ES21ES22ES23ES24ES25ES26ES27ES28ES29ESDiscuss the difference between the arithmetic performed by a computer and traditional arithmetic. Why is it so important to recognize the difference?2DQDiscuss the various different ways to round numbers.Discuss the difference between a number written in standard notation and one that is written in normalized decimal floating-point form. Provide several examples.1ES2ESThe Maclaurin series for the arctangent function converges for 1 x 1 and is given by arctanx=limnPn(x)=limni=1n(1)i+1x2i12i1. a. Use the fact that tan /4 = 1 to determine the number of n terms of the series that need to be summed to ensure that |4Pn(1) | 103. b. The C++ programming language requires the value of to be within 1010. How many terms of the series would we need to sum to obtain this degree of accuracy?4ES5ESFind the rates of convergence of the following sequences as n. a. limnsin1n=0 b. limnsin1n2=0 c. limn(sin1n)2=0 d. limn[ln(n+1)ln(n)]=0Find the rates of convergence of the following functions as h 0. a. limh0sinhh=1 b. limh01coshh=0 c. limh0sinhhcoshh=0 d. limh01ehh=18ES9ESSuppose that as x approaches zero, F1(x)=L1+O(x)andF2(x)=L2+O(x). Let c1 and c2 be nonzero constants and define F(x)=c1F1(x)+c2F2(x)andG(x)=F1(c1x)+F2(c2x). Show that if = minimum {, }, then, as x approaches zero, a. F(x) = c1L1 + c2L2 + O(x) b. G(x) = L1 + L2 + O(x).11ES12ES13ES14ESa. How many multiplications and additions are required to determine a sum of the form i=1nj=1iaibj? b. Modify the sum in part (a) to an equivalent form that reduces the number of computations.Write an algorithm to sum the finite series i=1nxi in reverse order.Construct an algorithm that has as input an integer n 1, numbers x0, x1, , xn, and a number x and that produces as output the product (x x0)(x x1) (x xn).Let P(x) = anxn + an1xn1 + + a1x + a0 be a polynomial and let x0 be given. Construct an algorithm to evaluate P(x0) using nested multiplication.4DQ5DQ6DQ1DQUse the Bisection method to find p3 for f(x)=xcosx = 0 on [0, 1].Let f(x) = 3(x +1)(x 12)(x 1) = 0. Use the Bisection method on the following intervals to find p3. a. [2, 1.5] b. [1.25, 2.5]Use the Bisection method to find solutions accurate to within 102 for x3 7x2 + 14x 6 = 0 on each interval. a. [0, 1] b. [1, 3.2] c. [3.2, 4]Use the Bisection method to find solutions accurate to within 102 for x4 2x3 4x2 + 4x + 4 = 0 on each interval. a. [2,1] b. [0, 2] c. [2, 3] d. [1, 0]Use the Bisection method to find solutions accurate to within 105 for the following problems. a. x 2x = 0 for 0 x 1 b. ex x2 + 3x 2 = 0 for 0 x 1 c. 2x cos(2x) (x + 1)2 = 0 for 3 x 2 and 1 x 0 d. x cos x 2x2 + 3x 1 = 0 for 0.2 x 0.3 and 1.2 x 1.36ES7ES8ES9ES10ES11ESLet f(x) = (x + 2)(x + 1)x(x 1)3(x 2). To which zero of f does the Bisection method converge when applied on the following intervals? a. [1.5, 2.5] b. [0.5, 2.4] c. [0.5, 3] d. [3, 0.5]Find an approximation to 253 correct to within 104 using the Bisection Algorithm. [Hint: Consider f(x) = x3 25.]Find an approximation to 3 correct to within 104 using the Bisection Algorithm. [Hint: Consider f(x) = x2 3.]A trough of length L has a cross section in the shape of a semicircle with radius r. (See the accompanying figure.) When filled with water to within a distance h of the top, the volume V of water is V=L[0.5r2r2arcsin(h/r)h(r2h2)1/2]. Suppose L = 10 ft, r = 1 ft, and V = 12.4 ft3. Find the depth of water in the trough to within 0.01 ft.16ESUse Theorem 2.1 to find a bound for the number of iterations needed to achieve an approximation with accuracy 104 to the solution of x3x1 = 0 lying in the interval [1, 2]. Find an approximation to the root with this degree of accuracy.18ES19ESLet f(x) = (x 1)10, p = 1, and pn = 1 + 1/n. Show that | f (pn)| 103 whenever n 1 but that |p pn | 103 requires that n 1000.The function defined by f(x) = sin x has zeros at every integer. Show that when 1 a 0 and 2 b 3, the Bisection method converges to a. 0, if a + b 2 b. 2, if a + b 2 c. 1, if a + b = 21DQ2DQIs the Bisection method sensitive to the starting value? Why or why not?Use algebraic manipulation to show that each of the following functions has a fixed point at p precisely when f(p) = 0, where f(x) = x4 + 2x2 x 3. a. g1(x) = (3 + x 2x2)1/4 b. g2(x)=(x+3x42)1/2 c. g3(x)=(x+3x2+2)1/2 d. g4(x)=3x4+2x2+34x3+4x1a. Perform four iterations, if possible, on each of the functions g defined in Exercise 1. Let p0 = 1 and pn+1 = g(pn), for n = 0, 1, 2, 3. b. Which function do you think gives the best approximation to the solution?Let f(x) = x3 2x + 1. To solve f(x) = 0, the following four fixed-pint problems are proposed. Derive each fixed point method and compute p1, p2, p3, p4. Which methods seem to be appropriate? a. x=12(x3+1),p0=12 b. x=2x1x2,p0=12 c. x=21x,p0=12 d. x=12x3,p0=12Let f(x) = x4 + 3x2 2. To solve f(x) = 0, the following four fixed-pint problems are proposed. Derive each fixed point method and compute p1, p2, p3, and p4. Which methods seem to be appropriate? a. x=2x43,p0=1 b. x=23x24,p0=1 c. x=2x43x,p0=1 d. x=23x2x3,p0=1The following four methods are proposed to compute 211/3. Rank them in order, based on their apparent speed of convergence, assuming p0 = 1. a. pn=20pn1+21/pn1221 b. pn=pn1pn13213pn12 c. pn=pn1pn1421pn1pn1221 d. pn=(21pn1)1/26ES7ES8ESUse Theorem 2.3 to show that g(x) = + 0.5 sin(x/2) has a unique fixed point on [0, 2]. Use fixed-point iteration to find an approximation to the fixed point that is accurate to within 102. Use Corollary 2.5 to estimate the number of iterations required to achieve 102 accuracy and compare this theoretical estimate to the number actually needed.Use Theorem 2.3 to show that g(x) = 2x has a unique fixed point on [13,1]. Use fixed-point iteration to find an approximation to the fixed point accurate to within 104. Use Corollary 2.5 to estimate the number of iterations required to achieve 104 accuracy and compare this theoretical estimate to the number actually needed.Use a fixed-point iteration method to find an approximation to 3 that is accurate to within 104. Compare your result and the number of iterations required with the answer obtained in Exercise 14 of Section 2.1.12ES13ES14ES15ESUse a fixed-point iteration method to determine a solution accurate to within 104 for x = tan x, for x in [4, 5].Use a fixed-point iteration method to determine a solution accurate to within 102 for x = 2 sin (x) + x = 0, for x on [1, 2]. Use p0 = 1.18ES19ES20ES21ESa. Show that Theorem 2.3 is true if the inequality |g(x)| k is replaced by g(x) k, for all x (a, b). [Hint: Only uniqueness is in question.] b. Show that Theorem 2.4 may not hold if inequality |g(x)| k is replaced by g(x) k. [Hint: Show that g(x) = 1 x2, for x in [0, 1], provides a counter example.]a. Use Theorem 2.4 to show that the sequence defined by xn=12xn1+1xn1,forn1, converges to 2 whenever x0 2. b. Use the fact that 0 (x0 2)2 whenever x0 2 to show that if 0 x0 2, then x1 2. c. Use the results of parts (a) and (b) to show that the sequence in (a) converges to 2 whenever x0 0.24ES25ESSuppose that g is continuously differentiable on some interval (c, d) that contains the fixed point p of g. Show that if |g(p)| 1, then there exists a 0 such that if |p0 p| , then the fixed-point iteration converges.1DQLet f(x) = x2 6 and p0 = 1. Use Newtons method to find p2.Let f(x) = x3 cos x and p0 = 1. Use Newtons method to find p2. Could p0 = 0 be used?Let f(x) = x2 6. With p0 = 3 and p1 = 2, find p3. a. Use the Secant method. b. Use the method of False Position. c. Which of part (a) or (b) is closer to 6?Let f(x) = x3 cos x. With p0 = 1 and p1 = 0, find p3. a. Use the Secant method. b. Use the method of False Position.5ES6ES7ES8ES9ES10ES11ES12ESThe fourth-degree polynomial f(x)=230x4+18x3+9x2221x9 has two real zeros, one in [1, 0] and the other in [0, 1]. Attempt to approximate these zeros to within 106 using the a. method of False Position b. Secant method c. Newtons method Use the endpoints of each interval as the initial approximations in parts (a) and (b) and the midpoints as the initial approximation in part (c).14ES15ES16ES17ES18ES19ES20ES21ES22ES23ES24ES25ES26ES27ESA drug administered to a patient produces a concentration in the bloodstream given by c(t) = Atet/3 milligrams per milliliter, t hours after A units have been injected. The maximum safe concentration is 1 mg/mL. a. What amount should be injected to reach this maximum safe concentration, and when does this maximum occur? b. An additional amount of this drug is to be administered to the patient after the concentration falls to 0.25 mg/mL. Determine, to the nearest minute, when this second injection should be given. c. Assume that the concentration from consecutive injections is additive and that 75% of the amount originally injected is administered in the second injection. When is it time for the third injection?29ES30ES31ES32ES1DQ2DQ3DQ4DQ1ES2ES3ES4ES5ES6ESa. Show that for any positive integer k, the sequence defined by pn = 1/nk converges linearly to p = 0. b. For each pair of integers k and m, determine a number N for which 1/Nk 10m.8ESa. Construct a sequence that converges to 0 of order 3. b. Suppose 1. Construct a sequence that converges to 0 of order .10ES11ES12ES13ES14ES1DQ2DQ3DQ4DQ1ES2ESLet g(x) = cos(x 1) and p0(0) = 2. Use Steffensens method to find p0(1).4ES5ES6ESUse Steffensens method to find, to an accuracy of 104, the root of x3 x 1 = 0 that lies in [1, 2] and compare this to the results of Exercise 8 of Section 2.2. 8. Use a fixed-point iteration method to determine a solution accurate to within 102 for x3 x 1 = 0 on [1, 2]. Use p0 = 1.8ES9ESUse Steffensens method with p0 = 3 to compute an approximation to 253 accurate to within 104. Compare this result with the results obtained in Exercise 12 of Section 2.2 and Exercise 13 of Section 2.1. 12. Use a fixed-point iteration method to find an approximation to 253 that is accurate to within 104. Compare your result and the number of iterations required with the answer obtained in Exercise 13 of Section 2.1. 13. Find an approximation 253 correct to within 104 using the Bisection Algorithm. [Hint: Consider f(x) = x3 25.]Use Steffensens method to approximate the solutions of the following equations to within 105. a. x = (2 ex + x2)/3, where g is the function in Exercise 13(a) of Section 2.2. b. x = 0.5(sin x + cos x), where g is the function in Exercise 13(f) of Section 2.2. c. x = (ex/3)1/2, where g is the function in Exercise 13(c) of Section 2.2. d. x = 5x, where g is the function in Exercise 13(d) of Section 2.2. 13. For each of the following equations, determine an interval [a, b] on which fixed-point iteration will converge. Estimate the number of iterations necessary to obtain approximations accurate to within 105 and perform the calculations. a. x=2ex+x23 b. x=5x2+2 c. x = (ex/3)1/2 d. x = 5x e. x = 6x f. x = 0.5(sin x + cos x)12ES13ES14ES15ES16ES17ES1DQ2DQ1ES2ES3ES4ES5ES6ES7ES8ES9ES10ES1DQ2DQ1DQ2DQ3DQFor the given functions f(x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct interpolation polynomials of degree at most one and at most two to approximate f(0.45) and find the absolute error.) a. f(x) = cos x b. f(x) = ln(x + 1) c. f(x)=1+x d. f(x) = tan x2ESUse Theorem 3.3 to find an error bound for the approximations in Exercise 1. 1. For the given functions f(x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct interpolation polynomials of degree at most one and at most two to approximate f(0.45) and find the absolute error.) a. f(x) = cos x b. f(x) = ln(x + 1) c. f(x)=1+x d. f(x) = tan x Theorem 3.3 Suppose x0, x1, ... , xn are distinct numbers in the interval [a, b] and Cn + 1[a, b]. Then, for each x in [a, b], a number (x) (generally unknown) between min{x0, x1,..., xn}, and the max{x0, x1,..., xn} and hence in (a, b), exists with f(x)=P(x)+f(n+1)((x))(n+1)!(xx0)(xx1)(xxn),(3.3) where P(x) is the interpolating polynomial given in Eq. (3.1).4ES5ES6ES7ESThe data for Exercise 6 were generated using the following functions. Use the error formula to find a bound for the error and compare the bound to the actual error for the cases n = 1 and n = 2. a. f (x) = e2x b. f (x) = x4 x3 + x2 x + 1 c. f (x) = x2 cos x 3x d. f (x) = ln(ex + 2) 6. Use appropriate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following: a. f(0.43) if f(0) = 1, f (0.25) = 1.64872, f(0.5) = 2.71828, f(0.75) = 4.48169 b. f(0) if f(0.5) = 1.93750, f(0.25) = 1.33203, f(0.25) = 0.800781, f(0.5) = 0.687500 c. f(0.18) if f (0.1) = 0.29004986, f (0.2) = 0.56079734, f (0.3) = 0.81401972, f (0.4) = 1.0526302 d. f (0.25) if f (1) = 0.86199480, f (0.5) = 0.95802009, f (0) = 1.0986123, f (0.5) = 1.29437679ES10ES11ES12ES13ES14ES15ES16ES17ES18ES19ESIt is suspected that the high amounts of tannin in mature oak leaves inhibit the growth of the winter moth (Operophtera bromata L., Geometridae) larvae that extensively damage these trees in certain years. The following table lists the average weight of two samples of larvae at times in the first 28 days after birth. The first sample was reared on young oak leaves, whereas the second sample was reared on mature leaves from the same tree. a. Use Lagrange interpolation to approximate the average weight curve for each sample. b. Find an approximate maximum average weight for each sample by determining the maximum of the interpolating polynomial.21ESProve Taylors Theorem 1.14 by following the procedure in the proof of Theorem 3.3. [Hint: Let g(t)=f(t)P(t)[f(x)P(x)].(tx0)n+1(xx0)n+1, where P is the nth Taylor polynomial, and use the Generalized Rolles Theorem 1.10.] Theorem 1.14 (Taylor's Theorem) Suppose f Cn[a, b], f(n + 1) exists on [a, b], and x0 [a, b], For every x [a, b], there exists a number (x) between x0 and x with f(x) = Pn(x) + Rn(x). where Pn(x)=f(x0)+f(x0)(xx0)+f(x0)2!(xx0)2+f(n)(x0)n!(xx0)n=k=0nf(k)(x0)k!(xx0)k23ES1DQIf we decide to increase the degree of the interpolating polynomial by adding nodes, is there an easy way to use a previous interpolating polynomial to obtain a higher-degree interpolating polynomial, or do we need to start over?1ES2ES3ESLet P3(x) be the interpolating polynomial for the data (0, 0), (0.5, y), (1, 3), and (2, 2). Find y if the coefficient of x3 in P3(x) is 6.Nevilles method is used to approximate f(0.4), giving the following table. Determine P2 = f(0.5).Nevilles method is used to approximate f(0.5), giving the following table. Determine P2 = f(0.7).Suppose xj = j, for j = 0, 1, 2, 3, and it is known that P0,1 (x) = 2x + 1, P0,2 (x) = x + 1, and P1,2,3(2.5) = 3. Find P0,1,2,3(2.5).8ESNevilles Algorithm is used to approximate f(0) using f(2), f(1), f(1), and f(2). Suppose f(1) was understated by 2 and f(1) was overstated by 3. Determine the error in the original calculation of the value of the interpolating polynomial to approximate f(0).10ES11ES12ES13ES1DQCan Nevilles method be used to obtain the interpolation polynomial at a general point as opposed to a specific point?Use Eq. (3.10) or Algorithm 3.2 to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f(8.4) if f(8.1) = 16.94410, f(8.3) = 17.56492, f(8.6) = 18.50515, f(8.7) = 18.82091 b. f(0.9) if f(0.6) = 0.17694460, f(0.7) = 0.01375227, f(0.8) = 0.22363362, f(1.0) = 0.65809197 Pn(x)=f[x0]+k=1nf[x0,x1,,xk](xx0)(xxk1)(3.10) ALGORITHM 3.2 Newtons Divided-Difference Formula To obtain the divided-difference coefficients of the interpolatory polynomial P on the (n + 1) distinct numbers x0, x1, xn, for the function f: INPUT numbers x0, x1, xn; values f(x0), f(x1), , f(xn) as F0,0, F1,0, , Fn,0. OUTPUT the numbers F0,0, F1,1, , Fn,n where Pn(x)=F0,0+i=1nFijj=0i1(xxj).(Fijisf[x0,x1,,xi].) Step 1 For i = 1, 2, , n For j = 1, 2, , i set Fi,j=Fi,j1Fi1,j1xixij,(Fi,j=f[xij,,xi].) Step 2 OUTPUT (F0,0, F1,1, , Fn,n); STOP.Use Eq. (3.10) or Algorithm 3.2 to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f(0.43) if f(0) = 1, f(0.25) = 1.64872, f(0.5) = 2.71828, f(0.75) = 4.48169 b. f(0) if f(0.5) = 1.93750, f(0.25) = 1.33203, f(0.25) = 0.800781, f(0.5) = 0.687500 Pn(x)=f[x0]+k=1nf[x0,x1,,xk](xx0)(xxk1)(3.10) ALGORITHM 3.2 Newtons Divided-Difference Formula To obtain the divided-difference coefficients of the interpolatory polynomial P on the (n + 1) distinct numbers x0, x1, xn, for the function f: INPUT numbers x0, x1, xn; values f(x0), f(x1), , f(xn) as F0,0, F1,0, , Fn,0. OUTPUT the numbers F0,0, F1,1, , Fn,n where Pn(x)=F0,0+i=1nFijj=0i1(xxj).(Fijisf[x0,x1,,xi].) Step 1 For i = 1, 2, , n For j = 1, 2, , i set Fi,j=Fi,j1Fi1,j1xixij,(Fi,j=f[xij,,xi].) Step 2 OUTPUT (F0,0, F1,1, , Fn,n); STOP.Use the Newton forward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f(13) if f(0.75) = 0.07181250, f(0.5) = 0.02475000, f(0.25) = 0.33493750, f(0) = 1.10100000 b. f(0.25) if f(0.1) = 0.62049958, f(0.2) = 0.28398668, f(0.3) = 0.00660095, f(0.4) = 0.24842440Use the Newton forward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f(0.43) if f(0) = 1, f(0.25) = 1.64872, f(0.5) = 2.71828, f(0.75) = 4.48169 b. f(0.18) if f(0.1) = 0.29004986, f(0.2) = 0.56079734, f(0.3) = 0.81401972, f(0.4) = 1.0526302Use the Newton backward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f(1/3) if f(0.75) = 0.07181250, f(0.5) = -0.02475000, f(0.25) = 0.33493750, f(0) = 1.10100000 b. f(0.25) if f(0.1) = 0.62049958, f(0.2) = 0.28398668, f(0.3) = 0.00660095, f(0.4) = 0.24842440Use the Newton backward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f(0.43) if f(0) = 1, f(0.25) = 1.64872, f(0.5) = 2.71828, f(0.75) = 4.48169 b. f(0.25) if f(1) = 0.86199480, f(0.5) = 0.95802009, f(0) = 1.0986123, f(0.5) = 1.2943767a. Use Algorithm 3.2 to construct the interpolating polynomial of degree three for the unequally spaced points given in the following table: ALGORITHM 3.2 Newtons Divided-Difference Formula To obtain the divided-difference coefficients of the interpolatory polynomial P on the (n + 1) distinct numbers x0, x1, xn, for the function f: INPUT numbers x0, x1, xn; values f(x0), f(x1), , f(xn) as F0,0, F1,0, , Fn,0. OUTPUT the numbers F0,0, F1,1, , Fn,n where Pn(x)=F0,0+i=1nFijj=0i1(xxj).(Fijisf[x0,x1,,xi].) Step 1 For i = 1, 2, , n For j = 1, 2, , i set Fi,j=Fi,j1Fi1,j1xixij,(Fi,j=f[xij,,xi].) Step 2 OUTPUT (F0,0, F1,1, , Fn,n); STOP.a. Use Algorithm 3.2 to construct the interpolating polynomial of degree four for the unequally spaced points given in the following table: b. Add f(1.1) = 3.99583 to the table and construct the interpolating polynomial of degree five. ALGORITHM 3.2 Newtons Divided-Difference Formula To obtain the divided-difference coefficients of the interpolatory polynomial P on the (n + 1) distinct numbers x0, x1, xn, for the function f: INPUT numbers x0, x1, xn; values f(x0), f(x1), , f(xn) as F0,0, F1,0, , Fn,0. OUTPUT the numbers F0,0, F1,1, , Fn,n where Pn(x)=F0,0+i=1nFijj=0i1(xxj).(Fijisf[x0,x1,,xi].) Step 1 For i = 1, 2, , n For j = 1, 2, , i set Fi,j=Fi,j1Fi1,j1xixij,(Fi,j=f[xij,,xi].) Step 2 OUTPUT (F0,0, F1,1, , Fn,n); STOP.a. Approximate f(0.05) using the following data and the Newton forward-difference formula: b. Use the Newton backward-difference formula to approximate f(0.65). c. Use Stirlings formula to approximate f(0.43).10ESThe following data are given for a polynomial P(x) of unknown degree. Determine the coefficient of x2 in P(x) if all third-order forward differences are 1.The following data are given for a polynomial P(x) of unknown degree. Determine the coefficient of x3 in P(x) if all fourth-order forward differences are 1.The Newton forward-difference formula is used to approximate f(0.3) given the following data. Suppose it is discovered that f(0.4) was understated by 10 and f(0.6) was overstated by 5. By what amount should the approximation to f(0.3) be changed?14ES15ES16ES17ES18ESShow that the polynomial interpolating the following data has degree three.20ES21ES22ES23ESCompare and contrast the various divided-difference methods you read about in this chapter.Is it easier to add a new data pair using divided-difference methods or the Lagrange polynomial in order to obtain a higher-degree polynomial?3DQUse Theorem 3.9 or Algorithm 3.3 to construct an approximating polynomial for the following data. ALGORITHM 3.3 Hermite Interpolation To obtain the coefficients of the Hermite interpolating polynomial H(x) on the (n + 1) distinct numbers x0, , xn for the function f: INPUT numbers x0, x1, , xn; values f (x0), ... , f (xn) and f (x0), ... , f (xn). OUTPUT the numbers Q0, 0, Q1, 1, , Q2n + 1, 2n + 1 where H(x)=Q0,0+Q1,1(xx0)+Q2,2(xx0)2+Q3,3(xx0)2(xx1)+Q4,4(xx0)2(xx1)2++Q2n+1,2n+1(xx0)2(xx1)2(xxn1)2(xxn). Step 1 For i = 0, 1, , n do Steps 2 and 3. Step 2 Setz2i=xi;z2i+1=xi;Q2i,0=f(xi);Q2i+1,0=f(xi);Q2i+1,1=f(xi). Step 3 If i 0 then set Q2i,1=Q2i,0Q2i1,0z2iz2i1. Step 4 For i = 2, 3, , 2n + 1 for j = 2, 3, ... , i set Qi,j=Qi,j1Qi1,j1zizij. Step 5 OUTPUT (Q0, 0, Q1, 1, , Q2n + 1, 2n + 1); STOP. Theorem 3.9 If f C1 [a, b] and x0, , xn [a, b] are distinct, the unique polynomial of least degree agreeing with f and f at x0, , xn is the Hermite polynomial of degree at most 2n + 1 given by H2n+1(x)=j=0nf(xj)Hn,j(x)+j=0nf(xj)Hn,j(x), where, for Ln, j (x) denoting the jth Lagrange coefficient polynomial of degree n, we have Hn,j(x)=[12(xxj)Ln,j(xj)]Ln,j2(x)andHn,j(x)=(xxj)Ln,j2(x). Moreover, if f C2n + 2 [a, b], then f(x)=H2n+1(x)+(xx0)2(xxn)2(2n+2)!f(2n+2)((x)), for some (generally unknown) (x) in the interval (a, b).Use Theorem 3.9 or Algorithm 3.3 to construct an approximating polynomial for the following data. ALGORITHM 3.3 Hermite Interpolation To obtain the coefficients of the Hermite interpolating polynomial H(x) on the (n + 1) distinct numbers x0, , xn for the function f: INPUT numbers x0, x1, , xn; values f (x0), ... , f (xn) and f (x0), ... , f (xn). OUTPUT the numbers Q0, 0, Q1, 1, , Q2n + 1, 2n + 1 where H(x)=Q0,0+Q1,1(xx0)+Q2,2(xx0)2+Q3,3(xx0)2(xx1)+Q4,4(xx0)2(xx1)2++Q2n+1,2n+1(xx0)2(xx1)2(xxn1)2(xxn). Step 1 For i = 0, 1, , n do Steps 2 and 3. Step 2 Setz2i=xi;z2i+1=xi;Q2i,0=f(xi);Q2i+1,0=f(xi);Q2i+1,1=f(xi). Step 3 If i 0 then set Q2i,1=Q2i,0Q2i1,0z2iz2i1. Step 4 For i = 2, 3, , 2n + 1 for j = 2, 3, ... , i set Qi,j=Qi,j1Qi1,j1zizij. Step 5 OUTPUT (Q0, 0, Q1, 1, , Q2n + 1, 2n + 1); STOP. Theorem 3.9 If f C1 [a, b] and x0, , xn [a, b] are distinct, the unique polynomial of least degree agreeing with f and f at x0, , xn is the Hermite polynomial of degree at most 2n + 1 given by H2n+1(x)=j=0nf(xj)Hn,j(x)+j=0nf(xj)Hn,j(x), where, for Ln, j (x) denoting the jth Lagrange coefficient polynomial of degree n, we have Hn,j(x)=[12(xxj)Ln,j(xj)]Ln,j2(x)andHn,j(x)=(xxj)Ln,j2(x). Moreover, if f C2n + 2 [a, b], then f(x)=H2n+1(x)+(xx0)2(xxn)2(2n+2)!f(2n+2)((x)), for some (generally unknown) (x) in the interval (a, b).The data in Exercise 1 were generated using the following functions. Use the polynomials constructed in Exercise 1 for the given value of x to approximate f (x) and calculate the absolute error. a. f (x) = x ln x; approximate f (8.4). b. f (x) = sin(ex 2); approximate f (0.9). c. f (x) = x3 + 4.001x2 + 4.002x + 1.101; approximate f (1/3). d. f (x) = x cos x 2x2 + 3x 1; approximate f (0.25). 1. Use Theorem 3.9 or Algorithm 3.3 to construct an approximating polynomial for the following data.The data in Exercise 2 were generated using the following functions. Use the polynomials constructed in Exercise 2 for the given value of x to approximate f (x) and calculate the absolute error. a. f (x) = e2x; approximate f (0.43). b. f (x) = x4 x3 + x2 x + 1; approximate f (0). c. f (x) = x2 cos x 3x; approximate f (0.18). d. f (x) = ln(ex + 2); approximate f (0.25). 2. Use Theorem 3.9 or Algorithm 3.3 to construct an approximating polynomial for the following data.5ESLet f (x) = 3xex e2x. a. Approximate f (1.03) by the Hermite interpolating polynomial of degree at most three using x0 = 1 and x1 = 1.05. Compare the actual error to the error bound. b. Repeat (a) with the Hermite interpolating polynomial of degree at most five using x0 = 1, x1 = 1.05, and x2 = 1.07.The following table lists data for the function described by f (x) = e0.1x2. Approximate f (1.25) by using H5(1.25) and H3(1.25), where H5 uses the nodes x0 = 1, x1 = 2, and x2 = 3 and H3 uses the nodes x0 = 1 and x1 = 1.5. Find error bounds for these approximations.8ES9ES