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numerical analysis

Questions and Answers of Numerical Analysis

The number e is defined byWhere n! = n(n ˆ’ 1) · · · 2 · 1 for n ‰ 0 and 0! = 1. Use four-digit chopping arithmetic to compute the following approximations to e, and determine the
Find the rates of convergence of the following sequences as n→∞. a. limn→∞ sin1/n = 0 b. limn→∞ sin 1/n2 = 0 c. limn→∞ (sin 1/n)2 = 0 d. limn→∞ [ln(n + 1) − ln(n)] = 0
Find the rates of convergence of the following functions as h → 0. a. limh→0 (sin h)/h = 1 b. limh→0 (1 − cos h)/h = 0 c. limh→0 (sin h − h cos h)h = 0 d. limh→0 (1 - eh)/h = −1
a. How many multiplications and additions are required to determine a sum of the formb. Modify the sum in part (a) to an equivalent form that reduces the number of computations.
Let P(x) = anxn + an−1xn−1 + · · · + a1x + a0 be a polynomial, and let x0 be given. Construct an algorithm to evaluate P(x0) using nested multiplication.
Let f (x) = (x+2)(x+1)2x(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.
Let f (x) = (x − 1)10, p = 1, and pn = 1 + 1/n. Show that |f (pn)| < 10−3 whenever n > 1 but that |p − pn| < 10−3 requires that n > 1000.
Let {pn} be the sequence defined byShow that {pn} diverges even though limn†’ˆž (pnˆ’pnˆ’1) = 0.
The function defined by f (x) = sin πx 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
Use the Bisection method to find solutions accurate to within 10−2 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 10−2 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 10−5 for the following problems. a. x − 2−x = 0 for 0 ≤ x ≤ 1 b. ex − x2 + 3x − 2 = 0 for 0 ≤ x ≤ 1 c. 2x cos(2x) −
Use the Bisection method to find solutions, accurate to within 10−5 for the following problems. a. 3x − ex = 0 for 1 ≤ x ≤ 2 b. 2x + 3 cos x − ex = 0 for 0 ≤ x ≤ 1 c. x2 − 4x + 4 −
a. Sketch the graphs of y = x and y = 2 sin x. b. Use the Bisection method to find an approximation to within 10−5 to the first positive value of x with x = 2 sin x.
a. Sketch the graphs of y = x and y = tan x. b. Use the Bisection method to find an approximation to within 10−5 to the first positive value of x with x = tan x.
a. Sketch the graphs of y = ex − 2 and y = cos(ex − 2). b. Use the Bisection method to find an approximation to within 10−5 to a value in [0.5, 1.5] with ex − 2 = cos(ex − 2).
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) =
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
For each of the following equations, use the given interval or determine an interval [a, b] on which fixed-point iteration will converge. Estimate the number of iterations necessary to obtain
Find all the zeros of f (x) = x2 +10 cos x by using the fixed-point iteration method for an appropriate iteration function g. Find the zeros accurate to within 10−4.
Let A be a given positive constant and g(x) = 2x − Ax2.a. Show that if fixed-point iteration converges to a nonzero limit, then the limit is p = 1/A, so the inverse of a number can be found using
a. Show that Theorem 2.2 is true if the inequality |g'(x)|≤k is replaced by g'(x) ≤ k, for all x∈(a,b). b. Show that Theorem 2.3 may not hold if inequality |g'(x)| ≤ k is replaced by g'(x)
a. Use Theorem 2.4 to show that the sequence defined by xn = 1/2xn−1 + 1/xn−1 , for n ≥ 1, converges to√2 whenever x0 >√2. b. Use the fact that 0 < (x0−√2)2 whenever x0 ≠√2 to show
a. 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
a. Show that if A is any positive number, then the sequence defined byxn = 1/2xn−1 + A/2xn−1, for n ≥ 1,converges to√A whenever x0 > 0.b. What happens if x0 < 0?
Replace the assumption in Theorem 2.4 that “a positive number k < 1 exists with |g'(x)| ≤ k” with “g satisfies a Lipschitz condition on the interval [a, b] with Lipschitz constant L <
Suppose 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 δ > 0 such that if |p0 − p| ≤ δ,
Suppose that g is continuously differentiable on some interval (c, d) that contains the fixed point p of g. Show that if |g'( p)| 0 such that if |p0 ˆ’ p| ‰¤ Î´, then the fixed-point
Let g ∈ C1 [a, b] and p be in (a, b) with g( p) = p and |g' ( p)| > 1. Show that there exists aδ > 0 such that if 0 < |p0 − p| < δ, then |p0 − p| < |p1 − p| . Thus, no matter how close the
Use Theorem 2.3 to show that g(x) = π + 0.5 sin(x/2) has a unique fixed point on [0, 2π]. Use fixed-point iteration to find an approximation to the fixed point that is accurate to within 10−2.
Use Theorem 2.3 to show that g(x) = 2−x has a unique fixed point on [ 1/3, 1]. Use fixed-point iteration to find an approximation to the fixed point accurate to within 10−4. Use Corollary 2.5 to
Repeat Exercise 6 using the method of False Position. a. ex + 2−x + 2 cos x − 6 = 0 for 1 ≤ x ≤ 2 b. ln(x − 1) + cos(x − 1) = 0 for 1.3 ≤ x ≤ 2 c. 2x cos 2x − (x − 2)2 = 0 for 2
Use all three methods in this Section to find solutions to within 10−5 for the following problems.a. 3xex = 0 for 1 ≤ x ≤ 2b. 2x + 3 cos x − ex = 0 for 0 ≤ x ≤ 1
Use all three methods in this Section to find solutions to within 10−7 for the following problems. a. x2 − 4x + 4 − ln x = 0 for 1 ≤ x ≤ 2 and for 2 ≤ x ≤ 4 b. x + 1 − 2 sin πx = 0
Use Newton's method to approximate, to within 10−4, the value of x that produces the point on the graph of y = x2 that is closest to (1, 0). [Hint: Minimize [d(x)]2, where d(x) represents the
The following describes Newton's method graphically: Suppose that f'(x) exists on [a, b] and that f'(x) ≠ 0 on [a, b]. Further, suppose there exists one p ∈ [a, b] such that f (p) = 0, and let p0
Use Newton's method to solve the equation 0 = 1/2 + 1/4 x2 − x sin x - 1/2cos 2x, with p0 = π/2.Iterate using Newton's method until an accuracy of 10−5 is obtained. Explain why the result seems
The fourth-degree polynomial f (x) = 230x4 + 18x3 + 9x2 − 221x - 9 has two real zeros, one in [−1, 0] and the other in [0, 1]. Attempt to approximate these zeros to within 10−6 using thea.
The function f (x) = tan πx − 6 has a zero at (1/π) arctan 6 ≈ 0.447431543. Let p0 = 0 and p1 = 0.48, and use ten iterations of each of the following methods to approximate this root. Which
The equation x2−10 cos x = 0 has two solutions,±1.3793646. Use Newton's method to approximate the solutions to within 10−5 with the following values of p0.a. p0 = −100 b. p0 = −50 c. p0 =
The equation 4x2 − ex − e−x = 0 has two positive solutions x1 and x2. Use Newton's method to approximate the solution to within 10−5 with the following values of p0. a. p0 = −10 b. p0 =
The function described by f (x) = ln(x2 + 1) − e0.4x cos πx has an infinite number of zeros.a. Determine, within 10−6, the only negative zero.b. Determine, within 10−6, the four smallest
A drug administered to a patient produces a concentration in the blood stream given by c(t) = Ate−t/3 milligrams per milliliter, t hours after A units have been injected. The maximum safe
Let f (x) = 33x+1 − 7 · 52x.a. Use the Maple commands solve and f solve to try to find all roots of f .b. Plot f (x) to find initial approximations to roots of f.c. Use Newton’s method to find
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 a. or b. is closer to √6?
Let f (x) = 33x+1 − 7 . 52x.a. Use the Maple commands solve and f solve to try to find all roots of f .b. Plot f (x) to find initial approximations to roots of f.c. Use Newton’s method to find
The logistic population growth model is described by an equation of the form P(t) = PL/(1 − ce−kt) , Where PL, c, and k > 0 are constants, and P(t) is the population at time t. PL represents the
The Gompertz population growth model is described by P(t) = (PLe−ce)−kt , Where PL, c, and k > 0 are constants, and P(t) is the population at time t. Repeat Exercise 31 using the Gompertz growth
In the design of all-terrain vehicles, it is necessary to consider the failure of the vehicle when attempting to negotiate two types of obstacles. One type of failure is called hang-up failure and
Use Newton's method to find solutions accurate to within 10−4 for the following problems. a. x3 − 2x2 − 5 = 0, [1, 4] b. x3 + 3x2 − 1 = 0, [−3,−2] c. x − cos x = 0, [0, π/2] d. x −
Use Newton's method to find solutions accurate to within 10−5 for the following problems. a. ex + 2−x + 2 cos x − 6 = 0 for 1 ≤ x ≤ 2 b. ln(x − 1) + cos(x − 1) = 0 for 1.3 ≤ x ≤
Repeat Exercise 5 using the Secant method. a. x3 − 2x2 − 5 = 0, [1, 4] b. x3 + 3x2 − 1 = 0, [−3,−2] c. x − cos x = 0, [0, π/2] d. x − 0.8 − 0.2 sin x = 0, [0, π/2]
Repeat Exercise 6 using the Secant method. a. ex + 2−x + 2 cos x − 6 = 0 for 1 ≤ x ≤ 2 b. ln(x − 1) + cos(x − 1) = 0 for 1.3 ≤ x ≤ 2 c. 2x cos 2x − (x − 2)2 = 0 for 2 ≤ x ≤ 3
Repeat Exercise 5 using the method of False Position. a. x3 − 2x2 − 5 = 0, [1, 4] b. x3 + 3x2 − 1 = 0, [−3,−2] c. x − cos x = 0, [0, π/2] d. x − 0.8 − 0.2 sin x = 0,
Use Newton's method to find solutions accurate to within 10−5 to the following problems. a. x2 − 2xe−x + e−2x = 0, for 0 ≤ x ≤ 1 b. cos(x +√2) + x(x/2 + √2) = 0, for −2 ≤ x ≤
Suppose p is a zero of multiplicity m of f, where f (m) is continuous on an open interval containing p. Show that the following fixed-point method has g'( p) = 0: g(x) = x − mf (x)/f'(x).
Show that the Bisection Algorithm 2.1 gives a sequence with an error bound that converges linearly to 0.
A zero of multiplicity m at p if and only if 0 = f ( p) = f'( p) = · · · = f (m−1)( p), but f (m)( p) ≠ 0.
The iterative method to solve f (x) = 0, given by the fixed-point method g(x) = x, wherehas g'( p) = g''( p) = 0. This will generally yield cubic (Î± = 3) convergence. Expand the analysis of
It can be shown (see, for example, [DaB], pp. 228-229) that if {pn}∞ n=0 are convergent Secant method approximations to p, the solution to f (x) = 0, then a constant C exists with |pn+1 − p| ≈
Use Newton's method to find solutions accurate to within 10−5 to the following problems. a. 1 − 4x cos x + 2x2 + cos 2x = 0, for 0 ≤ x ≤ 1 b. x2 + 6x5 + 9x4 − 2x3 − 6x2 + 1 = 0, for −3
Repeat Exercise 1 using the modified Newton's method described in Eq. (2.13). Is there an improvement in speed or accuracy over Exercise 1?
Repeat Exercise 2 using the modified Newton's method described in Eq. (2.13). Is there an improvement in speed or accuracy over Exercise 2?
Use Newton's method and the modified Newton's method described in Eq. (2.13) to find a solution accurate to within 10−5 to the problem e6x + 1.441e2x − 2.079e4x − 0.3330 = 0, for − 1 ≤ x
Show that the following sequences converge linearly to p = 0. How large must n be before |pn− p| ≤ 5 × 10−2? a. pn = 1/n , n ≥ 1 b. pn = 1/n2 , n ≥ 1
a. 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 < 10−m.
a. Show that the sequence pn = (10−2)n converges quadratically to 0. b. Show that the sequence pn = (10−n)k does not converge to 0 quadratically, regardless of the size of the exponent k > 1.
The following sequences are linearly convergent. Generate the first five terms of the sequence {n} using Aitken's (2 method. a. p0 = 0.5, pn = (2 − epn−1 + p2n−1)/3, n ≥ 1 b. p0 = 0.75, pn =
Use Steffensen's method to approximate the solutions of the following equations to within 10−5. a. x = (2 − ex + x2)/3, where g is the function. b. x = 0.5(sin x + cos x), where g is the
Use Steffensen's method to approximate the solutions of the following equations to within 10−5. a. 2 + sin x − x = 0, where g is the function. b. x3 − 2x − 5 = 0, where g is the function. c.
A sequence {pn} is said to be super linearly convergent to p ifa. Show that if pn †’ p of order Î± for Î± > 1, then {pn} is super linearly convergent to p.b. Show that pn = 1/nn is super
Suppose that {pn} is super linearly convergent to p. Show that
Let Pn(x) be the nth Taylor polynomial for f (x) = ex expanded about x0 = 0. a. For fixed x, show that pn = Pn(x) satisfies the hypotheses of Theorem 2.14. b. Let x = 1, and use Aitken's (2 method to
Use Steffensen's method to find, to an accuracy of 10−4, the root of x3 − x − 1 = 0 that lies in [1, 2], and compare this to the results of Exercise 6 of Section 2.2. In Section 2.2 Exercises 6
Find the approximations to within 10−4 to all the real zeros of the following polynomials using Newton's method. a. f (x) = x3 − 2x2 − 5 b. f (x) = x3 + 3x2 − 1 c. f (x) = x3 − x − 1 d. f
In 1224, Leonardo of Pisa, better known as Fibonacci, answered a mathematical challenge of John of Palermo in the presence of Emperor Frederick II: find a root of the equation x3 +2x2 +10x = 20. He
Find approximations to within 10−5 to all the zeros of each of the following polynomials by first finding the real zeros using Newton's method and then reducing to polynomials of lower degree to
Find the approximations to within 10−4 to all the real zeros of the following polynomials using Newton's method. a. f (x) = x3 − 2x2 − 5 b. f (x) = x3 + 3x2 − 1 c. f (x) = x3 − x − 1 d. f
Find approximations to within 10−5 to all the zeros of each of the following polynomials by first finding the real zeros using Newton's method and then reducing to polynomials of lower degree to
Use Newton's method to find, within 10−3, the zeros and critical points of the following functions. Use this information to sketch the graph of f . a. f (x) = x3 − 9x2 + 12 b. f (x) = x4 − 2x3
Use Maple to find a real zero of the polynomial f (x) = x3 + 4x − 4.
Use Maple to find a real zero of the polynomial f (x) = x3 − 2x − 5.
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
Use the following values and four-digit rounding arithmetic to construct a third Lagrange polynomial approximation to f (1.09). The function being approximated is f (x) = log10(tan x). Use this
Use the Lagrange interpolating polynomial of degree three or less and four-digit chopping arithmetic to approximate cos 0.750 using the following values. Find an error bound for the approximation.
Construct the Lagrange interpolating polynomials for the following functions, and find a bound for the absolute error on the interval [x0, xn]. a. f (x) = e2x cos 3x, x0 = 0, x1 = 0.3, x2 = 0.6, n =
Let f (x) = ex, for 0 ≤ x ≤ 2. a. Approximate f (0.25) using linear interpolation with x0 = 0 and x1 = 0.5. b. Approximate f (0.75) using linear interpolation with x0 = 0.5 and x1 = 1. c.
Repeat Exercise 11 using Maple with Digits set to 10. In Exercise 11 Use the following values and four-digit rounding arithmetic to construct a third Lagrange polynomial approximation to f (1.09).
a. The introduction to this chapter included a table listing the population of the United States from 1950 to 2000. Use Lagrange interpolation to approximate the population in the years 1940, 1975,
It 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
For the given functions f (x), let x0 = 1, x1 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. a. f
In Exercise 26 of Section 1.1 a Maclaurin series was integrated to approximate erf(1), where erf(x) is the normal distribution error function defined bya. Use the Maclaurin series to construct a
Prove Taylor's Theorem 1.14 by following the procedure in the proof of Theorem 3.3. [LetWhere P is the nth Taylor polynomial, and uses the Generalized Rolle's Theorem 1.10]
Show that maxxj≤x≤xj+1 |g(x)| = h2/4, where g(x) = (x − jh)(x − (j + 1)h).
The Bernstein polynomial of degree n for f ˆˆ C [0, 1] is given byWhere (n/k) denotes n!/k!(n ˆ’ k)!. These polynomials can be used in a constructive proof of the Weierstrass Approximation
Use Theorem 3.3 to find an error bound for the approximations in Exercise 1. In Exercise 1 For the given functions f (x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct interpolation polynomials of
Use Theorem 3.3 to find an error bound for the approximations in Exercise 2. In Exercise 2 For the given functions f (x), let x0 = 1, x1 = 1.25, and x2 = 1.6. Construct interpolation polynomials of
Use appropriate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following: a. f (8.4) if f (8.1) = 16.94410, f (8.3) = 17.56492, f (8.6) = 18.50515, f
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) =
The data for Exercise 5 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.

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