Jim Lambers MAT 460/560 Fall Semester Practice Final Exam
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1 Jim Lambers MAT 460/560 Fall Semester Practice Final Exam 1. Let f(x) = sin 2x + cos 2x. (a) Write down the 2nd Taylor polynomial P 2 (x) of f(x) centered around x 0 = 0. (b) Write down the corresponding Taylor remainder R 2 (x). (c) Suppose P 2 (x) is used to approximate f(x) on the interval [ π, π]. How large in magnitude can the absolute error in this approximation be? 1
2 2. Consider the expression 6 + h 6, h which is to be evaluated for some small positive constant h. (a) Compute the value of this expression with h = using five decimal digits of precision and chopping. Given that the exact value, to eight decimal places, is , explain why your computed value is so inaccurate. (b) Rewrite the expression and repeat the computation, again with five-digit chopping arithmetic. 2
3 3. Given that where lim f(h) = 1, h 0 f(h) = eh cos h h 2, h determine the rate of convergence of f(h) to 1. 3
4 4. Given f(x) = 9x 2 1, use the bisection method to compute a solution of f(x) = 0 that is accurate to within 1/16, given that a solution lies within the interval [0, 1/2]. 4
5 5. Given g(x) = e x/4 (a) Show that g(x) has a unique fixed point in the interval [1, 2]. (b) Perform three iterations of Fixed-point Iteration with x 0 = 1.5 to find an approximation to the fixed point. 5
6 6. Perform three iterations of Newton s Method to solve the equation f(x) = 0, where f(x) = x 3 3x Use x 0 = 1.9. Given that the exact root is x = 2, do the iterations appear to be converging quadratically? If so, explain why. If not, explain why not. 6
7 7. Suppose that fixed-point iteration is applied to the function g(x) = x 2 2x + 2. Given that the fixed point occurs at x = 1, state the rate of convergence and the asymptotic error constant. 7
8 8. Use the Newton divided-difference formula to compute the cubic interpolating polynomial p 3 (x) for the function f(x) = e x, using interpolation points x 0 = 0, x 1 = 1, x 2 = 2, and x 3 = 3. What is an upper bound for the error p 3 (0.5) f(0.5)? 8
9 9. Find the Hermite interpolating polynomial for the following data. i x i f(x i ) f (x i )
10 10. Find the cubic spline interpolant s(x) for the data i x i f(x i ) that satisfies free, or natural, boundary conditions. 10
11 11. Use Lagrange interpolation to derive a formula for approximating f (x 0 ) that uses the values of f(x) at x 0 h, x 0, and x 0 + 2h. This formula is accurate to order O(h k ), for some k. What is the value of k? Justify your answer. 11
12 12. Use Romberg integration to compute an approximation to that is sixth-order accurate. 1 1 e x cos 2x dx 12
13 13. Use Gaussian Quadrature with n = 3 to evaluate the integral 1 What is the degree of this quadrature rule? 0 sin 3 x dx. 13
14 14. Use the Composite Trapezoidal Rule with m = n = 2 to evaluate the double integral e (x+y) dy dx. 14
15 15. Use the Composite Simpson s Rule with n = 4 to evaluate the integral 1 0 cos x x dx. 15
16 16. Given the data points i x i y i find the linear function y = c 0 +c 1 x that best fits this data in the discrete least-squares sense. 16
17 Possibly Useful Formulas Taylor s Theorem states that if f is n times continuously differentiable on [a, b] and f (n+1) exists on [a, b], then for x, x 0 [a, b], f(x) = n j=0 where ξ(x) is between x 0 and x. f (j) (x 0 ) j! (x x 0 ) j + f (n+1) (ξ(x)) (x x 0 ) n+1 (n + 1)! The Intermediate Value Theorem states that if f is continuous on [a, b], then for any y between f(a) and f(b), f(c) = y for some c (a, b). A function g(x) has a unique fixed point on [a, b] if it is continuous on [a, b], maps [a, b] into [a, b], is differentiable on (a, b), and g (x) k for x (a, b), where k < 1. Fixed-point Iteration: Newton s Method: Secant Method: x k+1 = g(x k ) x k+1 = x k f(x k) f (x k ) x k+1 = x k f(x k)(x k x k 1 ) f(x k ) f(x k 1 ) A function f(h) converges to a limit f 0 with rate of convergence O(g(h)) if f(h) f 0 = O(g(h)). A sequence {x k } k=0 converges to order r to its limit x if x k+1 x lim k x k x r = C, where C is a positive constant known as the asymptotic error constant. Composite Trapezoidal Rule: b f(x) dx = h n 1 f(a) + 2 f(x j ) + f(b) 2 a j=1 where h = (b a)/n, x j = a + jh, and μ [a, b]. b a 12 h2 f (μ), 17
18 Composite Simpson s Rule: b a f(x) dx = h 3 [f(a) + 4f(x 1) + 2f(x 2 ) + 4f(x 3 ) + + 2f(x n 2 ) + 4f(x n 1 ) + f(b)] b a 180 h4 f (4) (μ), where h = (b a)/n, x j = a + jh, and μ [a, b]. Richardson Extrapolation: where the error in N j 1 (h) is O(h p ). N j (h) = N j 1 (h/2) + N j 1(h/2) N j 1 (h) 2 p, 1 Lagrange polynomials: L n,j (x) = n i=0,i =j x x i x j x i, j = 0, 1,..., n. Interpolation error: if x 0, x 1,..., x n [a, b], and p n (x) interpolates f(x) at x 0, x 1,..., x n, then where ξ [a, b]. f(x) p n (x) = f (n+1) (ξ) (n + 1)! n (x x i ), i=0 18
19 Gaussian quadrature nodes and weights on [ 1, 1]: Divided Differences: n Nodes Weights f[x i ] = y i, f[x i, x i+1 ] = y i+1 y i x i+1 x i, f[x i, x i+1,..., x i+k ] = f[x i+1,..., x i+k ] f[x i,..., x i+k 1 ] x i+k x i. Splines: If x j = x 0 + jh, j = 1, 2,..., n, then spline for f(x) on [x 0, x n ] is defined by s j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3, j = 0, 1,..., n 1, h(c j + 4c j+1 + c j+2 ) = 3 h (a j+2 2a j+1 + a j ), j = 0, 1,..., n 2 a j = f(x j ), j = 0, 1,..., n, b j = a j+1 a j h h 3 (2c j + c j+1 ), j = 0, 1,..., n 1, d j = c j+1 c j, j = 0, 1,..., n 1, 3h c 0 = c n = 0 (Free boundary conditions) h(2c 0 + c 1 ) = 3 h (a 1 a 0 ) 3f (x 0 ), (Clamped boundary conditions) h(c n 1 + 2c n ) = 3f (x n ) 3 h (a n a n 1 ) 19
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