all numerical analysis problem

Numerical analysis I, Spring 2024 01:640:373 1 February 21, 2024 Homework assignment 7 Problems 2,4,6 (4.7) from the textbook[10 points] Approximate the following integral Z 3.5 √ 3 x x2 − 4 dx using Gaussian quadrature with (a) n = 2, (b) n = 3, (c) n = 4, and compare your results to the exact values of the integrals. The table of roots of Legendre polynomial together with the coefficients can be taken from the textbook. 2 Problem 2 (b) (5.1) from the textbook[10 points] Show that the following initial-value problems has a unique solution and find the solution. Can Theorem 5.4 be applied in this case? y ′ (y) = t−2 (sin(2t) − 2ty(t)) , 1 ≤ t ≤ 2, y(1) = 2. 3 Problem 4 (a),(b) (5.1) from the textbook[10 points] For each choice of f (t, y) given in parts (a), (b): (i) Does f satisfy a Lipschitz condition on D = {(t, y) : 0 ≤ t ≤ 1, −∞ < y < ∞}. (ii) Can Theorem 5.6 be used to show that the initial-value problem y ′ (y) = f (t, y), t ∈ [0, 1], y(0) = 1 is well-posed? (a) f (t, y) = et−y , (b) f (t, y) = 1+y 1+t . 4 Problem 2 (c) (5.2) from the textbook[10 points] Use Euler’s method to approximate the solution of the following initial-value problems. y ′ (t) = −y(t) + ty(t)1/2 , 2 ≤ t ≤ 3, y(2) = 2, h = 0.25. 1 5 Problem 12 (4.7) from the textbook[10 points] Determine constants a, b, c, d, e and that will produce a quadrature formula Z 1 f (x) dx = af (−1) + bf (0) + cf (1) + df ′ (−1) + ef ′ (1) −1 that has degree of precision four. 6 Problem 14 (4.7) from the textbook[10 points] Show that the formula Q(P ) = n X ci P (xi ) i=1 R1 approximating −1 P (x) dx cannot have degree of precision greater than 2n − 1, regardless of the choice of x1 , x2 , . . . , xn and c1 , c2 , . . . , cn . [Hint: Construct a polynomial that has a double root at each of the xi ’s.] 7 Programming question[10 points] Write down a program for calculating the value of the Gaussian quadrature for arbitrary interval and n = 3, more precisely: Input: Function f , two numbers a, b. Rb Output: Value of the Gaussian quadrature with n = 3 for a f (x) dx. Then, use your program to approximate Z 3 1 dx. 1 + x3 1 Hint: Actually, you have to modify the program from the lecture to work for arbitrary interval of integration. 2