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

all numerical analysis problem

We offer the best custom writing paper services. We have answered this question before and we can also do it for you.

GET STARTED TODAY AND GET A 20% DISCOUNT coupon code DISC20

We offer the bestcustom writing paper services. We have done this question before, we can also do it for you.

Why Choose Us

  • 100% non-plagiarized Papers
  • 24/7 /365 Service Available
  • Affordable Prices
  • Any Paper, Urgency, and Subject
  • Will complete your papers in 6 hours
  • On-time Delivery
  • Money-back and Privacy guarantees
  • Unlimited Amendments upon request
  • Satisfaction guarantee

How it Works

  • Click on the “Place Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
  • Fill in your paper’s requirements in the "PAPER DETAILS" section.
  • Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
  • Click “CREATE ACCOUNT & SIGN IN” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page.
  • From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.