8. Use divided differences to find the interpolation polynomial for the data f (x)-1 -3 -2 4 f' (x) f"(x) 8. Use divided differences to find the interpolation polynomial for the data f (...
3. (30 points) Let f(x) = 1/x and data points Zo = 2, x,-3 and x2 = 4. Note that you can use the abscissae to find the corresponding ordinates (a) (8 points) Find by hand the Lagrange form, the standard form, and the Newton form of the interpolating polynomial p2(x) of f(x) at the given points. State which is which! Then, expand out the Newton and Lagrange form to verify that they agree with the standard form of p2...
This assignment is about polynomial interpolation. 1) The user should be able to enter: a. A function named f. b. A number of points (nodes) with their respective values. c. A point x0 2) The output should be: a. A Newton Divided Differences polynomial (function of x) that approximates the function with agreement in the points. b. An approximation of f(x0) by Newton Divided Differences polynomial. c. The approximation absolute and relative errors.
Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in...
2. a) Find Ts(x), the third degree Taylor polynomial about x -0, for the function e2 b) Find a bound for the error in the interval [0, 1/2] 3. The following data is If all third order differences (not divided differences) are 2, determine the coefficient of x in P(x). prepared for a polynomial P of unknown degree P(x) 2 1 4 I need help with both. Thank you.
For an nth-order Newton's divided difference interpolating polynomial fn(x), the error of interpolation can be estimated by Rn-| g(xmPX, , xm» ,&J . (x-x-Xx-x.) . . . (x-x.) | , where (xo, f(xo)), (xi, fx)).., (Xn-1, f(xn-1) are data points; g[x-,x,,x-.., x,] is the (n+1)-th finite divided difference. To minimize Rn, if there are more than n+1 data points available for calculating fn(x) using the nth-order Newton's interpolating polynomial, n+1 data points (Xo, f(xo)), (x1, f(x)), , (in, f(%)) should...
Consider the following set of data x f(x) 3 6 4 3 5 8 1. Use and order Newton polynomial to find f (4.5). 2. Use and order Lagrange polynomial to find f (4.5). You should get the same answer using both methods they are just different representations of a quadratic (i.e., 2nd order) interpolating polynomial.
Question 1 2 pts The Hermite Interpolation polynomial for 33 distinct nodes has Degree at most {Be Careful with the answer. Look at the Theorem and the Question Carefully; compare the given information} Question 2 2 pts If f € C4 [a, b] and p1, P2, P3, and p4 are Distinct Points in [a, b], Then 1. There are two 3rd divided differences 2. There is exactly one 3rd divided difference and it is equal to the value of f(iv)...
1. Using the Lagrange interpolation polynomial, estimate the value of f(4), knowing that f(-1) = 2; f(0) = 0; f(3) = 4 and f (7) = 7. (6 points)
19. Use Newton's divided difference formula to find the polynomial of degree less than or equal to four, that cos 2, at the interpolation points 0. π/2. π. 3π/2. 2π. Do not approximate π by a number interpolates f(x) with finite digits.
x. -1, 0, 1, 2, 3, 4 y. -14, -5, -2, 7, 34, 91 1. Use finite differences to determine the degree of the polynomial function that fits the data. 2. find the polynomial function. please show steps.