(b) Consider the Chebyschev interpolation points in the interval1,1 x,-cos lKjS n 2n The Chebysch...
6. We want to study the effect of different choices of interpolation points {X0,X1,..., 2n} on the function wn(x) = (x – xo)(x – 21)... (x – In) in the formula for the error in interpolation polynomials. In particular, we want to study evenly spaced points and Chebyshev points in the interval (-1, 1). Con- sider the following choices: 2i (a) x₂ = –1 + i = 0.....n n 7T (b) X; = - cos i = 0,...,n. n +1...
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...
5. Consider the signal x (t) = cos (2n . 500) + cos (2n . 1 500). Its spectrum X1c" consists of a pair of spectral lines at positive and negative frequencies. Use the MATLAB command fft to find and plot the signal's spectrum using various values of N.
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...
yin]-[x, -X]. Plot the magnitude of XIK] and YK] in Problem 1. Let x[n]=cos(2n/20*n)+randn( 1, 20) for n= 1:20 Matlab. x(k) is the fft of x zero padded to 40 data points. What is the relationship between X[k] and YK? data points. What is the relationship between XIK] and YIK]? yin]-[x, -X]. Plot the magnitude of XIK] and YK] in Problem 1. Let x[n]=cos(2n/20*n)+randn( 1, 20) for n= 1:20 Matlab. x(k) is the fft of x zero padded to 40...
Problem 5 (programming): Create a MATLAB function named lagrange interp.m to perform interpolation using Lagrange polynomials. Download the template for function lagrange interp.m. The tem Plate is in homework 4 folder utl TritonED·TIue function lakes in the data al nodex.xi and yi, as well as the target x value. The function returns the interpolated value y. Here, xi and yi are vectors while x and y are scalars. You are not required to follow the template; you can create the...
3. Consider the function f(x) = cos(x) in the interval [0,8]. You are given the following 3 points of this function: 10.5403 2 -0.4161 6 0.9602 (a) (2 points) Calculate the quadratic Lagrange interpolating polynomial as the sum of the Lo(x), L1(x), L2(x) polynomials we defined in class. The final answer should be in the form P)a2 bx c, but with a, b, c known. DELIVERABLES: All your work in constructing the polynomial. This is to be done by hand...
numerical methods 2+17), j = 0,1...... Problem 1: Recall that the Chebyshev nodes x0, 71,..., are determined on the interval (-1,1) as the zeros of Tn+1(x) = cos((n +1) arccos(x)) and are given by 2j +17 X; = cos in +12 Consider now interpolating the function f(x) = 1/(1+22) on the interval (-5,5). We have seen in lecture that if equispaced nodes are used, the error grows unbound- edly as more points are used. The purpose of this problem is...
MATLAB Code Question alpha = 2.3 beta = 4.3 zeta = 9.1 PROBLEM 4 (20 points). Consider three sinusoids with the following amplitudes and phases a.cs(2n(500t)) β.cos(2n(500t) +0.5r) x1n] x2[n] rn = cos(2(500t)0.75) Create a MATLAB program to sample each sinusoid and generate a sum of three sinusoids, that is using a sampling rate of 8,000 Hz over a range of 0.1 seconds Use the MATLAB function stem) to plot r[n] for the first 20 samples Use the MATLAB function...
Problem 1: Recall that the Chebyshev nodes 20, 21, ...,.are determined on the interval (-1,1) as the zeros of Tn+1(x) cos((n + 1) arccos(x)) and are given by 2; +17 Tj = COS , j = 0,1,...n. n+1 2 Consider now interpolating the function f(x) = 1/(1 + x2) on the interval (-5,5). We have seen in lecture that if equispaced nodes are used, the error grows unbound- edly as more points are used. The purpose of this problem is...