This is MatLab code prbolem, plz show the Matlab code in detail.

From lecture we saw integral ^b _a f (x) dx = b - a/2 integral ^1 _-1 f (a + b/2 + t b - a/2) dt holds from a simple linear change of variables x = a + b/2 + t b-a/2 . We are going to estimate log(1.9) with Gaussian quadrature and linear interpolation. We know integral ^1 _-1 f (x) dx = sigma ^n _j = 1 omega _j f (x_j) . For n = 8 we have w = [0.3626837833783620, 0.3626837833783620, 0.3137066458778873, 0.3137066458778873, 0.2223810344533745, 0.2223810344533745 0.1012285362903763, 0.1012285362903763] and x = [- 0.1834346424956498, 0.1834346424956498, - 0.5255324099163290 0.5255324099163290, - 0.7966664774136267, 0.7966664774136267, -0.9602898564975363, 0.9602898564975363] (a) Implement (1) for n = 8 and verify that your result works by computing log(1.9) using your code and the gausstable. m function. (b) Now assume that the integrand f (x) is only known on the grid x_j = a + j Delta x where Delta x = b - a/2^k . In order to evaluate our integrand at the points required by Gaussian quadrature we need to use interpolation. Find k such that the relative error for log(1.9) is near the relative error you found in problem 2a and remark on how the interpolation step changed the results