Question

from the previous two parts. E9.6 Consider the following quadratic function F(x) = *(3 21 - + [a 4] i. Sketch the contour plot for F(x). Show all work. ii. Take one iteration of Newton's method from the initial guess X0 = 0 0 iii. In part (ii), did you reach the minimum of F(x)? Explain. E9.7 Consider the function F(x) = (1 + 4 + 2.680 - 12

Answer 1

Sol.: Given quadratic form of a function F[X] = { x= [] 3] x + [44]x Where X = [] Hence F(x, y) = $(xyz [ 216]+[44[] F(x, y) = { Ir v2 [3* *2%] + [4x + 4y] F(x,y) = (3x + 4xy] + [4x + 4y] F(x,y) = 3x2 + 4xy + 4x + 4y Part(i): Contour plot of Flx.) Matlab programme to plot contour of F [x] = meshgrid(-5:5:5,-5:5:5); F = x.^2+4.*x.*y+4.*x+4.*y; [Ch] = contour(x..F); clabel(c.b) colormap cool xlabel("x")

vlabel('Y') legend('contour plot of F')| %output -53182 12273 -95455 21.36 56.909 34 091 99.545 31,364 2 55.909 -9.5455 34,091 12.273 12.273 34.091 9.5455 -9.5455 21ZZ -53.182 Part(ii): Newton method F(x,y) = 3x2 + 4xy + 4x + 4y Initial guess to find the minimum of function F using Newton method

Initial guess to find the minimum of function Fusing Newton method 1.1 = 0] First iteration of newton method is X. = 3) = [%:) - VF[7.]H[X.)-- Where Grad(F) at X. = VF[X] = ) and inverse of Hessian matrix = H[X]-- = VF = [6x + 4y + 4, 4x + 4] VF (1= [4 4] And #19 ay 10 0.251 22 al = = 14 od 10.25 -0.375 Hence x2 = (33) = 1.) – [4,4)[0.25 - 0:35] = [55]

Thus X2 = (23) Part(iii): Minimum of the function F is at point (-1. ).