Since g(x) = x²-x-20 has zeroes at x= -4 and x = 5, we expect zeroes of f(x) = x²-x-20.2 to be in the neighbourhood of -4 and 5. Concentrating on the positive root, we shall assume that it exists somewhere near 5 but definitely on the interval [0.2, 10.2]. Coarse Grid Search Beginning with a course grid, with left endpoints x0 = [0.2, 2.2, 4.2, 6.2, 8.2] and respective right endpoints x1 = [2.2, 4.2, 6.2, 8.2, 10.2], find the sub-interval in which the zero exists. The vector definition of x and x1 produces a vector of subintervals [(0.2, 2.2), (2.2, 4.2), (4.2, 6.2), (6.2, 8.2), (8.2, 10.2)]. The Intermediate Value Theorem will help to locate the zero in one of these subintervals. Set the left endpoint of this sub-interval = a and the right endpoint = b. Fine Search With a tolerance of 10-⁹, use the Regula Falsi method to compute the root of ƒ in [a, b]. The final value of the root must be stored in the variable fnroot, and the total number of steps must be stored in the variable numsteps. The code below is a modification of the bisection method script from the tutorial. User input is required wherever you see red ###. You are permitted multiple incorrect submissions without penalty. Script 1 %% INPUT: Define the tolerance 2 tol = ### 4 epsilon = 10^4; steps = 1; 6 %% found zero if f(c) <= tol %% An arbitrarily large value to begin with %% To keep track of the number of bisections 7 %% INPUT: Define the left endpoints of the coarse grid 8 x0 = ### 9 x1 = x0+2; 10 11 ye f(x0); 12 y1 = f(x1); %% x and x1 set the endpoints of a coarse grid %% for finding a zero. %% evaluate the function at the left endpoints %% evaluate the function at the right endpoints 13 %% INPUT: Compute the term-by-term product of the endpoint values 14 product = ### 15 index find (product<0); 16 a = x0(index); 17 b x1(index); 18 %% product of the endpoint values %% find the position where product < 0 %% define the left and right boundaries for %% the finer grid. 19 fprintf("The Regula-Falsi method will begin on the interval [%.1f, %.1f]\n",a,b); 20 while epsilon > tol 21 %% INPUT: insert the formula for computing c %% This is really |f(c)-0| fprintf("steps = %d, epsilon = %.10f, c = %.10f\n", steps, epsilon,c); %% Enter this if a root is found 22 #### 23 epsilon = abs(f(c)); 24 25 if epsilon <= tol 26 27 %% 28 %% total number of steps in numsteps. fprintf("With a tolerance of %.0e, after %d steps the root = %.10f\n", tol, steps,c); INPUT: store the final value of the root in fnroot and store the 29 % % These should be commands of the form variable = variable, not variable = value. 30 31 else 33 #### #### if f(b)*f(c) < 0 34 %% INPUT: Pay close attention to the if test above and decide where b=c and a=c should go. 35 36 ### else 37 %% INPUT: Pay close attention to the if test above and decide where b=c and a=c should go. 38 39 40 41 end 42 end 43 44 ### end steps steps + 1; function ans = f(x) 46 end ans = x############ 47 48 Save CReset MATLAB Documentation ► Run Script ?