Question

You are given the following function y = x^4 - 12x^3 + 49x^2 - 78x + 40 Plot this function and use graphic methods to initially estimate the roots of this function. Develop an M-file for false-position method based on figure 5.15 in our textbook. Estimate the third root using your program, until epsilon_a is smaller than 0.5% Fill up the following table for one of the roots FIGURE 5.15 Pseudocode for the modified false-position method. FUNCTION ModFalsePos(xl, xu, es, imax, xr, iter, ea) iter = 0 f] = f(xl) fu = f(xu) DO xrold = xr xr = xu - fu * (xl - xu) I (fl - fu) fr = f(xr) iter = iter + 1 IF xr <> 0 THEN ea = AbsKxr - xrold)/xr) * 100 END IF test = fl * fr IF test < 0 THEN xu = xr fu = f(xu) iu = 0 il = il + 1 If il greaterthanorequalto 2 THEN fl = fl/2 ELSE IF test > 0 THEN xl = xr fl = f(xl) il = 0 iu = iu + 1 IF iu greaterthanorequalto 2 THEN fu = fu/2 ELSE ea = 0 END IF IF ea < es OR iter greaterthanorequalto imax THEN EXIT END DO ModFalsePos = xr End MODFALSEPOS

Answer 1

% function biection method, parametres are function and limits.
function res = Bisect(f,a,b)
errArr = []
index = 1
if f(a)*f(b)>0
disp('Inputs wrong..')
else
res = (a + b)/2;
err = abs(f(res));
while err > 1e-5 %checking limit
errArr(index) = err % storing into array
fprintf('Root at iteration: %d is %f',index,res);
index = index + 1
if f(a)*f(res)<0
b = res;
else
a = res;
end
res = (a + b)/2; % result calculating every step
err = abs(f(res));
end
end
fprintf('Root calculated for equation is %f',res);
plot(errArr);
title('Error plot')
xlabel('iterations')
ylabel('Error')
end

f=(x) x^4 - 12*x^3 + 49*x^2 - 78*x + 40
ans = Bisect(f,-5,5)

TABLE

Root at iteration: 3 ---->3.750000 Root at iteration: 5 -- 4.062500 Root at iteration: 6 - 3.906250 Root at iteration: 7 ----> 3.984375 Root at iteration: 8 --> 4.023438 Root at iteration: 10 ----3.994141 Root at iteration: 11 ----3.999023 Root at iteration: 12 4.001465 Root at iteration: 13 ----4.000244 Root at iteration: 14 ---3.999634 Root at iteration: 15 ----3.999939 Root at iteration: 16 4.000092 Root at iteration: 17 4.000015 Root at iteration: 18 ----3.999977 Root at iteration: 19 ----3.999996 Root at iteration : 20--> 4.000006 Root calculated for equation is 4.000001ans 2 4.0000

Error plot 40 r 35 30 25 15 10 15 20 iterations