Question

question number 2

1. Write a computer program to solve the equation x = tan x by means of Newton's method. Find the roots nearest 4.5 and 7.7. ( 2. (Continuation) Write and test a program to compute the first ten roots of the equation tan x =x. (This is much more difficult than the preceding computer problem.) Cultural note: If a1, a2, ... are all the positive roots of this equation, then Zie A2 = 1/10. (Amer. Math. Monthly, Oct. 1986, p. 660.)

Answer 1

roots = [0]; % record the obvious root

% the function and its derivative

f = @(x) x - tan(x);

df = @(x) 1 - sec(x).^2;

% loop over which one we are finding

for k=1:9 x = (2*k+1 - 0.01) * pi/2; % starting point just left of (odd)*pi/2

% loop to do Newton iteration

for j=1:10

x = x - f(x) / df(x);

end

roots = [roots x];

end

roots = roots’

residuals = abs(f(roots))