Analysis by SBC

Name

송윤창

Subject

1장1(e)번 입니다.

File

hwk.jpg (35 KB) - Download : 110

% hwk1_1e.m
% Find all roots of f(x) = sin(x)-exp(-x)
%   using Bisection, Newton and Secant methods
%
function hwk1_1e
tol=10^(-7);
% Plot
ezplot(@f,[-1.0,10]); grid

% Guess intervals for roots
% [0.5,0.6], [3.0,3.1], [6.2,6.3], [9.4,9.5]

disp(' ')
disp('Bisect method')
% Call Bisection method
x = [];
x = [x, bisect(0.5,0.6,tol)];
x = [x, bisect(3.0,3.1,tol)];
x = [x, bisect(6.2,6.3,tol)];
x = [x, bisect(9.4,9.5,tol)];
xb=x;

disp(' ')
disp('Newton method')
% Call Newton method
x = [];
x = [x, newton(0.55,tol)];
x = [x, newton(3.05,tol)];
x = [x, newton(6.25,tol)];
x = [x, newton(9.45,tol)];
xn=x;

disp(' ')
disp('Secant method')
% Call Secant method
x = [];
x = [x, secant(0.5,0.6,tol)];
x = [x, secant(3.0,3.1,tol)];
x = [x, secant(6.2,6.3,tol)];
x = [x, secant(9.4,9.5,tol)];
xs=x;

disp(' ')
disp(' ')
disp('The roots by Bisection, Newton and Secant methods')

[xb; xn; xs]'

%%%%%% Define functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function y = f(x)
    y = sin(x)-exp(-x);
function y = fp(x)
    y = cos(x)+exp(-x);

%%%%%% Bisect Methods %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function c = bisect(a,b,tol)
  fprintf('------------------------------------------------------ \n')
  fprintf(' n        a            b            c         f(c) \n')
  fprintf('------------------------------------------------------ \n')
  n = 1; c = (a+b)/2;

  fprintf('%3.0f %12.8f %12.8f %12.8f %12.4e \n',n,a,b,c,f(c));

  while( abs(f(c)) > tol )
      if f(b)*f(c)>=0, b=c; else, a=c; end
      n=n+1; c = (a+b)/2;
  end
  fprintf('%3.0f %12.8f %12.8f %12.8f %12.4e \n',n,a,b,c,f(c))

%%%%%% Newton method%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  function x = newton(x,tol)
  fprintf('------------------------------------ \n')
  fprintf('  n       x          f(x) \n')
  fprintf('------------------------------------ \n')
  n = 1; x = x - f(x)/fp(x);

  fprintf('%3.0f %12.8f %12.4e \n',n,x,f(x));
  while ( abs(f(x)) > tol )
      n=n+1; x = x - f(x)/fp(x);
  end
fprintf('%3.0f %12.8f %12.4e \n',n,x,f(x))

%%%%%% Secant method%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  function x2 = secant(x0,x1,tol)
  fprintf('------------------------------------ \n')
  fprintf('  n       x2          f(x2) \n')
  fprintf('------------------------------------ \n')
  n = 1; x2= x1 - f(x1)*(x1-x0)/(f(x1)-f(x0));

  fprintf('%3.0f %12.8f %12.4e \n',n,x2,f(x2));
  while ( abs(f(x2)) > tol )
      x0 = x1; x1 = x2;
      n = n+1; x2 = x1 - f(x1)*(x1-x0)/(f(x1)-f(x0));
  end
fprintf('%3.0f %12.8f %12.4e \n',n,x2,f(x2))

<< 결 과 물 >>

Bisect method
------------------------------------------------------
n        a            b            c         f(c)
------------------------------------------------------
  1   0.50000000   0.60000000   0.55000000 -5.4263e-002
20   0.58853264   0.58853283   0.58853273 -1.3959e-008
------------------------------------------------------
n        a            b            c         f(c)
------------------------------------------------------
  1   3.00000000   3.10000000   3.05000000  4.4106e-002
16   3.09636230   3.09636536   3.09636383  9.7135e-008
------------------------------------------------------
n        a            b            c         f(c)
------------------------------------------------------
  1   6.20000000   6.30000000   6.25000000 -3.5110e-002
19   6.28504906   6.28504944   6.28504925 -2.5689e-008
------------------------------------------------------
n        a            b            c         f(c)
------------------------------------------------------
  1   9.40000000   9.50000000   9.45000000 -2.5298e-002
19   9.42469711   9.42469749   9.42469730 -4.9029e-008

Newton method
------------------------------------
  n       x          f(x)
------------------------------------
  1   0.58795982 -7.9478e-004
  3   0.58853274 -9.5479e-015
------------------------------------
  n       x          f(x)
------------------------------------
  1   3.09650297 -1.3261e-004
  2   3.09636393 -8.7314e-010
------------------------------------
  n       x          f(x)
------------------------------------
  1   6.28506129  1.2040e-005
  2   6.28504927 -2.6948e-013
------------------------------------
  n       x          f(x)
------------------------------------
  1   9.42469190  5.3500e-006
  2   9.42469725 -2.4913e-015

Secant method
------------------------------------
  n       x2         f(x2)
------------------------------------
  1   0.58892452  5.4327e-004
  3   0.58853274  3.9345e-010
------------------------------------
  n       x2          f(x2)
------------------------------------
  1   3.09634126  2.1624e-005
  2   3.09636393  3.6749e-009
------------------------------------
  n       x2          f(x2)
------------------------------------
  1   6.28503683 -1.2463e-005
  2   6.28504927  8.0893e-010
------------------------------------
  n       x2          f(x2)
------------------------------------
  1   9.42471280 -1.5543e-005
  2   9.42469724  1.4610e-008

The roots by Bisection, Newton and Secant methods

ans =

    0.5885    0.5885    0.5885
    3.0964    3.0964    3.0964
    6.2850    6.2850    6.2850
    9.4247    9.4247    9.4247

DATE: 2013.04.15 - 14:11

LAST UPDATE: 2013.04.15 - 14:29

연습 1-13 입니다