Bisection Method

◀
Enter a function to find its roots
\frac{\sin(x)}{x}
Lower bound of the interval
Upper bound of the interval
Tolerance of the method
Maximum number of iterations
Finds the root of a function using the bisection method

False Position Method

◀
Enter a function to find its roots
\frac{\sin(x)}{x}
Lower bound of the interval
Upper bound of the interval
Tolerance of the method
Maximum number of iterations
Finds the root of a function using the false position method

Newton-Raphson Method

◀
Enter a function to find its roots
x^2-2
Initial guess: x0
Tolerance of the method
Maximum number of iterations
Finds the root of a function using the Newton-Raphson method

Secant Method

◀
Enter a function to find its roots
x^2-2
First initial guess: x0
Second initial guess: x1
Tolerance of the method
Maximum number of iterations
Finds the root of a function using the secant method

Summary

  • Function x^2 - 2
  • Python Function x**2 - 2
  • Lower Bound None
  • Upper Bound None
  • Initial Guess 10
  • Second Initial Guess None
  • Tolerance 0.0001
  • Iteration Limit 100
  • Root 1.4142135968022693
  • Iteration Count 6
  • Error 9.738041173434908e-08
  • Condition True
  • Final X Prev 1.4145256551487377
  • Final X 1.4142135968022693
  • Final F Prime(X Prev) 2.8290513102974755
  • Final F(X Prev) 0.0008828290739657518
  • Final F(X) 9.738041173434908e-08

Iterations

  • Find the tangent line of the function at the initial guess

Iteration 1 of 6 with 10.0
↓
x0 = 10.0
f(x0) = f(10.0) = 98.0
f'(x0) = f'(10.0) = 20.0
x1 = x0 - f(x0) / f'(x0)
x1 = 10.0 - 98.0 / 20.0
x1 = 5.1
↓
|f(x1)| = |24.01| > 0.0001 = Tolerance
|x1 - x0| = |5.1 - 10.0| > 0.0001 = Tolerance

Iteration 2 of 6 with 5.1
↓
x1 = 5.1
f(x1) = f(5.1) = 24.01
f'(x1) = f'(5.1) = 10.2
x2 = x1 - f(x1) / f'(x1)
x2 = 5.1 - 24.01 / 10.2
x2 = 2.746078
↓
|f(x2)| = |5.540947| > 0.0001 = Tolerance
|x2 - x1| = |2.746078 - 5.1| > 0.0001 = Tolerance

Iteration 3 of 6 with 2.746078
↓
x2 = 2.746078
f(x2) = f(2.746078) = 5.540947
f'(x2) = f'(2.746078) = 5.492157
x3 = x2 - f(x2) / f'(x2)
x3 = 2.746078 - 5.540947 / 5.492157
x3 = 1.737195
↓
|f(x3)| = |1.017846| > 0.0001 = Tolerance
|x3 - x2| = |1.737195 - 2.746078| > 0.0001 = Tolerance

Iteration 4 of 6 with 1.737195
↓
x3 = 1.737195
f(x3) = f(1.737195) = 1.017846
f'(x3) = f'(1.737195) = 3.47439
x4 = x3 - f(x3) / f'(x3)
x4 = 1.737195 - 1.017846 / 3.47439
x4 = 1.444238
↓
|f(x4)| = |0.085824| > 0.0001 = Tolerance
|x4 - x3| = |1.444238 - 1.737195| > 0.0001 = Tolerance

Iteration 5 of 6 with 1.444238
↓
x4 = 1.444238
f(x4) = f(1.444238) = 0.085824
f'(x4) = f'(1.444238) = 2.888476
x5 = x4 - f(x4) / f'(x4)
x5 = 1.444238 - 0.085824 / 2.888476
x5 = 1.414526
↓
|f(x5)| = |0.000883| > 0.0001 = Tolerance
|x5 - x4| = |1.414526 - 1.444238| > 0.0001 = Tolerance

Iteration 6 of 6 with 1.414526
↓
x5 = 1.414526
f(x5) = f(1.414526) = 0.000883
f'(x5) = f'(1.414526) = 2.829051
x6 = x5 - f(x5) / f'(x5)
x6 = 1.414526 - 0.000883 / 2.829051
x6 = 1.414214
↓
|f(x6)| = |0.0| ≤ 0.0001 = Tolerance
|x6 - x5| = |1.414214 - 1.414526| ≤ 0.0001 = Tolerance
↓
Root = 1.4142135968022693
Error = ± 9.738041173434908e-08