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Algorithms for Each method.

Numerical Analysis Methods

This repository contains implementations of various numerical methods for solving mathematical problems like root-finding, integration, and differential equations.

Methods Included

  • Bisection Method
  • False Position Method
  • Newton-Raphson Method
  • Secant Method
  • Trapezoidal & Simpson’s Rules
  • Gauss-Seidel & Jacobi Methods
  • Euler's & Runge-Kutta Methods

Algorithms

Bisection Method Algorithm

  1. Input:

    • Two initial guesses: a and b such that ( f(a) ) and ( f(b) ) have opposite signs.
    • A tolerance value ε (the desired accuracy).
    • Maximum number of iterations max_iter.

  2. Step 1 (Check Bracketing):

    • Verify that ( f(a) \cdot f(b) < 0 ). If not, return an error as the root is not guaranteed within the interval.

  3. Step 2 (Initial Midpoint Calculation):

    • Calculate the midpoint: ( c = \frac{a + b}{2} ).

  4. Step 3 (Convergence Check):

    • If ( |f(c)| \leq ε ), return c as the root.

  5. Step 4 (Subinterval Selection):

    • If ( f(a) \cdot f(c) < 0 ), set b = c.
    • Otherwise, set a = c.

  6. Step 5 (Iteration Loop):

    • Repeat Step 2 to Step 4 until:
      • The absolute difference between a and b is less than ε, or
      • The maximum number of iterations is reached.

  7. Output:

    • The final value of c is the approximate root, or an error message if the method didn't converge within the allowed iterations.

False Position Method Algorithm

  1. Input:

    • Two initial guesses: x0 and x1 such that ( f(x0) \cdot f(x1) < 0 ).
    • A tolerance value ε (the desired accuracy).
    • Maximum number of iterations max_iter.

  2. Step 1 (Check Bracketing):

    • Verify that ( f(x0) \cdot f(x1) < 0 ). If not, return an error as the root is not guaranteed.

  3. Step 2 (Calculate the New Estimate):

    • Compute the new root estimate: [ x2 = x0 - \frac{f(x0) \cdot (x1 - x0)}{f(x1) - f(x0)} ]

  4. Step 3 (Convergence Check):

    • If ( |f(x2)| \leq ε ), return x2 as the root.

  5. Step 4 (Subinterval Selection):

    • If ( f(x0) \cdot f(x2) < 0 ), set x1 = x2.
    • Otherwise, set x0 = x2.

  6. Step 5 (Iteration Loop):

    • Repeat Step 2 to Step 4 until:
      • The function value ( f(x2) ) is close enough to zero, or
      • The maximum number of iterations is reached.

  7. Output:

    • The final value of x2 is the approximate root, or an error message if the method didn’t converge within the allowed iterations.

Newton-Raphson Method Algorithm

  1. Input:

    • An initial guess: x0 (a starting point close to the expected root).
    • A tolerance value ε (the desired accuracy, e.g., 0.0001).
    • Maximum number of iterations max_iter.

  2. Define the Function:

    • The function ( f(x) ) whose root is to be found.
    • The derivative of the function ( f'(x) ).

  3. Initialization:

    • Set iteration = 1.

  4. Process:

    • Step 1: Calculate ( f(x_0) ) and ( f'(x_0) ).
    • Step 2: Check for division by zero:
      • If ( f'(x_0) = 0 ), stop the process and report that the derivative is zero (Newton-Raphson cannot continue).
    • Step 3: Update the value using the Newton-Raphson formula: [ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} ]
    • Step 4: Check convergence:
      • If ( |f(x_1)| < \varepsilon ), stop the process and report ( x_1 ) as the root.
    • Step 5: Update x0 = x1 and increment iteration.

  5. Repeat:

    • Continue until either the tolerance ( \varepsilon ) is satisfied or the maximum number of iterations is reached.

  6. Output:

    • If the method converges: Output the root approximation ( x_1 ).
    • If the method does not converge after max_iter, report that the method failed to find a root within the given iterations.

Contributing

Contributions and improvements are welcome! Feel free to submit pull requests or open issues.