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Numerical_Analysis

Numerical Techniques, All Implemented from scratch in Python

1. Root Finding Algorithms for f(x)=0 :

  1. Bisection Method
  2. Secant Method
  3. Regula-Falsi Method
  4. Newton-Raphson Method
  5. Birge-Vieta Method
  6. Bairstow's Method
  • Functions can be any algebraic combinations of polynomials Pn(x), exp(x), pi, log2(x),log10(x), acos(x), asin(x) ,atan(x) , cos(x) ,sin(x), tan(x), acosh(x),asinh(x),atanh(x),cosh(x), sinh(x), tanh(x) ,gamma(x), lgamma(x)
  • All algorithms have independent implementations.
  • The iterative_methods.py file has all algorithms together, and can compare performance and convergence of all these methods.
  • Also implemented pretty printing for all functions, so functions can be printed in their mathematical notational form within the terminal.

Dependencies: Python 2.7, Numpy 1.16.1, Sympy 1.3

2. Matrix Decomposition Algorithms :

  1. LU Decomposition
    a) Doolittle Method, L[i,i]=1, Implemented in O(nxn)
    b) Crout's Method, U[i,i]=1, Implemented in O(nxn)
  2. L(L.T) Decompostion
    a) Cholesky's Method
  • All algorithms have independent implementations.
  • The decomposition.py file has all algorithms together, and can compare performance and convergence of all these methods.

Dependencies: Python 3.6.7, Numpy 1.16.1

3. Matrix Equation Solvers for Ax=b :

  1. Gauss-Jacobi Method
  2. Gauss-Seidel Method
  • All algorithms have independent implementations.
  • The solving_matrix_equations.py file has all algorithms together, and can compare performance and convergence of methods.

Dependencies: Python 3.6.7, Numpy 1.16.1

4. Matrix Inversion

  • O(n^3) Algorithm.
  • Prints the inverse of the matrix if it is invertible, 'INVALID' otherwise. Also handles 0 pivot and NaN exceptions.

Dependencies: E