In this tutorial, we will analyze some of the most fundamental algorithms in numerical computing. 

Algorithms Overview

   Roots of polynomial equations:

• Bisection Method
• Regula-Falsi Method
• Fixed Point Iteration Method
• Newton-Raphson Method

   Numerical Integration:

• Trapezoidal Rule
• Simpson's 1/3rd Rule
• Simpson's 3/8th Rule
• Weddle's Rule

    The system of Linear Equations:

• LU Decomposition Method( Cholesky's Decomposition
• Gauss-Seidel Method
• Jacobi's Method

    Numerical Solution of Ordinary Differential Equations:

• Euler's Method
• Modified Euler's Method
• Runga-Kutta Method( of 4th Code)    
  

Note: After reading the algorithms don't forget to analyze your solution with the exact answer of the equations.

Hopefully, you have an idea about How to solve the equations not using the Numerical Methods.

        As our answer consists of a decimal form of irrational numbers or non-terminating rational number just have a look on Approximate Values, Significant figures, Rounding of a number and least possible error attended by rounding, as we are dealing with the approximate solution.