Roots (or Zeros) of a function f(x) are values of x that produces an output of 0. Roots can be real or complex numbers. Finding the root of {f(x)-g(x)}{f(x)-g(x)} is the same as solving the equation {f(x)=g(x)}f(x)=g(x). Solving an equation is finding the values that satisfy the condition specified by the equation.
A key feature of algorithms for solving simultaneous equations is called a “pivoting strategy” that is used to reduce round-off error that occurs in the solutions of these equations.
Curve-fitting is when you have a dataset of scattered points and find a line (or curve) that best fits the general shape of the data. Interpolation is when you have two points of data and want to know what a value between the two would be. Half way between would be their average, but if you want to know only a quarter of the way between the two you'd have to interpolate.
Numerical integration is the method to calculate the approximate value of the integral by using numerical techniques. There are various useful and interesting methods for numerical integration such as trapezoidal rule, Simpson's rules, Gauss's, Newton-Leibnitz rules etc.
ODE is a differential equation whose unknowns consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
Partial differential equations or (PDE) are equations that depend on partial derivatives of several variables. It is a special case of an ordinary differential equation. There are three-types of second-order PDEs in mechanics. They are Elliptic PDE, Parabolic PDE and Hyperbolic PDE.