Last Updated on September 23, 2026 by Maged kamel
Introduction To Numerical Analysis Part 1.
Difference between Numerical and analytical methods.
Introduction to Numerical Analysis: numerical methods are techniques that solve mathematical problems using repeated arithmetic operations.
As an introduction to numerical analysis, it is important to distinguish between Numerical and analytical methods and understand polynomials. Numerical methods approximate the problem, while analytical methods give exact solutions that are more time-consuming. The next slide image shows the difference between analytical and numerical methods.
Numerical methods, such as finite difference and finite element methods, only give solutions at certain points; these methods are considered Numerical because they produce results at specific points.
Definition of the Quadratic function.
A quadratic function is a function that can be expressed in the form f(x)=ax^2+bx+c. There are three cases in which real solutions are needed. This depends on the relation between b^2 and 4ac. The complete illustration is shown in the next slide image.
Refer to this Link for more information. What is discrimination? Therefore, the discriminant formula for the general quadratic equation is D = b^2 – 4ac, where
a is the coefficient of x2, b is the coefficient of x, and c is the constant term.
Definition of the polynomial function.
A polynomial function of degree n has a leading coefficient an and a constant a0, and can be written as f(x) = anx^n + a (n-1) x^ (n-1) + a (n-2) x^ (n-2) +… + a0.
Roots for y=x^2-4.
If we have a graph, y = x^2 – 4, we draw the x and y coordinates and the intersection with the x-axis.
The graph will give us the Value of f(x) = 0. Supposed to cross at +2 & -2. These two points are zeros. How do we get these two points? We let f(x),or y = 0. We make bracketing (x^2-4 ), which will give us a multiplication of (x+2) *(x-2). We equate to zero, so( x= -2).
The first solution is x = -2, and the other solution is x = 2. We have two solutions: x = -2 or x = +2. Are they the Zeros? The second or last step is to check that our solution is correct. x= +2, this equation y =x^2-4 ,we put x=+2 so y=2 to power of 2=4-4 ,gives us zero .
For the other solution, x = -2, we write -2^2 – 4, which gives us zero. Once we have satisfied f (x) = 0,
We have these two zeros, + 2 and -2. In synthetic division, we sometimes guess the Value of x using the so-called rational roots to find the function’s zeros. In the next slide image, a representation of the function y = x^2- 4 is required to determine its roots.
What is the solution to an equation?
This is a graphical representation of the function’s solution, meaning it has one or more roots. If one root is shown in the graph, one point is shown for the curve of the function touching the x-axis with y=f(x)=0.
On the other hand, if there is no solution, then the function’s graph will not intersect with the x-axis. This is a graphical representation of the function’s solution; based on the number of intersections with the x-axis, the solution may be multiple points or a single point.
Root-finding.
As an introduction to numerical analysis, we will review the concept of root-finding of a function. The roots of a function, or zeros, are the points where the function crosses the x-axis, where f(x)=0; for analytical methods, we use synthetic division, and for quadratic functions, we use the following equation to find the solution.
The remainder theorem.
The remainder theorem is a formula for finding the remainder when a polynomial is divided by a linear polynomial.
The difference between the case where -a is not a factor of the polynomial and the other case when it is a factor of the polynomial.
When a certain number of things are divided into groups with an equal number of things in each group, the leftover things are called the remainder. It is something that “remains” after division. Let us learn the concept of the remainder theorem. Please refer to the next slide image that shows the remainder Value of -1.
You can download the PDF data for this Post from the following document.
The next Post is What is synthetic division? Numerical Analysis Part 2.
A very intersection source is Holistic Numerical Methods.