What are numerical methods? The area of mathematics which is in charge of creating effective numerical solutions to solve mathematical problems is called numerical analysis. Numerical methods help in solving complex mathematical problems. Computers can also understand and produce results for these numerical approaches.
Why should we use Numerical techniques? The majority of mathematical challenges in science and engineering are extremely challenging, and sometimes there is no straightforward solution. When traditional analytical techniques fail, numerical approaches are used in their stead. We can adopt numerical methods when there is a time constraint because they are very simple to solve. To make a difficult mathematical problem easier to solve, observation is therefore crucial. Numeracy has grown in popularity as a modern tool for scientists and engineers as computing technology has advanced.
Numerical procedures are effective to get approximate solutions to mathematical equations rather than exact solutions. There are several answers to a problem. As a result, adopting the right strategy is crucial for obtaining an accurate answer in no time.
We use numerical methods to solve algebra and calculus issues, especially differential equations. People commonly face difficulty remembering techniques for solving differential equations. This numerical approach helps in reaching the solution with less effort. The only point to remember is the formula and its proper usage. An equation may have any number of roots or solutions based on the degree of the equation. These numerical methods will help us find only one root of the equation.
Types of numerical methods The numerical approach is involved in solving algebraic and transcendental equations. A transcendental equation is an equation involving trigonometric or exponential or logarithmic functions. The most popularly used methods to solve algebraic and transcendental equations are the bisection method, regula-falsi method, iteration method, and Newton-Raphson method. Numerical methods namely the Gauss-Jacobi method and the Gauss-seidel method are employed in solving the system of linear equations. The trapezoidal rule and Simpson's rule are used in solving integral equations numerically. Taylor's method, Picard's method, Euler's method, and the Runge-Kutta method are the appropriate numerical methods to solve ordinary differential equations numerically.
Accuracy of Numerical methods If we compare the solutions obtained by numerical methods and analytical methods, we achieve answers with very minute differences. The solution obtained in the analytical method is called an actual solution. The solution obtained in the numerical method is called the calculated solution. The difference between the actual solution and the calculated solution is called the error which is quite negligible. The numerical method with minimum error is considered the best method to solve that particular problem.
As computer technology is advancing, numerous ways of tackling complicated mathematical problems in a limited number of steps are being developed. Based on the numerical approach, an algorithm is initially written. The required parameters are then taken as input from the user, and the logic of the algorithm runs to generate an estimated answer. According to the needs of the user, this approximation is rounded to a few decimal points. The accuracy requirements in decimals have to be taken care of.
A new field known as numerical computing has started emerging due to the tremendous applications of mathematics in the field of technology. The extensive use of mathematics in technology has led to the emergence of a new discipline known as numerical computing. Because numerical techniques have a wide range of applications in the fields of science and mathematics, students at the graduate level are also taught about them.
A nice article explaining the methodology used in numerical analysis. We are generally habitual of solving problems by direct methods but sometimes it is not possible to do so directly and then we take help of calculators and computers for the same. Today the methods of the numerical analysis are used in computer algorithms for calculating the solution of a problem with a good degree of certainty.
Let us understand this by solving a simple problem to find value of x in the equation x^3 + 4 = 31
If we use the direct method then we would say that -
x^3 = 31 - 4
so x^3 = 27
and hence x = 3
It was so easy to find x in the above case.
Sometimes problem might be a complex one and it would not be possible to solve it in a direct way. For example if x^3 = 11 (which also means that x^3 - 11 = 0) then it is not easy to find x directly. In such cases the numerical analysis helps us and one of the most common method in numerical analysis goes like -
let f(x) = x^3 - 11
Now we can see that x is more than 2 but less than 3 as 2^3 = 8 which is less than 11 and 3^3 = 27 which is more than 11.
Let us iterate the value of x by increasing it from 2 and decreasing it from 3 and get some values of f(x) at those points. The value of f(x) will slowly converge to zero and we would get a solution for x somewhere in between 2 and 3. It actually comes approximately as x = 2.2235
The interesting point is that more number of iterations we do we will get a more accurate value of x somewhere between 2.223 to 2.224. But in practice we cannot go for an infinite number of iterations and we have to accept a value of x with some minimal error. That is what many computer programmes do and we get an approximate solution as x = 2.2235 which is a number between 2.223 and 2.224
Numerical analysis is definitely a great tool for solving complex mathematical problems.