Numerical Analysis: Key Methods and Computation

Numerical Analysis is the Mathematics branch responsible for designing effective ways to find numerical solutions to complex Mathematical problems. Most Mathematical problems from science and engineering are very complex and sometimes cannot be solved directly. Therefore, measuring a complex Mathematical problem is very important to make it easier to solve. Due to the great advances in computational technology, numeracy has become very popular and is a modern tool for scientists and engineers. As a result many software programs are being developed such as Matlab, Mathematica, Maple etc. the most difficult problems in an effective and simple way. These softwares contain functions that use standard numeric methods, in which the user can bypass the required parameters and obtain the results in a single command without knowing the numerical details.

The Numerical Analysis method is mainly used in the area of Mathematics and Computer Science that creates, analyzes, and implements algorithms for solving numerical problems of continuous Mathematics. Such types of problems generally originate from real-world applications of algebra, geometry and calculus, and they also involve variables that vary continuously. These problems occur throughout the natural sciences, social sciences, engineering, medicine, and the field of business. Introduction of Numerical Analysis during the past half-century, the growth in power and availability of digital computers has led to the increasing use of realistic Mathematical models in science and engineering. Here we will learn more about numerical method and analysis of numerical methods.

Numerical Method

Numerical methods are techniques that are used to approximate Mathematical procedures.  We need approximations because we either cannot solve the procedure analytically or because the analytical method is intractable (an example is solving a set of a thousand simultaneous linear equations for a thousand unknowns). 

Different Types of Numerical Methods

The numerical analysts and Mathematicians used have a variety of tools that they use to develop numerical methods for solving Mathematical problems. The most important idea, mentioned earlier, that cuts across all sorts of Mathematical problems is that of changing a given problem with a 'near problem' that can be easily solved. There are other ideas that differ on the type of Mathematical problem solved.

An Introduction to Numerical Methods for Solving Common Division Problems Given Below:

• Euler method - the most basic way to solve ODE

• Clear and vague methods - vague methods need to solve the problem in every step

• The Euler Back Road - the obvious variation of the Euler method

• Trapezoidal law - the direct method of the second system

• Runge-Kutta Methods - one of the two main categories of problems of the first value.

Numerical Methods

• Newton method

Some calculations cannot be solved using algebra or other Mathematical methods. For this we need to use numerical methods. Newton's method is one such method and allows us to calculate the solution of f (x) = 0.

• Simpson Law

The other important ones cannot be assessed in terms of integration rules or basic functions. Simpson's law is a numerical method that calculates the numerical value of a direct combination.

• Trapezoidal law

A trapezoidal rule is a numerical method that calculates the numerical value of a direct combination. The other important ones cannot be assessed in terms of integration rules or basic functions.

Numerical Computation

The term “numerical computations” means to use computers for solving problems involving real numbers. In this process of problem-solving, we can distinguish several more or less distinct phases. The first phase is formulation. While formulating a Mathematical model of a physical situation, scientists should take into account the fact that they expect to solve a problem on a computer. Therefore they will provide for specific objectives, proper input data, adequate checks, and for the type and amount of output. 

Once a problem has been formulated, then the numerical methods, together with preliminary error analysis, must be devised for solving the problem. A numerical method that can be used to solve a problem is called an algorithm. An algorithm is a complete and unambiguous set of procedures that are used to find the solution to a Mathematical problem. The selection or construction of appropriate algorithms is done with the help of Numerical Analysis. We have to decide on a specific algorithm or set of algorithms for solving the problem, numerical analysts should also consider all the sources of error that may affect the results. They should consider how much accuracy is required. To estimate the magnitude of the round-off and discretization errors, and determine an appropriate step size or the number of iterations required.

The programmer should transform the suggested algorithm into a set of unambiguous that is followed by step-by-step instructions to the computer.  The flow chart is the first step in this procedure. A flow chart is simply a set of procedures, that are usually written in logical block form, which the computer will follow. The complexity of the flow will depend upon the complexity of the problem and the amount of detail included. However, it should be possible for someone else other than the programmer to follow the flow of information from the chart. The flow chart is an effective aid to the programmer, they must translate its major functions into a program. And, at the same time, it is an effective means of communication to others who wish to understand what the program does. 

Numerical Computing Characteristics

• Accuracy: Every numerical method introduces errors. It may be due to the use of the proper Mathematical process or due to accurate representation and change of numbers on the computer. 

• Efficiency: Another consideration in choosing a numerical method for a Mathematical model solution efficiency Means the amount of effort required by both people and computers to use the method.

• Numerical instability: Another problem presented by a numerical method is numerical instability. Errors included in the calculation, from any source, increase in different ways. In some cases, these errors are usually rapid, resulting in catastrophic results.

Numerical Computing Process

• Construction of a Mathematical model.

• Construction of an appropriate numerical system.

• Implementation of a solution.

• Verification of the solution.

Trapezoidal Law

In Mathematics, trapezoidal law, also known as trapezoid law or trapezium law, is the most important measure of direct equity in Numerical Analysis. Trapezoidal law is a coupling law used to calculate the area under a curve by dividing the curve into a small trapezoid. The combination of all the small trapezoid areas will provide space under the curve. Let's understand the trapezoidal law formula and its evidence using examples in future sections.

Numerical and Statistical Methods

Numerical methods, as said above, are techniques to approximate Mathematical procedures. On the other hand, statistics is the study and manipulation of data, including ways to gather, review, analyze, and draw conclusions from the given data. Thus we can say, statistical methods are Mathematical formulas, models, and techniques that are used in the statistical analysis of raw research data. The application of statistical methods extracts information from research data and provides different methods to assess the robustness of research outputs. Some common statistical tools and procedures are given below :

• Descriptive

• Mean (average)

• Variance

• Skewness

• Kurtosis

• Inferential

• Linear regression analysis

• Analysis of variance

• Null hypothesis testing

Introduction to Finite Element Method

The various laws of physics related to space and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). If we have the vast majority of geometries and problems, these PDEs cannot be solved using analytical methods. Instead of that, we have created an approximation of the equations, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. Thus, the solution to the numerical model equations is, in turn, an approximation of the real solution to the PDEs. The finite element method is used to compute such approximations.

The finite element method is a numerical technique that is used for solving problems that are described by partial differential equations or can be formulated as functional minimization. A domain of interest is represented by the assembly of finite elements. Approximating functions in finite elements are determined in terms of nodal values of a physical field. A continuous physical problem is transformed into a discretized finite element problem with the help of unknown nodal values. For a linear problem, a system of linear algebraic equations must be solved. We can recover values inside finite elements using the nodal values.

Two Features of the Fem are Mentioned below:

• Piecewise approximation of physical fields on finite elements provides good precision even with simple approximating functions (i.e. increasing the number of elements we can achieve any precision).

• Locality of approximation leads to sparse equation systems that are mainly used for a discretized problem. With the help of this, we can solve problems with a very large number of nodal unknowns.

Typical Classes of Engineering Problems That Can be Solved Using Fem are:

• Structural mechanics

• Heat transfer

• Electromagnetics

• Diffusion

• Vibration

Finite Element Method MATLAB

Finite element analysis is a computational method for analyzing the behaviour of physical products under loads and boundary conditions. A typical FEA workflow in MATLAB includes 

• Importing or creating geometry.

• Generating mesh.

• Defining physics of the problem with the help of load, boundary and initial conditions.

• Solving and visualizing results.

The design of experiments or optimization techniques can be used along with FEA to perform trade-off studies or to design an optimal product for specific applications.

MATLAB is Very Useful Software and is Very Easy to Apply Finite Element Analysis Using MATLAB. It Helps Us in Applying Fem in Several Ways:

• Partial differential equations (PDEs) can be solved using the inbuilt Partial Differential Equation Toolbox.

• In MATLAB, with the help of Statistics and Machine Learning Toolbox, we can apply the design of experiments and other statistics and machine learning techniques with finite element analysis.

• Also, the optimization techniques can be applied to FEM simulations to come up with an optimum design with Optimization Toolbox.

• Parallel Computing Toolbox speeds up the analysis by distributing multiple Finite element analysis simulations to run in parallel.