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Published: May 31, 2024
In this article, we are going to discuss how computer software has helped students and users in numerical data analysis with practical issues like languages used in programming. In addition, the interaction between numerical computation and symbolic computation will also be reviewed.
Mathematical modeling and numerical analysis have been very important in current life affairs. Pragmatic numerical analysis software has been integrated into most software packages, such as programs in the spreadsheet, enabling people to perform mathematical modelling without prior knowledge of the processes involved. This, therefore, demands the installation of numerical analysis software which can be relied on by the analysts. Designing of Problem solving environments (PSE) enables us to solve and model many situations. Interface for graphical users has made PSE for modeling a given situation easy with good models for mathematical theories. There have been modern applications of numerical analysis in computer software of late in many fields. For example, computer-aided manufacturing and computer-aided design in the engineering sector have led to improved PSEs to be developed for both CAD and CAM. Mathematical models in this field are based on the basic laws of newton on mechanics. Algebraic expressions and ordinary differential equations are mostly involved in mathematical models. Manipulation of the mixed systems of these models is very difficult but very important in the modeling of mechanical systems such as car simulators, plane simulators and other engine moving mechanicals needs real-time solving of differential-algebraic systems.
Atmospheric modeling is important in understanding the effects of human activities on the atmosphere. A great number of variables such as the velocity of the atmosphere in a given point at a given time, temperature, and pressure need to be calculated. In addition, chemicals in the atmosphere such as carbon dioxide, which is a pollutant, and their reactions need to be studied—studying velocity, pressure, and time which are defined with partial differential equations and the kinetic, chemical reactions which are defined using ordinary differential equations which very complex needs sophisticated software to handle. Businesses have incorporated the use of optimization methods in decision-making on efficient resource allocation. Locating manufacturing and storage facilities, inventory control problems and proper scheduling are some of the problems which require numerical analysis of optimization. (Brinkgreve, R. B. J. (1996).)
Fortran has remained the widely used programming language, and it keeps on being updated to meet the required standards, with Fortran 95 as the latest version. Other useful languages include C++, C, and java. In numerical data analysis, there are several numerical analysis software packages used. The following is a list of the packages used in data analysis with computer software. 1. Analytica software is a wide range of tools used in analyzing and generating numerical models. This is a language programmed visually and linked with influence diagrams. 2. FlexPro program is used in data analyzing and presenting measurement data. It has an excellent interface similar to Excel program with a vector programming language built in it. 3. GNU Octave-this is a high-end language used in the computation of numbers. It has a command- line interface that is used in numerically solving nonlinear and linear problems. Numerical experiments are solved with a language that is mostly suited with MATLAB. There are several newly developed programs of Linux such as cantor and KAlgebra, which offers Octave a GUI front ends. 4. Jacket is a MATLAB GPU tool that enables offloading of computations for MATLAB to the GPU to visualize data and acceleration purposes. 5. Pandas. It is a BSD- licensed python programming language tool used for the provision of data structures and data analysis. 6. Torch provides support for manipulation, analysis of statistics, and tensor presentation. 7. TK Solver is a commercialized tool by the universal technical system used in problem-solving and mathematical modeling software basically on rule-based and declarative language. 8. fit is a statistical analysis and curve-fitting plugin to excel. 9. GNU MCSim is a package for numerical integration and simulation with fast Markov chain Monte Carlo and Monte Carlo capabilities. 10. Sysquake is an application based on MATLAB-compatible language for computing environment with interactive graphics engineering, mathematics, and physics.( Conte, S. D., & De Boor, C. (2017).)
There have been efficient tools in the types of programming languages that creates computer solutions. The following are some of the basic qualities that a mathematical programming language should possess. First, a syntax that enables accurate and fast transformation from mathematical formulae into program statements should be possessed by the language. Also, the language should be rooted on primitives next similar to the basic concepts of mathematics. Lastly, tools for efficient and fast execution should be included in the language. The programming languages have been categorized into different generations: First Generation languages 1954-1958 (Fortran I, ALGOL 58, Flowmatic, and IPL V). Second-generation languages 1959-1961 (Fortran II, ALGOL 60, COBOL, and LISP). Third Generation Languages 1962-1970 (PL/1 (Fortran +COBOL+ALGOL), ALGOL 68, PASCAL, SIMULA, and APL. The Generation Gap 1970-1980. This had many languages which were different. (Bartholomew- Biggs, M. C. (2000).)
There are three classes for software. (1) Compilers for language and graphic packages, which are basic tools. (2) Tools which can solve the users’ problems, such as systems in structuring engineering. (3) Widely applicable generic tools such as mathematical systems and compiler generators.
A course of several actions has to be undertaken for a numerical solution to be attained, which include: (1) using the existing black-box package. Those packages include PAFEC for elementary work and GENSTAT used in the analysis of statistics. (2) Application of library routines like IMSL, NETLIB, and NAG after splitting the problem in hand to components well defined. (3) Writing a whole purpose-built program sometimes which needs deep computing and analytical knowledge.
Numerical libraries mainly perform normal numerical linear algebra operations, disintegration of singular value, and transformation of Fast Fourier, optimization of nonlinear problems, linear programming, curve fitting, quadrature, and performing special functions.
In May 1970, a group of six UK universities centers for computing decided to create a numerical routine library. A year later, they released Mark 1 of the NAG library, which contains 98 routines documented in it. In 1989 mark 12 had 688 routines and many other library versions created in Algol 60, Pascal, and Algol 68. In addition, there are specific library versions used by for computers from cray, CDC, Data General, Harris, Telefunken, Xerox, and Philips. The Philosophy of NAG of giving maximum efficiency indicates that the software is trying to excellently calculate mathematical problems within the algorithm domain solution. It also strives to signal and reject a condition erred and returning a best upper bound of the erred condition where possible to the user-supplied tolerance.
This contains a big mathematical software library. Its main aim is achieving success commercially with the lowest cost and a resulting high volume. Over 350 subroutine-specific software types are readily compatible for computers from Data General, Xerox, DEC, Hewlett- Packard, and Burroughs. (Wang, J. Y., & Garbow, B. S. (1981).)
This is a Fortran-based subroutine library that is assumed to be easy to use than IMSL and NAG which are commercial libraries. Peapack was designed in the 1980s with a documents for teaching numerical analysis introduction. This package contains routines for most basics, polynomial roots, interpolation, and ordinary differential equation calculations.
Supercomputers contain software libraries like floating point systems kind of machines. In 1989 July, Brad Carlile gave a report to the NA Digest, which informed that computing for FPS had announced "at-cost" FPSmath availability which was a standard library de facto software for scientific algorithms and engineering systems. This speeds the application research and development by allowing institutions to have the same mathematical tools across all their environment for computing a nominal cost hence guaranteeing portability and taking merits of supercomputer and features of acceleration.
Users are provided with various documentation categories, including condensed information, encyclopedic, detective, and specialized information.
The following factors affect the choice of which algorithm to choose: efficiency, storage costs, generality, and reliability of solving all problems.
A computer algebra system is a package for software used in mathematical formulae manipulation. The computer algebra system's main purpose is to automate tedious calculations and solve hard algebraic expressions. Computer algebraic systems' ability to solve equations symbolically makes the main difference between it and the traditional calculator. Computer Algebra systems provide the user with a programming language to define her procedures and graphing equations facilities. Some of the widely used systems include Mathematica, MathCAD, and maple. They are used in rational functions simplifications, factor polynomials, solving equations, and many other calculations. While Newton and Leibniz's algorithmic processes of solving calculus are very hard and tedious to solve, computer Algebra systems perform these tasks and outdo the man from the processes. The following is a summary of how these computer algebra systems work.
Speakeasy-this was developed in the 1960s with its main being manipulation of matrix, and within the process of evolution, it geared the most common paradigms tools with typing dynamically the structured data objects, collection of garbage and dynamic allocation, overloading operators, and connecting added modules by the groups of users.
Sampath-this is software that is an open-source containing a unified python interface. Examples are proprietary general and open source purposes CAS and other programs such as GP, Magma, GAP, and Maple.
PARI-matrices, algebraic numbers, and polynomials are computed in this computer algebra system.
Mathematica-this has computer algebra capabilities and programming languages that are designed for number theory computations.
Mathcad which has WYSIWYG interface which aid in mathematical equations publication quality.
Trivino - It is an open-source, object-oriented collection of libraries applied in engineering and science, and it solves linear and parallel algebra algorithms. (Wester, M. J. (1999).)
Computer-assisted software for qualitative data analysis like MAXQDA gives the solutions to given data problems without directly giving interpretations to the user. Qualitative data software tools give way to structuring, sorting, and analyzing big data, which facilitates evaluation and interpretation management. Quality data analysis depends on methods of organizing, systemizing, and analyzing non-numeric like those used in the analysis of qualitative content, analysis of mixed methods, group discussions, case and field studies, and Grounded theory. Computer-assisted data analysis packages should facilitate and support methods of sorting, analyzing, and structuring data content despite which approach the researcher chose. Data in the form of image files, video, audio material, and data from social media can also be included in these packages with sophisticated computer-assisted data analysis software authorizes for transcribing and importing the content to the program direct. Software for QDA, such as MAXQDA, provides support to the whole process of analyzing by providing overviews and relationships visualization. It also provides space for the addition of memos to the various analytical processes which aid the user in understanding them better. The first version of MAXQDA computer-assisted data analysis software was created in 1989, and this makes it a pioneer software program in the area. From collecting data to publishing the final report regardless of the approach used, the program provides support to the user. Coding or systematic assignment of portions of texts to themes and probability of making notes and associations are the central elements of MAXQDA. In MAXQDA, evaluation and interpretation of data are performed by sorting materials into portions using a system of hierarchical coding through variables defining, tabular overviews provision, and colors assigning to segments of text. The procedures can be easily tracked, and within a few steps, the results are easily accessed. Creating stunning visualizations helps the user to view the data from a completely different perspective and be able to test theories. The results can be projected and exported to many programs so as to be included in the final publication through these visualizations. (Chitu, C., & Song, H. (2019).)
The speed of current developments in cyber physical systems and IoT leads to new challenges for business owners and data analysts to come up with new techniques for analyzing big data. Cyber-physical systems is the integration of systems of computer connected to the physical world. The systems process and display signals for the problem in the hand of the user.
The most important aspect to consider when in the journey of understanding data is to be able to distinguish the different types of data. Therefore, the implementation of a machine learning algorithm in order is very important. Variables are either numerical or categorical types. The categorical data is subdivided into two subcategories: nominal, which means there is no meaningful order, and ordinal type, which shows there is an obvious order. Numerical data is counts or measurements and is grouped into two other types: discrete or integers and continuous data. There are several types of data analysis using computer software as listed here: qualitative analysis, hierarchical analysis, graph analysis, spatial analysis, and textual data analysis.
This is a system where the interested area behavior is determined by coupling processes of distinct characteristics in specific discrete dynamics and coupling them continuously. They generate signals consisting of discrete-valued signals and continuous signals. The signals depend on variables that are independent such as time. Hybrid models are used in the control of automotive engines by solving control algorithms that are implemented through embedded controllers, which reduce gas consumption and pollutant emissions with neutral car performance.
This is an approach used in the separation of soft computing knowledge grounded on intelligence on computation from hard computing skills grounded on artificial intelligence computation. Hard computing has characteristics of formality and precision, and it is channeled toward analyzing and designing systems and physical processes. It handles crisp systems, probability theories, mathematical programming, binary logic, approximation theory, and differential equations. Soft computing is used in analyzing and designing intelligent systems, and it handles problems related to fuzzy logic, probabilistic reasoning, and neural networks.
For a program in a computer to manipulate an equation symbolically, the equation has first to be stored in certain computer memory. At the center of any computer algebra system, there is a single data structure or a combination of many data structures responsible for mathematical equation describing. Equations might have other functions references, or be rational functions, and sometimes exist in several variables. Hence, there is no certain specific solution of an equation to a structure of data presentation. A presentation can be Complex in space and time; hence it becomes inefficient, but it may be easy to program. An efficient presentation to a certain mathematical problem does not mean it is efficient to others; hence there is no specific answer to a given problem.
COMSOL- This is a simulation and solver software for doing engineering and physics applications, specifically coupled phenomena.
Baudline- this is used for scientific visualization and numerical signal analysis, and it is a time-frequency browser.
Dataplot, which is provided by the NIST.
Euler Mathematical toolbox. It is a laboratory-powerful numerical programming language which can solve complex, interval, and real numbers, matrices, and vectors.
Hermes which is a library tool for C++ of improved finite algorithms for solving partial differential equations and problems coupled with multiphysics.
DADiSP. It is a DSP-focused program for combining MATLAB numerical capability with a spreadsheet interface.
Flexpro. It is a program used commercially for an automated and interactive presentation and analysis of measurement data. Other programs include IGOR Pro, FEniCS project, Fityk, and Labplot.
ADMB. It is a C++ software that uses automatic differentiation to model nonlinear statistics.
AcsIX. It is an application software for evaluating and modeling the continuous system performance described by nonlinear differential equations and time-dependent.
AMPL is a language for mathematical modeling for solving and describing problems of high complexity for optimization on a large scale.
Armadillo which is a C++ program for linear algebra which has factorizations, decompositions, and functions of statistics.
APMonitor. It is a language for mathematical modeling used for solving and describing physical system representations inform of algebraic and differential equations.
Clojure. It contains numeric Neanderthal libraries, ClojureCL, and ClojureCUDA to handle linear algebraic functions and optimized matrix functions on GPU and CPU.
R is a system for data manipulation and statistical analysis in which the SAS language is implemented.
SAS is statistics software that includes a matrix programming language.
VisSim. It is a nonlinear dynamic simulation and a visual block-diagram program which supports fast ordinary differential equations with the simulation of real-time complex large scale models.
World Programming Systems (WPS), which supports python mixing SAS and R languages using a single-user program for analysis of statistics and evaluation of data. Other language-oriented software includes Julia, Madagascar, O-Matrix, Optim, GAUSS, pearl data language, and many others.
Due to the current trend in transforming the world in many aspects of life, computers have to be included in the numerical data analysis systems to ease faster and accurate data simulations. Industrialization increased business capital, future reverences demands, increasing populations, and other life affairs have led to the demand for computer-aided data analysis systems in the numerical analysis field, as discussed in this article.
References
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