1.
MATLAB
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MATLAB is a multi-paradigm numerical computing environment and fourth-generation programming language. Although MATLAB is intended primarily for numerical computing, an optional toolbox uses the MuPAD symbolic engine, an additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. In 2004, MATLAB had around one million users across industry, MATLAB users come from various backgrounds of engineering, science, and economics. Cleve Moler, the chairman of the science department at the University of New Mexico. He designed it to give his students access to LINPACK and EISPACK without them having to learn Fortran and it soon spread to other universities and found a strong audience within the applied mathematics community. Jack Little, an engineer, was exposed to it during a visit Moler made to Stanford University in 1983, recognizing its commercial potential, he joined with Moler and Steve Bangert. They rewrote MATLAB in C and founded MathWorks in 1984 to continue its development and these rewritten libraries were known as JACKPAC. In 2000, MATLAB was rewritten to use a set of libraries for matrix manipulation. MATLAB was first adopted by researchers and practitioners in control engineering, Littles specialty and it is now also used in education, in particular the teaching of linear algebra, numerical analysis, and is popular amongst scientists involved in image processing. The MATLAB application is built around the MATLAB scripting language, common usage of the MATLAB application involves using the Command Window as an interactive mathematical shell or executing text files containing MATLAB code. Variables are defined using the assignment operator, =, MATLAB is a weakly typed programming language because types are implicitly converted. It is a typed language because variables can be assigned without declaring their type, except if they are to be treated as symbolic objects. Values can come from constants, from computation involving values of other variables, for example, A simple array is defined using the colon syntax, init, increment, terminator. For instance, defines a variable named array which is an array consisting of the values 1,3,5,7 and that is, the array starts at 1, increments with each step from the previous value by 2, and stops once it reaches 9. The increment value can actually be left out of this syntax, assigns to the variable named ari an array with the values 1,2,3,4, and 5, since the default value of 1 is used as the incrementer. Indexing is one-based, which is the convention for matrices in mathematics, although not for some programming languages such as C, C++. Matrices can be defined by separating the elements of a row with blank space or comma, the list of elements should be surrounded by square brackets. Parentheses, are used to access elements and subarrays, sets of indices can be specified by expressions such as 2,4, which evaluates to

2.
GNU Octave
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GNU Octave is software featuring a high-level programming language, primarily intended for numerical computations. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is compatible with Matlab. It may also be used as a batch-oriented language, since it is part of the GNU Project, it is free software under the terms of the GNU General Public License. Octave is one of the free alternatives to Matlab, others being FreeMat. Scilab, however, puts emphasis on syntactic compatibility with Matlab than Octave does. The project was conceived around 1988, at first it was intended to be a companion to a chemical reactor design course. Real development was started by John W. Eaton in 1992, the first alpha release dates back to January 4,1993 and on February 17,1994 version 1.0 was released. Version 4.0.0 was released on May 29,2015, the program is named after Octave Levenspiel, a former professor of the principal author. Levenspiel is known for his ability to perform quick back-of-the-envelope calculations, in addition to use on desktops for personal scientific computing, Octave is used in academia and industry. For example, Octave was used on a parallel computer at Pittsburgh supercomputing center to find vulnerabilities related to guessing social security numbers. Octave is written in C++ using the C++ standard library, Octave uses an interpreter to execute the Octave scripting language. Octave is extensible using dynamically loadable modules, Octave interpreter has an OpenGL-based graphics engine to create plots, graphs and charts and to save or print them. Alternatively, gnuplot can be used for the same purpose, Octave includes a Graphical User Interface in addition to the traditional Command Line Interface, see #User interfaces for details. The Octave language is a programming language. It is a programming language and supports many common C standard library functions. However, it does not support passing arguments by reference, Octave programs consist of a list of function calls or a script. The syntax is matrix-based and provides functions for matrix operations. It supports various data structures and allows object-oriented programming and its syntax is very similar to Matlab, and careful programming of a script will allow it to run on both Octave and Matlab

3.
Mathematical
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times