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Anthony Vuolo III

POCKETBRAIN: MATRIX

CALCULATOR APPLICATION

Abstract ......................................................................................................................................................... 2

Literature Review .......................................................................................................................................... 3

Introduction ............................................................................................................................................... 3

The User Interface ..................................................................................................................................... 3

The Basic Attributes of the User Interface ............................................................................................ 4

Matrices .................................................................................................................................................... 5

The Basics ............................................................................................................................................. 5

Examples of Complex Functions .......................................................................................................... 7

Accuracy ................................................................................................................................................... 9

Conclusion .............................................................................................................................................. 11

Charter ........................................................................................................................................................ 12

Materials and Procedure ............................................................................................................................. 13

Results ......................................................................................................................................................... 14

Conclusions ................................................................................................................................................. 19

References ................................................................................................................................................... 21

Appendices .................................................................................................................................................. 23

Appendix 1: Acknowledgements ............................................................................................................ 23

Appendix 2: List of Functions ................................................................................................................ 24

Appendix 3: Limitations and Assumptions ............................................................................................. 25

Appendix 4: Sample Code ...................................................................................................................... 27

Appendix 5: Notes File ........................................................................................................................... 35

Appendix 6: Field Competitors Handheld Graphing Calculators ........................................................ 44

Abstract

A matrix is a two-dimensional set of values used for storing numerical information, depicting transformations on a coordinate plane, computing statistical data, or even encoding neural networks. Matrix computations are taught in linear algebra and advanced modeling courses in college curricula, in preparation for use in real occupations. Current free matrix calculators are often inaccurate, function under a user interface that is difficult to navigate, or lack important traits such as exportable data. The goal of the project was to engineer an application that performs most college-level matrix calculations while maximizing computational accuracy and usability. The program was coded to include a list of attributes that functioned as guidelines for optimizing compared to other programs using function accuracy tests and a scoring matrix that included user interface assessments. A free matrix calculator with maximal usability and exact computational results will make calculations faster and easier, allowing matrices to truly become the standardized format for data storage. 3

Literature Review

Introduction

Matrices are used in many different areas of math. They are useful for calculating changes in two-dimensional motion and in making changes to large groups of data. This method of storage is compact, reliable, and versatile, as it can be converted into many different forms. However, even basic calculations with matrices can be tedious; operations as basic as multiplication require realigning the two factor matrices and multiplying long strings of elements for each section of the product matrix. For this reason, it is more efficient to use programmed calculators to perform operations on these data types. In addition to the long list of mathematical methods one must code to provide an adequate calculator, one must engineer a user interface that a human being ca between user and program should be organized and visually appealing, implementing an operation system that allows data to be easily entered and neatly displayed in different forms. Many matrix calculators have interfaces that do not allow easy navigation or have inefficient systems for accepting input. For this reason, there is a need for an application that can run accurate matrix calculations in a user interface that can easily support the display requirements of any size matrix in an organized navigation system.

The User Interface

The user interface, abbreviated as UI, is the set of programs that allows a human being to find the resources needed from a website or application. The interface is important because it represents all interactions between the code and the user; only a well-designed UI will allow easy access to all the functions the program has to offer. Coding languages built for this part of an 4 application include HTML and CSS; however, other languages do support graphical images and other visual effects.

The Basic Attributes of the User Interface

The menu is the basic UI component that allows any and all interaction with a computer application to take place. Although often overlooked, the state of the menu is what has the most influence on the usability of the program. A properly functional menu is uncondensed, allowing the user to skim it for the functions or submenu of interest (Pérez-Montoro, 2017). The interface should feature a description of each option to permit the user to preview a menu before choosing a function. An optional feature is to allow a list of functions to move around the screen; for some applications, this design could allow information on a page to be placed in any area and not remain permanently obscured (Oldenburg, 1989). This ability may also allow windows to overlap and reduce screen clutter for optimal use (Kerkez, 2000). Preferably, the information about the matrices in each menu would also be shared with other locations in the app (He, Shen, Li, Shi, and Zhao, 2010). The subsections of a menu contain functions or submenus that are too complicated or require too much visual space to be contained on the screen. Menu options should be organized in a sort of hierarchy or tree format, which can allow multiple function categories to be contained in a single screen (Tate, Goto, and Takeuchi, 1996). Another important detail about a user interface is its ability to undo an accidental action made by the user. Although most calculators do not implement this sort of code, it would make for a more navigable interface. The most basic function would be a set of forward and backward buttons for scrolling through program history, as these would allow the user to travel back and forth between menu layers (Pérez-Montoro, 2017). In the functions themselves, an undo and redo 5 button would allow a user to nullify any previous action made while developing a matrix without having to remove every single element.

Matrices

The Basics

The basic form of a matrix is a two-dimensional list of polynomials, although they usually use single numbers. A matrix can store lists of variable sets such as coordinate points. For such data points, square matrices can represent translations, rotations, dilations, or other coordinated movements across a coordinate grid. Matrices are also used in economic programs and in neural networks, which aid machine-learning algorithms (Christensen, 2003). Matrices have strange mathematical properties because of how their basic operations to a matrix, every element is multiplied by the scalar. However, the dot product uses a different mathematical technique. For computation, the product of an m-by-n matrix and a p-by-q matrix has dimensions m-by-q. For each element in the array, a row of the first matrix and a column of the second are lined up; elements are paired together, multiplied within their pairs, then summed to produce the new element. Because each value must have a pair with which to multiply, the number of columns in the first matrix must match the number of rows in the second. Dot product multiplication is associative but rarely commutative; switching the order of the matrices may produce a very different sum of dot products that can change the elements of the new array. This computation allows for a standardized division function, and certain matrix decompositions are computable by converting the dot product into a system of equations. The determinant of a matrix is what confirms the existence of a matrix inverse. When using the normal process of calculating the inverse of a square matrix, the final step is to divide 6 the matrix by its determinant; if the divisor is 0, then the inverse cannot be found using that method. Finding a determinant involves iteratively taking the determinants of smaller submatrices and summing them along rows in the matrix. When a value a is divided by a value b, the other way of interpreting the mathematical operation is noting that a is multiplied by the multiplicative inverse of b, or ଵ ௕. With matrices, division is performed differently because several terms are multiplied and added to form each element. Thus, multiplication by an inverse is the standard method of matrix division. The inverse of matrix M can be found by first creating a matrix A formed by the determinants of each M subsection to form the matrix of minors. Certain values in a grid pattern are then multiplied by -1 to form the matrix of cofactors. Once the values in this step have been transposed along the upper-left-to-lower-right diagonal the adjoint the whole matrix is divided by its determinant. Aside from standard division, the inverse of a coefficient matrix can be used to solve linear systems. The matrix inverse is one of a few specific matrices that can be commutatively multiplied by the original, and when this occurs the result is an identity matrix (Wang and Dai, 2010). This matrix is formed when a null matrix, or array of zeros, has ones placed along its main diagonal. The identity is essentially a multiplicative identity and behaves much like the number one does in the set of all complex numbers. It is its own multiplicative inverse, and when multiplied by any matrix M, the resulting product is the same array M. 7

Examples of Complex Functions

Linear systems of equations can be solved by using matrices. A series of linear equations can be modeled as an equality that sets the product of a coefficient matrix and an unknown column matrix equal to a constant column matrix. Fig. 1: A linear system set up to be solved by a matrix system 2014) This system can be solved by multiplying both sides of the equation by the inverse of the coefficient matrix; this would form an identity on the left with the unknowns and a column matrix on the right, whose values would then be equivalent to each corresponding unknown. Thus, by evaluating the inverse one has solved a linear system. Matrices can represent movement in a Cartesian coordinate grid; however, it is worth noting that complex numbers do the same. Thus, the two can be interchanged. Similarly, rotations using matrices can be performed using an arrangement of trigonometric identities. Eigenvalues are numbers used to designate the magnitude of Cartesian plane movements that a matrix performs on one or more coordinate points. When the eigenvalues of a matrix are added 8

to its diagonal elements, the determinant of the resulting matrix is 0; this subtraction can occur if

the product of an eigenvalue and a properly-sized identity matrix is subtracted from the original. Fig. 2: A visual of the characteristic polynomial being formed given the input matrix (Gundlach). The characteristic polynomial is the determinant of the resulting matrix; this will be in the form of a polynomial of degree n, where n is the size of the matrix. The roots of the polynomial will be the eigenvalues. When a matrix is multiplied by one of its eigenvalues, there is always a matrix n such that when multiplied by the original matrix produces the same result.

This is known as an eigenvector.

Matrix decomposition is the name given to a set of complicated operations. However, most decompositions have a common characteristic: they often produce multiple matrices. For example, lower-upper decomposition strips an array into two components: a reformed identity matrix where any nonzero values appear below the main diagonal, and an upper triangular matrix where all nonzero numbers are at or above the diagonal. A complicated system of equations, split using Gauss-Jordan elimination, is solved to produce all nonzero values so that the product of the two matrices is the original. When representing a linear system, coefficient matrices with more zeros are easier to solve because there are fewer variables per equation. Thus, many 9 decompositions of a matrix result in a simpler set of coefficient arrays to encode variables. As with inverses, some matrices do not have a lower-upper decomposition.

Accuracy

A problem with calculators, common between handheld models and web applications alike, is that they often perform calculations using decimals. Decimals provide very accurate answers that allow the calculator to continue with its computations using numbers that usually produce the right results. Some similar variable types, such as floating-point values, allow a computer to work at faster speeds (). However, decimal answers are almost always approximate. Irrational numbers are truncated or rounded, and rational numbers are sometimes represented the same way, upsetting simple calculations. Floating-points use

Ichimura,

Katagiri, Ozaki, Ogita, and Nagai, 2018). Erroneous rounding can cause large or obvious deviations between answers; calculator programs such as Desmos and BlueBit do not always produce an identity matrix when they perform multiplication on a matrix and its inverse because of errors in significant figures. Even Java programs with primitive decimal-based data types can be prone to errors where exact values, such as terminating decimals, are recorded incorrectly. For certain calculators, a solution to the accuracy problem could be to use fractions. Since a fraction can represent any rational number with no error or rounding, any calculator that does not require irrational numbers in its output matrix can implement a fraction-based variable storage system. In an experiment, a few matrix calculators were tested for error using their inverse function. A Java program and an online calculator called Wims use fractions for storage, while Desmos and BlueBit use decimals. For each trial, a matrix was inputted into a calculator. The application would compute the inverse, and then multiply it by the original matrix. The 10 resulting product would be a matrix close to, if not congruent to an identity matrix. The residual errors of each element as deviating from the respective elements of the identity matrix were taken and logged. As is shown in Fig. 3, the fraction-based programs experienced no residual error. The output from Desmos had enough deviation to be visible in the output matrix. The resulting matrix from BlueBit, since the numbers could not be stored directly in the program and had to be manually copied from another output screen, had the largest error; this amount of inaccuracy would be likely to accumulate through multiple computations and lead to wrong results. Overall, this experiment supports the use of fractions over that of decimals because fractions maintain exact values. Table 1: A recording of residual error for the identity matrix given from each calculator. Matrix Calculator Operation Input Matrix Residual Error

Java Program Inverse 3 5 -3

4 2 1

11 6 7

Wims Inverse Ͳ

However, some types of decompositions and other matrix computations require using the irrational roots of integers. Fractions cannot represent such values. In many cases, this problem can be solved with radical numbers. Radicals function as a unique variable type; a radical and a non-radical cannot be added into a single term. However, the existence of radicals allows irrational numbers to exist in their exact forms. Quadratic equations can be solved without the need for decimals by using radicals of root 2, which allow some quantities, such as the eigenvalues of small matrices, to be found in exact value. Some radical exponents have simple simplification methods; any exponent power of 2 can be iteratively squared until all roots have been eliminated. This means that such radicals can be removed from the denominator of any 11 fraction, which maintains a constant variable format and makes mathematical operations easier to perform.

Conclusion

The modeling capabilities of matrices are useful for a variety of scientific studies. However, calculations using matrices are tedious and are completed quickly on a computer. Accuracy in these calculations can be maintained if a radical number format is used; values can remain in their exact form and there will be no rounding errors as would occur with a decimal- based system. Because of the amount of data that must be contained in a matrix, the program must have a system that allows easy entering of information. All functions that use the input data must be easily accessible from a menu network. A matrix calculator that can implement both the interface and the number type outlined above would be one of the most capable calculators of its kind. 12

Charter

Problem Statement: Free calculator apps rarely have all mathematical operations commonly used by students in college-level mathematics courses. Few existing apps that perform matrix calculations maintain numerical accuracy and an easy-to-use user interface. Solution: The goal of this project is to engineer an application that performs most matrix calculations used in college-level mathematics courses while maximizing computational accuracy and usability. 13

Materials and Procedure

Materials:

- Dell Inspiron 13-7378 laptop computer - Android Studios - IntelliJ IDEA Java IDE editor - Nexus 4 Android Device Emulator

Procedure:

A series of rational number-based matrix calculation programs were completed in Java code (see Appendix 1). Research was conducted to find a mobile device emulator that was free, available on a Windows device, and compatible with Java. Android Studios was selected as the preferred editor. Research was done on the attributes of a user interface to generate a list of its ideal qualities. Notes were taken on existing matrix calculators to list their qualities; these were sorted by positive or negative, depending on whether they were viewed as helpful or detrimental to the usability of the application. Data was taken to log the accuracy of each calculator by performing calculations that required the storage of repeating decimals in the matrix calculator systems. The decimal accuracy of each calculator was logged for comparison. Designs were sketched to incorporate the preferable attributes into the interface. As drafts of each design were created, the designs were coded into Android Studio interactive pages, starting with the homepage. The mathematical programs were assimilated into the interface code duringquotesdbs_dbs20.pdfusesText_26

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