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A number of modules have been designed and developed by the paper writer, using LabVIEW for numerical analysis to the design of a virtual interactive graphical user interface (GUI) software to solve QR Factorization, and LU Factorization. The focus of this study is to provide virtual models to solve systems of linear Equations by using QR Factorization, LU methods. The significant finding reached by the researchers, is that the virtual model can be used to solve any systems of linear equation. Based on the previous finding, the researcher recommends that other systems of linear equation also can follow the same analogy to solve numerical equation method.

There are several numerical computation packages that serve as educational tools and are also available for commercial use. Of the available packages, LabVIEW is the most widely used in GUI [

This paper will design and develop Interactive Virtual modules (VIs) for studying 1) QR Factorization, 2) LU Factorization. The linear system of equations using LU and QR methods is used because the sample students study a course of the linear system of equations solved by LU and QR methods.

Using the LabVIEW, program has been designed and tested, and the application of the solution of a system of linear equations by 1) QR Method A is

The main purpose of this paper is to help unspecialized people in the computer science. It is for those who use the numerical linear system. The LU is preferred to Matlab because LABVIEW is a graphical user interface (GUI) and it can easily be understood by the students. It is essential in the third word.

BackgroundThis background concurs with Virtual laboratories, Solutions of Systems of Linear Equations (The QR factorization technique and LU factorization technique), and LU factorization technique.

Virtual laboratories typically originate from computation and simulation software such as Matlab [

Solutions of Systems of Linear Equations [

The equations in the system can be linear or non-linear. This paper adopts the systems of linear equations. The problem can be expressed in narrative algebraic form.

The QR factorization technique factors the input matrix A as the product of two matrices, such that A = QR, where R is an upper triangular matrix. QR factorization can be used with both square and rectangular matrices. When the matrix A is square and non-singular, its QR factorization is unique, and all the diagonal elements of R are positive [

A number of algorithms are possible for QR factorization, such as the Householder transformation, the Givens transformation, the Fast Givens transformation, and the Gram-Schmidt method.

The LU factorization technique factors the input square matrix A as the product of two triangular matrices, such that A = LU, where L is a lower triangular matrix with ones on the diagonal, and U is an upper triangular matrix. This factorization A = LU exists if all the N leading principal sub-matrices of A have non-zero determinants, where N is the number of rows or columns in A. If this factorization exists and A is non-singular, the LU factorization is unique.

Typically, in order to receive a more precise calculation, LU factorization is performed using the format PA = LU, where P is a permutation matrix. You can use this format to perform the LU factorization on any square matrix.

LabVIEW [

LabVIEW can be used to address the needs of various courses in technology and science curriculum [

In this paper, the description of how to design and implement a Virtual lab using the LabVIEW program for the solution of a system of linear equations by: QR Method and the LU factorization Method is given.

In this section examples of virtual modules in QR Method are presented: 1) QR Method A is

These will be dealt with successively.

The basis of the QR Method for calculating the Eigenvalues of A is the fact that a

We will begin by first solving a linear system. Let us consider QR Method A is

Solve by Use LabVIEW

Example VI to perform QR Method A is

We will begin by second solving a linear system. Let us consider QR Method A is

Solve by Use LabVIEW

Example VI to perform QR Method A is

Solve by Use LabVIEW

The LU factorization technique factors the input square matrix A as the product of two triangular matrices, such

that A = LU, where L is a lower triangular matrix with ones on the diagonal, and U is an upper triangular matrix. This factorization A = LU exists if all the N leading principal sub matrices of A have non-zero determinants, where N is the number of rows or columns in A. If this factorization exists and A is non-singular, the LU factorization is unique.

Typically, in order to receive a more precise calculation, LU factorization is performed using the format PA = LU, where P is a permutation matrix. This format can be used to perform the LU factorization on any square matrix.

Example VI to perform LU Decomposition.

Solve by Use LabVIEW

The VI module presented in this paper to solve numerical models, has been experimented and has given positive results as shown by the table from (1 to 4).

Using LabVIEW program has been designed and tested, and the application of the solution of a system of linear equations by:

The linear equations are written in Matrix Form (Ax = B). Form A and B (known vector) are supplied as inputs to the VI.

The VI solves the QR Method. A is

The VI solves for the QR Method A is

Example 1 Matrix A is 6 × 6 | Find (R) Factorization | ||
---|---|---|---|

Find (Q) Factorization | Solve QR Method |

Example Matrix A is 3 × 6 | Find (R) Factorization | ||
---|---|---|---|

Find (Q) Factorization | Solve QR Method |

The VI solves for the QR Method A is

The VI solves for the LU decomposition and displays the results as shown in

This study has attempted to apply modern methods of solution of linear equations form Ax = b. The study started by providing an account about the analysis of linear systems and the types of matrices, especially those

Example 2 Matrix A is 6 × 3 | Find (R) Factorization | ||
---|---|---|---|

Find (Q) Factorization | Solve QR Method |

Example 1 Matrix A is 3 × 3 | Find (U) Solve the upper-triangular system | ||
---|---|---|---|

Solve LU-factorization | Find (L) Solve the lower-triangular system |

of two-dimensional form, which are used in the solution of these systems. The modern methods used to solve these linear equations are QR and LU methods.

A number of modules have been developed using LabVIEW for numerical analysis and engineering problem solving; these linear equations are QR and LU methods courses.

Abd Elrazig Awadelseed Edries Suliman, (2016) Applications of Virtual Modules in Numerical Analysis. Open Access Library Journal,03,1-8. doi: 10.4236/oalib.1102299