Scientific Computing I- Deliverable 3: Research Paper Draft

In fulfillment of CISC 600-50

Summer 2019

Project Title:

How to Calculate the Singular Value Decomposition and the Eigen Decomposition.

Problem Statement:

This research project is trying to calculate the singular value decomposition and the eigendecomposition.

Singular Value Decomposition:

Singular Value Decomposition can be defined as SDV. The SDV of any matrix is the factorization of that matrix into the result of three matrices. That can be defined as UDV T where both columns are Ortho normal along with the matrix D which is diagonal with positive real entries. The SVD is helpful in several tasks. Here we talk about several examples. Initially, in many applications, the information surrounding substance is close to a matrix of low rank as well as it is constructive to discover a low-rank matrix that is a superior estimation of the data matrix. Also, singular value decomposition that is distinct for all matrices is different from the more frequently used spectral decomposition in Linear Algebra.

Eigen Decomposition:

Matrix decompositions are a helpful tool o decrease a matrix to their ingredient fractions. It is done to make simpler a variety of extra complex processes. Perchance the majority times used the sort of any matrix decay or decomposition is the Eigen decomposition. Eigen decomposition decays a matrix into the Eigenvectors as well as Eigenvalues. This rotting process to plays a task in techniques used in machine learning (ML), such as in the Principal Component Analysis technique or PCA (Hadrien, 2018). The Eigen decomposition is one of the basic forms of matrix decomposition. Decomposition of a matrix means a particular way to discover the product of matrices namely equivalent to the original matrix. For Eigen decomposition, first, we have to decay the original matrix into some value that is equivalent to the product value of its Eigenvectors and Eigenvalues.

Proposed solution and proposed research methodology/strategy:

Initially, we have an idea about the MATLAB commands to find out the Singular Value Decomposition and also the value of Eigen decomposition. Using Matlab, we take the commands for SVD and Eigen decomposition for granted.

MATLAB Syntax for Singular Value Decomposition (SDV):

[A,D,B] = svd(Y)

MATLAB Syntax for Eigen Decomposition:

e = eig(C)

[X,S] = eig(C)

[X,S,W] = eig(C)

e = eig(C,D)

[X,S] = eig(C,D)

[X,S,Y] = eig(C,D)

[___] = eig(C,balance option)

[___] = eig(C,D,algorithm)

[___] = eig(___,eigvalOption)

By doing some paper research we get knowledge about the Mathworks commands those are used to find out the values of SVD as well as Eigen decomposition. In this project work, we will try to find out how the SVD and/or the Eigen decomposition are essentially calculated. So in this project proposal, we will propose the backward/forward error analysis properties. We will make sure the high relative accuracy even when the SDV and/or Eigenvalues are miniature in the norm, etc (Jaimes, 2010). The forward error is constructed by the approximation. It is an error analysis process and shows the difference between the approximated and the actual solution but it is not possible to use this method every time so the backward error is being used. The backward error has the distinguishably of being assessable but not our correct aim when solving a given predicament. Backward error is given by the quantity of the problem statement would have to modify to understand the given estimation of its clarification and solutions. To compute the values of SVD and Eigen Decomposition we may need to calculate the condition number. The condition number of any problem can be defined by the ratio of the amount of the solution changes to the number of its statement alterations under small perturbations. This can be defined as c.

C = ==

Forward and backward errors signify the accuracy of a solution. So we use this analysis technique to find out the value of Singular Value Decomposition and the value of eigendecomposition in our research project.

References:

Jaimes, D. A. A. (2010). Development and applications of two and three-component particle image velocimetry techniques for simultaneous measurement in multi-phase flows and automative fuel sprays. The University of Leeds.