Computational Time for Forward Elimination Steps of Naive Gaussian Elimination on a Square Matrix

Problem Statement

How much computational time does it take to conduct the forward elimination part of the Naïve Gauss Elimination method on a square matrix?

CTdecomposition

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Clearing up the confusion about diagonally dominant matrices – Part 2

In a previous post, we discussed the confusion about the definition and associated properties of diagonally dominant matrices.  In this blog, we answer the next question.

What is a weak diagonally dominant matrix?

The answer is simple – the definition of a weak(ly) diagonally dominant matrix is identical to that of a diagonally dominant matrix as the inequality used for the check is a weak inequality of greater than or equal to (≥).  See the previous post on Clearing up the confusion about diagonally dominant matrices – Part 1 where we define a diagonally dominant matrix.

Other blogs on diagonally dominant matrices
Clearing up the confusion about diagonally dominant matrices – Part 1

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Computational Time to Find Determinant Using Gaussian Elimination

The time it would take to find the determinant of a matrix using the Gaussian Elimination is many-many orders less than when the cofactor method is used.  In this blog, we derive the formula for a typical amount of computational time it would take to find the determinant of a nxn matrix using the forward elimination part of the Naive Gauss Elimination method.  The time is compared with that using the cofactor method.
 Computational time to find determinant

Computational time to find determinant

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu, the textbook on Numerical Methods with Applications available from the lulu storefront, the textbook on Introduction to Programming Concepts Using MATLAB, and the YouTube video lectures available at http://numericalmethods.eng.usf.edu/videos.  Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.

Computational Time to Find Determinant Using CoFactor Method

The time it would take to find the determinant of a matrix using the cofactor method can be daunting.  A student may not realize this as they may be limited to finding determinants of matrices of order 4×4 or less by hand.  In this blog, we derive the formula for a typical amount of computational time it would take to find the determinant of a nxn matrix using the cofactor method.
 Computational time to find determinant

Computational time to find determinant

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu, the textbook on Numerical Methods with Applications available from the lulu storefront, the textbook on Introduction to Programming Concepts Using MATLAB, and the YouTube video lectures available at http://numericalmethods.eng.usf.edu/videos.  Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.

Computational Time for Forward Substitution

In the previous blog, we found the computatational time for back substitution. This is a blog that will show you how we can find the approximate time it takes to conduct forward substitution, while solving simultaneous linear equations. The blog assumes a AMD-K7 2.0GHz chip that uses 4 clock cycles for addition, subtraction and multiplication, while 16 clock cycles for division. Note that we are making reasonable approximations in this blog. Our main motto is to see what the computational time is proportional to – does the computational time double or quadruple if the number of equations is doubled.

Forward Substitution Time
Forward Substitution Time

The pdf file of the solution is also available.

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Computational Time for Back Substitution

This is a blog that will show you how we can find the approximate time it takes to conduct back substitution, while solving simultaneous linear equations using Gaussian elimination method. The blog assumes a AMD-K7 2.0GHz chip that uses 4 clock cycles for addition, subtraction and multiplication, and 16 clock cycles for division. Note that we are making reasonable approximations in this blog. Our main motto is to find how the computational time is related to the number of equations.

The pdf file of the solution is also available.

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How do I solve simultaneous linear equations given in equation form?

Many students ask me how do I do this or that in MATLAB.  So I thought why not have a small series of my next few blogs do that.  In this blog, I show you how to solve simultaneous linear equations given in equation form.

  • The MATLAB program link is here.
  • The HTML version of the MATLAB program is here.
  • DO NOT COPY AND PASTE THE PROGRAM BELOW BECAUSE THE SINGLE QUOTES DO NOT TRANSLATE TO THE CORRECT SINGLE QUOTES IN MATLAB EDITOR.  DOWNLOAD THE MATLAB PROGRAM INSTEAD

%% HOW DO I DO THAT IN MATLAB SERIES?
% In this series, I am answering questions that students have asked
% me about MATLAB.  Most of the questions relate to a mathematical
% procedure.

%% TOPIC
% How do I solve a set of simultaneous linear equations
% given in equation form?

%% SUMMARY

% Language : Matlab 2008a;
% Authors : Autar Kaw;
% Mfile available at
% http://numericalmethods.eng.usf.edu/blog/sle_equations.m;
% Last Revised : August 22, 2009;
% Abstract: This program shows you how to solve a set of
%     simultaneous linear equations given in equation form?
%           .
clc
clear all
clf

%% INTRODUCTION

disp(‘ABSTRACT’)
disp(‘   This program shows you how to solve a’)
disp(‘   set of simultaneous linear equations given in equation form’)
disp(‘ ‘)
disp(‘AUTHOR’)
disp(‘   Autar K Kaw of https://autarkaw.wordpress.com’)
disp(‘ ‘)
disp(‘MFILE SOURCE’)
disp(‘   http://numericalmethods.eng.usf.edu/blog/sle_equations.m’)
disp(‘ ‘)
disp(‘LAST REVISED’)
disp(‘   August 22, 2009’)
disp(‘ ‘)

%% INPUTS
% Enter the equations
eqn1=’12*a+23*b+39*c=29′;
eqn2=’13*a+17*b+19*c=37′;
eqn3=’21*a+23*b+29*c=59’;
%% DISPLAYING INPUTS
disp(‘  ‘)
disp(‘INPUTS’)
disp(‘________________________’)
disp(‘Equations’)
disp(‘________________________’)
disp(eqn1)
disp(eqn2)
disp(eqn3)

%% THE CODE
% The solution
X=solve(eqn1,eqn2,eqn3);
% Assigning the output
a=double(X.a);
b=double(X.b);
c=double(X.c);
%% DISPLAYING OUTPUTS
disp(‘  ‘)
disp(‘OUTPUTS’)
disp(‘________________________’)
disp(‘Solution Vector’)
disp(‘________________________’)
fprintf(‘\nValue of a= %g’,a)
fprintf(‘\nValue of b= %g’,b)
fprintf(‘\nValue of c= %g’,c)
disp(‘  ‘)
disp(‘________________________’)

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu, the textbook on Numerical Methods with Applications available from the lulu storefront, and the YouTube video lectures available at http://numericalmethods.eng.usf.edu/videos and http://www.youtube.com/numericalmethodsguy

Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.