Matrix Algebra: Gauss-Seidel Method

Many university STEM major programs have reduced the credit hours for a course in Matrix Algebra or have simply dropped the course from their curriculum.   The content of Matrix Algebra in many cases is taught just in time where needed.  This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra. In this series of blogs, we bring to you ten topics that are of immediate and intermediate interest for Matrix Algebra. Here is the eighth topic where we talk about solving a set of simultaneous linear equations using the Gauss-Seidel method.  Learn this iterative method of solving simultaneous linear equations, and its pitfalls and advantages. Get all the resources in form of textbook content, lecture videos, multiple choice test, problem set, and PowerPoint presentation. Gauss-Seidel Method
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Matrix Algebra: LU Decomposition Method

Many university STEM major programs have reduced the credit hours for a course in Matrix Algebra or have simply dropped the course from their curriculum.   The content of Matrix Algebra in many cases is taught just in time where needed.  This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra. In this series of blogs, we bring to you ten topics that are of immediate and intermediate interest for Matrix Algebra. Here is the seventh topic where we talk about solving a set of simultaneous linear equations using the LU decomposition method.  First, the LU decomposition method is discussed along with its motivation.  The LU decomposition method to find the inverse of a square matrix is discussed. Get all the resources in form of textbook content, lecture videos, multiple choice test, problem set, and PowerPoint presentation. LU Decomposition Method
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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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