Deriving trapezoidal rule using undetermined coefficients

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Implications of diagonally dominant matrices

In the previous blogs (Part 1, Part 2, Part 3, Part 4), we clarified the difference and similarities between diagonally dominant matrices, weakly diagonal dominant matrices, strongly diagonally dominant matrices, and irreducibly diagonally dominant matrices.  In this blog, we enumerate what implications these classifications have.

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If a square matrix is strictly diagonally dominant

  • then the matrix is non-singular [1].
  • then if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite [1].
  • then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge [2].
  • then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge [2].
  • then if the diagonal entries of the matrix are positive, the real parts of the matrix eigenvalues are positive [1].
  • then if the diagonal entries of the matrix are negative, the real parts of the matrix eigenvalues are negative [1].
  • then if the matrix is column dominant, no pivoting is needed for Gaussian elimination [2].
  • then if the matrix is column dominant, no pivoting is needed for LU factorization [2].

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If a square matrix is irreducible diagonally dominant

  1. then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge.
  2. then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge.
  3. the matrix is non-singular [2].

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If a square matrix is diagonally dominant (also called weakly diagonally dominant)

  1. then if the matrix is column dominant, no pivoting is needed for Gaussian elimination [3].
  2. then if the matrix is column dominant, no pivoting is needed for LU factorization [3].

References

  1. Briggs, Keith. “Diagonally Dominant Matrix.” FromMathWorld–A Wolfram Web Resource, created by Eric W. Weisstein.  http://mathworld.wolfram.com/DiagonallyDominantMatrix.html
  1. Diagonally Dominant Matrix, see https://en.wikipedia.org/wiki/Diagonally_dominant_matrix, Last accessed on November 4, 2016.
  1. “Lecture 4: A Gaussian Elimination Example”, see http://www.cs.yale.edu/homes/spielman/BAP/lect4.pdf, last accessed on November 4, 2016.

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WordPress does not update image

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Till recently I used to place images within my WordPress blog at my own website.  I would do this as I was using the free WordPress.com site where storage is limited.  Well, I found out that if I update the image on my personal website, WordPress will take an unknown time to update it at the blog.

As of this date, October 21, 2016, 2:50PM EDT, the two images are different (the image is updated now as of October 30, 2016).  It is because I updated the image on my website but it did not get updated on the wordpress.com site.  I have cleared my cache, used different computers and browsers in the house that do not belong to me, used my desktop computer at school, but the result is the same – the two images are still different after a week of updating.

WordPress support has been helpful and ultimately says that it is an image cacheing problem and there is no set time for it to be cleared.  I have resolved the problem by avoiding it.  I now upload images to the WordPress site as I started using the paid premium site.

What do you think?

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Clearing up the confusion about diagonally dominant matrices – Part 4

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You can view the above document as a pdf file as well.

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

Clearing up the confusion about diagonally dominant matrices – Part 2

Clearing up the confusion about diagonally dominant matrices – Part 3

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Clearing up the confusion about diagonally dominant matrices – Part 3

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You can view the above document as a pdf file as well.

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

Clearing up the confusion about diagonally dominant matrices – Part 2

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Clearing up the confusion about diagonally dominant matrices – Part 2

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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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Clearing up the confusion about diagonally dominant matrices – Part 1

You can view the above document as a pdf file as well.

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MOOC Released:Introduction to Numerical Methods – Part 2 of 2

After the rigorous and comprehensive development and assessment of the NSF-funded innovative open courseware on Numerical Methods since 2002, we are offering a FREE Massive Open Online Course (MOOC) in Numerical Methods – Part 2 of 2 at https://learn.canvas.net/courses/1189

This part of the MOOC covers the mathematical procedures of interpolation, regression, integration and ordinary differential equations.

The Part 1 of 2 of the course is also available and is at https://learn.canvas.net/courses/1065

Start your journey today whether you are learning numerical methods for the first time or just need a refresher.  Unlike other MOOCs, you have a lifetime (mine) access to the course and you can pace yourself. Ask questions within the course and we will keep the conversation going!

part2of2

About: Numerical methods are techniques to approximate mathematical procedures (an example of a mathematical procedure is an integral).  Approximations are needed because we either cannot solve the procedure analytically (an example is the standard normal cumulative  distribution function) or because the analytical method is intractable (an example is solving a set of a thousand simultaneous linear equations for a thousand unknowns).

Materials Included: Textbook Chapters, Video Lectures, Quizzes, Solutions to Quizzes

How Long to Complete: About 20 hours of lectures need to be watched and estimated  time to read the textbook and do quizzes is 40 hours.  It is a typical 7-week semester length course.

Course Structure: For each section, you have video lectures, followed by a textbook chapter, a quiz and its complete solution, and automatically graded online quizzes.

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Unresolved CANVAS LMS bug in algorithmic quizzes

I have been using CANVAS Learning management system for algorithmic quizzes for the last two years at University of South Florida and recently in a MOOC for a course in Numerical Methods.

Due to a bug introduced via an update in the CANVAS software, integers now are followed by a decimal point and a zero.  For example, what should be written as 5, now gets written as 5.0.  Some may believe that this is a minor hassle, but see what happened to statements within the quizzes.

  • 3 bits becomes 3.0 bits
  • 5 terms becomes 5.0 terms
  • N=5 becomes N=5.0
  • Represent integer 2 becomes Represent integer 2.0
  • Solve 37 simultaneous linear equations becomes Solve 37.0 simultaneous linear equations
  • Number of zeros after 2 steps becomes Number of zeros after 2.0 steps
  • A binary number is 1.001 becomes A binary number is 1.0.0.0.0.1.0

We already have students who come to a course with many misconceptions; it is discouraging that a bug in a learning management system will create more misconceptions, and it has hence resulted in I not using the quizzes at all.

So far, CANVAS has acknowledged the bug but what has driven me up the wall is that they have made no promise of solving it, let alone a timeline of resolving the bug.  Our university officials have not been able to make much headway either.  What do you think?

UPDATE (Sunday June 20, 2016):  From our university personnel: “It’s been escalated all the way up the chain. From what I can tell it has been put on the top 10 bug list and it has been assigned an engineer.  I should have more information tomorrow morning.”

UPDATE (Sunday June 27, 2016):  From CANVAS: “We’ve deployed a fix for this issue in our Beta environment. If all goes well, it will make it to the live Production environment on July 16, 2016. This ticket will remain in an “On-Hold” status until then.”

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A MOOC on Numerical Methods Released

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After the rigorous and comprehensive development and assessment of the NSF funded innovative open courseware on Numerical Methods since 2002, we are offering a FREE Massive Open Online Course (MOOC) in Numerical Methods – Part 1 of 2 at https://www.canvas.net/browse/usflorida/courses/numerical-methods

The MOOC is Part 1 of a two-part course in Numerical Methods.  The course covers the mathematical procedures of differentiation, nonlinear equations and simultaneous linear equations.  We had the MOOC on Udemy but we are migrating it to CANVAS in two stages. CANVAS has a broader appeal for free MOOCs, it has a user friendly interface, looks familiar for many students using CANVAS, and has the capability of online quizzes that are algorithmic.

Start your journey today whether you are learning numerical methods for the first time or just need a refresher.  Unlike other MOOCs, you have a lifetime access to the course and you can pace yourself. Ask questions within the course and we will keep the conversation going!

canvasnumerical

About: Numerical methods are techniques to approximate mathematical procedures (an example of a mathematical procedure is an integral).  Approximations are needed because we either cannot solve the procedure analytically (an example is the standard normal cumulative  distribution function) or because the analytical method is intractable (an example is solving a set of a thousand simultaneous linear equations for a thousand unknowns).

Materials Included: Textbook Chapters, Video Lectures, Quizzes, Solutions to Quizzes

How Long to Complete: About 20 hours of lectures need to be watched and estimated  time to read the textbook and do quizzes is 40 hours.  It is a typical 7-week semester length course.

Course Structure: For each section, you have video lectures, which are followed by a textbook chapter, a quiz and its complete solution, and automatically graded online quizzes.

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