Category: Numerical Methods

Reconciling secant method formulas

Depending on how you derive the secant method, one gets the final formulas that are the same but the form is different.  This blog shows one how to reconcile the formulas.

Reconciling secant method formulas
Reconciling secant method formulas
Reconciling secant method formulas

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A Grant to Study Flipped (Inverted) Classrooms

I will be leading a 4-institution team (USF, AAMU, ASU and Univ of Pitt) to improve and assess student learning in an inverted STEM classroom setting.  This is a 3-year study funded by National Science Foundation, and we will compare a hybrid classroom to the flipped classroom for a course in Numerical Methods.

Summary: Numerical methods are taught by many engineering disciplines throughout the nation and therefore more engaging and effective teaching of these methods could result in better prepared engineering graduates. When teaching numerical methods it is critical to cover not only theoretical concepts required for algorithm development and selection but also the precise language of computer programming required for algorithm implementation. This project is evaluating the efficacy of using an inverted classroom methodology to teach these diverse concepts.

Results from this project will include on-line materials that encourage and support use of active in-class pedagogical methods and assessments of these materials and methods. These results will inform the education community regarding the efficacy of teaching numerical methods using an Inverted classroom methodology. The materials will be readily available to engineering departments nationally and assessment results will help faculty better understand what aspects of numerical methods can be taught using an inverted methodology and also will give them the resources needed to teach in similar ways.

Multiple universities with diverse student populations are implementing the methods and materials of this project. Assessment and evaluation is performed by another university not affiliated with the implementers; therefore, evaluation is independent of the instructors. Evaluation of student performance is partially being assessed using a numerical methods concept inventory which is being developed by the program.

Successful results in this research could transform the way engineering programs teach numerical methods nationally.

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Inverse Factorial

Problem:
It is easy to find the factorial of a positive integer by using the factorial command in MATLAB? But how would one find the value of an integer n in MATLAB such as n!>m, where m is a real number. Such inequalities need to be found in problems such as finding how many terms of the Taylor series are needed to assure that the true error is less than a certain number. To see an example of such a need, see Example 6 of this document: http://bit.ly/12Fktwm

Solution

clc
clear all
m=32034567.89
n=0
while (factorial(n)<=m)
n=n+1
end
n

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The Learning Management System Canvas (Instructure) Lacks Key Feature in Quizzes

Canvas lacks a key feature in its quiz option which many STEM (science, technology, engineering and mathematics) instructors have requested. In fact, I consider the lack of this feature as a bug. The error margins in setting up quiz problems based on a formula are based on true error and not on RELATIVE true error.

In Spring 2014, the university I teach at, University of South Florida is migrating from the current learning management system of Blackboard to Canvas.  It has been a welcome change but Canvas lacks a key feature in its quiz option which many STEM (science, technology, engineering and mathematics) instructors have requested.  In fact, I consider the lack of this feature as a bug.

To give you an example, one of the options in making a quiz is called the formula question.  This option is attractive to STEM instructors as one can develop a question whose correct answer is numeric but based on a formula.

For example, one may ask the question what is x/3 and the instructor can choose a range of input values of x (say 1 to 100,000).  The quiz option allows the instructor to generate up to 200 combinations with x being chosen randomly in the selected range.   The quiz option asks for an error margin for the combinations.  But here is the problem: The error margins are based on true error and not on RELATIVE true error.  An error margin of +-1 may be acceptable for a question that asks – “What is 10000/3?” but not for “What is 1/3?”.  This issue could be easily resolved by making the error margin to be a RELATIVE error margin.

The above example is simple to rectify as one may argue that one could use an error margin of +-0.001 and it would work reasonably well for all possible value of x, but for problems with many intermediate steps, where students could be carrying varying number of significant digits, the prescribed error margins can create issues with a correct answer being deemed incorrect.

I have tweeted to Instructure founders, submitted tickets to Canvas help, asked our LMS division in our university to communicate this issue, but I need your help in making this feature request popular.  If you have your own CANVAS account (it is free to make one) or are using CANVAS as a student or a faculty member at your university, go to http://help.instructure.com/home, login and click on ME TOO button on this link: http://help.instructure.com/entries/21499124-Express-the-Error-Margin-in-Formula-Question-as-a-of-the-Result

Thank you.

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A MOOC on Introduction to Numerical Methods

After the rigorous and comprehensive development and assessment of the NSF funded innovative open courseware on Numerical Methods between 2002 and 2012, we are offering a FREE Massive Open Online Course (MOOC) in Numerical Methods at https://canvas.instructure.com/enroll/KYGTJR

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. Ask questions within the course and we can keep the conversation going!

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About: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).  Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative  distribution function) or because the analytical method is intractable (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 40 hours of lectures need to be watched and estimated  time to read textbook and do quizzes is 80 hours.  It is a typical 15-week semester length course.

Course Structure: For each section, you have video lectures, followed by a textbook chapter, a quiz and solutions to quizzes.

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This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com, 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://nm.MathForCollege.com/videos. Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.

The decimal point display in TI30Xa calculators

The decimal point on the TI 30 Xa calculator is hardly noticeable. I prescribe this calculator for all my students in a Numerical Methods course as a programmable calculator would defeat the purpose of the course. The decimal point used to be noticeable in the older versions. I am saddened that TI has done this most probably to shave off a few cents. For a student taking a 50 minute test and doing homework with such a calculator is quite a challenge.  TI has replied my query on this issue and has forwarded it to their product developers.  In the meanwhile, buy the TI 30Xa solar version of the calculator.  The decimal point is more visible in this version.

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This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com, 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://nm.MathForCollege.com/videos. Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.

Making sense of the Big Oh!

Many students are challenged to understand the nature of Big Oh in relating it to the order of accuracy of numerical methods.  In this exercise, we are using the central divided difference approximation of the first derivative of the function to ease some of the mystery surrounding the Big Oh.



You can visit the above example by opening a pdf file.

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com, 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://nm.MathForCollege.com/videos.  Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.

Using Taylor polynomial to approximately solve an ordinary differential equation

Taylor polynomial is an essential concept in understanding numerical methods. Examples abound and include finding accuracy of divided difference approximation of derivatives and forming the basis for Romberg method of numerical integration.

In this example, we are given an ordinary differential equation and we use the Taylor polynomial to approximately solve the ODE for the value of the dependent variable at a particular value of the independent variable. As a homework assignment, do the following.
1) compare the approximate solution with the exact one, and
2) get another approximate solution by using a third order Taylor polynomial.

Taylor polynomial approximation of solving ordinary differential equations

You can visit the above example by opening a pdf or video file.

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com, 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://nm.MathForCollege.com/videos.  Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.

2012 in review

The WordPress.com stats helper monkeys prepared a 2012 annual report for this blog.

Here’s an excerpt:

19,000 people fit into the new Barclays Center to see Jay-Z perform. This blog was viewed about 150,000 times in 2012. If it were a concert at the Barclays Center, it would take about 8 sold-out performances for that many people to see it.

Click here to see the complete report.

Proving the denominator of the linear regression formula for its constants is greater than zero.

In a previous blog, we wrote without proof that the denominator of the constants of the linear regression formulas are greater than zero.  In this blog, we show the proof.


This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com, 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 athttp://nm.mathforcollege.com/videos.  Subscribe to the blog via areader or email to stay updated with this blog. Let the information follow you.