Author: autarkaw

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.

Prove that the least squares general straight-line model gives the absolute minimum of the sum of the squares of the residuals?

Many regression models when derived in books only show the first derivative test to find the formulas for the constants of a linear regression model.  Here we show a thoroughly explained derivation.



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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.

Effect of Significant Digits: Example 2: Regression Formatting in Excel

In a series of bringing pragmatic examples of the effect of significant digits, we discuss the influence of using default and scientific formats in the trendline function of Microsoft Excel.  This is the second example (first example was on a beam deflection problem) in the series.

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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.

Effect of Significant Digits: Example 1: Beam Deflection

In a series of bringing pragmatic examples of the effect of significant digits, we discuss the exact solution of finding the deflection in a simply supported beam with a uniform load and a tensile axial load.

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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.

Length of curve

This is a solution to the problem I gave to the class in a recent test.

A robot follows a path generated by a quadratic interpolant from x=2 to x=4.  The interpolant passes through three consecutive data points (2,4), (3,9) and (4,16) and is given by y=x2.  Find the best estimate of the length of the interpolant path from x=2 to x=4. 

The solution is given as an audio pdf (also called pencast – need Adobe X reader or higher). Three methods are shown. I expected one to use Method 2 or Method 3.
http://www.eng.usf.edu/~kaw/class/EML3041/livescribe/length_of_curve.pdf

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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.

Example: How many significant digits are correct in my answer?

In the previous post I showed you the theorem that tells us how many significant digits are correct in our numerical solution that is iterative in nature.  In this post, we illustrate this with an example.

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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.

 

How many significant digits are correct in my answer?

To find how many significant digits are correct in my answer in a numerical method that gives iterative values, one finds the absolute relative approximate  percentage error defined as
|(Current approximation-Previous approximation)/Current approximation|*100
If the absolute relative approximate percentage error is less than or equal to 0.5*10^(2-m), then m significant digits are at least correct in the answer.
For example, if you want

  • at least 1 signficant digit to be correct in your answer, your absolute relative approximate error should be less than or equal to 5%
  • at least 2 signficant digit to be correct in your answer, your absolute relative approximate error should be less than or equal to 0.5%
  • at least 3 signficant digit to be correct in your answer, your absolute relative approximate error should be less than or equal to 0.05%
    and so on.

In the next blog, we will illustrate this concept via an example.
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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.

Do we have to setup all 3n equations for the n quadratic splines for (n+1) data points?

QUESTION ASKED BY STUDENT: For linear splines, we still find each line individually. In quadratic, we are forced to find every equation at once, correct? Which is where the matrix comes from?

ANSWER: That sounds right, but one can find quadratic equations individually also .
How?
If the first spline is linear, then that spline can be found by the two points it goes through.
Then the second spline goes thru two points, and that gives two equations.  The slope of the second spline is same as the slope of first spline at the common interior point, and the first spline is known. This gives the third eqn.  This process can be repeated for each following spline.

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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.