Author: autarkaw

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://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.

A MATLAB program to find quadrature points and weights for Gauss-Legendre Quadrature rule

Recently, I got a request how one can find the quadrature and weights of a Gauss-Legendre quadrature rule for large n.  It seems that the internet has these points available free of charge only up to n=12.  Below is the MATLAB program that finds these values for any n.  I tried the program for n=25 and it gave results in a minute or so.  The results output up to 32 significant digits.
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% Program to get the quadrature points
% and weight for Gauss-Legendre Quadrature
% Rule
clc
clear all
syms x
% Input n: Quad pt rule
n=14;
% Calculating the Pn(x)
% Legendre Polynomial
% Using recursive relationship
% P(order of polynomial, value of x)
% P(0,x)=1; P(1,x)=0;
% (i+1)*P(i+1,x)=(2*i+1)*x*P(i,x)-i*P(i-1,x)
m=n-1;
P0=1;
P1=x;
for i=1:1:m
    Pn=((2.0*i+1)*x*P1-i*P0)/(i+1.0);
    P0=P1;
    P1=Pn;
end
if n==1
    Pn=P1;
end
Pn=expand(Pn);
quadpts=solve(vpa(Pn,32));
quadpts=sort(quadpts);
% Finding the weights
% Formula for weights is given at
% http://mathworld.wolfram.com/Legendre-GaussQuadrature.html
% Equation (13)
for k=1:1:n
    P0=1;
    P1=x;
    m=n;
    % Calculating P(n+1,x)
    for i=1:1:m
        Pn=((2.0*i+1)*x*P1-i*P0)/(i+1.0);
        P0=P1;
        P1=Pn;
    end
    Pn=P1;
    weights(k)=vpa(2*(1-quadpts(k)^2)/(n+1)^2/ …
                                   subs(Pn,x,quadpts(k))^2,32);
end
    fprintf(‘Quad point rule for n=%g \n’,n)
disp(‘  ‘)
disp(‘Abscissas’)
disp(quadpts)
disp(‘  ‘)
disp(‘Weights’)
disp(weights’)_______________________________________________________ 

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.

YouTube Videos on Numerical Methods Cross 1-Million Views Mark

In a short 2.5 years since starting the numericalmethodsguy YouTube channel in January 2009, this month the channel crossed the benchmark of receiving 1 million video views.  Currently the channel gets between 2,500-3,500 video views per day.  Although we have playlists on the channel, the playlist for all the available topics are given on single webpage at http://nm.mathforcollege.com/videos/index.html

Complete resources on each topic of available numerical methods including textbook chapters, videos, multiple-choice tests, PPTs, and worksheets are  given at http://nm.mathforcollege.com/topics/textbook_index.html
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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.

A Wolfram demo on how much of a floating ball is under water

Here is another Wolfram demo. This one shows how much of a floating ball would be submerged under water.  The demonstration uses the specific gravity and radius of the ball as inputs.  Using calculus and Archimedes’ principle, the level to which the ball is submerged turns out to be the solution of a cubic equation.  To play with the demo, download the free CDF player from Wolfram first.
  
 
 
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.

Order of accuracy of central divided difference scheme for first derivative of a function of one variable

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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.

A Wolfram demo on converting a decimal number to floating point binary representation

Here is another Wolfram demo. This one converts a decimal number to a floating point binary representation.  To play with the demo, download the free CDF player first.
 
The total number of bits used for the representation =
       one bit for the sign of the number +
       one bit for the sign of the exponent +
       number of bits for the exponent +
       number of bits for the mantissa +
       As an example, how would 54.75 be represented in a 9-bit register where the first bit is used for the sign of the number, second bit is used for sign of exponent, next three bits are used for the exponent, and the last four bits are used for the mantissa?
Both the number and the exponent are positive. 
As the number is normalized to lie between 1 and 2 (the interval being half-closed at the bottom and half-open at the top), the leading binary digit is always 1. So we do not actually use it in the representation of the mantissa. Hence the mantissa bits are 1011. Moreover the exponent bits are 101, the sign of the number bit is 0, and the sign of the exponent bit is 0.
Therefore the representation is .
 

Reference: Floating Point Representation

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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.

A Wolfram Demo for Numerical Differentiation

We are in the process of developing Wolfram Demonstrations for Numerical Methods.  In this demo, we show approximations of derivatives by finite difference formulas.  We compare three difference approximations with the exact value.  To play with the demo, download the free CDF player first.
 
 

Reference: Approximation of First Derivatives by Finite Difference Approximations

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Taylor Series in Layman’s terms

The Taylor series for a function f(x) of one variable x is given by
f(x+h) = f(x) + f '(x)h +f ''(x) h^2/2!+ f ''' (x)h^3/3! + .............

What does this mean in plain English?
As Archimedes would have said (without the fine print), “Give me the value of the function at a single point, and the value of all (first, second, and so on) its derivatives, and I can give you the value of the function at any other point”.
It is very important to note that the Taylor series is not asking for the expression of the function and its derivatives, just the value of the function and its derivatives at a single point.

Now the fine print: Yes, all the derivatives have to exist and be continuous between x (the point where you are) to the point, x+h where you are wanting to calculate the function at. However, if you want to calculate the function approximately by using the n^{th} order Taylor polynomial, then 1^{st}, 2^{nd} ……., n^{th} derivatives need to exist and be continuous in the closed interval [x, x+h] , while the n+1^{th} derivative needs to exist and be continuous in the open interval (x, x+h).

Reference: Taylor Series Revisited

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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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