December 06, 2017 Wednesday
Bedtime Story
Divided Difference for the polynomial
The process of divided difference is a way
of handling polynomials.
Let us consider some abstract points on an
x, y graph.
We are considering here k + 1 data points.
So the points on the graph will be labeled
as following:
(x0, y0), (x1,
y1)…(xk, yk)
The divided differences can be either
forward or backward.
To simplify things for you (and for me), I
will not use the variable k but simply take 3 concrete date points on our x, y
graph.
(x1, y1), (x2,
y2) and (x3, y3)
In this case the forward difference of 5 would
be:
[y3] := y3
[y3,…,y3+5] :=
([y3+1,…y3+5] – [y3,…,y3+5-1])/(x3+5
– x3)
Obviously all this sounds like mumbo jumbo
and that is very understandable since most of us are illiterate when it comes
to the language of mathematics.
But it was mathematics like this that was
the basis of difference engine.
The actual mathematics that was deployed in
the difference engine goes by the name of Newton’s method of divided
differences.
Yes, it is named so because Isaac Newton
contributed to this method of calculating the coefficients of polynomials.
Let us try with an example if we can make a
sense of this divided difference with an example.
We can take a simple quadratic polynomial.
p(x) = 2x2 – 3x + 2
The goal would be to get the value of p(0),
p(1), p(2) and so on.
Now we shall construct a table that will
have four columns.
The first one will have the serial values
of variable x.
The second will have the value of
polynomial p(x).
The third will have the differences of the
two left columns to its left.
The fourth one will have the differences of
two adjacent numbers of the third column.
You will get it easily once you study the
column.
x
p(x) diff1(x)= diff2(x)=
(p(x+1) – p(x)) diff1(x+1) – diff1(x)
0
2 -1 4
1
1 3 4
2
4 7 4
3
11 11 4
4
22
Stay tuned to the
voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is Professor
Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I recommend
this large collection of Halloween Songs for Kids:
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