



Back
in ancient times (before about 1975) it was very valuable



to
be able to manually calculate the values of things like 


trig
functions, logs, and roots. 





Suppose
we wanted to calculate the square root of 8.25? 


How
would we do it? 





One
way would be to say that since 2
^{2} was 4 and 3
^{2} was 9, 


that
the answer should be somewhere between the two. 


So let's
guess 2.5. 


2.5
squared is 6.25. That's too low. Try 2.75. 


2.75
squared is 7.5625. That's closer, but still too low. 





You
can see that we might be here for a while 


to
get to say, 3 decimal point accuracy. 





There
is, however, a much slicker method. 





That
old math wizard Isaac Newton came up with a formula. 


What
we need to start with, 


is
an equation for our little square root problem (without any roots). 





OK,
how about X ^{2}
= 8.25 





Next
we need to put it in the form ... 





Stuff = 0






OK,
that would be ... 





X^{2}
 8.25 = 0






Nothing
tricky so far ... 





Next we
do our little guessing game. 


This time
we do it with a formula that gives us our next guess 


and zeros
in on the answer real fast. 


The
formula is called Newton's Method. 


It is ... 











In
words, this means: 


Take the
old guess and subtract a fraction made by dividing the old guess 


by the
value of the derivative of the old guess. 


The
answer you get is the new and closer guess. 


Just keep
putting the numbers through the formula 


until the
number coming out is about the same 


as the
number going in. 


The
really amazing thing, is that the first answer 


doesn't
even need to be close! 





So
let's make a guess at the square root of 8.25. 


Let's
also not make the guess too good. 


Just
so we can see what happens. 


Make
our first guess 2 ... 











Now
we use this to make a new guess ... 











Looks
like one more time might do it ... 











That
was fast! 





When
each calculation we do gets closer to one particular number 


math
types say it CONVERGES on that number. 


If
each new calculation gives us a much bigger or much smaller number, 


or
if the answer bounces all over the place 


math
types say it "diverges". 





Here
is where Newton's Method comes from ... 





The
graph of the equation we used in the example (F(X)
= X
^{2}  8.25) , 


looks
like this ... 











The
values we want to find are where F(X)
= 0. 


In
other words, X ^{2}
 8.25 = 0. 


A
line tangent to the equation at our first guess for X (X = 2), 


looks
like this ... 











call
this X guess X_{1}. 





The
equation for this line can be found using the old algebra equation
... 





Y
 F(X_{1})
= m(X  X_{1})






but
the slope (m) is also the derivative, 


so
we can write ... 





Y
 F(X_{1})
= F'(X_{1})(X
 X_{1})






Now
the X value we want to solve for 


is
the one that makes Y = 0, 


so
we set Y = 0 and solve for X ... 











Since
X1 was just a guess, X is just a guess too (just a better one). 


This
new X guess value (call it X_{2}) 


is
used to create an even better guess (call it X_{3}) 











In
general form (if it matters to you) is written ... 











The
reason this thing works is in the ... 











Since
we're looking for F(X)
= 0, 


and
since F(X)
is one power greater than F'(X), 


F(X)
gets small faster than F'(X). 


The
closer we get to the X value where F(X)
= 0, 


the more
microscopic the function fraction gets. 


As
it does, it keeps nudging the value to a better answer. 





Now
that we have $3 calculators that do roots, logs, and trig functions 


you
might think that all this Newton's Method stuff 


should
go to some museum. 


For
more than 90% of the stuff people used it for in the 1960's 


you'd
be right. 


But
there's still a few percent of the time 


where
stuff like this is the only game in town. 





copyright 2005 Bruce Kirkpatrick 
