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You're Getting Warmer
Newton's Method

 

 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 ...
 

 X2 - 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 X1.
 
 The equation for this line can be found using the old algebra equation ...
 

 Y - F(X1) = m(X - X1)

 
 but the slope (m) is also the derivative,
 so we can write ...
 

 Y - F(X1) = F'(X1)(X - X1)

 
 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 X2
 is used to create an even better guess (call it X3)
 

 
 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

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