To answer your questions (A little more about the problem).

Using Timer 1 on a 16C64 Micro, I wish to generate an acceleration profile.
This profile is governed by the equation:

N = 1/2 * a * t * t

which is the standard positional equation based on time.

I'm interested in obtaining the time that I have to wait before a whole unit
of movement (ie until N = N + 1).  This can be worked out to the equation
below:

dt = sqrt(2/a) * sqrt(N) - sqrt(N-1)

Where N is the position and N-1 is the previous position.

My current best uses Newton's method.  Newton's method involves an iterative
solution to the find the root of an equation.  If we want to set up a
standard newton's method to approximate the square root, we would set up an
equation similar to:

f(x) = x * x - r

the derivative of which is

f'(x) = x.

Newton's method will converge at the root of a function, and is a recursive
function as follows:

Let A be the current approximation and Ap be the previous approximation,

A = Ap - f'(Ap)/f(Ap)

If you plug in the above to equations this yeilds the iterative
approximation

A = (Ap + r/Ap) / 2

so A is the square root of r.

If I used the floating point math package, this would work, but I'd like to
accomplish this in under 380 instruction cycles if possible.

To eliminate floating point math, I decided to distribute the 32768
multiplier.  This results in the need to do integer math (division,
addition, and subtraction) on 40-bit (minimum) math.  I'm not sure whether I
can do this in my desired amount of instructions.

-----Original Message-----
From: Bob Buege <BobBuege@AOL.COM>
To: PICLIST@MITVMA.MIT.EDU <PICLIST@MITVMA.MIT.EDU>
Date: Wednesday, November 05, 1997 1:11 PM
Subject: Re: Algorithm Help


>In a message dated 97-11-05 14:01:04 EST, Keith Ballantyne wrote:
>
>> I'm currently working on a project using a PIC16C64.  I need to solve the
>>  following equation:
>>
>>  T = sqrt(2/A) * ( sqrt(N) - sqrt(N-1) )
>>
>>  I've seen a number of spiffy square root routines, mostly derived from
>>  newton's method, but some other, more innovative ones as well.
>>
>>  Any suggestions as to the most effecient way to do this.
>>
>>  - Keith
>>
>Could you please send a few details about this equation?
>
>Are the numbers 8 bits, 16 bits, or larger?
>Are A and N integers or fractions?
>Are they positive or do you need to consider imaginary numbers?
>What precision do you need for T?
>What is the range for A and N?
>Is there anything about the problem which will allow the substitution of an
>easier equation which approximates
>the actual equation for the range needed?
>How much memory do you have left in your PIC for this equation?
>Is speed, accuracy, or program space the major concern?
>
>Bob