I am writing a software function to solve for an interest rate by
interpolation. The high level requirement is to mimic the
functionality of the RATE() function in Microsoft Excel.

The details of which are found here:
http://support.microsoft.com/default.aspx?scid=kb;en-us;Q123757


There are 3 scenarios for which I need to solve for. The differences
lies in the available inputs to the formulas:
Nper:    is the total number of payment periods in an annuity.
Pmt :    is the payment made each period and cannot change over the
life of the annuity. Typically, pmt includes principal and interest
but no other fees or taxes. If pmt is omitted, you must include the fv
argument.
Pv:      is the present value? the total amount that a series of
future payments is worth now.
Fv:      is the future value, or a cash balance you want to attain
after the last payment is made. If fv is omitted, it is assumed to be
0 (the future value of a loan, for example, is 0).
Type:    is the number 0 or 1 and indicates when payments are due.

	Scenario 1	Scenario 2	Scenario 3
nper	60	        36	        24
pmt	482.92	        682.26	        0.00
PV	-25,130.15	-44,858.77	-28,840.33
FV	0.00	        26,997.00	31,795.45
Type	0	        1	        1

Scenario 1: Straight forward example of solving by basic
interpolation. I actually have something for this that works fairly
well. It needs to be more robust.
Scenario 2: Not so straightforward for me. It has a FV. 
Scenario 3: Murkier more still because it has no pmt.

I implemented what I found here:
http://mathforum.org/dr.math/faq/faq.interest.html#install
This helped with scenario 1 but it is not as robust of an example as I
need to solve all 3 completely.

I need formulas and algorithms to solve these scenarios by
interpolation taking all the inputs including Type.

I would like the solution to be explained as I see them at the
mathforum above or in a similar example. I need to translate them into
a program language so if you could provide an algorithm to accompany
the solution that would be huge.
I will start the request with 50 bucks. If more money is needed to
solve please give me notice. I don?t have time to research myself.

Thanks

An Engineering Economics text book will show you the calculations for
each of these scenerios.

For example: http://www.amazon.com/gp/product/0070238340/103-9004346-1556623?v=glance&n=283155&n=507846&s=books&v=glance

I think you may appreciate the solutions for each scenereo if you
study the math behind them.

I am a little rusty on this, and don't have my Engineering Economics
book on hand. But this is what I remember:

For Case 2 and 3, you need to take the FV and solve for PV (call this
PV2 for this explaination), which will be a function of the unknown
interest. Add the calculated PV2 to the given PV to get the net PV.
Then you can solve for the interest rate using the net PV value. (Case
3 is the same as Case 2... the difference is that Case 3 has the
payment (M) = 0.

Note: The numerical technique suggested on the DrMath link you
provided works for solving the equation, but I would consider another
numerical method that converges quicker and with more confidence.  I
am pretty sure most Engineering Economics text books cover better
numerical methods ...

Thanks, I have ordered that book and will review. I didn't notice that
the math form did not converge with high accuracy. Sometimes not at
all.

I meant I did notice... :)

Are you still interested in an answer?  I can provide one if you are
still interested.