04-19-2015, 04:38 AM
Post: #35
 bshoring Member Posts: 266 Joined: Dec 2013
RE: HP-80
(04-18-2015 02:19 PM)Thomas Klemm Wrote:
(04-16-2015 10:04 PM)bshoring Wrote:  I believe for solving for i (interest rate) which must use Newton's Method, the single storage register is also used.

The algorithm used to calculate i when n, PMT and PV are given can be found in APPENDIX A of A Pocket-Sized Answer Machine for Business and Finance.

Quote:For a variety of reasons the best technique to try is the Newton-Raphson one and this was used for the annuity functions.

There's also flow chart in Fig. 3 but the initial value i0 has a typo: the number 2 should be in the numerator:
$i_0 = \frac{2(n-P)}{n(n+1)}$

A similar flow-chart can be found in Fig. 32 of United States Patent 3863060 on page 25 where it is correct.

I was wondering how they arrived at that initial guess:
Quote:First we sum the first three terms of the binomial expansion for f(i) and solve for i.

It turns out they just used the Taylor series as can be seen using Wolfram|Alpha:

Series[P-(1-(1 + r)^(-n))/r, {r, 0, 1}]

$$(P-n)+\frac{1}{2}n(n+1)r+O(r^2)$$

Solve[(P-n)+1/2 n (n+1) r, r]

$$r=\frac{2n-2P}{n^2+n}$$ and $$n(n+1)\neq0$$

Cheers
Thomas

Thanks, Thomas. You provided a lot of good information and the two links you referenced are worth reading for anyone interested in the HP-80 or any of the financial calculators that followed. Also, in case anyone missed it, there is a great article by Max Stone at:
with a lot of good stuff.

The HP-80 really did set the standard for all the financial calculators. The basic operation is pretty much incorporated in the legendary HP-12C, though with many improvements. I am still amazed at how much functionality HP managed to pack into this instrument back in 1973, with such limited ROM and RAM, and no independent financial registers.

Even though the machine has no keys for logarithms & anti-logs, there are some clever ways to get both common and natural logs using the financial keys, outlined in "Application Notes" entitled "Calculating Logs, anti-logs and roots of numbers." If anyone's interested, I can give specifics.

Also the Trend Line function is actually quite effective, if X-Values are evenly spaced, as you can only input the Y values, as the X-Values are implied as 1,2,3,etc. But once you do the compute function, you can predict a Y value for any X value, including fractional values. So there is more there than meets the eye.

Really a great, and well thought out machine !

Bob
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