41C/CV root finders

05202015, 07:07 PM
Post: #1




41C/CV root finders
I threw together a TVM program  mostly a conversion of the 32S one I did recently, with some local alpha label and flag 22 tricks to make it imitate a 12C  and I'm at the point where I need a numeric solver to compute i%.
I've played around with integrating two different solver routines: Equation Solver for the HP41C/CV/CX Root Finder for the HP41C/CV/CX (From the Standard Applications module.) Both of them seem to deteriorate in various ways if the initial guesses aren't very good, given how rapidly the TVM equation starts to grow as i% increases, along with regions where certain subexpressions can be undefined. The first one seemed not too bad if supplied with fairly accurate initial guesses. Does anybody have any recommendations for good root finders, or techniques for obtaining initial guesses for i%? I know the Advantage module has a general purpose solver that's based on the 15C, but it's also got a TVM solver, which would eliminate the point in writing my own anyway. 

05202015, 07:36 PM
Post: #2




RE: 41C/CV root finders
(05202015 07:07 PM)Dave Britten Wrote: Does anybody have any recommendations for good root finders, or techniques for obtaining initial guesses for i%? This post in a recent thread might give you the answer. Cheers Thomas 

05202015, 07:44 PM
Post: #3




RE: 41C/CV root finders
(05202015 07:07 PM)Dave Britten Wrote: Does anybody have any recommendations for good root finders, or techniques for obtaining initial guesses for i%? Welcome back to the wonderful world of TVM solvers. ;) This subject has been discussed here numerous times, including some basic facts that most probably will not change: there is no simple way of getting a decent guess for all possible combinations of the four other TVM variables. Their signs, and even their values, influence the way a first guess for the interest rate may be determined. Last time this subject has been discussed in December 2014: have look at this thread, especially post #7 and #14 ff. That thread also includes a link to the old forum where the problems of solving for the interest rate has been discussed in detail, even with graphs showing different scenarios and their special problems. Here is a link. I fear things haven't changed much since then. ;) Dieter 

05202015, 08:08 PM
Post: #4




RE: 41C/CV root finders
(05202015 07:36 PM)Thomas Klemm Wrote:(05202015 07:07 PM)Dave Britten Wrote: Does anybody have any recommendations for good root finders, or techniques for obtaining initial guesses for i%? Thanks Thomas, that linked journal article looks useful. I'll pore over it more closely this evening. Whatever they were doing on the HP80 must have been designed specifically for a slow, spaceconstrained machine, so that would be a great start. (05202015 07:44 PM)Dieter Wrote:(05202015 07:07 PM)Dave Britten Wrote: Does anybody have any recommendations for good root finders, or techniques for obtaining initial guesses for i%? Yeah, I think I was reading some of those threads last night while playing around with the calculation. It's the e^(ln(1+i%)*n) parts that seem to grow rapidly, so I tried solving just those for i%, coming up with e^(x/n)1=i%, then used x=1 and x=10 to get starting values for i%. The "Susan problem" converged very rapidly, but solving for i% in my own mortgage caused it to oscillate wildly around the real interest rate over a range of about two orders of magnitude. It converged eventually, but took an unusually long time. But I think both of the solvers I tried were based on the secant method. Maybe NewtonRaphson performs better (its usage in the HP80 seems to suggest so). On a side note, it's rather strange that the 41C has exotic functionality like ln1+X, e^x1, mod, and extensive alpha capability, but lacks a root finder, present on earlier models like the 34C. Not to mention the lack of combinatorics and hyperbolic trig (good thing it already has e^x1, since I don't know a good way to calculate it without sinh). 

05202015, 08:40 PM
(This post was last modified: 05202015 08:41 PM by Massimo Gnerucci.)
Post: #5




RE: 41C/CV root finders
(05202015 08:08 PM)Dave Britten Wrote: On a side note, it's rather strange that the 41C has exotic functionality like ln1+X, e^x1, mod, and extensive alpha capability, but lacks a root finder, present on earlier models like the 34C. Not to mention the lack of combinatorics and hyperbolic trig (good thing it already has e^x1, since I don't know a good way to calculate it without sinh). The nice thing with the 41 is that you don't need to have everything on board: if you want something exotic just key in the program from a Users' Library Solutions book or, better, plug in the required module. ;) Greetings, Massimo +×÷ ↔ left is right and right is wrong 

05202015, 08:49 PM
Post: #6




RE: 41C/CV root finders
(05202015 08:40 PM)Massimo Gnerucci Wrote:(05202015 08:08 PM)Dave Britten Wrote: On a side note, it's rather strange that the 41C has exotic functionality like ln1+X, e^x1, mod, and extensive alpha capability, but lacks a root finder, present on earlier models like the 34C. Not to mention the lack of combinatorics and hyperbolic trig (good thing it already has e^x1, since I don't know a good way to calculate it without sinh). Yup, that's the big advantage. If it didn't have the extensive memory and expandability, I probably wouldn't give it a second thought, and just stay with my 32S. It's still odd that some of those basics were omitted, though. Linear regression is missing as well. I wonder if they had to keep ROM size down to allow for enough RAM at a reasonable price point. 

05212015, 11:12 AM
(This post was last modified: 05212015 11:16 AM by Didier Lachieze.)
Post: #7




RE: 41C/CV root finders
(05202015 08:08 PM)Dave Britten Wrote: On a side note, it's rather strange that the 41C has exotic functionality like ln1+X, e^x1, mod, and extensive alpha capability, but lacks a root finder, present on earlier models like the 34C. Well, while being the last HP calculator with red LED display, the 34C was introduced after the 41C, not before: it was introduced along with the 33C and the 38C on August 15, 1979, one month after the 41C introduction on July 16, 1979 (source: PPC Journal V6N4 & V6N5). 

05212015, 11:40 AM
(This post was last modified: 05212015 11:50 AM by Dieter.)
Post: #8




RE: 41C/CV root finders
(05202015 07:07 PM)Dave Britten Wrote: I've played around with integrating two different solver routines: FTR: both programs use a regula falsi method. The first one uses the simple standard method, the second one seems to implement a modified version, the Illinois method. The latter should converge to a solution within an interval if (!) the two initial guesses bracket the solution, i.e. f(x1) and f(x2) have opposite signs. Further modifications can be found in the German Wikipedia article. (05202015 07:07 PM)Dave Britten Wrote: ...given how rapidly the TVM equation starts to grow as i% increases, along with regions where certain subexpressions can be undefined. That's what I already said in earlier discussions: there are various ways to write the TVM equation. Some are more suitable for solving, others less so. This point has even been discussed in the HP Journal article on the thennew HP92 (cf. Oct 1977 issue, p. 24/25). Here the Newton method was chosen, along with an initial guess that is claimed to be "correct to five places over all ranges of PV, FV, PMT, and i, and with n as large as 10^{8}". Which sounds truly amazing. Dieter 

05212015, 12:51 PM
Post: #9




RE: 41C/CV root finders
I knocked out a simple lazy Newton's method solver (lazy in that it approximates the 1st derivative using f(1.000001x) and f(.999999x)), and it seems functional but definitely suboptimal. Here's the code for it, if you want to have a chuckle at my background obviously not being in numeric methods.
Code:
LBL 09 is where it would check for extrema or discontinuities if I had implemented that part yet. I fed it numbers similar to my mortgage loan, and it obtained i%, but it wasn't exactly fast. Evaluating the function three times for every iteration really takes its toll on speed. I'll have to study the HP80 article more so as to have a better approximation of its algorithm. There's also a 19C/29C i% solver that uses Newton's method that's on the museum DVD (part of the general application book, I think), and I might be able to use some of that. I don't even necessarily need a general purpose solver, just one that's suited for handling the TVM equation. 

05212015, 07:36 PM
Post: #10




RE: 41C/CV root finders
(05212015 12:51 PM)Dave Britten Wrote: I knocked out a simple lazy Newton's method solver (lazy in that it approximates the 1st derivative using f(1.000001x) and f(.999999x)), and it seems functional but definitely suboptimal. Well.... ;) (05212015 12:51 PM)Dave Britten Wrote: I'll have to study the HP80 article more so as to have a better approximation of its algorithm. I would not recommend the HP80 as a reference. It used a different TVM paradigm with unsigned variables that was changed with the HP92, which essentially used the method that was later implemented in many classic HP financial calculators. (05212015 12:51 PM)Dave Britten Wrote: I don't even necessarily need a general purpose solver, just one that's suited for handling the TVM equation. I already suggested the method in post #7 of the older thread linked in my previous post. Here is a quick and dirty implementation of that idea. Try it and see what you get. Code: Registers: This is essentially untested code, so beware. ;) But it seems to work. Try the standard example: 10 STO 01 (n) 50 STO 03 (PV) –30 STO 04 (PMT) 400 STO 05 (FV) XEQ"I%" => 14,4359 The program evaluates a first guess of 9,8361% and finally returns 14,4359% after five iterations. With flag 01 set (BEGIN) the result is 10,2051%. Dieter 

05222015, 08:34 AM
(This post was last modified: 05222015 08:42 AM by Dieter.)
Post: #11




RE: 41C/CV root finders
(05212015 07:36 PM)Dieter Wrote: I already suggested the method in post #7 of the older thread linked in my previous post. Here is a quick and dirty implementation of that idea. Try it and see what you get. FTR, here is another version with a different definition of the TVM equation. It is shorter, runs faster and may require slightly less iterations here and there. Maybe you want to try this as well. Lines 01...29 are the same as before, here is the new code from line 30: Code: .. ... The standard problem was solved with just four iterations (instead of five before) in about 10 seconds. Dieter 

05222015, 01:14 PM
(This post was last modified: 05262015 05:09 AM by Ángel Martin.)
Post: #12




RE: 41C/CV root finders
(05202015 07:07 PM)Dave Britten Wrote: Does anybody have any recommendations for good root finders, or techniques for obtaining initial guesses for i%? Here's the formula I used in the TVM Module for the first derivative (as it uses Newton's method): f ’ (i) = (PMT / i^2 ) * [ (1+i)^(n)  1 ] + n * [PMT (1 + ip)/i – FV ] * (1+i)^(n+1) with initial value: i0 = [ abs( PV + n*PMT + FV) ]^1/n Seems to hold up quite well but again this is a MCODE implementation using the internal 13digit precision routines from the OS. Note: edited to add the missing exponent signs. Cheers, AM 

05222015, 07:07 PM
(This post was last modified: 05232015 06:39 PM by Dieter.)
Post: #13




RE: 41C/CV root finders
(05222015 01:14 PM)Ángel Martin Wrote: Here's the formula I used in the TVM Module for the first derivative (as it uses Newton's method): Hmmm... looks like there are some characters missing. ;) How did you arrive at the formula for the inital guess? For the record, here are the formulas I used for the posted 41C programs. In all cases, k=0 (END) resp. 1 (BEGIN). The initial guess is calculated as follows: Code: s = 2*k  1 The first program uses a TVM formula close to the definition in the 12C manual. It is evaluated this way: Code: q = 1 + i The second, improved version uses the following formulas: Code: q = 1 + i This way slow operations like powers and logs are reduced to a minimum. Of course the actual implementations use e^{x}–1 and ln(1+x) for better accuracy. Addendum: Here's another way of calculating a first guess for the interest rate. Again, it's simply the first approximation using Newton's method starting at i=0. But this time based on the second TVM definition: Code: s = 2*k  1 For the test case this leads to an initial estimate of 17,647% (END) resp. 13,044% (BEGIN) while the final solutions are 14,436% resp. 10,205%. Dieter 

05242015, 07:10 PM
Post: #14




RE: 41C/CV root finders
(05222015 01:14 PM)Ángel Martin Wrote: Seems to hold up quite well but again thit is a MCODE implementation using the internal 13digit precision routines from the OS. Ángel, how does your MCODE version behave if there is no solution? Does it loop forever, did you provide an iteration counter, or what will happen? And – please excuse my dumb question – is MCODE interruptable by the user? You may try the standard example (n=10, PV=50, PMT=–30, FV=400) with n=8 or less. In this case no solution exists. Some time ago I tried to handle such cases in an extensive 34s program that was able to detect whether a solution exists or not (well, at least in most cases ;)). This included finding a local extreme of the TVM function and checking for sign changes, so that even multiple solutions could be found. But covering at least the most common cases is not trivial at all, cf. earlier discussions in the old forum. Dieter 

05252015, 05:35 AM
(This post was last modified: 05252015 05:35 AM by Ángel Martin.)
Post: #15




RE: 41C/CV root finders
(05242015 07:10 PM)Dieter Wrote: Ángel, how does your MCODE version behave if there is no solution? Does it loop forever, did you provide an iteration counter, or what will happen? And – please excuse my dumb question – is MCODE interruptable by the user? Yes it is interruptible (the code checks for a key press event) but no, there is no counter due to available space restriction (it is also in the SandMath, which was very crowded already). So the example above just keeps iterating until the user stops it. Not perfect, but then... who said "the better is the enemy of the good"? As to whether there are missing terms in the formula for the derivative, well I hope not; barring a sinister transcription error that's the expression used in the code. I'll double check my notes. Cheers, 'AM 

05252015, 07:19 PM
(This post was last modified: 05252015 07:26 PM by Dieter.)
Post: #16




RE: 41C/CV root finders
(05252015 05:35 AM)Ángel Martin Wrote: Yes it is interruptible (the code checks for a key press event) but no, there is no counter due to available space restriction (it is also in the SandMath, which was very crowded already). So the example above just keeps iterating until the user stops it. Not perfect, but then... who said "the better is the enemy of the good"? Voltaire? (05252015 05:35 AM)Ángel Martin Wrote: As to whether there are missing terms in the formula for the derivative, well I hope not; barring a sinister transcription error that's the expression used in the code. I'll double check my notes. I think there are some power symbols "^" missing. Both in the derivative and in the initial guess. I assume the latter should read [abs(PV + n*PMT + FV)]^{1/n}. For the sample case in this thread this yields an initial i of 1,65 or 165%. ?!? Also the value can never be negative. That's why I asked how you developed this formula. BTW if space is an issue there is a very simply solution: simply set the inital i to a value very close to zero, e.g. 10^{–4}. A value of exactly zero will not work because the standard TVM formulas divide by i, but a value very close to this would return (approximately) the estimate I posted  right after the first iteration. So all you lose is a bit of execution speed – i.e. the difference between the time required for one iteration loop and the calculation of the initial estimate. ;) Dieter 

05262015, 05:06 AM
(This post was last modified: 05262015 05:09 AM by Ángel Martin.)
Post: #17




RE: 41C/CV root finders
(05252015 07:19 PM)Dieter Wrote: I think there are some power symbols "^" missing. Both in the derivative and in the initial guess. You're right  sorry I didn't notice those before, The exponent signs were not handled well in the copypaste action; I have corrected the expressions in the initial post now. The initial guess just look intuitively appropriate to me, I confess I didn't derive it from any algorithm or deductive process... but it just looked right  yes I know, math heresy but sometimes one must follow the hunch even in math. (05252015 07:19 PM)Dieter Wrote: BTW if space is an issue there is a very simply solution: simply set the inital i to a value very close to zero, e.g. 10^{–4}. A value of exactly zero will not work because the standard TVM formulas divide by i, but a value very close to this would return (approximately) the estimate I posted  right after the first iteration. So all you lose is a bit of execution speed – i.e. the difference between the time required for one iteration loop and the calculation of the initial estimate. ;) I fail to see how reducing the initial guess to a nearzero value will solve the problem when there is not a solution to be found...which is the only problem left. Again, the initial guess I use works rather well for all other *solvable* cases, it's sweet and short to obtain and doesn't require too many iterations to converge. The SandMath manual has some further details in case it helps. Thanks Dieter for the suggestions. 'AM 

05262015, 05:39 PM
(This post was last modified: 05262015 05:40 PM by Dieter.)
Post: #18




RE: 41C/CV root finders
(05262015 05:06 AM)Ángel Martin Wrote: The exponent signs were not handled well in the copypaste action; I have corrected the expressions in the initial post now. OK. But back to the initial guess: does it really evaluate to 165% for the sample case, or did I understand something wrong? (05262015 05:06 AM)Ángel Martin Wrote: I fail to see how reducing the initial guess to a nearzero value will solve the problem when there is not a solution to be found...which is the only problem left. No, of course an estimate close to zero will not solve cases where no (real) solution exists. But it saves memory space at the cost of merely one more iteration. (05262015 05:06 AM)Ángel Martin Wrote: Again, the initial guess I use works rather well for all other *solvable* cases, it's sweet and short to obtain and doesn't require too many iterations to converge. Please forgive me if I get back to this point once again, but I simply want to know: How many iterations are required for the example posted in this thread until the result (14,436%) is returned? Does the algorithm actually return the lower one of the two solutions (the other one being 53,172%), although the first estimate is as high as 165%? Since PV+n*PMT+FV usually is > 1, the estimate will (almost) always be > 100%. And finally... may I ask this...? Would you mind trying your algorithm with a first estimate of 1E–4? How does it behave? Dieter 

05262015, 09:26 PM
Post: #19




RE: 41C/CV root finders
(05262015 05:39 PM)Dieter Wrote: OK. But back to the initial guess: does it really evaluate to 165% for the sample case, or did I understand something wrong? Not sure I follow, I didn't try the example before  but I just ran it an here are the results: 10 = (n) 50 = (PV) –30 =(PMT) 400 = (FV) I=14.43587035% in end mode, and I=10.20509732% in begin mode you can also run it yourself on V41 of course... (05262015 05:39 PM)Dieter Wrote: How many iterations are required for the example posted in this thread until the result (14,436%) is returned? Does the algorithm actually return the lower one of the two solutions (the other one being 53,172%), although the first estimate is as high as 165%? Since PV+n*PMT+FV usually is > 1, the estimate will (almost) always be > 100%. I'd need to put breakpoints in the code to read the initial guess and the number of iterations, let me work on that and I'll get back to you tomorrow. Cheers, 'AM 

05262015, 10:08 PM
Post: #20




RE: 41C/CV root finders
(05262015 09:26 PM)Ángel Martin Wrote: Not sure I follow, I didn't try the example before  but I just ran it an here are the results: That's close to the exact results (14,43587133% and 10,20509807%). I still wonder why an estimate > 100% leads to these results instead of the second, larger solutions. (05262015 09:26 PM)Ángel Martin Wrote: you can also run it yourself on V41 of course... I could do so if I was more familiar with V41. ;) (05262015 09:26 PM)Ángel Martin Wrote: I'd need to put breakpoints in the code to read the initial guess and the number of iterations, let me work on that and I'll get back to you tomorrow. Thank you very much. :) Dieter 

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