SandMath routine of the week: Inverse Gamma Function - Printable Version +- HP Forums (https://archived.hpcalc.org/museumforum) +-- Forum: HP Museum Forums (https://archived.hpcalc.org/museumforum/forum-1.html) +--- Forum: Old HP Forum Archives (https://archived.hpcalc.org/museumforum/forum-2.html) +--- Thread: SandMath routine of the week: Inverse Gamma Function (/thread-240692.html) SandMath routine of the week: Inverse Gamma Function - Ángel Martin - 03-15-2013 As a belated celebration of pi-day, here's a short routine to calculate the Inverse Gamma function (positive arguments only) - Simply using Newton's method and the SandMath, it doesn't get any shorter - and on the CL it quickly converges after a few iterations... worth seeing it in action! ```01 LBL "IGMMA" 02 STO 01 03 LN initial guess 04 STO 00 05 LBL 00 06 RCL 01 07 RCL 00 08 GAMMA 09 / 10 CHS 11 1 12 + 13 RCL 00 14 PSI 15 / 16 ST- 00 17 VIEW 00 18 ABS 19 1 E-8 20 X 0,4428773962 Check: GAMMA -> 2.000000001 Cheers, ÁM Note: it's a rather crude first-pass, go ahead and improve on it if you're interested... Edited: 15 Mar 2013, 10:48 a.m. Re: SandMath routine of the week: Inverse Gamma Function - Walter B - 03-15-2013 Buenas tardes, Ángel, what do you use PSI for in your function set? TIA for enlightenment. d:-? Re: SandMath routine of the week: Inverse Gamma Function - Ángel Martin - 03-15-2013 The above example shows one usage. PSI is a handy trick to obtain the derivative of Gamma, as well as being very useful in probability problems. Re: SandMath routine of the week: Inverse Gamma Function - Gerson W. Barbosa - 03-15-2013 Very nice short routine! Here is the HP-50g version: ```%%HP: T(3)A(D)F(.); \<< DUP LN DO DUP2 GAMMA / NEG 1. + OVER Psi / - LASTARG DROP OVER UNTIL == END NIP \>> 2. --> 0.442877396485 --> 2. However, GAMMA overflows for x=3 and x=4, for instance. This will work for x > 0.886, but it's not a small program anymore: %%HP: T(3)A(D)F(.); \<< DUP LN OVER INV 1. UNROT 3. \->ARRY OVER 1.E12 < { [ 3.793 .4539 -1.952 ] } { [ 12.84 .2618 -388400000000. ] } IFTE DOT DO DUP2 GAMMA / NEG 1. + OVER Psi / - LASTARG DROP OVER - ABS UNTIL .0000000001 < END NIP \>> x INVGAMMA(x) GAMMA(INVGAMMA(x)) t(s) 1/2*sqrt(pi) 1.50000000008 0.886226925455 0.51 1. 2.00000000001 1. 0.41 2. 3. 2. 0.37 4. 3.66403279721 4.000000000002 0.40 120. 6. 120. 0.31 87178291200. 15. 87178291200. 0.34 1.26416966261E12 15.9876543210 1.26416966261E12 0.89 1.E60 48.3499572940 9.99999999872E59 0.46 7.77777777777E200 121.991689534 7.77777777648E200 3.42 ``` Cheers, Gerson. Re: SandMath routine of the week: Inverse Gamma Function - Paul Dale - 03-15-2013 We should have included PSI in the 34S too... - Pauli Re: SandMath routine of the week: Inverse Gamma Function - Ángel Martin - 03-16-2013 Glad it piqued your curiosity, Gerson. Interestingly enough the SandMath implementation does converge for x=3 and x=4, albeit it takes a long time. It's somewhat of a miracle because the estimations become negative for a while and we all know what happens to Gamma in the negative plane. I guess there must be a clever way to tweak the calculation to reduce the number of iterations, perhaps keeping the action in the positive plane. In a google search I found almost no references to the InverseGamma function, so I guess it's totally uninteresting (which makes it attractive to my eyes :-) Best,'AM Edited: 16 Mar 2013, 6:21 a.m. Re: SandMath routine of the week: Inverse Gamma Function - Walter B - 03-16-2013 Something left for the 43S ;-) Anyway it's included as a library function AFAICS. d:-) Re: SandMath routine of the week: Inverse Gamma Function - Paul Dale - 03-16-2013 Yes, it is in the standard library. It was also coded as an internal function but it got left out :-( The 43S will definitely have this one. A native inverse gamma is tempting too :-) - Pauli Re: SandMath routine of the week: Inverse Gamma Function - Gerson W. Barbosa - 03-16-2013 ¡Hola, Ángel! Quote: In a google search I found almost no references to the InverseGamma function, References to InverseGamma are indeed hard to find. I think I had already stumbled onto your first link last year when I was trying to implement this function. The best I was able to to gave only three digits. The method you have presented, using Psi, is the best I've ever seen. Thanks again for posting! Here is a shorter and faster RPL version. The number of iterations is 6 or 7 for most arguments. It will work in the range 1 < x <= 1.34830732701E488. Best regards, Gerson. ```InvGamma: %%HP: T(3)A(D)F(.); \<< DUP LN DUPDUP \v/ 2.21 * OVER .16 * + .194 + / / DUP 2. < { DROP 2. } IFT DO DUP2 GAMMA / NEG 1. + OVER Psi / - LASTARG DROP OVER - ABS UNTIL .0000000001 < END NIP \>> x InvGamma(x) Gamma(InvGamma(x)) t(s) 1.00000000001 2.00000000001 1.00000000001 0.07 1.2 2.34593737867 1.2 0.43 1.5 2.66276634532 1.5 0.51 2. 3. 2. 0.53 4. 3.66403279721 4.00000000002 0.53 24. 5. 24. 0.40 120. 6. 120. 0.35 12345678. 11.5153468559 12345678.0005 0.35 87178291200. 15. 87178291200. 0.33 1.26416966261E12 15.9876543210 1.26416966261E12 0.35 2.E30 29.5598564460 2.00000000026E30 0.34 3.E50 42.4792187916 1.99999999970E50 0.31 1.E60 48.3499572940 9.99999999872E59 0.29 5.E70 54.6170625140 4.99999999915E70 0.34 6.E100 71.3783728942 6.00000000063E100 0.51 9.91677934871E149 97. 9.91677934871E149 0.66 7.77777777777E200 121.991689534 7.77777777648E200 0.41 8.E300 168.326353359 7.99999999942E300 0.56 1.E357 193.196868201 9.99999998531E356 0.83 9.E400 212.260312687 9.00000000902E400 1.09 1.23456789012E450 233.199708617 1.23456789318E450 1.47 1.34830732701E488 249.172968355 1.34830732373E488 1.79 ``` Re: SandMath routine of the week: Inverse Gamma Function - Ángel Martin - 03-16-2013 Hi Gerson, I really have trouble to read RPL, can you provide the RPN equivalent for the calculation of the initial guess? Looks like a substantial improvement to the initial idea! This is what I figured out: ```01 LBL "IGMMA" 02 STO 01 03 ENTER^ 04 ENTER^ 05 LN 06 ENTER^ 07 2.21 08 * 09 1/X 10 ,16 11 * 12 + 13 ,194 14 + 15 / 16 / 17 STO 00 18 LBL 00 19 RCL 01 20 RCL 00 21 GAMMA 22 / 23 CHS 24 1 25 + 26 RCL 00 27 PSI 28 / 29 ST- 00 30 VIEW 00 31 ABS 32 1E-8 33 XY 20 STO 00 21 LBL 00 ... ``` Cheers, Gerson. ---------------- (*) The following is part of a curve fitting made with help of an old version of DataFit. ``` Model Definition: Y = x/(a+b*x+c*sqr(x)) Number of observations = 172 Number of missing observations = 0 Solver type: Nonlinear Nonlinear iteration limit = 250 Diverging nonlinear iteration limit =10 Number of nonlinear iterations performed = 23 Residual tolerance = 0,0000000001 Sum of Residuals = -1,97982012453611E-02 Average Residual = -1,1510582119396E-04 Residual Sum of Squares (Absolute) = 0,071461272245453 Residual Sum of Squares (Relative) = 0,071461272245453 Standard Error of the Estimate = 2,05632625029686E-02 Coefficient of Multiple Determination (R^2) = 0,9994754674 Proportion of Variance Explained = 99,94754674% Adjusted coefficient of multiple determination (Ra^2) = 0,99946926 Durbin-Watson statistic = 0,181006272327966 Regression Variable Results Variable Value Standard Error t-ratio Prob(t) a 0,194180061078165 6,75779102290044E-02 2,873425065 0,00458 b 0,160140574611016 5,79167114529964E-04 276,5014977 0,0 c 2,20860298743044 1,42205819324635E-02 155,3103099 0,0 ------------------------------------------------------------------------------------------------------------------------------------- X Value Y Value Calc Y Residual % Error Abs Residual Min Residual Max Residual 1 0 0 0 0 0 0 -0,09225268066 0,09044327801 2 0,6931471804 0,2310490602 0,3233017409 -0,09225268066 -39,92774547 0,09225268066 3 1,791759469 0,4479398673 0,5212429458 -0,07330307854 -16,36449084 0,07330307854 4 3,17805383 0,635610766 0,6848643811 -0,0492536151 -7,749021529 0,0492536151 5 4,787491743 0,7979152904 0,8263771459 -0,02846185547 -3,567027204 0,02846185547 6 6,579251212 0,9398930303 0,9517404203 -0,01184739003 -1,260504084 0,01184739003 7 8,525161361 1,06564517 1,064573069 0,001072101061 0,10060582 0,001072101061 ... 167 691,1834011 4,114186911 4,091157824 0,02302908714 0,5597481991 0,02302908714 168 696,3073651 4,120161924 4,096382192 0,02377973199 0,5771552776 0,02377973199 169 701,4372638 4,126101552 4,101568196 0,02453335547 0,5945892306 0,02453335547 170 706,5730622 4,132006212 4,106716318 0,02528989421 0,6120487945 0,02528989421 171 857,9336698 4,289668349 4,241416568 0,04825178142 1,124837109 0,04825178142 172 1128,523771 4,514095084 4,423651805 0,09044327801 2,003574943 0,09044327801 ------------------------------------------------------------------------------------------------------------------------------------- ``` In the X column, ln(InvGamma(x)); in the Y column, ln(InvGamma(x))/x. This was the best fit out of 57 models. Edited: 16 Mar 2013, 9:52 p.m. Re: SandMath routine of the week: Inverse Gamma Function - Ángel Martin - 03-17-2013 Hi Gerson, many thanks for the RPL help and the additional details on the data fit - yes, that's a good idea that has prompted me to look for other methods to get the initial estimation, with the goal to reduce the number of iterations even further (albeit your method is already very good, perfect for fast machines like the 50G and the CL :-) I thought about some value based on the Stirling approximation, and that took me straight to the Lambert function. Now I can see how things fit in the first URL, so I'm using David Cantrell's formula as initial guess. This works for x greater than x0=1.461632, the zero of PSI (aka the local minimum of Gamma for x>0). I've added a branch to deal with xY 2 05 X>Y? X<>Y 06 GTO 02 X<=Y? 07 LN SF 01 08 X=0? ENTER^ 09 1 LN 10 STO 00 FS? 01 11 GTO 00 GTO 02 12 LBL 02 ENTER^ 13 ,036534 SQRT 14 + 2.21 15 PI * 16 ST+ X RCL Y 17 SQRT ,16 18 / * 19 LN + 20 ENTER^ ,194 21 ENTER^ + 22 1 / 23 E^X / 24 / LBL 02 25 WL0 X=0? 26 / 1 27 ,5 STO 00 28 + LBL 00 29 STO 00 RCL 01 30 LBL 00 RCL 00 31 RCL 01 GAMMA 32 RCL 00 / 33 GAMMA CHS 34 / 1 35 CHS + 36 1 RCL 00 37 + PSI 38 RCL 00 / 39 PSI ST- 00 40 / VIEW 00 41 ST- 00 ABS 42 VIEW 00 1E-8 43 ABS X 1, returning values > 2), since Psi(x) get closer to ln(x) as x grows. Thus, an HP-42S version is possible. There will be occasional differences of one unit it the last significant digits for arguments < 30 or so. There is a couple of unnecessary operations in the RPL program above (also in the RPN version). Fix below. Cheers, Gerson. ---------------------------------------------------------- ```InvGamma: %%HP: T(3)A(D)F(.); \<< DUP LN DUP \v/ 2.21 * SWAP .16 * + .194 + DUP 2. < { DROP 2. } IFT DO DUP2 GAMMA / NEG 1. + OVER Psi / - LASTARG DROP OVER - ABS UNTIL .0000000001 < END NIP \>> ``` ```01 LBL "IGMMA" 02 STO 01 03 ENTER^ 04 LN 05 ,16 06 X<>Y 07 * 08 LASTX 09 SQRT 10 2,21 11 * 12 + 13 ,194 14 + 15 2 16 XY 18 STO 00 19 LBL 00 ... HP-42S version: 00 {65-Byte Prgm } 19 LBL 00 01 LBL "IGMM" 20 RCL 01 02 STO 01 21 RCL 00 03 ENTER 22 GAMMA 04 LN 23 / 05 0,16 24 +/- 06 X<>Y 25 1 07 * 26 + 08 LASTX 27 RCL 00 09 SQRT 28 LN 10 2,21 29 / 11 * 30 STO- 00 12 + 31 ABS 13 0,194 32 1E-10 14 + 33 XY 36 END 18 STO 00 2 XEQ IGGM --> 3 ( ~6 s) 5040 XEQ IGGM --> 8 ( ~3 s) 248 GAMMA --> 2.09409007702E485 XEQ IGGM --> 248 ( ~12 s) ``` P.S.: This HP-42S version uses only the stack. There are no noticeable differences in the running times however. The conversion to the wp34s should be straightforward. ```00 {74-Byte Prgm } 19 Rv 01 LBL "IGMM" 20 Rv 02 ENTER 21 STO ST Z 03 LN 22 X<>Y 04 SQRT 23 STO ST T 05 0.16 24 GAMMA 06 X<>Y 25 / 07 * 26 +/- 08 X<> ST L 27 1 09 2,21 28 + 10 RCL+ ST L 29 RCL ST Z 11 * 30 LN 12 0,194 31 / 13 + 32 STO- ST Z 14 2 33 ABS 15 XY 35 X ST T 36 GTO 00 18 LBL 00 37 R^ 38 END ``` -------------------- Edited to remove my comment about your use of Lambert W. Somehow I imagined it was inside of loop. I think I hadn't woken up yet :-) Edited: 18 Mar 2013, 12:52 a.m. Re: SandMath routine of the week: Inverse Gamma Function - Ángel Martin - 03-18-2013 Very nice Gerson, I think we've nailed it - save the "left" branch of course.To be continued... :) I went ahead and timed the execution times for both versions, see the table below. Using the more complex initial estimation yields consistently shorter times as you can see. Times in seconds.Never mind the actual values (on V41, standard speed) but the deltas tell the story: ```x D. Contrell DataFit 1 2.370024 2.9339976 1.5 15,4800000 17,6000040 2 17,96998 17.219989 2.5 11.85998 17.469972 3 10.98 17.66 3.5 10.36008 15.289992 4 10.47996 14.72004 4.5 10.179972 15.17004 5 10.110024 14.7900024 10 9.34992 14.230008 15 8.740008 13.86 20 9.36 14.349996 ``` Cheers, ÁM Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-18-2013 Hi Ángel, Significantly faster! Better use the curve fit equation only in programs for calculators that lack the Lambert W then. Here is a WP 34S version. It's very for arguments near 1, probably because I am using LN instead of Psi. Lambert W, Gamma, pi and e are all there, only Psi is missing. (Psi is available in a library, but I have yet to read the WP 34S Blue Book and find out how to use it in a program). Cheers, Gerson. ```WP 34S version: 001 LBL A 017 # 1/2 002 # pi 018 + 003 # 086 019 <> XZZX 004 / 020 GAMMA 005 RCL+ Y 021 / 006 # pi 022 +/- 007 STO+ X 023 INC X 008 SQRT 024 RCL Z 009 / 025 LN 010 LN 026 / 011 ENTER^ 027 y<> T 012 ENTER^ 028 - 013 # eE 029 CNVG? 00 014 / 030 RTN 015 Wp 031 BACK 012 016 / 032 END 1 A --> 2.000000000000012 ( 5.5 s) 2 A --> 3.000000000000002 (3.2 s) 5040 A --> 7.999999999999998 (1.8 s) 205 GAMMA --> 1.326057243693621E384 A --> 205 ( 1.0 s) x time(s) 0.89 20.6 0.90 12.5 1.00 5.5 1.50 3.4 2.00 3.2 2.50 2.7 3.00 2.8 3.50 2.8 4.00 2.8 4.50 2.8 5.00 2.9 10.00 2.2 15.00 2.2 20.00 2.3 100.00 2.2 500.00 2.0 5000.00 1.8 1.0E006 2.0 1.0E020 1.5 1.0E050 1.3 1.0E100 1.0 1.0E200 1.3 1.0E350 1.1 ``` Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Paul Dale - 03-18-2013 Quote:Psi is available in a library, but I have yet to read the WP 34S Blue Book and find out how to use it in a program. ``` XEQ'[PSI]' ``` This can be entered via the keyboard and the g-shift Greek letters or via CAT and navigating there and pressing XEQ. - Pauli Edited: 19 Mar 2013, 2:27 a.m. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Paul Dale - 03-18-2013 Nice use of some of the 34S's new features by the way. Can I add this routine to the library? - Pauli Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-18-2013 Of course! Again, thanks Ángel for presenting us the method :-) Gerson. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Ángel Martin - 03-19-2013 Gerson you're a gentleman - my pleasure of course, and thanks to you for the improvements. Re: SandMath routine of the week: Inverse Gamma Function - peacecalc - 03-21-2013 Hello all, nevertheless it is a interesting algorithm, but the hp50g can solve the problem with it's own built in function "ROOT", a very nice and short RPL proggie: ``` \<< \-> W @Input y-value \<< 'tX' GAMMA W = @Equation which should be solved 'tX' 2 ROOT @Command for solving with initial value 2 'tX' PURGE @Delete temp. variable \>> \>> ``` Interesting feature: If you use a 1 as a initial guess, you get the negative solutions. Greetings peacecalc Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-22-2013 Compatible with SSIZE8 and one step shorter: ```+ LBL'[GAMMA][^-1]' + # [pi] + # 86 + / + RCL+ Y + # [pi] + STO+ X + SQRT + / + LN + [<->] XXYY + # eE + / + W[sub-p] + / + # 1/2 + + + LBL 00 + [<->] XZZX + GAMMA + / + +/- + INC X + RCL Z + LN + / + [<->] XZZY + - + CNVG? 00 + RTN + GTO 00 + END ``` I guess [<->] XZZY takes longer than y[<->] T, but I haven't noticed any difference when the argument is 0.89 (21 second with a chronometer, about the same time I obtained with TICK in the previous version). DBLON would require the CVNG?02, but I don't see a way to do this without extra instructions inside to loop. Simply changing the parameter to 02 would cause endless loop for certain arguments. Gerson. Re: SandMath routine of the week: Inverse Gamma Function - Gerson W. Barbosa - 03-22-2013 Hi, Interesting technique for using the numeric solver in a program. Although the generic built-in solver can be used to compute this and other functions, like Lambert's W, for instance, a specific solver will usually be faster. Cheers, Gerson. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Paul Dale - 03-22-2013 Gerson, thanks for this update. I suspect there isn't much time difference between the shuffle command and the swap. The other overheads of interpreting instructions will likely far exceed the time to do either. What about using CNVG? 03. This picks CNVG? 00 in single precision mode and CNVG? 02 in double. - Pauli Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-23-2013 Thanks! I'd found this information in page 91 of the Blue Book: `"3 = Choose the best for the mode set, resulting in taking 0 for SP and 2 for DP (see Appendix H)."` This is plain clear, but the reference to Appendix H somehow diverted me. I was expecting to find more information about CNVG?, but Walter was meaning more information about SP and DP, I realize now. So, let it be CNVG? 03 then. Gerson. Re: SandMath routine of the week: Inverse Gamma Function - peacecalc - 03-24-2013 Hello Gershon, ```Original from Gerson W. Barbosa: ... a specific solver will usually be faster ``` Of course that is right, but it is easy to use (without a great bunch of knowledge in numeric analysis). And I've stated an interesting behavior of my "brutal force" method: With huge arguments my 50g needs let's say some seconds for the solution, and with small arguments in the scale of 10, it is very fast compared to the values of post #15 that posted Gershon. Certainley the duration of computation depends on the displaying mode because I use the built in function "ROOT". Greetings peaceglue Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-24-2013 Thanks! As I suspected, the use of LN instead of Psi is responsible for the overall slowness, especially for arguments in the beginning of the range. Calling the library digamma program would take up only one step, but it doesn't preserve the stack. This would have to be taken care of, which would require extra steps. Since Psi need not be exact in this application, I decided to use an approximation: ```Psi(x) ~ ln(x) - 1/(2*x) For the sake of crediting, the first character in Ángel's name is unreadable in the documentation, perhaps because of the diacritic. Would 7 extra steps compensate for the faster execution? Gerson. [pre] ---------------------------------------------- LBL'[GAMMA][^-1]' # [pi] # 86 / RCL+ Y # [pi] STO+ X SQRT / LN [<->] XXYY # eE / W[sub-p] / # 1/2 + LBL 00 [<->] XZZX GAMMA / +/- INC X # 1/2 [<->] TXYZ / RCL L LN STO- Y x[<->] L [<->] YZTX +/- / [<->] XZZY - CNVG? 03 RTN GTO 00 END ---------------------------------------------- x time(s) first updated version version 0.89 20.6 4.4 0.90 12.5 3.5 1.00 5.5 2.3 1.50 3.4 1.7 2.00 3.2 1.6 2.50 2.7 1.4 3.00 2.8 1.5 3.50 2.8 1.5 4.00 2.8 1.5 4.50 2.8 1.5 5.00 2.9 1.4 10.00 2.2 1.3 15.00 2.2 1.4 20.00 2.3 1.4 100.00 2.2 1.3 500.00 2.0 1.3 5000.00 1.8 1.2 1.0E006 2.0 1.4 1.0E020 1.5 1.1 1.0E050 1.3 1.1 1.0E100 1.0 0.9 1.0E200 1.3 1.1 1.0E350 1.1 0.9 ---------------------------------------------- ``` Edited: 25 Mar 2013, 12:20 a.m. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Paul Dale - 03-25-2013 Quote:For the sake of crediting, the first character in Ángel's name is unreadable in the documentation, perhaps because of the diacritic. The diacritic is gone now. Quote:Would 7 extra steps compensate for the faster execution? In a library routine, I think so. If anyone objects, they can always key in the earlier version or delete the routine completely. - Pauli Re: SandMath routine of the week: Inverse Gamma Function - Gerson W. Barbosa - 03-25-2013 Here is another HP-42S version. Psi(x) is approximated as ln(x) - 1/(2*x) instead of simply ln(x) to speed execution up, especially for small arguments. ```00 {86-Byte Prgm } 23 STO ST Z 01 LBL "InvGam" 24 GAMMA 02 ENTER 25 / 03 LN 26 +/- 04 SQRT 27 1 05 0.16 28 + 06 X<>Y 29 R^ 07 * 30 RCL+ ST T 08 X<> ST L 31 1/x 09 2.21 32 RCL ST T 10 RCL+ ST L 33 LN 11 * 34 STO- ST Y 12 0.194 35 X<> ST L 13 + 36 Rv 14 2 37 +/- 15 XY 39 STO- ST Z 17 X<> ST T 40 ABS 18 LBL 00 41 1E-9 19 Rv 42 XY 45 END x InvGam(x) GAMMA(InvGam(x)) t(s) 1 2 1 1.0 1.2 2.34593737867 1.2 4.1 1.5 2.66276634533 1.50000000001 4.2 2 3 2 4.4 4 3.66403279721 4.00000000002 4.2 24 5 24 3.2 120 6 120 3.1 5040 8 5040 2.2 12345678 11.5153468559 12345678.0005 3.1 87178291200 15 87178291200 3.0 1.26416966261E12 15.9876543210 1.26416966261E12 3.1 2.E30 29.5598564460 2.00000000026E30 3.1 2.E50 42.4792187916 1.99999999970E50 2.1 1.E60 48.3499572940 9.99999999872E59 2.5 5.E70 54.6170625140 4.99999999915E70 2.9 6.E100 71.3783728942 6.00000000063E100 4.2 9.91677934871E149 97. 9.91677934871E149 5.9 7.77777777777E200 121.991689534 7.77777777648E200 3.5 8.E300 168.326353359 7.99999999942E300 4.8 1.E357 193.196868201 9.99999998531E356 6.9 9.E400 212.260312687 9.00000000901E400 9.0 1.23456789012E450 233.199708617 1.23456789318E450 11.9 1.34830732701E488 249.172968355 1.34830732373E488 14.6 ``` Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-28-2013 Saving two instructions now, so five extra steps actually: ``` LBL'[GAMMA][^-1]' # [pi] # 86 / RCL+ Y # [pi] STO+ X SQRT / LN [<->] XXYY # eE / W[sub-p] / # 1/2 + LBL 00 [<->] XZZX GAMMA / +/- INC X RCL T LN RCL* L STO+ X DEC X RCL/ T / STO+ X [<->] XZZY - CNVG? 03 RTN GTO 00 END Running time comparion among versions from the older to the more recent one, measured with TICKS x t (s) 0.89 20.6 4.4 4.5 0.90 12.5 3.5 3.6 1.00 5.5 2.3 2.3 1.50 3.4 1.7 1.7 2.00 3.2 1.6 1.7 2.50 2.7 1.4 1.4 3.00 2.8 1.5 1.5 3.50 2.8 1.5 1.5 4.00 2.8 1.5 1.5 4.50 2.8 1.5 1.5 5.00 2.9 1.4 1.5 10.00 2.2 1.3 1.3 15.00 2.2 1.4 1.4 20.00 2.3 1.4 1.5 100.00 2.2 1.3 1.3 500.00 2.0 1.3 1.3 5000.00 1.8 1.2 1.3 1.0E006 2.0 1.4 1.4 1.0E020 1.5 1.1 1.2 1.0E050 1.3 1.1 1.1 1.0E100 1.0 0.9 0.9 1.0E200 1.3 1.1 1.0 1.0E350 1.1 0.9 1.0 ``` Gerson. Edited: 28 Mar 2013, 7:19 p.m. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gene Wright - 03-28-2013 Fabulous work, Gerson! Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-29-2013 Not quite! I completely forgot to test it in SSIZE8 mode. When I finally did it this afternoon, I noticed it was taking much longer than when in SSIZE4 mode. Fortunately the fix is very easy: just replace the only RCL T instruction with RCL Z: ``` ... +/- INC X RCL Z LN RCL* L ... ``` Occasional stack-size-wise incompatibilities are a side effect of having two stack sizes. I only change to SSIZE8 when doing many calculations involving complex number (which lately I have no need to). I would prefer a parallel four-level stack for doing this, though, as in the HP-15C. But there's a reason this couldn't be implemented on the wp34s, so the 8-level stack for such situations is a blessing (but only in this case, IMHO). I wonder what would be the consequences of user-settable stack size, as some have suggested... A really fabulous work was yours, adapting matrices routines to work on the wp34s! Regards, Gerson. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Walter B - 03-29-2013 Obrigado, Gerson, por a sua programa! Quote: I only change to SSIZE8 when doing many calculations involving complex number (which lately I have no need to). I would prefer a parallel four-level stack for doing this, though, as in the HP-15C. But there's a reason this couldn't be implemented on the wp34s, so the 8-level stack for such situations is a blessing (but only in this case, IMHO). All a matter of habitude: I run on SSIZE8 all the time. I love RPN calculating free of stack-overflow worries. d:-) Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-30-2013 Quote: Obrigado, Gerson, por a sua programa! De nada! Just a couple of corrections: por + a = pela. But is this case it should be "pelo seu programa" (por + o = pelo) because "programa" is a masculine noun. The gender of words ending in -a are most always feminine, as you know, but many nouns ending in -a derived from Greek are masculine, like programa, drama, diagrama, anagrama, for instance. Quote: All a matter of habitude: I run on SSIZE8 all the time. Yes, I know there is a number of wp34s user who prefer the 8-level stack. That's why I realized all library programs should ideally run under both stack sizes, without changing the user setting. It would be easier just to save the user settings in the beginning of the program, set the proper stack size and restore the user settings when the program returns, but I don't know whether this is possible. At least I haven't found a way to do it (but I haven't read all the book yet). Regards, Gerson. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Paul Dale - 03-30-2013 Making a program behave like a built in command is fairly easy. It is much easier in XROM but that isn't available to users. ``` LBL'ABC' // Allocate local variables, save the stack and the current mode settings LocR 09 STOS .01 STOM .00 // Do what ever you want to the machine's state and produce a result in X . . . // Recall the system stack, move the result into the saved stack and set up Last X RCLM .00 x<> .01 STO L RCLS 01 END ``` And despite Walter's comment an eight level stack can still run out, especially when programming :-) - Pauli Edited: 30 Mar 2013, 2:35 a.m. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Walter B - 03-30-2013 Pauli, Quote: And despite Walter's comment an eight level stack can still run out, especially when programming If you want to sabotage a system you can always, of course ;-) As stated above, I was talking about calculating not programming. Some minimum intelligence given, you cannot overload an eight-level stack in real calculations IMHO. d:-) Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-30-2013 Thanks! I would use this technique in complex and long programs. Preserving the stack only is a nice bonus. However, in a relatively short program like Gamma-1 seven extra steps might be too many. Quote: And despite Walter's comment an eight level stack can still run out, especially when programming :-) I don't remember ever getting out of stack when doing calculations, but then again mine usually are not complicated. The 8-level stack should be hand for calculating this one, though:-) Gerson. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Walter B - 03-30-2013 Quote: The 8-level stack should be hand for calculating this one, though:-) Hmmmh - let me see: ```input x y z t 350 ENTER 350 350 661.5 661.5 350 / x^2 .2 0.2 [...]^2 * 1 + (...) 3.5 y^x 1 - [...] = factor 1 25500 ENTER 25500 25500 [f1] 6.875e-6 6.875e-6 25500 [f1] * ... [f1] 1 x<>y ... 1 [f1] - [..] [f1] -5.2656 y^x [..]^(-5.3) [f1] * {...} 1 + (...) 0.286 y^x 1 - [...] 5 * SQRT ``` Unless I'm mistaken, three levels are sufficient here. d:-? Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-30-2013 Quote: Unless I'm mistaken, three levels are sufficient here. Indeed. Your result is correct: `0.835724531752514` I remember there is an expression somewhere in the archives which is not solvable within four stack registers, unless intermediate results are stored in a numbered register, but I cannot find it right now. This example is from one of many discussions about RPN versus AOS:Edited: 30 Mar 2013, 6:16 p.m. Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Thomas Klemm - 03-30-2013 Quote: I remember there is an expression somewhere in the archives which is not solvable within four stack registers, unless intermediate results are stored in a numbered register, but I cannot find it right now. But there was an earlier thread: It Can Be Done! Kind regards Thomas Re: SandMath routine of the week: Inverse Gamma Function (WP 34S version) - Gerson W. Barbosa - 03-31-2013 Thanks, Thomas! Yes, that was the expression you presented in message #14 in the first link. Of course no problem with it on the WP-34S with SSIZE8 mode on. Your comment is very impressive: "Whichever solution you might choose you will always run into a stack-overflow and what's worse you probably won't even notice." Walter does have a point :-) Cheers, Gerson.