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1. 2 3 1 Hyperbolic Functions Yes there are many unanswered questions in the universe but certainly one of them is why oh why didn t HP MotherGoose provide a decent set of hyperbolic functions in the otherwise pathetic MATH PAC and worse yet adding insult to injury how come that error wasn t corrected in the Advantage ROM For sure we ll never know so it s about time we move on and get on with our lives whilst correcting this forever and ever The first incarnation of these functions came in the AECROM module I believe programmed by Nelson C Crowle a real genius behind such ground breaking module but it was also somehow limited to 10 digit precision The versions in the SandMath all use internally13 digit routines __ Function Author Description _ O Z o gt Z o o H HSIN Angel Martin Hyperbolic Sine H HCOS_ Angel Martin ___ Hyperbolic Cosine _ _ _ _ _ o Z oS H HASIN Angel Martin _ Inverse Hyperbolic Sine H HACOS_ Angel Martin _ Inverse Hyperbolic Cosine _ __ o o lt The use of the function launcher permits convenient access to these six functions without having to assign them to any key in USER mode Efficient usage of the keyboard which can double up for other launchers or the standard USER mode assignment if that s also required Combining the XFL and the SHIFT keys does the trick in a clean and logical way The formulas used are well
2. Jalar cos am J__ x Kf TI als Tal sin a7 2 sin am Ya 2 These expressions are valid for any real number as order with the same issues as the first kind functions above when the order is integer To avoid the singularities and to reduce the calculation time the following expressions are used for integer orders mY x 2 y Ln x 2 Ja Z CD fk n x Zfg n x 2 Ka x C1 2 fy Ln x 2 CD Lf fi 0 x ZCI gx 0 x where y is the Euler Mascheroni constant 0 5772 and g n X x 2 n k 1 k 3 k 0 1 2 n 1 f 10 x x 2 H k H n k k n k 5 k 0 1 2 and H n is the Aarmonic number defined as H n X 1 k k 1 2 n Where Y 1 Yn and Kn x K x note that for x lt 0 Y n x and K n x are complex numbers c Angel M Martin Revision 44_E Page 58 SandMath_44 Manual The graphics below plot the Bessel functions of the second kind Y x and their modified K x for integer orders a 0 1 2 Note that KNBS and YNBS are FOCAL programs that use dedicated MCODE functions specially written for the calculations BS and BS2 Their entries are located in the sub functions FAT thus won t be shown in the main CAT listings in case you wonder about their whereabouts Getting Spherical are we The spherical Bessel functions jn and yn and are very closely related to the ordinary Besse
3. with special guests Riemann s ZETA and Lambert s W both branches Function Author Description D IBS Ange Martin _ Bessel I n x of thefirstkind pi JBS Angel Martin _ Bessel J n x of the firstkind ____ A5 KBS Angel Martin __ Bessel K n x of the second kind _____ _ SIBS Angel Martin __ Spherical Bessel i n x pi SJBS_ Angel Martin __ Spherical Bessel j n x Dig SYBS_ Angel Martin _ Spherical Bessel nx Dei WLo Angel Martin Lambert s W mainbranch _ _ WL1i Argel Martin _ Lambert s W secondary branch TAJ YBS Angel Martin _ Bessel Y n x of the second kind ____ _ _ TA ZETA Angel Martin _ Riemann s Zeta directmethod o ZETAX JM Baillard ______ Riemann s Zeta Borwein algorithm _ __ _ The SandMath Module includes a set of functions written with the harmonic analysis in mind specifically to facilitate the calculation of the Bessel functions in their more general sense for any real number for order and argument Bessel functions of the First kind I n x and J n x The formulae used are as follows ox _ 1 as r dm J SL z Xu miT m a 1 2 aa 1 op 2m4 a x i Jalir PETTE 9 Where denotes the Gamma function These expressions are valid for any real number as order although there are issues for negative integers due to the singularities in the poles of the gamma function
4. In fact that s exactly what the functions do in RUN mode upon completion the fraction is converted to a decimal number then D gt F presents the final output That s why the display settings determine the accuracy of the conversions even if it s not obviously seen This has the advantage that the result is always reduced to the best possible fit For instance when calculating 2 4 plus 18 24 in program mode with the four values in the stack the result will be 120 in Y and 96 in X thus 120 96 However on RUN mode or SST ing the program will show the reduced fraction 574 USER RAG Oo 2 4 If you want to see the reduced result from a program execution you ll need to add program steps to perform the division and add a conversion to fraction after the fraction math operation step The code Snippet below describes this see lines 10 and 11 01 LBL TEST 02 2 03 ENTER 04 4 05 ENTER 06 18 07 ENTER 08 24 09 F D gt F 12 END INT and FRC are two more test functions which criteria is the integer or fractional nature of the number in X Having them available comes very handy for decision branching in FOCAL programs The Fractions section of the module is the natural placement for them The answer is YES NO depending on whether the condition is true or false In program mode the following line is skipped it the test is false c ngel M Martin Revision 44_E Page 24 SandMath_44 Manual
5. l mly m Pressing A will calculate an estimation of the function for the argument in X using the fourier temrs calculated previously In this case X 5 XEQ A gt f x 23 254423 X 1 XEQ A gt f x 0 154639 which obviously misses the point Typically the functions used are related to the harmonic analysis though Here s an nteresting one the Christmas Tree function and its Fourier representation for different number of terms c ngel M Martin Revision 44_E Page 75 SandMath_44 Manual Appendix 10 Fourier Coefficients by brute force Since the coefficients are basically integrals of the functions combined with trigonometric functions nothing besides common sense stops us from using INTEG to calculate them This brute force approach is just a work around considering the time requirements for the execution but it can be useful to calcuate a single term randomly as opposed to the sequential approach used by FFOUR So here the idea is to calculate the n th Coefficient independently which responds to the following definig equation n TA a Notice that the module SIROM 7 Solve and Integrate ROM contains 2 RON not only FROOT and FINTG but i PROMPT zi _ also the program FOURN in its 7 APPLIED section so you can use peT that 4k rom instead of the Advantage ma that ll also save you from having to type i
6. 04 STO 02 16 RCL 00 05 1 17 RCL 02 06 STO 03 18 YX 07 CLX 19 08 STO 00 20 09 LBL 01 21 X Y 10 RCL 01 22 GTO 01 11 RCL 03 23 END 12 c ngel M Martin Revision 44_E Page 73 SandMath_44 Manual Fourier Series In mathematics a Fourier series decomposes periodic functions or periodic signals into the sum of a possibly infinite set of simple oscillating functions namely sines and cosines or complex exponentials The study of Fourier series is a branch of Fourier analysis The partial sums for f are trigonometric polynomials One expects that the functions XN f approximate the function f and that the approximation improves as N tends to infinity The infinite sum ag oo an cos nx bn sin nz 2 n l is called the Fourier series of f The Fourier series does not always converge and even when it does converge for a specific value x0 of x the sum of the series at x0 may differ from the value f x0 of the function It is one of the main questions in harmonic analysis to decide when Fourier series converge and when the sum is equal to the original function FFOUR Calculates the Fourier coefficients for a periodic function F x defined as ee l pee fd st A i o gt f Gn l f x cos A Ja x NXOR i f t f y l mx fy b tx sin ix aa a L J with the following characteristics centered in x x0 with period 2L on an interval x0 x0 2L with a give
7. 1 000000005 0 000100000 0 000100000 0 000100000 11013 23287 10 00000000 11013 23292 10 00000000 0 999999996 10 00271302 By now you ve become an expert in the HYP launcher and for sure appreciate its compactness lots of keystrokes With a couple of exceptions it s a100 accuracy and really the only sore point is in the point 0 001 for the HACOS But don t worry there s no bugs creating havoc here it s just the nature of the beast bound to occur with the limited precision used even 13 digits in the Coconut CPU No wonder you re going to repeat the same table for the trigonometric functions and see how it stacks up right While you re at it go ahead and calculate the power of two of the square root pressing Ex 9 7 2 j but don t call HP to report a bug For very small arguments the accuracy of SINH and COSH will also start to show incorrect digits However HTAN and HATAN use an enhanced formula that will hold the accuracy regardless of how small the argument is c Angel M Martin Revision 44_E Page 26 SandMath_44 Manual The SandMath Module includes a set of functions written to extend the native RCL functionality mainly in the direct math operations missing when compared to the STO equivalents but also increasing its versatility and ease of use There are five new RCL Math functions plus a launcher to access them in a convenient and useful way RCL __ A
8. 5 ENTER XEQ MFCT gt Sf 41 type 41 UFL SFCT gt Sf 100 type 100 EFL SFCT gt To complete this trinity of factorials Occasionally the hyperfactorial of n is considered It is written as H n and defined by H n JJ k 1 2 33 n 1 1 n k 1 The figures below show a plot for both the hyperfactorial and its logarithm itself a convenient scale change very useful to avoid numeric range problems Note that they re extended to all real arguments and not only the natural numbers also called the K function Hin Ini Hind T 100 3 B0 60 a 7 6 8 1 See below a couple of simple FOCAL program to calculate the hyperfactorial which runs beyond the numeric range dramatically soon and its logarithm written by JM Baillard Understandably slow and limited as these programs are you can visit his web site for a comprehensive treatment using dedicated MCODE functions for the many different possible cases 01 LBL HFCT 01 LBL LOGHF 02 1 02 0 03 LBL 01 03 LBL 01 04 RCL Y 04 RCL Y 05 ENTER 05 ENTER 06 Y X 06 LOG 07 07 08 DSE Y 08 09 GTO 01 09 DSE Y 10 END 10 GTO 01 11 END c ngel M Martin Revision 44_E Page 44 SandMath_44 Manual Logarithm Multi Factorial The product of all odd integers up to some odd positive integer n is often called the double factorial of n even though it only involves about half the factors of the
9. Fr w E Ti T Ten re a ee TAE Ke E Mi A im ae Mo me is N W ea a kan A mo il fare aA Jea i S je 1 a uf T i a a az m ee ee b tn Wav ava avalehal etal chal ar af nn cf Dee va vas mah d i OG a OE pa af See va Nas E a LI a as i K ar a iris T a i oe i a par ni PE L ie ae te Fe ii Ps pei a e nal T Fi i a Y i Ls F ee et RT in FA 1E al Lt pi Fa e diy DF ps r Pa i VA F Fi rod Png 4 r 1i a AAU os Ma aye Wai Tin ai a a ms j SE h ue a at a ee ay ne oats a i i Leader e244 Moe PE a ai i i n a We 56 gt buriani i c ngel M Martin Revision 44_E Page 32 SandMath_44 Manual Combinations and Permutations two must have classics Nowadays would be unconceivable to release a calculator without this pair in the function set but back in 1979 when the 41 was designed things were a little different So here there are finally and for the record NPR calculates Permutations defined as the number of possible different arrangements of N different items taken in quantities of R items at a time No item occurs more than once in an arrangement and different orders of the same R items in an arrangement are counted separately The formula is n n k NCR calculates Combinations defined as the number of possible sets or N different items taken in quantities or R items at a time No item occurs more than once in a set and different orders of the same R items is a se
10. er 108 LBLO8 46 109 RCL 03 47 a0 2 a2 3 27 a1 a2 6 110 COS 48 iii 49 E a f 50 cbrt x R3 2 413 RCL 02 a2 3 C 51 f T 114 F 52 RCL 02 a2 3 f 115 X s H F 116 54 x1 117 OOOO 55 ARCLX 3 118 56 119 ARCL X 3 57 STO 00 real root 120 58 RCL 01 cbrt x R3 2 121 STO IND05 59 RCL 03 cbrt x R3 2 122 120 60 123 ST 03 61 3 1244 ISG05 02O 62 SQRT 125 GTO 08 O i 63 126 END c Angel M Martin Revision 44_E Page 21 SandMath_44 Manual 2 1 4 Additional Tests Rounded and otherwise Ending the first section we have the following additional test functions Function Author Description _ Z gt Z S X 1 Melson C Crowle Is X exactly equal to 1 Is X equal to or greater than Y Angel Martin Rounded Comparison FRC Angel Martin Is X a fractional number F Angel Martin Is X an integer number Angel Martin Is X equal to or greater than zero They follow the general rule returning YES NO in RUN mode and skipping a program line if false in a program Their criteria are self explanatory for the first three These functions come very handy to reduce program steps and improve the legibility of the FOCAL programs X gt Y compares the values in the X and Y registers skipping one line if false X gt 0 compares with zero the value in the X register skipping one line if false These functions are arguably missing on the mainframe set a fact partially correct
11. Alpha Integer Part ATAN2 Dual argument ATAN BS gt D Base to Dec CBRT Cubic Root CEIL Ceil function CHSYX CHSY by X CROOT Cubic Equation Roots CVIETA Driver for CROOT D gt BS Dec to Base D gt H Dec to Hex E3 E 1 00X FLOOR Floor Function GEU Euler s Constant H gt D Hex to Dec HMS HMS Multiply HMS HMS Divide LOGYX LOG b of X MANTXP Mantissa MKEYS Mass Key Assgn P gt R Complete P R QREM Quotient Reminder QROOT 2nd Degree Roots QROUT Outputs Roots R gt P Complete R P R gt S Rectangular to Spherical S gt R Spherical to Rectangular STLINE Straight Line from Stack T gt BS__ Dec to Base VMANT View Mantissa X 3 X 3 X 1 Is X 1 X Y R Is X Y rounded X gt 0 is X gt 0 X gt Y is X gt Y Y41 X Xth Root of Y YAAX Extended Y X YX Modified Y X FRC_ Fraction Math Launcher D gt F Decimal to Frac F Fraction Addition F Fraction Subtract F Fraction Multiply F Fraction Divide FRC is X fractional INT Is X Integer HYP_ Hyberbolics Launcher HACOS Hypebolic ACOS O CON DUM BWN FP OT num A aA A PHP PP BPP HPWBWWWWWWWwWWWNNNNNNNNNNPRP RP RP RP RP RP RP RP PRP PB e O O WON DUN PWN PRP CO HO WON DU BPWNPRP ODO HW WON DU PWNHRP DUO WON DU PWN FP OO Name Description LHL MATH _ Secton Header 1 GMF Reciprocal Gamma Cont Frc 2FL Main Function Launcher FL Launcher by Name Z FLH Launcher by index BETA Beta Function Cl Cosine Integral El Exponential Integral ELIPF Eliptic Integral 1st Kind ERF
12. Anger functions WEBAN In mathematics the Anger function introduced by C T Anger 1855 is a function defined as 1 iT J z cos v zsin d T The Weber function introduced by H F Weber 1879 is a closely related function defined by 1 jf E z sin v zsin d WT Jo If v is an integer then Anger functions Jv are the same as Bessel functions Jv and Weber functions can be expressed as finite linear combinations of Struve functions Hn and Ln With n and x in the stack WEBAN will return both J n x and E n x in the Y and X stack registers respectively The figures below show four of these functions for 4 orders 0 0 5 1 and 1 5 Anger on the left plots and Weber on the right Check J 0 0 1 and E 0 0 1 Note that WEBAN will return both values to the stack REGISTERS ROO thru R06 FLAGS none Stack Y Example 2 SQRT PI FL WEBAN gt E sqrt 2 x 0 315594385 X lt gt Y gt J sqrt 2 x 0 366086559 Or Dial H W c Angel M Martin Revision 44_E Page 81 SandMath_44 Manual 3 5 2 Hankel Struve and similar functions The second sub function launcher is the Hankel group It s loosely centered on the Hankel functions plus related sort The launcher prompt is activated by pressing H at the main XFL prompt and offers the following 14 choices in two line ups controlled by the SHIFT key Note the different leadings on each s
13. Digamma and the Bessel functions LOGYX is the base b Logarithm defined by the expression logio x E log 2 bgs gnl log 6 where the base b is expected to be in register Y and the argument in register X QREM Calculates the Remainder R and the Quotient Q of the Euclidean division between the numbers in the Y dividend and X divisor registers Q is returned to the Y registers and R is placed in the X register The general equation is Y Q X R where both Q and R are integers CBRT calculates the cubic root of a number Note that this is identical to the mainframe function X Y with Y 1 3 for positive values of X but unfortunately that results in DATA ERROR when X lt 0 and therefore the need for a new function Obviously CBRT x CBRT x for x gt 0 Y 1 X and X 3 are purely shortcut functions obviously equivalent to 1 X Y X and to X42 LASTx respectively but with additional precision due to the 13 digit intermediate calculations c Angel M Martin Revision 44_E Page 16 SandMath_44 Manual Y X is used to calculate powers exceeding the numeric range of the calculator simply returning the base in X and the exponent in Y The result is shown in ALPHA in RUN mode For instance calculate 8569 to obtain Pga4hbb tt E 133 USER RAL Hiz 4 YX is a modified form of the native Y X function with the only difference being its tolerance to the 0 0 case which results in DATA ERROR
14. Error Function FFOUR Fourier Series GAMMA Gamma Function Lanczos HCI Hyperbolic Cosine Integral HGF Generalized Hypergeometric Function HSI Hyperbolic Sine Integral IBS Bessel In Function ICBT Incomplete Beta Function ICGM Lower Incomplete Gamma Function JBS Bessel Jn Function KBS Bessel Kn Function LINX Polylogarithm LNGM Logarithm Gamma Function PSI Digamma Function PSIN Polygamma PP2 Point to Point Dist POCH Pochhammer Symbol SI Sine Integral SIBS Spherical Bessel SJBS Spherical J Bessel SYBS Spherical Y Bessel WLO Lambert W Function WL1 Lambert W Function YBS Bessel Yn ZETA Zeta Function Direct method ZETAX Zeta Function Borwein ZOUT Output Complex to ALPHA DECX Decrease X DECY Decrease Y INCX Increase X INCY Increase Y PRB STS _ T Percentual CORR Correlation Coefficient COV Sample Covariance DSP Display Digits EVEN is X Even GCD Greatest Common Divisor LCM Least Common Multiple LGMF Log Multi Factorial LR Linear Regression LRY LR Y value NCR Permutations c Angel M Martin Revision 44 _E Page 12 SandMath_44 Manual 52 53 54 55 56 57 58 59 60 61 62 63 HASIN HATAN HCOS HSIN HTAN RcL_ Extended Recall AIRCL RCL RCL RCL RCL RCL Hyperbolic ASIN Hyperbolic ATAN Hyperbolic COS Hyperbolic SIN Hyperbolic TAN ARCL Integer Part Recall Power Recall Add Recall Subtract Recall Multiply Recall Divide Functions in blue ar
15. Integral S Notable examples of multi purposed function are also the Carlson Integrals themselves a generator for several other functions like the Elliptic Integrals More about this in the corresponding paragraphs later on Exponential Integral and associates The first sub section covers the Exponential Logarithmic Trigonometric and Hyperbolic integrals They re all calculated using their expressions using the Generalized Hyper geometric function in a clear demonstration of the usefulness or the adopted approach For real nonzero values of x the exponential integral Ei x is defined as 4 e Rifas dt co t Integrating the Taylor series for exp t and extracting the logarithmic singularity we can derive the following series representation for real values g oo Fi a y In x X x 0 k k k 1 where we substitute the series by its Hyper Geometric representation Yt x k kk x F 1 1 2 2 x The logarithmic integral has an integral representation defined for all positive real numbers by the definite integral z dt o Int li The function li x is related to the exponential integral Ei x via the equation iia Elie j which is the one used to program it in the SandMath module c Angel M Martin Revision 44_E Page 69 SandMath_44 Manual The different trigonometric and hyperbolic integral definitions and their relations with the Hyper Geometric funcion for the rel
16. M Lambert s W SST Incomplete Gamma N FCAT sub functions CATalog O Carlson Launcher R Exponential integral S Error Function T Polygamma PsiN V Cosine Integral W Spherical Y n x X Incomplete Beta Z Sine Integral Spherical J n x Fractions Launcher R S View Mantissa ZB Cancels out to the OS ALPHA Sub function Launcher 0N Turns the calculator OFF A green H on the overlay prefixing the function name represents the Hyperbolic functions This also includes the Hyperbolic Sine and Cosine integrals in addition to the three standard ones Using the SHIFT key will toggle between the direct and inverse functions Pressing lt will take you back to the main XFL prompt The Fraction functions are encircled by a red line on the overlay at the bottom and left rows of the keyboard They include the fraction math plus a fraction Viewer and fraction Integer tests The Hankel and Carlson launchers will present their choices in their prompts and will be covered later in the manual Note that the RCL Math functions are also linked to the main launcher to invoke them use the Yell launcher sort of Hyper RCL thus need to press MA1 HYP to get the RCL _ _ prompt c Angel M Martin Revision 44_E Page 10 SandMath_44 Manual Appendix 1 Launcher Maps The figures below provide a better overview illustrating the hierarchy between laun
17. M Martin Revision 44_E Page 30 SandMath_44 Manual 3 Upper Page Functions in detail It s time now to move on to the second page within the SandMath holding the Special Functions and the Statistical and Probability groups Let s see first the Statistical section easier to handle and of much less extension and later on we ll move into high level math taking advantage of the extended launchers and additional functionality described in the introduction of this manual The following functions are in this general group Some of them are plain catch up with the aim to complete the set of basic functions Some others are a little more advanced reaching into the high level math as well _ Function Author s Description O Z OZ oO O DSP Angel Martin _ Number of decimal places EVEN Angel Martin _ Tests whether x is an even number _ __ o o GCD Angel Martin Greatest Common Divider LCM Angel Martin _ Least Common Multiple O O O SS MFCT_ Ballard _____ Multifactorial o O NCR_____ Angel Martin ____ Combinations of N elements taken in groups of R _ __ NPR ngelMartin Permutations of N elements taken in groups of R ODD ngel Martin Tests whether x in an odd number PDF ngelMartin Normal Probability Density Function Prime Factorization PRIME Jason DeLooze Primality Test finds one factor RAND Hakan Th rgren Random
18. ROM DF routine As per the fraction arithmetic functions there s not much to say about them apart from the fact that they use the four stack levels to enter both fractions components the inputted values are expected to be all integers and return the numerator and denominator of the result fraction in registers Y and X respectively In RUN mode the execution continues to show the fraction result in ALPHA according to the currently set number of decimals see below The fraction arithmetic functions can be used in chained calculations there s no need to re enter the intermediate results and the Stack enabled makes unnecessary to press ENTER Notice that fractions are entered using the Numerator first To re calculate the fraction after changing the decimal settings just press the divide key followed by D gt F to re generate the fraction values For example calculate 2 7 over 4 13 then add 9 17 to the result 2 ENTER 7 ENTER 4 ENTER 13 F 9 ENTER 17 F gt 347 238 in FIX 6 mode 34 238 USER RAG iz 4 c Angel M Martin Revision 44_E Page 23 SandMath_44 Manual Needless to say the fractional representation display will not be produced in PRGM mode but it ll have a silent execution instead Note that the fraction math functions operate on integer numbers in the stack returning also the numerator and denominator as integers To get the decimal number just execute to divide them
19. a numeric one which could be misinterpreted The indirect addressing is actually quite simple as it simply consists of an offset added to the register number in the index All the function code must do is remove it from the entry data provided by the OS and the task is done The offset value is hex 80 or 128 decimal We ll revisit this when discussing the RCL launcher And the third objective is provided for free by the OS as well no need for extra code at all just using the appropriate prompting bits in the function s name c Angel M Martin Revision 44_E Page 27 SandMath_44 Manual Stack arguments are more involved than the indirect addressing No attempt has been made to use the mainframe internal routines to accommodate this case so stack prompts are excluded Note that even if the Stack arguments are not directly allowed controlled by the prompting bits it is unfortunately possible to use the decimal key in an indirect register sequence that is after pressing the SHIFT key This won t work properly in the current design so must be avoided ORT RAND STO y 2 4 2 RCL Launcher the Total Rekall aay The basic idea of a launcher is a function capable of calling a set of other functions The grouping in this case will be for the five RCL Math functions described above plus logically the standard RCL operation inclusive its indirect registers addressing Other enhancements include the prompt l
20. e calculating the function value is reduced to a single MCODE function The following trivial looking program it really can t get any simpler uses SOLVE within the Advantage Pack no less Starting with zero obvious guess values are the previous root and the root incremented by one Successive repetitions will unearth all those roots just make sure you have the turbo mode enabled on V41 or equivalent emulator Enjoy The first few roots j n k of the Bessel function Jn x are given in the following table for small nonnegative integer values of n and k See also http cose math bas bg webMathematica webComputing BesselZeros jsp 1 LBL ITJBS 2 XY 3 STOO00 4 XoY 5 0 E X lt gt 7 JB 8 INTEG q RTN 10 LBL JZER 11 HLMATH 12 STOP 13 STO00 14 0 15 16 17 18 19 20 SOLVE 71 STOP 22 INCX 23 GTO 00 24 26 RCL OO 26 X lt Y 27 JBS 28 END Orders k dof Oy 1 24048 38317 2 55201 7 0156 3 86537 10 1735 4 11 7915 13 3237 5 14 9309 16 4706 Fix mH O 2 Ja x Fy x 5 1356 6 3602 64172 9 7670 11 6198 13 0152 14 7960 16 2235 17 9598 19 4094 Ja x 7 5883 11 0647 14 3725 17 6160 J x 6 7715 12 3386 15 7002 18 9801 20 8269 22 2178 PY I y A ey A A hte b4 Note that the program listing also includes code to calculate the Integral of JBS defined as incomplete function with the argument in the upper integrati
21. handle requiring quite a number of coefficients to be calculated for good accuracy result Moreover that calculation involves a lot of registers to store the values since there isn t any iterative approach based on recursion m O The expression below is definitely too inaccurate only three or four digits are correct to deserve a dedicated MCODE function Tr 1277 43697 348077 A isl O A aE 7 g 1l ne Ta ove z 12 480 10320 5806080 182476800 A paper from 1968 by A Strecok lists the first 200 coefficients of a power series that represents the inverse error function While using this approach it became clear that at least 30 of them are needed for a 10 digit accuracy for 0 lt x lt 0 85 This only gets worse as x approaches 1 getting into a clear example of the law of diminishing results A better method for the vicinity of 1 is probably to use an asymptotic expansion such as 2 2 log log los _ 74 y2 s og log J 1 4 oh A combination of both approaches would seem to be the best compromise depending on the argument Typing the 30 coefficients is not fun however thus the best is no doubt to use a data file in X Memory to keep them safe erf z x c Angel M Martin Revision 44_E Page 72 SandMath_44 Manual How many logarithms say again LINX calculates the polylogarithm function also known as Jonqui re s function a spec
22. implementation Also provision for the case x 0 was added trivially returning the character 0 for any base The prompt can be filled using the two top keys as shortcuts from 1 to 10 A J or the numeric keys 0 9 Ha e BASE 35 USER RAD Hic q4 USER RAD ji 4 Because of its importance the hexadecimal conversions have the dedicated MCODE functions D gt H and H gt D Use them to convert the number in X to its Hex value in Alpha and vice versa The maximum number allowed is 0x2540BE3FF or 9 99999999 E9 decimal much smaller than with T gt BS so there s a price to pay for convenience These functions were written by William Graham and published in PPCJ V12N6 p19 enhancing in turn the initial versions first published by Derek Amos in PPCCJ V12N1 p3 c ngel M Martin Revision 44_E Page 18 SandMath_44 Manual 2 1 3 First Second and Third degree Equations A MCODE implementation of these offers no doubt the ultimate solution even if it doesn t involve any high level math or sophisticated technique The Stack is used for the coefficients as input and for the roots as output No data registers are used Function Author Description STLINE _ Angel Martin _ Calculates straight line coefficients from two data points QROOT Angel Martin _ Calculates the two roots of the equation QROUT AngelMartin ___ Displays the rootsin Xand Y o o Z o CROOT Calculates the t
23. includes more general purpose functions re visiting the usual themes Fractions Base conversion Hyperbolic functions RCL Math extensions as well as simple but neat little gems to round off the page In sum all the usual suspects for a nice ride time The second page delves into deeper territory touching upon the special functions field and Probability Statistics Some functions are plain catch up for the 41 system sorely lacking in its native incarnation whilst others are a divertment into a tad more complex math realms All in all a mixed and matched collection that hopefully adds some value to the legacy of this superb machine for many of us the best one ever I am especially thankful for the essential contributions from Jean Marc Baillard more than 3 4ths of this module are directly attributable to his original programs one way or another Wherever possible the 13 digit OS routines have been used throughout the module ensuring the optimal use of the available resources to the MCODE programmer This prevents accuracy loss in intermediate calculations and thus more exact results For a limited precision CPU certainly per today s standards the Coconut chip still delivers a superb performance when treated nicely The module uses routines from the Page 4 Library a k a Library 4 Many routines in the library are general purpose system extensions but some of them are strictly math related as auxiliary code reposi
24. is always given for additional information that can and should also be consulted The Module starts with an assorted group of functions providing simple but important additions to the native function set 2 1 1 Elementary Math functions Even the most complex project has its basis simple enough but reliable so that it can be used as solid foundation for the more complex parts The following functions extend the HP 41 Math function set and many of them will be used either as MCODE subroutines or directly in FOCAL programs _ Function Author Description __ O O Z O 2 X 1_ Angel Martin __ Self descriptive faster and better precision than FOCAL Z1 N______ Angel Martin _ Harmonic Number H n ATAN2 AngelMartin _ Two argumentarctangent _ _ O o Z o 2 X 1 provides a more accurate result for smaller arguments than the FOCAL equivalents It will be used in the ZETAX program to calculate the Zeta function using the Borwein algorithm 1 N calculates the Harmonic number of the argument in X as is the sum of the reciprocals of the natural numbers lower and equal than n 1 k i It will be used to calculate the Bessel functions of the second kind K n x and Y n x Also related to the same problem and in general relevant to the summation of alternating series is the function CHSYX an extension of CHS but dependent of the number in Y Its expression is CHS Y X x 1 y retu
25. kind F is defined as P p da F y k F p k F sing k which in terms of the Carlson Symmetric form Re it results F k singAp cos 1 k sin D 1 c ngel M Martin Revision 44_E Page 77 SandMath_44 Manual ELIPF is implemented as a MCODE function which simply calls CRG with the appropriate input parameters All the heavy lifting is thus performed by CRG which together with CRJ do all the hard work in the calculation for the Elliptic Integrals if first second and third kinds The figure below shows the first and third kinds in comparison Elliptic Of the First Kind Elliptic Of the Third Kind 0 5 n 0 o 1 25 0 75 n 0 5 m 1 25 1 25 n 0 25 p n 2 p n 2 n 0 75 p n 2 ellint_1 k phi ellint_3 k n phi This is a perhaps a good moment to define the Carlson symmetric forms The Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced They are a modern alternative to the Legendre forms The Legendre forms may be expressed in terms of the Carlson forms and vice versa The Carlson Symmetric Elliptic integrals of the First and Third kinds are defined as 7 a Re 2z y z t L x t yit T z R3 x Y z p t p t a y z CRF and CRJ are the functions in the SandMath that calculate their values The arguments are expected to be in the stack r
26. of sets from a sample of 335 objects taken in quantities of 167 Type 335 ENTERN 167 XEQ NCR gt 3 0443587 99 Example How many different arrangements are possible of five pictures which can be hung on the wall three at a time Type 5 ENTER 3 XEQ NPR gt 60 00000000 The execution time for these functions may last several seconds depending on the magnitude of the inputs The display will show RUNNING during this time c Angel M Martin Revision 44_E Page 33 SandMath_44 Manual Linear Regression Let s not digress The following four functions deal with the Linear Regression the simplest type of the curve fitting approximations for a set of data points They complement the native set which basically consists of just MEAN and SDEV Function Author Description f z CORR IM Baillard Correlation Coefficient of an X Y sample Covariance of an X Y sample sx LR JM Baillard __ Linear Regression of an X Y sample Y value for an X point Linear regression is a statistical method for finding a straight line that best fits a set of two or more data pairs thus providing a relationship between two variables By the method of least squares LR will calculate the slope A and Y intercept B of the linear equation Y Ax B se 15 r 5 a 5 tat 10 i k 4 fie TO e da ra d as a ke a atta oe C a ie P a i The results are placed in Y and X reg
27. of the associated Legendre functions respectively Legendre polynomials are the associated Legendre functions of order O These functions may actually be defined for general complex parameters and argument 1 z AOD pie l z 2fi A A 1 1 u 5 for l z lt 2 The figures below give a couple of graphical plots for the Legendre Polynomials Associated Legendre Polynomials Pix Pii r i i z Paix Pix Pal REGISTERS ROO thru R05 FLAGS Stack Input Output Examples 0 4 ENTER 1 3 ENTER 0 7 XFL ALF gt P1 3 0 4 0 7 0 274932821 0 6 ENTER 1 7 ENTER 4 8 XFL ALF gt P1 7 0 6 4 8 10 67810281 Or Sig H SHIFT L instead c ngel M Martin Revision 44_E Page 90 SandMath_44 Manual Toronto Function In mathematics the Toronto function T m n r is a modification of the confluent hypergeometric function defined by Heatley 1943 as pen mt oT 2i s m gt T n 1 Which to untrained eyes just appears to be a twisted cocktail of the Kummer function adding the exponential tot he mix and scaling it with Gamma T m n T 1F m 3 n4 l r DATA REGISTERS ROO thru R04 Stack Input Output Flags none Barok Se ra 2 SQRT 3 SQRT PI XFL TMNR gt gt gt gt T sqrt 2 sqrt 3 n 0 963524225 Or Sig H SHIFT R instead Poisson Standard Distribution PSD is another Statistical function which calculates the Po
28. polling points SEEDT will take the fractional part of the number in X as seed for RNG storing it into the buffer If x 0 then a new seed will taken using the Time Module really the only real random source within the complete system RAND will compute a RNG using the current seed using the same popular algorithm described in the PPC ROM and incidentally also used in the CCD module s function RNG Both functions were written by Hakan Thorngren an old hand and MCODE expert and published in PPC V13N4 p20 PRIME Determines whether the number in the X register is Prime i e only divisible by itself and one If not t returns the smallest divisor found and stores the original number into the LASTX register PRIME Also acts as a test YES or NO are shown depending of the result in RUN mode When in a program the execution will skip one step if the result is false i e not a prime number enabling so the conditional branching options This gem of a function was written by Jason DeLooze and published in PPCCJ V11N7 p30 Example program The following routine shows the prime numbers starting with 3 and using diverse Sandbox Math functions 01 LBL PRIMES 05 PRIME 09 INCX 02 3 06 VIEW X lt yes gt 10 GTO 00 03 LBL 00 07 X Y lt no gt 11 END 04 RPLX 08 LASTX See other examples later in the manual relative to prime factorization programs Fa s gur Fa w pa dA D m eai i 7 n Tos ma Fi y i 1 F Wit if y
29. reset counter 69 MSUA 25 RDN 70 CLA 27 RCL 01 previous prime factor 72 28 X lt gt Y 73 29 PRIME 74 30 SF 00 75 31 X Y different PF 76 32 TO 02 YES 77 33 ISG 00 increase counter 78 34 NOP SAME pf 79 35 FS C 00 was it prime 80 36 GTO 03 skip if Prime 81 37 ST L 82 38 LASTX 83 ST 00 39 GTO 00 84 FC 10 40 85 mp 41 86 FC 10 42 87 GTO 06 43 88 RCL 00 44 f 89 PROMPT 45 STO 01 90 END c Angel M Martin Revision 44_E Page 39 SandMath_44 Manual Distance between two points The Euclidean distance between points p and q is the length of the line segment connecting them In the Euclidean plane if p p1 p2 and q q1 q2 then the distance is given by d p q vy pi q1 pe g2 PP2 expects the coordinates of the two points stored in the stack y1 x1 y2 x2 in T Z Y and X or vice versa The distance will be placed in X upon completion Example Calculate the distance between the points a 3 5 and b 6 2 from the figure below s Type 5 ENTER 3 ENTER 2 ENTER 6 ENTER XEQ PP2 gt 9899494937 E 3 5 Note A similar function exists in the 41Z module ZWDIST which basically calculates the same thing albeit done in a complex number context Note PP2 is the only geometry function within the SandMath The VECTOR ANALYSIS module contains many more as well as a full featured 3D Vector Calculator see overlay below It is a 4k module that can be
30. s reused by both modules so far Its hot keys and actions are listed below R S halts the enumeration SST BST moves the listing one function up down SHIFT sets the direction of the listing forwards backwards XEQ direct execution of the listed function or entered in a program line ENTER moves to the next previous section depending on SHIFT status lt back arrow cancels the catalog One limitation of the sub functions scheme that you ll soon realize is that contrary to the standard functions they cannot be assigned to a key for the USER keyboard Typing the full name or entering its index at the XFL prompt is always required This can become annoying is you want to repeatedly execute a given sub function A work around this consists of writing a micro FOCAL program with just the sub function as a single pair of program lines and then assign it to the key of choice Not perfect but it works Note Make sure the revision G or higher of the Library 4 module is installed c Angel M Martin Revision 44_E Page 14 SandMath_44 Manual 2 Lower Page Functions in detail The following sections of this document describe the usage and utilization of the functions included in the SandMath_44 Module While some are very intuitive to use others require a little elaboration as to their input parameters or control options which should be covered here Reference to the original author or publication
31. when the solution is a complex number like J 0 5 0 5 or I 0 5 0 5 There s no way around that save in some particular cases of the order You can always use the versions available in the 41Z Module for a full complex range OUT OF RANGE when the calculator numeric range is exceeded This typically occurs for large indexes during the power exponentiation step ALPHA DATA indicates alphabetic data in registers X or Y May also trigger DATA ERROR iP c Angel M Martin Revision 44_E Page 62 SandMath_44 Manual Appendix 7 FOCAL program used to calculate the Bessel Functions of the second kind As you can see it s just a simple driver of the MCODE functions with the additional task of orchestrating the logic for the different cases Note the usage of the sub functions from the auxiliary FAT as well as other SandMath functions 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 b a e a J O O O O O J o Ss a s O I O O O LBL KBS BL YBs g CF 01 LBL 00 X 0 RTN single case x 0 e HALFX x 2 swap things n CTO 02 ___ __ yes _divert to section _____ F CHS n X lt gt Y x 2 RAD SPFC Multi Function Launcher 1 Recurrence Sum BS CHS J n x STO 02 partial result RCL 01 n RCL 03 x 2 SPFC Multi Function Lau
32. with the standard function but here returns 1 This has practical applications in FOCAL programs where the all zero case is just to be ignored and not the cause for an error 2 1 2 Number Displaying and Coordinate Conversions A basic set of base conversions and diverse number displaying functions round up the elementary set Author _ Description MANTEXP AINT elegantly solves the classic dilemma to append an index value to ALPHA without its radix and decimal part eliminating the need for FIX 0 and CF 29 instructions taking extra steps and losing the original calculator settings Note that HP added AIP to the Advantage module and the CCD has ARCLI to do exactly the same MANTEXP and VMANT are related functions that deal with the mantissa and exponent parts of a number MANTEXP places the mantissa in X and the exponent in Y whereas VMANT shows the full mantissa for a few instants before returning to the normal display form or permanently if any key Is pressed and held during such time interval similar to the HP 42S implementation of SHOW R gt P and P gt R are modified versions of the mainframe functions R P and P R The difference lies in the convention used for the arguments in Polar form which here varies between 0 and 360 as opposed to the 180 180 convention in the mainframe Continuing with the coordinate conversion R gt S and S gt R can be used to change between rectangular and spherical coordinates The co
33. 0521E 10 4 17535E 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c Angel M Martin Revision 44_E SandMath_44 Manual 3 3 2 Reciprocal Gamma function y L igamma s oie 2 The reciprocal Gamma function is the function al fe l Ar 2 D z where z denotes the Gamma function Since the Gamma function is meromorphic and nonzero 4 everywhere in the complex plane its reciprocal is an entire function The reciprocal is sometimes used as a Starting point for numerical computation of the Gamma function and a few software libraries provide it separately from the regular E F ry F 4 Gamma function Plot of Tix along the real axis Taylor series expansion around 0 gives zp yzi fO ee eT eer 2 12 T z The SandMath however uses the expression based in continuous fractions according to which r x x x 1 2 sqrt 2r exp x 1 12 x 1 30 x 53 210 x 195 371 x Comparing the results obtained by GAMMA using Lanczos and continuous fractions it appears that the precision is generally better in the Lanczos case which also happens to be faster due to its polynomial like form and the absence of loops to adjust the result for smaller arguments Note the special case for x 0 which is not a pole for this function but it is a singularity for all the others that used the common subroutines therefore the dedicated check in the
34. 29 Page 3 SandMath_44 Manual 3 2 Factorials A timid foray into Number Theory 41 Pochhammer symbol rising and falling empires 42 Multifactorial Superfactorial and Hyperfactorial 43 Logarithm Multi Factorial 45 Appendix 5 Primorials a primordial view 46 3 3 High Level Math The case of the Chameleon function in disquise 49 Gamma Function and associates 50 Lanczos Formula 51 Appendix 6 Comparison of Gamma results 52 Reciprocal Gamma function 53 Incomplete Gamma function 53 Logarithm Gamma function 54 Digamma function 55 Euler s Beta function 56 Incomplete Beta function 56 Bessel Functions and Modified 57 Bessel functions of the 1 Kind 57 Bessel functions of the 2 Kind 58 Getting Spherical are we 59 Programming Remarks 60 Appendix 7 FOCAL program for Yn x Kn x 63 Riemann Zeta Function 64 Appendix 8 Putting Zeta to work Bernoulli numbers 66 Lambert W Function 67 3 4 Remaining Special Functions in Main FAT Exponential Integral and associates 69 Errare humanum est 71 The unsung Hero 71 Appendix 9 Inverse Error function coefficients galore 72 How many logarithms say what 73 Fourier Series 74 Appendix 10 Fourier Coefficients by brute force 76 3 5 More Special Functions in Secondary FAT 3 5 1 Carlson Integrals and associates The Launcher The Elliptic Integrals 77 Carlson Symmetric Form 78 Airy Functions 79 Fresnel integrals 80 Weber and Anger Functions 81 c ngel M Martin R
35. 4 3 x 9 The below shows Ai and Bi for 15 lt x lt 5 tet MANNA NS eal AVN AV fe 0 30 REGISTERS ROO thru R04 FLAGS none Stack Output Example 0 4 EFL AIRY gt Ai 0 4 0 254742355 or Mal H Y X lt gt Y gt Bi 0 4 0 801773001 c ngel M Martin Revision 44_E Page 79 SandMath_44 Manual Fresnel Integrals Fresnel integrals S x and C x are two transcendental functions named after Augustin Jean Fresnel that are used in optics They arise in the description of near field Fresnel diffraction phenomena and are defined through the following integral representations S x l sin t2 dt C x cos t2 dt The function CSX will calculate both S x and C x for the argument in X returning the results in Y and X respectively It is a short FOCAL program that uses yes you guessed it the Generalized Hyper geometric function according to the expressions S x 1 x 6 1Fa 3 4 3 2 7 4 n x 16 and C x x F 1 4 1 2 5 4 n x7 16 The figure below shows both functions plotted for 0 lt x lt 5 REGISTERS ROO thru R04 FLAGS none Stack Output n a S x X X C x Examples 1 5 XFL CSX gt C 1 5 0 445261176 X lt gt Y S 1 5 0 697504960 4 XFL CSX gt C 4 0 498426033 X lt gt Y S 4 0 420515754 Or al H C c Angel M Martin Revision 44_E Page 80 SandMath_44 Manual Weber and
36. 628800 1 66667E 10 4 16667E 10 0 1 11111E 09 3 96825E 10 7 44048E 10 5 591146E 10 2 5573E 10 2 50521E 10 1 04384E 09 4 81771E 10 1 000000001 1 000000001 2 000000001 6 000000002 24 120 720 0000001 5040 40319 99999 362879 9998 3628800 018 39916800 01 479001598 3 6227020798 1E 09 1E 09 5SE 10 3 33333E 10 0 0 1 38889E 10 0 2 4802E 10 5 0115E 10 4 96032E 09 2 00521E 10 3 549E 09 3 2118E 10 Enhanced 13 digit Implementation error O 1 000000001 2 6 000000004 24 120 720 5039 9999990 40 320 00001 362880 3628800 error 0 1E 09 0 6 66667E 10 1 9841E 10 2 48016E 10 0 1 000000012 1 000000012 2 000000024 6 000000071 24 00000028 120 0000014 720 0000087 9040 00006 40320 00048 362879 9988 3628800 05 39916800 9 479001580 2 6227020957 Result 1 1 1 999999999 9 999999997 23 99999999 120 0000014 719 9999996 9 039 999998 40 319 99998 362 880 3 628 800 1 2E 08 1 2E 08 1 2E 08 1 18333E 08 1 16667E 08 1 16667E 08 1 20833E 08 1 19048E 08 1 19048E 08 3 30688E 09 1 37787E 08 2 20469E 08 4 1336E 08 2 92127E 08 Lanczos k 6 Continuous Fractions Windschitl Stirling Result error 1 0 0 5E 10 5E 10 4 16667E 10 1 16667E 08 5 55556E 10 3 96825E 10 4 96032E 10 0 0 39916800 479001600 6227020800 39916800 479 001 600 6 227 020 800 39916800 479001600 6227020800 39 916 99 99 479 001 599 8 6 227 020 800 2 5
37. 98 an error of err 4 E 10 where FL can be called using the main launcher gl ALPHA Example 2 calculate the Omega constant W 1 1 gt RUNNING followed by 0 567143290 Example 3 Calculate both branches of W for x 1 2e 1 EX CHS ST X 1 X gt Wo 1 2e 0 231960953 LASTX XEQ WL1 gt W 1 2e 2 678346990 And here s a 3D representation of the complex Lambert to end this section with a graphical splash Enough to make you want to start using your 41Z Module isn t it c Angel M Martin Revision 44_E Page 68 SandMath_44 Manual 3 4 Remaining Special Functions in the Main FAT The third and last chapter of the Special functions in the main FAT comprises other Hyper geometric derived functions plus a couple of notable exceptions not easy to associate FFOURN and LINX __ Function Author Description o gt Z o oO Di CI M Baillard Cosine Integral ooo Di EI M Baillard Exponential Integral O S RF ELIPF ngel Martin _ Eliptic Integral ooo Ai ERF M Baillard Error Function _ _ SSS FFOUR Angel Martin _ Fourier coefficientsfor x o o H HCI MM Baillard _ Hyperbolic Cosine Integral o HGF MM Baillard _ Generalized Hyper geometric Function __ _ H HSI M Bailard_______ Hyperbolic Sine Integral _ o o LINX AngelMartin ____ Dilogarithmfunction _ _ _ _ A SI M Baillard Sine
38. DATA ERROR 05 1 2 0 671 30 431 Z 0 671 J0 431 Z 0 552 J0 979 Z 0 552 J0 979 Z 0 541 J0 990 Z 0 541 J0 990 Z 0 429 J2 608 Z 0 429 J2 608 05 1 Z 0 431 J0 671 Z 0 431 J0 671 Z 0 959 J0 111 Z 0 959 J0 111 0 5 EE DATA ERROR O o oaao OOOO OOOO Sm Se A Me Where we see that for negative arguments integer and non integer orders both the result of the Bessel function of the second kind is itself a complex number therefore the DATA ERROR message Note also the symmetric nature of the values for each of the function pairs H1 with H2 and SH1 with SH2 c Angel M Martin Revision 44_E Page 84 SandMath_44 Manual Struve functions Struve functions are solutions y x of the non homogenous Bessel s differential equation 2 o l ty dy 2 n 42 2 vag tts 2 OW Frag hy Ge T mIa 3 Struve functions H n x and Modified Struve Functions L n x have the following power series forms in 1 r 2mt oatl H 2 gt Bpi L L a ep a etl 1 wy 2k L 3 Xerar la a B 3 2 er eg ea ee The figure below shows a few Struve functions or integer order n 1 to 5 for 10 lt x lt 10 7 y L x n 3 H 100 nan x e i ad gt L f ae j EE er ee El AO A a N Up ta 5 10 iia ee i Tp 210 oe 5 10 1 25 50 fd 4 n 4 Struve functions of any order can be expressed in terms of the Generalized Hypergeometric function 1F2 which is not the Gauss Hypergeometri
39. Dig PSI Angel Martin ____ Digamma Psi function o Dia PSIN JM Bailard Polygamma function __ SSS In mathematics the Gamma function represented by the capital Greek letter is an extension of the factorial function with its argument shifted down by 1 to real and complex numbers If n is a positive integer then T n n 1 showing the connection to the factorial function For a complex number z with positive real part the Gamma function is defined by z J tte dt BATATA en eeene Things become much more interesting in the negative semi plane as can be seen in the plot on the right for real arguments The Gamma function has become standard in pocket calculators either as extended factorials or as proper gamma definition It s already available in the HP 11C and of course on the 15C and that has continued to today s models Implementing it isn t the issue but achieving a reasonable accuracy is the challenge A popular method uses the Stirling approximation to compute Gamma This is relatively simple to program but its precision suffers for small values of the argument A version suitable for calculators is as follows Valid for Re z gt 0 and with reasonable precision when Re z gt 8 c Angel M Martin Revision 44_E Page 50 SandMath_44 Manual For smaller values than that it s possible to use the recurrence functional equation taking it to the safe region and
40. Integral also a quick application of the EI function using the formula Li x Ei In x see description for EI before Note how LI starts as a MCODE functions that transfers into the FOCAL code calculating EI so strictly speaking it s a sort of hybrid natured function SAE is a direct application of th Carlson Symmetric Integral of second kind RG to calculate the surface aerea of an escalene ellipsoid i e not of revolution r y z2 etet which formula is Area 470 Rc ab a c be c Angel M Martin Revision 44_E Page 92 SandMath_44 Manual Let s now continue with the not so simple functions left where some of them will not surprisingly be based on the Hyper geometric functions again Debye Function The family of Debye functions is defined by nm fe t dt j et pn The functions are named in honor of Peter Debye who came across this function with n 3 in 1912 when he analytically computed the heat capacity of what is now called the Debye model The formula used for n positive integers and X gt 0 is n k db x n E po e x k n x k pe ix 40 N N w ee aii _ 40 Despite being a FOCAL program DEBYE pretty much behaves like an MCODE function no data registers are used only the stack and ALPHA and the original argument preserved in LASTx credit is due to JM Baillard once more
41. It also uses a few others SandMath functions as additional 37 RCLM bonus 33 CHS 34 35 END c Angel M Martin Revision 44_E Page 66 SandMath_44 Manual 3 3 10 Lambert W function The last function deals with the implementation of the Lambert W function Oddly enough its definition is typically given as the inverse of another function as opposed to having a direct expression This makes it a bit backwards looking initially but in fact it is significantly easier to implement than the Riemann Zeta seen before The Lambert W function named after Johann Heinrich Lambert also called the Omega function or product log is the inverse function of f w w exp w where exp w is the natural exponential function and w is any complex number The function is denoted here by W gE 4 For every complex number z z i 0 5 a _ The Lambert W function cannot be expressed in terms of elementary functions It is useful in P Lo combinatory for instance in the enumeration of far 7 i trees Fi a It can be used to solve various equations involving exponentials and also occurs in the SL nn eee eee solution of delay differential equations The graph of W x for Aes xs 4 The Taylor series of W around 0 can be found using the Lagrange inversion theorem and is given by oo n Wolz 5 2 nl a TL where n is the factorial However this series oscillates between ever larger positi
42. M 13 17 221 LCM 12 18 36 LCM 15 33 165 RGSORT sorts the contents of the registers specified in the control number in X defined as bbb eee where bbb is the begin register number and eee is the end register number If the control number is positive the sorting is done in ascending order if negative it is done in descending order This function was written by HaJo David and published in PPCCJ V12N5 p44 STSORT sorts in descending order the contents of the four stack registers X Y Z and T No input parameters are required This function was written by David Phillips and published in PPCCJV12N2 p13 RGMAX finds the maximum within a block of consecutive registers which will be placed in X returning also the register number to Y The register block is defined with the control word in X as input with the same format as before bbb eee It was written by Jean Marc Baillard ST lt gt z exchanges the contents of the statistical registers and the stack Use it as a convenient method to review their values when knowing their actual location is not required ODD And EVEN are simple tests to see is the number in X is odd or even The answer is YES NO and in program mode the following line is skipped it the test is false The implementation is based on the MOD function using MOD x 2 0 as criteria for evenness INCX DECX INCY and DECY are convenient substitutes for 1 and 1 These functions will decrement or incr
43. M prize so adjust your expectations accordingly The Riemann zeta function is a function of complex argument s that analytically continues the sum of the infinite series Bia 1 _ wt R s 1 rosz du n Dix b e l n 1 or the integral form The Riemann zeta function satisfies the functional equation 1S C s 2a sin r 1 s 1 s valid for all complex numbers s excluding 0 and 1 which relates its values at points s and 1 s The plots below of the real Zeta function show the negative side with some trivial zeros as well as the pole at x 1 ir etx itl at ih EA i x e ne 1f Hi lTi The direct implementation in the SandMath module uses the alternative definitions shown below in a feeble attempt to get a faster convergence which in theory it does although not very noticeably given the long execution times involved The summations are called the Dirichlet Lambda and Eta functions respectively 1 B Io Sen T IEF ir a Ft _ i l il re x 2 n ly i s ls L p 7 H Go ahead and try ZETA with FIX 9 set in the calculator you ll see the successive iterations being shown for each additional term until the final result doesn t change Be aware than MCODE or not t7 take a very long time for small arguments approaching infinite as x approaches zero For values lower than 1 we make use of the following x relationship a sort of
44. Number from Seed in buffer RGMAX JM Baillard Maximum ina register block Sorts a block of registers SEED with Timer il sre gt z Neon amp ome SREG exchange whs STSORT David Phillips _ Stack Sort ooo O Statistical Menu Another type of Launcher Pressing ae twice will present the STAT PROB functions menu allowing access to 10 functions using the top row keys A J Two line ups are available toggled by the SHIFT key xx Default Lineup Linear Regression xx Shifted Lineup Probability i wa FW FF oa LR fa L wi Es p USER SHIFT O12 4 Note the inclusion of the mainframe functions MEAN and SDEV in the menus for a more rounded coverage of the statistical scope With the manus up you just select the functions by pressing the key under the function abbreviated name Use SHIFT to toggle back and forth between both lineups and the back arrow key to cancel out to the OS Obviously the data pairs must be already in the XREG registers for these functions to operate meaningfully c Angel M Martin Revision 44_E Page 31 SandMath_44 Manual Alea jacta est It s a little known fact that the SandMath module also uses a buffer to store the current seed used for random number generation The buffer id is 9 and it is automatically created by SEEDT or RAND the first time any of them is executed and subsequently upon start up by the Module during the initialization steps using the
45. Output Y n n db n x Lb a DB 0 7 3 6 406833597 Example 3 ENTER 0 7 XFL DEBYE gt Page 93 Revision 44 E c Angel M Martin SandMath_44 Manual Dawson Integral The Dawson function or Dawson integral named for John M Dawson is either F z D x e e dt or D_ z e f e dt 0 DAW computes F x by a series expansion F x e x x3 3 x5 5 2 x7 7 3 The figures below show both functions in graphical form TE oT oT TA TO Here as well no data registers are used X x aS Examples 1 94 XFL DAW gt F 1 94 0 3140571659 10 R S gt F 10 0 05025384716 15 R S gt F 15 0 03340790676 For x gt 15 there will be an OUT OF RANGE condition c Angel M Martin Revision 44_E ng E ee D x Page 94 SandMath_44 Manual Hyper geometric Functions HGF and RHGF are the ordinary and the Regularized Hyler geometric functions The Gaussian or ordinary hypergeometric function 2F1 a b c z is a special function represented by the hypergeometric series that includes many other special functions as specific or limiting case It is defined for z lt 1 by the power series a nlb n z 2Fi a bcz oe nl o Ch 7 provided that c does not equal 0 1 2 Here q n is the Pochhammer symbol which is defined by fil tn 0 a n g g 1 q n 1 ifn gt 0 Many of the common mathematica
46. SandMath_44 Manual INIMATH HY Moth Extensions for the HFH i aa User s Manual and Quick Reference Guide PRGM ALPHA xi o yttx LOGYx ej ely LOS ENII Aa p C M G Ey S pee sin l Hcos vtan li ba a E J kr SIN 4 COS TAN J Ss a H c ka pe GRT SEED STO RCL SST L a e CAT WMT CHS x ENTER CHS p i oO T BS D H Written and programmed by ngel M Martin F WIEW November 2012 R S HP 41SandMath m E Ssh EEE Revision 44_E Page 1 SandMath_44 Manual This compilation revision 4 44 44 really Copyright 2012 Angel Martin Acknowledgments Documentation wise this manual begs steals and borrows from many other sources in particular Jean Marc Baillard s program collection on the web Really both the SandMath and this manual would be a much lesser product without Jean Marc s contributions There are multiple graphics and figures taken from Wikipedia and Wolfram Alpha notably when it comes to the Special Functions sections I m not aware of any copyright infringement but should that be the case I ll of course remove them and create new ones using the SandMath function definition and PRPLOT Just kidding Original authors retain all copyrights and should be mentioned in writing by any party utilizing this material No commercial usage of any kind is allowed Screen captures taken from V41 Windows based emulator developed by Warren Furlow See ww
47. T 67 RCLO03 cbrt x R3 2 05 ST Z 1 a 1 a 3 inZ f eB 06 69 2 07 STO 00 a0 a 0 a 3 70 V 08 RDN 71 CHS 09 STO 01 a1 a 1 a 3 F 72 RCL 02 a2 3 F 10 RDN a2 a 2 a 3 r 733 ee real part OOOO 11 3 r 74 12 f 75 ISTO T 0 flag it as Complex i F 13 STO 02 a2 3 76 RDN Z 0 indicates it 14 xag 77 415 ST X 3 2 q243 27 78 STO 01 16 RCL 01 a1 T 79 xo 17 RCL 02 a2 3 80 STO 02 T 18 a1 a2 3 81 RTN T 19 2 a2 3 27 a1 a2 3 82 LBLO1 all real roots 20 83 DEG r 21 84 LASTX 22 85 CHS 23 STO 03 a0 2 a2 3 27 a1 a2 6 86 SQRT T 24 xm a0 2 a2 3 27 a1 a2 6 2 87 ST X 3 F 25 RCL 01 a1 88 X 0 F 26 RCL 02 a2 3 89 1 X 27 Xm a2 2 9 90 RCL 03 a0 2 a2 3 27 a1 a2 6 23 53 T 91 ST X 3 a0 2 a243 27 a1 a2 3 T 29 a2 2 3 92 CHS 30 o a1 a2 2 3 F 93 gt 31 STO 01 a1 a2 2 3 94 ACOS r 32 l3 95 8 r Ba 1 3 a1 a242 3 96 T 34 xa 97 STO 03 r 35 1 27 al a2 2 3 3 a0 2 a2 3 27 al a2 98 LASTX 36 Xss0 _ Pe 99 E3 E 37 GTO 01____ yes allrealroots_ __ _____ 100 STO 05 1 003 38 SQRT complexroots 101 RCL Ol a1 a2 2 3 F 39 IENTERA f 102 3 40 JENTERA O ReX OOO 103 a1 3 a2 2 9 F 41 RCL 03 a0 2 a2 3 27 a1 a2 6 104 CHS a2 2 9 a1 3 a l 105 SQRT 43 a 106 ST X 3 44 cbrt x R3 2 107 2 SQR a2 2 9 a1 3
48. The FOCAL programs below list the simple code snippets to program the three pairs of functions just covered as follows 1 Hankel functions HK1 and HK2 2 Spherical Bessel functions SJBS and SYBS 3 Spherical Hankel functions SHK1 and SHK2 Note the symmetry in the code for the spherical programs making good use of the stack efficiency derived from the utilization of the MCODE JBS function The plots on the left show the Spherical Hankel 1 function for orders 1 and 2 for a short range of the argument x x from 10 to 10 Explicitly the first few are real part imaginary part ni 1 i j ye ren es i Ay l iz g o ay gb diz lt 3 hi z ie l ix from 4 to 4 T TP 2 6iz 152 151 real part h G e imaginary part c ngel M Martin Revision 44_E Page 83 SandMath_ 44 Manual i XEQ SJBS STO 04 STO 04 g RCL Z RCL 00 l 10 RCL Z i 10 RCLO 10 neve 11 ST x xt 11 STAX 11 FC 03 12 YBS 12 AEG SYBES 12 CHS n 2 13 FO C 03 13 FC FC 04 13 Meo 14 CHS 14 CHS 14 JES 15 RCL O4 15 RCL O4 15 FO 03 16 ZAVIEW 16 ZAVIEW 16 GTO00 17 END 17 END 17 RCLOO H 18 INCX H Plot of H 1H 1 x 19 CHSYX feat aS x from 10 to 10 real part imaginary part if Examples Calculate H1 H2 SH1 and SH2 for the following values in the table Pon x 2 0 440 J0 781 Z 0 440 J0 781 Z 0 301 J1 382 Z 0 301 J1 382 pa ff __pav eexOR
49. ags Used FO1 Output Z m fy Example 2 SQRT 3 SQRT PI XFL LOML gt sil sqrt 2 sqrt 3 n 3 003060384 Or gl H L instead c Angel M Martin Revision 44_E Page 86 SandMath_44 Manual Lerch Transcendent Function In mathematics the Lerch zeta function sometimes called the Hurwitz Lerch zeta function is a special function that generalizes the Hurwitz zeta function and the polylogarithm It is named after Mathias Lerch The Lerch zeta function L and a related function the Lerch Transcendent are given by exp fmi An L A 0 8 D E W 2 8 0 nao a nao n mar Special cases The Lerch Transcendent generates other special functions as particular cases as it s shown in the table below The Hurwitz zeta function C s a _ qx s P 1 s CE 1 5 1 The polylogarithm Li z zP z s 1 n s 1 s 1 The figures below depict the representation for x given the other two constant Re Piz 1 0 Re P z 0 1 2 2 l l 0 0 1 1 2 2 4 4 a3 2 4 4 9 z a The SandMath implementation LERCH is for the Lerch Transcendent function It is a short MCODE routine originally written by Jean Marc Baillard which calculates the series terms and adds them until they don t have a contribution to the final result It is a slow converging series and therefore the execution time can be rather long at normal CPU speeds Data input follow
50. an independent life of their own in separate modules more on that to come This manual is a more or less concise document that only covers the normal use of the functions c Angel M Martin Revision 44_E Page 7 SandMath_44 Manual Conventions used in this manual All throughout this manual the following convention will be used in the function tables to denote the availability of each function in the different function launchers jat assigned to the keyboard by HMKEYS is direct execution from the main launcher FL H executed from the hyperbolics launcher HYP F executed from the fractions launcher FRC Xe executed from the RCL launcher RCL CR executed from the Carlson Launcher no separate function exists HK executed from the Hankel launcher no separate function exists xx executed from the Statistics Menu ST PRB sub function in the secondary FAT LFL HMKEYS uses the value in X as a flag to decide whether to assign or to remove the mass key assignments If x 0 the assignments will be removed any other value will make them There are a total of 25 functions assigned refer to the SandMath overlay for details Note for Advanced Users Even if the SandMath is an 8k module it is possible to configure only the first lower page as an independent ROM This may be helpful when you need the upper port to become available for other modules like mapping the CL s MMU to another modul
51. as there s always a term for which m n 1 equals zero or negative integers all of them being problematic To avoid this we use the following expression for negative integer orders E si J alz 1 J 2 Whilst _ x I x for every real number order This definition is also valid for negative values for X as there s no singularity for any x value The SandMath implementation uses a recurrence formula instead of the one shown above It has the clear advantage of not having to calculate Gamma for each term in the sum contributing to a much faster and robust algorithm The iterative relationships are as follows J n x X U k k 1 2 x 2 T n 1 where U k U k 1 x 2 42 k k n with U 0 1 c Angel M Martin Revision 44_E Page 57 SandMath_ 44 Manual The graphics below plot the Bessel functions of the first kind J x and their modified I x for integer orders a 0 1 2 10 Note that for large values of the argument the order or both these algorithms will return incorrect results for J n x This is due to the alternating character of the series which fools the convergence criteria at premature times and fouls the intermediate results Unfortunately there isn t an absolute criteria for validity but a practical rule of thumb is to doubt the result if n x it greater than 20 Bessel functions of the Second kind K n x and Y n x The formulae used are as follows
52. back calculating the result with the appropriate adjusting factor C z 1 zT z Incidentally this method can be used for any approximation method not only for Stirling s The method used on the SandMath is the Lanczos approximation which lends itself better to its implementation and can be adjusted to have better precision with careful selection of the number of coefficients used For complex numbers on the positive semi plane Re z gt 0 the formula used is as follows go 75122 6331530 q 80916 6278952 oa q 36308 2951477 T z g 5 5 pre g 8687 24529705 e v0 qa 1168 92649479 gs 83 8676043424 qgs 25066282 Although the formula as stated here is only valid for arguments in the right complex half plane it can be extended to the entire complex plane by the reflection formula on T oh ila ial das sin wz An excellent reference source is found under http www rskey org gamma htm written by Viktor T Toth Let s mention that this method yields good enough a precision that doesn t require using the functional equation to adjust it for small values of the argument The obvious advantage is that without the required program loop the execution time is shorter and constant for any input This becomes of extreme importance when Gamma is used as a subroutine of more complex cases like the Bessel J and I functions where the cumulative additional time is very noticeable Ir c ng
53. benefits from faster execution on the calculator Non integer orders utilize the special MCODE function GAMMA with shorter execution times than equivalent FOCAL programs but still longer than FACT when integers Besides that for integer orders the execution time is further reduced by ca culating simultaneously the two infinite sums involved in the first kind and the second kind terms This assumes that the convergence occurs at comparable number of terms which fortunately is the case given their relative fast convergence Note that in order to obtain similar expressions for both Yn and Kn and so getting simpler program code we can re write Kn as follows 1 2 Ky x 2 Ly Ln x 2 LA Df fk n x CD CD gx n x Dedicated MCODE Functions To further decrease the execution time of the programs two dedicated functions have been written implemented as MCODE routines as follows Function Flag 00 Clear Flag 00 Set BS 2 Ux n x K 0 1 2 where Uk Ux x 2 2 k k n BS2 V fk n x K 0 1 2 or LDLi gy n x K 0 1 n 1 The first function BS is used equally in the calculation of the first kind and the second kind of non integer orders Function Integer Non integer JBS IBS YBS 2x BS2 KBS As it was said before the summation will continue until the contribution of the newer term is negligible to the total sum value All calculations are done using the full 13 digit precision of the calc
54. c function 2F1 This is the expression used in the SandMath implementation 2 24 ay Hal ara 43 0 sie 8 2 fA in other words referred to the Rationalized Generalized Hypergeometric function which with such a long name it definitely must be a formidable function but it s just the same divided by Gamma H x x 2 F 1 3 2 n 3 2 x 4 Li x x 2 F 1 3 2 n 3 2 x7 4 Examples Compute H 1 2 3 4 and L 1 2 3 4 1 2 ENTER 3 4 XxXFL HNX gt H 1 2 3 4 1 113372657 gt L 1 2 3 4 4 649129471 1 2 ENTER 3 4 EFL LNX and Ma1 H SHIFT H for LNX Or Ma1 H H for HNX Revision 44_E Page 85 c ngel M Martin SandMath_44 Manual Lommel functions The Lommel differential equation is an inhomogeneous form of the Bessel differential equation d 2 2 p 2S tz lz y 2 vs vs Two solutions are given by the Lommel functions su v z and Su v z introduced by Eugen von Lommel 1880 E a Y 2 z Jaz dz JL z z Y z dz Sp 2 spr 2 A Jol2 c0s n u 2 2 where Jv z is a Bessel function of the first kind and Yv z a Bessel function of the second kind Using the Generalized Hypergeometric function the expressions for s1 m n x is sh n x x m 1 n Fz 1 m n 3 2 m n 3 2 x7 4 LOMMEL calculates s1 m n x here are the specifics DATA REGISTERS ROO thru ROY temp Fl
55. chers and their interconnectivity For the most part it is always possible to return to the main launcher pressing the back arrow key improving so the navigation features rather useful when you re not certain of a particular function s location The first one is the Main SandMath Launcher The first mapping doesn t show all the direct execute function keys Use the SandMath overlay as a reference for them names written in BLUE aside the functions Launchers invoked trom the Main FL Note that XFL will require pressing ALPHA a second time in order to type the sub function name And here s the Enhanced RCL MATH group RCL Math Launchers Here all the prompts expect a numeric entry The two top rows keys can be used as shortcuts for 1 10 Note that No STK functionality is implemented even if you can force the prompt at the IND step Typically you ll get a DATA ERROR message Rather not try it c Angel M Martin Revision 44_E Page 11 SandMath_44 Manual Function index at a glance And without further ado here s the list of functions included in the module First the main functions 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Name Description 2 X 1 Powers of 2 1 N Harmonic Numbers ZDGT Sum of mantissa digits 2 N X Geometric Sums AINT
56. creen keeping the choices constant regardless HK INH LIRIK GHK E INIH LIRIK USER USER SHIFT The table below shows in the first column the letter used for each of the functions within this group HK Function Author Description O Z o O S 1 HK1 Angel Martin Hankel Function o S 2 HK2 Angel Martin Hankel2 Function _ o S w WOL AngelMartin LambertWO __ S H HNX MBailard_ Struve H Function _ __ SSS L LOML MBailard_ Lommel s4 function _ ___ S R LERCH M Baillard Lerch Transcendental function _ BE ESS SEE SSS 4 SHK2 Angel Martin Spherical Hankelt S 2 SHK2 AngelMartin SphericalHankel2 __ oo Ww WiL AngelMartin Lambet Wi S H LNX_ MBailard_ Struve Ln Function __ S L ALF MBaillard_______ Associated Legendre function 1st kind Pam x R TMNR_ MBailard Toronto function _ __ SSS K KUMR AngelMartin Kummer Function _ _ _ _ S Here we finally find both branches of the Lambert W function WOL and W1L described previously in thie manual Observant users would have noticed that the name in the previous section was WLO instead this is not a dyslexya induced error but an intended difference to tell both FAT entries apart That and also the limitation in the secondary FAT to have function names ending with numeric characters but that s another story So your several choices in terms of launchers are as follows a Fun
57. ction WLO in main FAT sat WLO the ordinary method page M shortcut using the main launcher XFL WLO since XFL also finds functions in the main FAT Mal ALPHA WLO b Functions WOL and W1L in secondary FAT Dae H W pial H SHIFT W ZFL 032 ZFL 033 FL WOL SFL WIL PAi ALPHA WOL PAi ALPHA WiL Now that s what I d call both a digression and multiple ways to skin this cat c Angel M Martin Revision 44_E Page 82 SandMath_44 Manual Hankel functions yet a Bessel third kind Another important formulation of the two linearly independent solutions to Bessel s equation are the Hankel functions Ha 1 x and Ha 2 x defined by H x Jol iYa a H x Jaz iYa 2 where i is the imaginary unit These linear combinations are also known as Bessel functions of the third kind and it s just an association of the previous two kinds together This definition allows for relatively simple programming only using the real domain Bessel programs assuming the individual results for J and Y are not complex The small program in the next page shows the FOCAL code to just drive the execution of both JBS and YBS piercing them together via ZOUT or ZAWIEW in the 41Z module Getting Spherical are we Finally there are also spherical analogues of the Hankel functions as follows py x z Jn X ar 1Yn T i x Jn x _ 1Yn X
58. d with the same decimal number after the two executions T gt BS Ten to Base is the MCODE equivalent to D gt BS much faster and more elegant due to its prompt where in RUN mode you input the destination base The result is show in the display and also left in ALPHA so it could be also used by BS gt D once the base is in X Note that the original argument decimal value is left in X unaltered so you can use T gt BS repeated times changing the base to see the results in multiple bases without having to re enter the decimal value T gt BS is programmable In PRGM mode the prompt is ignored and the base is expected to be in the Y register much the same as its FOCAL counterpart D gt BS Obviously using zero or one for the base will result in DATA ERROR The maximum base allowed is 36 and the SASE gt 36 error message will be shown it that s exceeded note that larger bases would require characters beyond Z The maximum decimal value to convert depends on the destination base since besides the math numeric factors it s also a function of the Alpha characters available up to Z and the number of them length in the display 12 For b 16 the maximum is 9999 E9 or 0x91812D7D600 T gt BS is an enhanced version of the original function also included in Ken Emery s book MCODE for Beginners The author added the PRGM compatible prompting as well as some display trickery to eliminate the visual noise of the original
59. dMath_44 Manual Kummer Function Kummer s equation has two linearly independent solutions M a b z and U a b z dw dz Kummer s function of the first kind M also called Confluent Hypergeometric function is a generalized hypergeometric series introduced in Kummer 1837 given by dap b z aw 0 ge M a b z gt in i F a b z nO Where a is the rising factorial defined as a ala Ij a 2 a n 1 The figures below depict two particular cases for a 2 b 3 and a 2 b 3 Re A 2 3 2 Re F 2 3 2 20 10 The SandMath implementation is got to be one of the simplest application ot HGF possible which renders acceptable accuracy to the results DAT REGISTERS a R00 b RO1 Stack X X M a b x Examples Compute M 2 3 r and M 2 3 7 2 STO01 3 STO02 PI CHS XFL KUMR gt M 2 3 n 0 166374562 2 STOO01 3 STO02 PI XFL KUMR gt M 2 3 n 10 24518011 Or Sig H SHIFT K instead c ngel M Martin Revision 44_E Page 89 SandMath_44 Manual Associated Legendre Functions In mathematics the Legendre functions P A Q A and associated Legendre functions Pu A and Qu A are generalizations of Legendre polynomials to non integer degree Associated Legendre functions are solutions of the Legendre equation 1 2 y Qey past 74S y 0 where the complex numbers A and u are called the degree and order
60. e all in MCODE Functions in black are MCODE entries to FOCAL programs 52 53 54 55 56 57 58 59 60 61 62 63 NPR ODD PDF PFCT PRIME RAND RGMAX RGSORT SEEDT ST lt gt STSORT XFACT Combinations Is X Odd Probability Distribution Function Prime Factorization in Alpha Is X Prime Random Number Block Maximum Register Sort Stores Seed for RNDM Exchange ST amp XREG Stack Sort Extended FACTorial And now the sub functions within the Special Functions Group deeply indebted to Jean Marc s contribution and not the only section in the module Note there are two sections within this auxiliary FAT you can use the FCAT hot keys to navigate the groups Description Cat header does FCAT Aux routine All Bessel Aux routine 2nd Order Integers Airy Functions Ai x amp Bi x Author Angel Martin Angel Martin Angel Martin JM Baillard Associated Legendre function 1st kind Pnm x JM Baillard Inverse Lambert Carlson Integral 1st Kind Carlson Integral 2nd Kind Carlson Integral 3rd Kind Fresnel Integrals C x amp S x Dawson integral Debye functions Eliptic Integral Hypergeometric function Hankel1 Function Hankel2 Function Struve H Function Integral if IBS Integral of JBS Kelvin Functions 1st kind Kummer Function Lerch Transcendent function Logarythmic Integral Struve Ln Function Lommel s1 function Regularized hypergeometric function Surface Area of an Ellipsoid S
61. e force and use PFD in an INTEG like scenario or course The relation to use is F x 1 07 gt 1er ER Example program The routine below calculates CDF Enter u o and x values in the stack 01 LBL CDF 08 02 RCLZ 09 ERF 03 11 INCX 04 X lt gt Y 12 2 05 13 06 2 14 END 07 SQRT PDF expects the mean and standard deviation in the Z and Y stack registers as well as the argument x in the X register Upon completion x will be saved in LASTx and f u o x will be placed in X It has an all MCODE implementation using 13 digit routines for increased accuracy PDF is a function borrowed from the Curve Fitting Module which contains others for different distribution types With the Normal distribution being the most common one it was the logical choice to include in the SandMath c ngel M Martin Revision 44_E Page 36 SandMath_44 Manual And what about prime factorization Function PFCT will do a very fast and simple prime factorization of the number in X using PRIME To look for the successive divisors until 1 is found PFCT uses the ALPHA registers to present the results grouping the repetitions of a given factor in its corresponding exponent For example for x 126 the three prime factors are 2 3 and 7 with 3 repeated two times For large numbers or when many prime factors exist the display will scroll left to a maximum length of 24 characters This is sufficient for the majorit
62. e found 6 AINT 37 RCL 00 T 38 K 1 B PRIME 9 SF 00 10 AINT first prime factor 11 K 17 12 GTO 01 13 FS 7C O0 if prime we re done 14 GTO 01 15 STO 01 save for grouping 16 ST L 17 LASTX reduced number 18 19 E 20 STO 00 reset counter 21 RDN 22 23 RCL 01 previous prime factor 24 KY 25 PRIME 26 SF 00 27 MAY different PF 28 GTO 02 YES 29 ISG 00 increase counter 60 AVIEW 30 NOP SAME pf 61 ANUM 31 FS 7C 00 Was it prime 62 END c Angel M Martin Revision 44_E Page 38 SandMath_44 Manual Below is the Enhanced version allowing for any number of different prime factors and repetition indices all stored in a n x 2 matrix file in extended memory PRMF Note how the program structure is basically the same despite the addition of the matrix handling Since the Advantage module is required we ve used AIP instead of AINT totally interchangeable as they re basically the same function LBL PRMF 46 RCLOO 2 PRMF 47 MSE 3 48 RDN 4 49 ST L 5 50 LASTX 6 CLX 51 7 MSA O 52 8 RDN 53 9 CF 00 54 X lt gt Z 10 INT 55 MSRt OO 11 ABS 56 FS C 00 was it prime 12 PRIME 57 GTO 01 yes wrap up 13 SF 00 58 X lt gt Y no swap things 14 59 GTO 05 and do next PF 16 GTO 01 61 SF 04 17 FS C 00 if prime we re done 62 SF 10 18 GTO 01 63 PRMF 19 STO 01 save for grouping 64 20 ST L 65 E 21 LASTX reduced number 66 FC 10 23 E 68 STO 00 re build the number 24 STO 00
63. e others only perform a FAT entry placeholder function and could be removed replaced by others in future versions of the module 5288 Z J33 14e USER RAD USER RAD 01 4 c Angel M Martin Revision 44_E Page 22 SandMath_44 Manual FRE USER RAD 2 2 1 Fraction Arithmetic and Displaying A rudimentary set of fraction arithmetic functions is included in the SandMath including the four basic operations plus a fraction viewer and two test functions Function Author__ Description Fractions Launcher Calculates a fraction that gives the number in X Fraction addition Fraction subtraction Angel Martin Fraction multiplication Angel Martin Fraction division FRC ngel Martin Is X a fractional number Is X an integer number D gt F is the key function within this group Shows in the display the smallest possible fraction that results in the decimal number in X for the current display precision set Change the display precision as appropriate to adjust the accuracy of the results This means the fraction obtained may be different depending on the settings returning different results For example the following approximations are found for n m 104348 33215 in FIX 9 FIX 8 and FIX 7 m 355 113 in FIX 6 FIX 5 and FIX 4 m 333 106 in FIX 3 m 22 7 in FIX 2 FIX 1 and FIX 0 This function was written by Frans de Vries and published in DataFile DF V9N7 p8 It uses the same algorithm as the PPC
64. e temporarily or permanently if you don t care about the High Level Math Special Functions and Statistics sections Think however that the FAT entries for the Function launchers are in the upper page so they ll be gone as well if you use the reduced foot print 4k version of the SandMath Upper Pad High Level Math Stats pP ge Function Launchers Lower Page tower Page SandMath_44 FRC HYP RCL Math Note that it is not possible to do it the other way around that is plugging only the upper page of the module will be dysfunctional for the most part and likely to freeze the calculator do not attempt Final Disclaimer With just an EE background the author has had his dose of relatively special functions from college to today However not being a mathematician doesn t qualify him as a field expert by any stretch of the imagination Therefore the descriptions that follow are mainly related to the implementation details and not to the general character of the functions This is not a mathematical treatise but just a summary of the important aspects of the project highlighting their applicability to the HP 41 platform c Angel M Martin Revision 44_E Page 8 SandMath_44 Manual Getting Started Accessing the Functions There are about 160 functions in the SandMath Module With each of its two pages containing its own function table this would only allow to index 128 functions where are the others a
65. e user needn t remember where a specific function sought for was located In fact XFL will also find a function from any plugged in module in the system even outside of the SandMath module Main Launcher and Dedicated Launchers The Module s Main launcher is DAAI Think of it as the trunk from which all the other launchers stem providing the branches for the different functions in more or less direct number of keystrokes With a well thought out logic in the functions arrangement then it s much easier to remember a particular function placement even if its exact name or spelling isn t know without having to type it or being assigned to any key Despite its unassuming character the XFL prompt provides direct access to many functions Just press the appropriate key to launch them using the SandMath Overly as visual guide the individual functions are printed in BLUE with their names set aside of the corresponding key They become active when the F _ prompt is in the display HLPHH Besides providing direct access to the most common Special Functions XFL will also trigger the dedicated function launchers for other groups like HYP FRC CR HK STAT and lt FL itself Think of these groupings as secondary menus and you ll have a good idea of their intended use The following keys activate the secondary menus A activates the STAT PRB menus H activates the HK _ launcher O activates the CR _ lau
66. ecution form of this is BT x 04 Number in Egg is divided by the number in the X register and the quotient is placed mto Rog The display execution form of this is st c Angel M Martin Revision 44_E Page 29 SandMath_44 Manual Primary Data Storage Registers The standard HP AIC has 63 registers that can be allocated to data storage or program memmory in ary combination As you add HP memory Modules up to four the total number of registers can increase to 319 64 registers for each memory module When allocated data storage registers numbered Ro through Roo are Primary Data Storage Registers J i i i i oe A ee ee ee i ee ee M M O ea Extended Data Storage Registers When allocated data storage registers numbered Ryj99 through Rgn are extended Data Storage Registers tow Program Memory All registers that are not allocated as Primary of Extended data J storage registers are part of program memory When allocated as i pro memory the standard HP 41C registers provide space for 3 200400 fully merged lines of programs When allocated as z program memory each memory module adds 200 400 limes The E total can be 1000 2000 lines when alle 319 registers are allocated z J as program memory Variations in storage capacity depend on the kinds of functions stored in program memory Note Make sure the revision G or higher of the Library 4 module is installed c Angel
67. ed with the indirect comparison functions of the CX model XK gt NN7 but unfortunately not quite the same X 1 is a quick and simple way to check whether the value in X equals one As usual program execution skips one step if the answer is false X YR establishes the comparison of the rounded values of both X and Y according to the current decimal digits set in the calculator Use it to reduce the computing time albeit at a loss of precision when the algorithms have slow convergence or show unstable results for larger number of decimals INT and FRC are two more test functions which criteria is the integer or fractional nature of the number in X Having them available comes very handy for decision branching in FOCAL programs The Fractions section of the module is the natural placement for them The remaining functions shown in the table below really are displaced in that their entries are in the upper page but certainly have nothing to do with High Level math Call it a misdemeanour if you want and allow me to include them now and get them out of the way _ Function Author Description O O OZ oO O INCX Ken mery _ Increases X by one __ INCY Angel Martin Increases Y by one Angel Martin Decreases X by one DECY Angel Martin Decreases Y by one ZOUT Angel Martin Combines the values in Y and X into a complex result Of these only ZOUT has been used in FOCAL programs in the SandMath so th
68. egisters and the result will be placed in x upon completion The term symmetric refers to the fact that in contrast to the Legendre forms these functions are unchanged by the exchange of certain of their arguments The value of Rf is the same for any permutation of its arguments and the value of Ry is the same for any permutation of its first three arguments The Carlson Symmetric Elliptic integral of the 2 Kind is defined as 1 f t r n z And is calculated using the following expression involving CRF and CRJ 2 Re 3 y 2 Z Re x 5y 2 X 2 y Z 3 Ro xsy3z x y z c ngel M Martin Revision 44_E Page 78 SandMath_44 Manual Airy Functions For real values of x the Airy function of the first kind is defined by the improper integral Ip Ailt cos 4t at dt 0 which converges because the positive and negative parts of the rapid oscillations tend to cancel one another out as can be checked by integration by parts The Airy function of the second kind denoted Bi x is defined as the solution with the same amplitude of oscillation as Ai x as x goes to oo which differs in phase by n 2 Bi a lexp 3t at sin 3t at dt The expressions used to program them are again based on HGF as follows Ai x 3 7 T 2 3 oF 1 2 3 x9 x 37 T 1 3 oF 1 4 3 x 9 Bi x 37 T 2 3 oF 1 2 3 x 9 x 3 8 T 1 3 oF 3
69. el M Martin Revision 44_E Page 51 SandMath_44 Manual Appendix 6 Accuracy comparison of different Gamma implementations The tables below provide a clear comparison between three methods used to calculate the Gamma function 1 Lanczos formula with k 6 2 Continuous fractions and 3 Windschitl Stirling Each of them implemented using both standard 10 digit and enhanced 13 digit precision routines The results clearly show that the best implementation is Lanczos and that the 13 digit routines provide a second order of magnitude improvement to the accuracy or in other words that it cannot compensate for the deficiencies of the used method We re lucky in that the more accurate method is faster that the second best albeit not as fast as Stirling s Obviously the extrapolation from integer case to the general case for the argument is assumed to follow the same trend albeit not shown in the summary tables Standard 10 digit Implementation Lanczos k 6 Continuous Fractions Windschitl Stirling Result error Result error Result Reference x 1 120 720 5040 40320 362880 3628800 39916800 479 001 600 6 227 020 800 Reference x 1 120 720 5040 40320 362880 3628800 CON OOF WD K CON OOF WD 1 000000001 1 2 9 999999999 24 00000001 120 720 0000008 9040 000002 40320 00003 362880 0002 3628800 001 39916799 99 479001599 5 6227020803 120 5040 40320 362880 3
70. ement by one the current content of the X or Y registers Also used instead of ISG X and DSE X when there s no desire to branch the program execution even if the boundary condition is reached this saves a NOP line placed right after the conditional instruction DSP returns in X the number of decimal places currently set in the display mode 0 regardless whether it s FIX SCI or END Little more than a curiosity it can be used to restore the initial settings after changing them for displaying or formatting purposes c Angel M Martin Revision 44_E Page 35 SandMath_44 Manual Normal Probability Distribution Function In probability theory the normal or Gaussian distribution is a continuous probability distribution that has a bell shaped probability density function Known as the Gaussian function or informally as the bell curve 1 f z 1 07 oyr The parameter u is the mean or expectation location of the peak and o 2 is the variance o is known as the standard deviation The distribution with u O and o 2 1 is called the standard normal distribution or the unit normal distribution Ey Probability density function Cumulative distribution function The read curve is the standard normal disinbution The figure above shows both the density functions as well as the cumulative probability function for several cases The Error function ERF in the SandMath can be used to calculate the CDF no need to apply brut
71. engthener to three fields for registers over 99 albeit this is de facto limited to 128 as we ll see later on The keyboard mapping for RIAT is as follows Numeric keypad or Top rows to perform the standard RCL SHIFT for Indirect register addresses EEX for the prompt lengthener to three places Math keys and to invoke the RCL Match functions Back arrow to cancel out to the OS Note that RIAT is not programmable This is done by design so that t can be used in a program to enter any of the RCL Math functions directly as a program line ignoring the corresponding prompt The drawback is of course that the standard RCL operation won t be registered in a program you must use the standard RCL function instead Notice also that indirect addressing is indeed supported by this scheme just add hex 80 that is decimal 128 to the register number you want to use as indirect register As simple as that So RCL IND 25 will be entered as the following two program lines RCL followed by 153 This however effectively limits the usefulness of the prompt lengthener to the range R100 to R127 because from R128 and on the index is interpreted as an indirect register address instead However the function will allow pressing SHIF and EEX for a combination of IND and prompt lengthener which will work as expected provided that the 128 limit isn t reached enough to make your head spin a little bit Example Store 5 in regi
72. evant integral in the definition are as follows z sinht Shi x f dt 0 i T x F 1 2 3 2 3 2 x 4 x 4F 1 2 3 2 3 2 x 4 cost 1 cosht 1 oy ine f dt Chi x 7 Ina f i 0 i x 4 2F 3 1 1 2 2 3 2 x7 4 x 4 oF3 1 1 2 2 3 2 x7 4 Examples 1 4 XEQ SI gt Si 1 4 1 256226733 or B5 Z 1 4 XEQ CI gt Ci 1 4 0 462006585 or iz V 1 4 XEQ SHI gt Shi 1 4 1 561713390 or ois 1 4 XEQ CHI gt Chi 1 4 1 445494076 or Daal 1 Z 1 V SiG l ga i ee amm a l Ci x NS 5 10 IS a ee eee i Even if it s not covered by the SandMath the following relation between the Exponential and Trigonometric Integrals is available E ix i 3r Si ax Cile 2 gt 0 c Angel M Martin Revision 44_E Page 70 SandMath_44 Manual Errare humanum est In mathematics the error function also called the Gauss error function is a special function non elementary of sigmoid shape which occurs in probability statistics and partial differential equations Its definition and the expression based on the Hyper geometric function via ascending series are given in the table below erf x 2x n exp x 1Fi 1 3 2 x The complementary error function denoted erfc is defined as erfc 1 erf x Both functions are shown below for an overview er
73. evision 44_E Page 4 SandMath_44 Manual 3 5 2 Hankel Struve and others The Launcher A Lambert relapse 82 Hankel functions yet a Bessel 3 Kind 83 Getting Spherical are we 83 Struve Functions 85 Lommel functions 86 Lerch Trascendent function 87 Kelvin functions 88 Kummer Functions 89 Associated Legendre functions 90 Toronto Function 91 Poisson Standard Distribution 91 3 5 3 Orphans and Dispossessed Tackle the Simple ones First 92 Debye Function 93 Dawson Integral 94 Hypergeometric Functions 95 Integrals of Bessel functions 96 Appendix 11 Looking for Zeroes 97 END 98 c Angel M Martin Revision 44_E Page 5 SandMath_44 Manual Note Make sure the revision G or higher of the Library 4 module is installed c Angel M Martin Revision 44_E Page 6 SandMath_44 Manual SandMath_44 Module Revision E Math Extensions for the HP 41 System 1 Introduction Simply put here s a compilation of mostly MCODE Math functions to extend the native function set of the HP 41 system At this point in time way over 30 years after the machine s launch it s more than likely not realistic to expect them to be profusely employed in FOCAL programs anymore yet they ve been included for either intrinsic interest read challenging MCODE or difficult to realize or because of their inherent value for those math oriented folks This module is an 8k implementation The first 4k
74. f x erfc x lt 3 zl O 1 2 3 The unsung hero HGF If we re to believe that behind a great man there is often an even greater woman then the greatest idea behind all these functions is the implementation of the Generalized Hyper geometric function A general purpose definition requires the use of data registers for the parameters al am and b1 bn and expects the argument x in the X register and the number of parameters m and n is stored in Z and Y for the generic expression mrp Aig doesn bj bo b x 01 2 arx ar ceee Am id b1 k b2 k Dp x x7 Ifm p 0 HGF returns exp x The program doesn t check if the series are convergent or not Even when they are convergent execution time may be prohibitive press any key to stop Stack register T is saved and x is saved in L register ROO is unused The alpha register is cleared The original HGF was written by Jean Marc Baillard Only small changes have been made to the version in the SandMath optimizing the code and checking for ALPHA DATA in all registers used as well as for the argument x c Angel M Martin Revision 44_E Page 71 SandMath_44 Manual Appendix 9 Inverse Error Function coefficients galore The inverse error function can be defined in terms of the Maclaurin series 20o 2k 1 17 Ck va af 2 ty Se k 0 Where c0 1 and Coy Ck il m T 127 a 2 ea e hha gore This really is a bear to
75. first of the polygamma functions Its relationship to the harmonic numbers is shown in that for natural numbers T n H i where Hn is the n th harmonic number and y is the Euler Mascheroni constant As can be seen in the figure on the left plotting the digamma function it s an interesting behavior showing the same poles and other singularities to worry about It should be possible to find an approximation valid for all the definition range of the function It has been implemented on the SandMath using the formulas derived from the called Gauss digamma theorem although further simplified in the following algorithm 1 4 1 1 1 W x log z 5 7553 aon 25275 P programmed as uA2 u 2 20 1 21 u 2 1 10 u 2 1 12 Ln u u 2 The implementation also makes use of the analytic continuation to take it to arguments greater than 9 same as it s done for LogGamma using the following recurrence relation to relate it to smaller values Wir 1 V x 7 Which naturally can be applied for negative arguments as well c Angel M Martin Revision 44_E Page 55 SandMath_44 Manual 3 3 6 Euler s Beta function The beta function also called the Euler integral of the first kind is a special function defined by 1 B y ey tdt Rela Re y gt 0 ri The beta function was studied by Euler and Legendre and was given its name by Jacques Binet The most common way to formula
76. hallenged by the available space in ROM the internal accuracy and the speed of the CPU It is therefore understandable that more commonality and re usable components identified will make it more self contained and powerful overcoming some of the inherent design limitations The Generalized Hyper geometric function is one of those rare instances that works in our favor as many of the special functions can be expressed as minor variations of the general case Indeed there are no less than 20 functions implemented as short FOCAL programs really direct applications of the general case saving tons of space and contributing to the general consistency and common approach in the SandMath We have Jean Marc Baillard to thank for writing the original HGF the Generalized Hyper geometric function real cornerstone of the next two sections The SandMath has an enhanced MCODE implementation that optimizes speed and accuracy thanks again to internal usage of 13 digit OS routines The reuse made of it more than pays off for its lengthy code A few examples will illustrate this Af erf x VA iF 4 8 r dalt a5 oFi a 1 iz H z Chibi Fy 1 3 2 a 3 2 z 4 2a a 3 2 Naturally this is not the case for any special function and even it there s such an expression it may be more appropriate to use the direct definition instead or an alternative one for the implementation This is the case of the Bessel functi
77. hree roots of the equation CVIETA Driver program for CROOT STLINE is a simple function to calculate the straight line coefficients from two of its data points P x1 y1 and P2 x2 y2 The formulas used are Y ax h with a Y2 y1 X2 X1 and b y 4a X1 It is trivial to obtain the root once a and b are known using Xo b a Example Get the equation of the line passing through the points 1 2 and 1 3 3 ENTER 1 ENTER 2 ENTER 1 gt Y 2 500 X 0 500 and to obtain its root I gt X 5 000 I will be shown in RUN mode only For the second and third degree equations use functions QROOT and CROOT The general forms are 2 ax bx ec 0 witha o Given the quadratic equation above QROOT calculates its two solutions or roots Input the coefficients into the stack registers Z Y X using a ENTER b ENTER c The roots are obtained using the well known formula X1 2 b 2a sqrt b 2a 2 c a Upon execution x1 will be left in Y and x2 will be left in X If the argument of the square root is negative then the roots z and z are complex and conjugated symmetrical over the X axis with Real and Imaginary parts defined by Re Z b 2a Z Re z i Im z Im Z sqrt abs b 2a 2 c a Z2 Re z i Im z Upon execution Im z will be left in Y and Re z will be left in X 2 2 m axt br ecr d witha 0 For the cubic equation case input the fo
78. ial function defined by the infinite sum or power series 2 z zt iis gt is z ks Js 3s Only for special values of the order s does the polylogarithm reduce to an elementary function such as the logarithm function The above definition is valid for all complex orders s and for all complex arguments z with z lt 1 it can be extended to z 1 by the process of analytic continuation For particular cases the polylogarithm may be expressed in terms of other functions see below Particular values for the polylogarithm may thus also be found as particular values of these other functions For integer values of the polylogarithm order the following explicit expressions are known Li z In 1 z mo z 7 0 m j iW 1 si o H Lia 2 Gaya 3 _ 2 1 2 i Li_a z z 7 a 0 5 l z 1 4z 4 z o Li_3 z 1 z o 4 z 14 231141024 z oe oe Li_4 z a Ea ea Lita Liste The SandMath implementation is a direct series summation adding terms until their contribution to the sum is negligible Convergence is very slow especially for small arguments Its usage expects n to be in register Y and x in register X The result is saved in X and X is moved to LastX The program below gives a FOCAL equivalent note the clever programming done by JM Baillard to only do one Y X to reduce the execution times significantly 01 LBL LIN 13 STO 03 02 STOO1 14 ISG 00 03 X lt gt Y 15 CLX
79. ich uses the same formula but with different steps subtracting k instead of 2 from the original number thus n n n l a 2 2 1 n n n 2 n 4 n nin 3 in 0 where the products run through positive integers Obviously for k 1 we have the standard FACT One can define the k th factorial denoted by nu recursively for non negative integers as 1 FUS nk nE p nin k ifn gt k Another extension to the factorial is the Superfactorial It doesn t use any step size as variant rather it follows a similar formula but using the factorial of the numbers instead of the numbers themselves Th sf n k KE 1 2 1 3 n 1 ent k k Both the multifactorial and specially the superfactorial will exceed the o i calculator numeric range rather quickly m f so the SandMath functions use a E i separate mantissa and exponent ale f approach using registers X and Y respectively Gin 2 i 4 i fo Pi Nevertheless the functions will put up a Lag Pa consolidated combined representation a in the display using the letter E to E a separate both amounts Make sure to pe adjust the FIX settings as approriate 484689 E3928 USER RAD 4 c ngel M Martin Revision 44_E Page 43 SandMath_ 44 Manual Examples _ Calculate the multi and superfactorials given below 234511111 type 6 ENTER XEQ MFCT gt 1234 type
80. isson Standard Distribution In probability theory and statistics the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and or space if these events occur with a known average rate and independently of the time since the last event A discrete stochastic variable X is said to have a Poisson distribution with parameter A gt 0 if for k 0 1 2 the probability mass function of X is given by ee fik A Pr X k kl Poisson distribution A 3 Its inputs are k and A in stack registers f Y and X PSD s result is the probability aiii corresponding to the inputs 020 0 15 0 7190 gg DAD e ee es 1 Fa 3 4 5 i f i 1m 1 Fs Note that this function is not in the Hankel launcher technically it belongs to the Statistics section c Angel M Martin Revision 44_E Page 91 SandMath_44 Manual 3 4 3 Orphans and dispossessed The last group of sub functions include those not belonging to any particular launcher for no other particular reason that there s no more available space in the ROM Keep in mind that the only way to execute them is using the FL and XFL launchers _ Function Author _ Description O O i i SC s s sCSCSC sS SPFNC_ ngelMartin Cat header does FCAT i Z o o BS_ ngelMartn Aux routine All Bessel O Z O Z O O i y O ngel Marti
81. isters respectively When executed in RUN mode the display will show the straight line equation similar to the STLINE function described before e COV will calculate the sample covariance CORR will return the correlation coefficient and YLR the linear estimate for a given x Example find the y intercept and slope of the linear approximation od the data set given below x p w o w Assuming all data pairs values have been entered using Y value ENTER X value s we type XEQ LR gt 0 038650000 and X lt gt Y gt 0 038650000 producing the following output in FIX 2 c Angel M Martin Revision 44_E Page 34 SandMath_44 Manual Ratios Sorting and Register Maxima T is a miniature function to calculate the percent of a number relative to another one its reference The formula is T y x 100 x y Example the relative percent of 4 over 25 is 16 GCD and LCM are fundamental functions also inexplicably absent in the original function set They are short and sweet and certainly not complex to calculate The algorithms for these functions are based on the PPC routines GC and LM conveniently modified to get the most out of MCODE environment If a and b are not both zero the greatest common divisor of a and b can be computed by using least common multiple Icm of a and b a b cd a b ian lem a b Examples GCD 13 17 1 primes GCD 12 18 6 GCD 15 33 3 Examples LC
82. known and don t require any special consideration to program e e sinh z 2 i ete Vie EO AAE cosh _ ji 2 et tanhzr er eT y The SINH code is also used as a subroutine for the Digamma function c ngel M Martin Revision 44_E Page 25 SandMath_44 Manual The direct functions are basically exponentials _ 3 whilst the inverses are basically logarithms arsinh x In 2 VE 1 Both cases are well covered with the mainframe arcoshz ln z Vx 1 2x gt 1 internal math routines without any need to worry about singularities or special error handling 1 l z artanha In be 1 2 l r For all hyperbolic functions the input value is expected in X and the return value will also be left in X The original argument is saved in LASTx No data registers are used Examples Complete the table below calculating the inverses of the results to compare them with the original arguments Use FIX 9 to see the complete decimal range SIN SIN HCEODS L WHTAN I SIN H COS TAN i HMKEYS assigns to the SHIFT key for convenience x HSIN HASIN HCOS HACOS HTAN HATAN 1 175201194 1 000000000 1 543080635 1 000000000 0 761594156 0 761594156 1 001 1 176744862 1 001000000 1 544256608 1 001000000 0 762013811 1 001000000 0 010000167 0 010000000 1 000050000 0 009999958 0 009999667 0 010000000 0 0001 0 000100000 0 000100000
83. l functions Jn and Yn by jal Y Jale yale Y3 Yarla 1 naa Which graphical representation naturally very JBS ish looking is show below xX X spherical Bessel functions of 1st kind JC for 0 1 h Spherical Bessel functions of 2nd kind y x for 0 1 a Notice that there really isn t any Spherical i n x properly defined but there s one in the SandMath just the same using the same relationship as for j n x and y n x Once again remember than as n x increases the accuracy of the results decreases specially for J n x Y n x and the spherical counterparts where the returned value can be completely incorrect if n x gt 20 a practical rule not an absolute criterion c Angel M Martin Revision 44_E Page 59 SandMath_44 Manual Programming Remarks The basic algorithms use the summation definition of the functions calculating the successive values of the sum until there s convergence for the maximum precision 10 decimal places on the display Therefore the execution time can take a little long a fact that becomes a non issue on the CL Or when using 41 emulator programs like V41 setting the turbo mode on There are different algorithms depending on whether the order is integer or not This speeds up the calculations and avoids running into singularities as mentioned before Note that for integer indexes the gamma function changes to a factorial calculation which
84. l functions can be expressed in terms of the hypergeometric function or as limiting cases of it Some typical examples are In 1 z z9F 1 1 2 z 1 z 2F a 1 1 z 2 arcsin z Zof 5 gt 2 z The relation 2F a b c x 1 x a 2F a c b c x 1 x is used if x lt 0 The Regularized Hypergeometric function has a similar expression for each summing term just divided by Gamma of the corresponding Pochhamer symbol plus the index n REGISTERS RO1 thru R03 They are to be initialised before executing HGF or RGHF ROO is not used RO1 a R02 b R03 01 a R02 b R03 c Stack Output 2F a b c x HGF Examples e 1 2 STOO1 2 3 STO02 3 7 STO 03 0 4 XFL HGF gt 1 435242953 3 XFL HGF gt 0 309850661 RHGF Examples e 2 STOOI 3 STO02 7 STO 03 0 4 XFL RHGF gt 5353330 290 3 XFL RHGF gt 2128 650875 c Angel M Martin Revision 44_E Page 95 SandMath_44 Manual Integrals of Bessel Functions One of the usual approaches is to use the following recurrent relations for the calculation T i f J L t di 25 Fuvtor i le I kool With Re n gt 0 More specifically for positive integer orders n 1 2 we have T 7 m Ja t dt hdt 2V Jona 2 j aie gt and also T i Jan tf dt n Jal o 2 Jo 2 3 2 c There s however another approach based yes here as well on the Generalized Hypergeometric function HGF In fac
85. lot For the nth prime number pn the primorial pn is defined as the product of the first n primes where pk is the kth prime number primes lt n ls mn Pn I Pr np I Pi Prin FF k l i 1 The FOCAL programs below can be used to calculate both flavors of primorials Note the primordial pun intended role of function PRIME which effectively makes this a simple application as opposed to a full fledge program from the scratch Examples Calculate both primorials for the first 20 natural numbers c Angel M Martin Revision 44_E Page 46 SandMath_44 Manual Table of primorials n n Pn Pn 0 1 no prime i l l 2 2 2 3 3 6 5 30 4 6 7 210 5 30 11 2310 6 30 13 30030 7 210 17 510510 8 210 19 9699690 9 210 23 223092870 10 210 29 6469693230 11 2310 31 200560490130 12 2310 37 7420738134810 13 30030 41 304250263527210 14 30030 43 13082761331670030 15 30030 47 614889782588491410 16 30030 53 32589158477190044730 17 510510 59 1922760350154212639070 18 510510 61 117288381359406970983270 19 9699690 67 7858321551080267055879090 20 9699690 71 557940830126698960967415390 01 LBL NPRML 01 LBL PPRML 02 ABS 02 ABS 03 INT 03 INT 04 E 04 E 05 X gt Y 05 X gt Y 06 RTN 06 RTN 07 X lt gt Y 07 STO Z LBL 00 09 PRIME LBL 00 INCX 10 GTO 01 10 PRIME 1 X lt gt L li GTO 01 GTO 03 12 X lt gt L 1 GTO 00 gi 7 GTO 00 17 GTO 00 18 X lt gt Y 18 RCL Z 19 RTN 19 END Both routi
86. ls for n 0 1 2 3 4 and 2 lt x lt 2 im A n 4 n 3 n Function POCH calculates the rising factorial It expects n and x to be in the Y and X registers respectively i e the usual convention For large values of n the execution time may be very long you can hit any key to stop the execution at any time The falling factorial is related to it a k a Pochhammer symbol by xn 1 x The usual factorial n Is related to the rising factorial by n Whereas for the falling factorial the expression is n py Examples Calculate the rising factorial for n 7 x 4 and the falling factorial for n 7 x 7 7 ENTER 4 XEQ POCH gt 604 800 0000 7 ENTER 7 CHS XEQ POCH 7 XEQ CHSYX gt 5 040 000000 c Angel M Martin Revision 44_E Page 42 SandMath_44 Manual Multifactorial Superfactorial and Hyperfactorial This section covers the main extensions and generalizations of the factorial There are different ways to expand the definition depending on the actual sequences of numbers used in the calculation The double factorial of a positive integer n is a generalization of the usual factorial n defined by ne a 2 5 3 1 n gt 0odd nlledn n 2 60 4 2 n gt Oeven l n 0 Even though the formulas for the odd and even double factorials can be easily combined into nll r 2i i hian The double factorial is a special case of the multifactorial wh
87. n Aux routine 2nd kind Integer orders AWL ngelMartin Inverse Lambert O O Z S DAW mM Baillard Dawson integral oo DBY Ballard Debye functions gt Z o O o o HGF sMBaillard Hyper geometricfunction O Z ITI Angel Martin Integral if IBS ITJ Angel Martin Integral of IBS LE Angel Martin Logarithmic Integral S PSD ngel Martin __ Poisson Standard Distribution RHGF JM Baillard _ _ Regularized Hyper geometric function _ SAE JMBaillard Surface Area of an Ellipsoid FCAT AngelMartin Function Catalogue O S Let s tackle the simpler ones on the list first FCAT and SP FNC are usability enhancements for the admin and housekeeping It invokes the sub function CATALOG with ot keys for individual function launch and general navigation Users of the POWERCL Module will already be familiar with its features as it s exactly the same code which in fact resides in the Library 4 and it s reused by both modules so far BS and BS2 are auxiliary functions used in the FOCAL programs for the Bessel functions of 2 Kind KBS and YBS They were explained in more detail in the Bessel Functions paragraph AWL is the Inverse Lambert W function an immediate application of the W definition involving just the exponential but with additional accuracy when using the 13 digit routines in MCODE AWL W exp W LI is the Logarithm
88. n XFL prompt and offers the following seven choices The table below shows in the first column the letter used for each of the functions within this group CR Author Description __ O Z o E ELIPF AngelMartin Elliptic Integral F CRF JMBailard___ Carlson Integral ist Kind _ _ ooo o G CRG IM Baillard Carlson Integral 2nd Kind _ __ _ _ o CRJ JMBailard___ Carlson Integral 3rd Kind _ ___ C CSX_ JMBaillard________ Fresnel Integrals C x amp S x W WEBAN JmBaillard Weber and Anger functions Airy Functions Ai x amp Bi x The Elliptic Integrals In integral calculus elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse They were first studied by Giulio Fagnano and Leonhard Euler Modern mathematics defines an elliptic integral as any function f which can be expressed in the form f a f R t PE dt C where R is a rational function of its two arguments P is a polynomial of degree 3 or 4 with no repeated roots and c is a constant The most common ones are the incomplete Elliptic Integrals of the first second and third kinds Besides the Legendre form given below the elliptic integrals may also be expressed in Carlson symmetric form which has been the basis for the implementation in the SandMath completely based on the JMB_MATH ROM The incomplete elliptic integral of the first
89. n precision for calculations significant decimal places FFOUR is a rather large FOCAL program despite having a MCODE FAT entry It calculates all integrals internally not making use of general purpose numeric integrators like INTEG IG etc so it s totally self contained The function must be programmed in main memory under its own global label The program prompts for the first index to calculate and the number of desired coefficients The program also calculates the approximate value of the function at a given argument applying the Summation of the terms using the obtained coefficients if T nx A wy anx x ag Ty COS Dy in f iat a n tA To use it simply enter the value of x and press A XEQ A in user mode on this assumes that no function is assigned to the key The approximation will be more correct when a sufficient number of terms is included The goodness is also dependent on the argument itself c Angel M Martin Revision 44_E Page 74 SandMath_44 Manual Example calculate the first six coefficients for F x x 2 assuming a period T 2pi centered in x0 0 As it s known XA2 4 3 n 2 XL 4 cos nx n 2 4r sin nx n n 0 1 Using an accuracy of 6 decimal places the program returns the following results p a0 13 1595 b0 0 al 4 b1 12 566 a2 1 b2 6 5797 a3 0 4444 b3 4 1888 a4 0 250 b4 3 1415 a5 0 160 b5 2 513
90. n the program 9 STO 00 10 FIX 4 ST X l l l ii Simply enter the information asked at T NDEN F the prompts including the precision 13 desired number of decimal digits T STO 0 RER function name and its chosen period 15 CF 00 27 16 AEG 00 SIN P E The screenshot below shows the 19 FEN ILPER output of the process 20 0 21 RCL 00 22 INTEG 23 RCL OO 24 25 ST XK 26 g 27 FS OO 28 h Example f x x 3 141593 79 nf 2 000000 31 AINT 2 END RUN 33 op jo PREC 33 ARCLY P a RUN 34 PROMPT 35 FC OO 7 000000 RUN 36 RTN 37 GTO E Using this program we ll calculate the coeffcients for the 7 and 9 terms for f x x 2 a7 0 081633 b7 1 795196 and a9 0 049383 b9 1 396263 c Angel M Martin Revision 44_E Page 76 SandMath_44 Manual 3 5 More Special Functions in the Secondary FAT We ve finally come to the last part of the manual covering those functions included in the Sub functions group with entries in the secondary hidden FAT Go ahead and review the accessibility information from the introduction for a quick refresher if needed Let s divide and conquer using the Carlson and Hankel launchers as grouping criteria 3 5 1 Carlson Integrals and associates The first sub function launcher is the Carlson group It s loosely centered on the Carlson s integrals plus related functions The launcher prompt is activated by pressing O at the mai
91. nals of BS2 perform quite an involved procedure utilizing multiple resources within the SandMath module Furthermore BS2 is called twice within the FOCAL program to calculate KBS or YBS once for the first infinite summation and a second time for the second finite sum The status of User Flag 00 controls the calculation made That was done to save one FAT entry when the limiting factor was the maximumm nuber of functions per page i e 64 functions Now they have been pushed eveh further off to the secondary FAT used for the sub functions group Bessel Function Summed Functions by BS2 Flag00 FlagO1 Set Yn x fk n x cear DE gK N x e cia Note also that for this case integer orders there s two infinite summations involved for the Bessel functions of the second kind This is done simultaneously within BS2 when user flag 02 is set as both series converge in very similar conditions i e with the same number of terms Main functions IBS JBS KBS and YBS The first kind pair IBS and JBS are 100 written in MCODE including exception handling and special cases This is the only version known to the author of a full MCODE implementation on the 41 platform and it is however a good example of the capabilities of this machine No data registers are used for IBS and JBS but both the stack and the Alpha registers are used The number of terms required for the convergence is stored in register N up
92. ncher radix activates the FRC launcher switches into the hyperbolic choices Use SHIFT to toggle direct inverse SHIFT ALPHA activates the XFL sub functions launcher lt back arrow cancels it or returns to it from a secondary menu c Angel M Martin Revision 44_E Page 9 SandMath_44 Manual Like the native implementation the name of the launched function will be shown in the display while you hold the corresponding key and NULLED if kept pressed This provides visual feedback on the action for additional assurance Typically the secondary launchers have the possible choices in their prompt we ll see them later on The STAT menu differs from the others in that it consists of two line ups toggled with the SHIFT key providing access to 10 functions using the keys in the top row directly below the function symbol This is a good moment to familiarize yourself with the 3g launcher Go ahead and try it using it also in PRGM mode to enter the functions as program lines Note that when activating XFL you ll need to press ALPHA a second time to spell the sub function name Direct access function keys in alphabetical order A Stat Prob MENUS B Euler s Beta C Digamma PSI D Rieman s Zeta E Gamma Natural log F Inverse Gamma G Euler s Gamma H Hankel s Launcher I Bessel I n x J Bessel J n x Hyperbolics Launcher K Bessel K n x L Bessel Y n x
93. ncher 1 Recurrence Sum BS STO 00 save J n x here used by Hankel FC 0t _ fis KBS Z LLun J IGTO 01 TT eee ee RCL 01 n PI COS LBL 01 RCL 02 partial result 4 RCL 01 n PI SIN FS 0t jis VBS IGTO03 gt i 2 AA RC TE OA ERM ek ee OO E E RN ND HALFX PI CHS 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 bo n a a a O O O O O O O O o O a I O a O a S O O O O O O O a 43 Y 1 J ST X 3 ST 02 RCL 02 orden argument swapped default case is it negative reset counter and partial sum skip if n 0 selects B2 amp gk n x k 0 1 n 1 BS2 abs n x 2 selects B1 DL fk n x k 0 1 2 BS2 partial result showing off partial result is it YBS c Angel M Martin Revision 44 E Page 63 SandMath_44 Manual 3 3 9 Riemann Zeta function Perhaps one of the most studied functions in mathematics it owes its popularity to its deep rooted connections with prime numbers theory Not having an easy approximation to work with its implementation on the 41 will be a bit of a challenge mainly due to the very slow convergence of the series representation used to program it Be assured that this numeric calculation won t help you prove the Riemann hypothesis and collect the 1
94. nd how can they be accessed The answer is called the Multi Function groups Multi Functions XFL and xFL provide access to an entire group of sub functions grouped by their affinity or similar nature The sub functions can be invoked using either its index within the group using XFL or its direct name using XFLS This is implemented in such a way that they are also programmable and can be entered into a program line using a technique called non merged functions You may already be familiar with this technique originally developed by the HEPAX programmers In the HEPAX there were two of those groups one for the XF M functions and another for the HEPAX A extensions The PowerCL Module also contains its own and now the SandMath joins them this time applied to the mathematical extensions particularly for the Special Functions group A sub function catalog is also available listing the functions included within the group Direct execution or programming if in PRGM mode is possible just by stopping the catalog at a certain entry and pressing the XEQ key The catalog behaves very much live the native ones in the machine you can stop them using R S SST BST them press ENTER to move to the next sub section cancel or resume the listing at any time As additional bonus the sub function launcher XFL will also search the main module FAT if the sub function name is not found within the multi function group so th
95. nes only use the stack no data registers or user flags are used Clearly the numeric range will again be the weakest link reaching it for n 54 for PPRML and n 251 for NPRML c Angel M Martin Revision 44_E Page 47 SandMath_44 Manual A glimpse of what s ahead sil Hy f oF A N C S 7 Relationship between common special functions Taken from John Cook s web site http www johndcook com special_ function diagram html c Angel M Martin Revision 44_E Page 48 SandMath_44 Manual A word about the approach The Hyper Geometric Function as a generic function generator or the case of the chameleon function in disguise Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis functional analysis physics or other applications frequently as solutions to differential equations There is no general formal definition but the list of mathematical functions contains functions which are commonly accepted as special Some elementary functions are also considered as special functions The implementations described in this manual do nothing but scratching the surface or more appropriately gingerly touching it of the Special Functions field where one can easily spend several life times and still be just half way through Implementing multiple special functions in a 41 ROM is clearly c
96. ngel Martin _ RCL Math Launcher RCL _ Angel Martin _ RCL Plus RCL __ Angel Martin _ RCL Minus R ike RCL __ Angel Martin RCL Division o e RCL __ Angel Martin RCL Power SS ike AIRCL__ Angel Martin _ ARCL integer Part of number in Register nn _ __ 2 4 1 Individual Recall Math functions The five RCL Math new functions cover the range of four arithmetic operations like STO does plus a new one added for completion sake The functions would recall the number in the register specified by the prompt and will perform the math using the number in register X as first argument and the recalled number as the second argument Design criteria for these were 1 should be prompting functions 2 should support indirect addressing SHIFT 3 should utilize the top 2 rows for index entry shortcut The first condition is easy to implement in RUN mode as it s just a matter of selecting the appropriate prompting bits in the function MCODE name But gets very tricky when used under program mode This has been elegantly resolved using a method first used by Doug Wilder by means of using the program line following the instruction as the index argument Somewhat similar to the way the HEPAX implemented it although here there s some advantages in that the length of the index argument doesn t need to be fixed dropping leading zeroes and even omitting it altogether if it s zero assuming the following line isn t
97. nvention used is shown in the figure below defining the origin and direction of the azimuth and polar angles as referred to the rectangular axis The SandMath implementation makes use of the fact that appropriate dual P R conversions are equivalent to Spherical and vice versa c Angel M Martin Revision 44_E Page 17 SandMath_44 Manual HMS and HMS complement the arithmetic handling of numbers in HMS format adding to the native HMS and HMS pair As it s expected the result is also put in HMS format as well xDGT is a small divertiment useful in pseudo random numbers generation It simply returns the sum of the mantissa digits of the argument at light blazing speed using just a few MCODE instructions More about random numbers will be covered in the Probability Stats section later on Entering the base conversion section The following functions are available in the SandMath Author Description George Eldridge Base to Decimal George Eldridge Decimal to base in Y William Graham Decimal to Hex William Graham Hex to Decimal T gt BS__ Base Ten to Base prompting version The first two are FOCAL programs taken from the PPC ROM They are the generic base b to from Decimal conversions The Direct conversion D gt BS expects the base in Y and the decimal number in X returning the base b result in Alpha The inverse function BS gt D uses the string in Alpha and the base in X as arguments You can chain them to en
98. on limit Granted it isn t the fastest one in town but such isn t an issue on a modern day emulator and the economy of code cannot be stronger EAG dt a Cl w OO EAG dt sri Which allegedly satisfies the equation o Jn t dt 2 Jn 2 Jni3 X Jn 5 X seeeso c ngel M Martin Revision 44_E SandMath_ 44 Manual This completes this manual Don t forget to check Jean Marc Baillard extensive and authoritative references on the web despite its unassuming web site name located at http hp41programs yolasite com A treasure chest awaits you enjoy the ride c Angel M Martin Revision 44_E Page 98
99. on termination The second kind pair KBS and YBS is implemented using a FOCAL driver program for BS and BS2 Notably more demanding than the previous two their expressions require additional calculations that exceed the reasonable MCODE capabilities Although they re not normally supposed to be used outside of the Bessel program BS and BS2 could be called independently Both use the same input parameters index in Y and half of the argument in X Pay close attention to the status of user flags 00 and 01 as they directly influence their result c Angel M Martin Revision 44_E Page 61 SandMath_44 Manual Other functions used in the Bessel calculations which aren t part of the native HP 41 function set are as follows GEU Euler s constant 0 577215665 CHSYX CHS repeated n times in X register multiplied by Y register INT and FRC Conditional based on value in X being integer or fractional E3 E Self explanatory divides X by 1 000 then adds one Examples J 1 1 0 440050586 I 1 1 0 565159104 J 1 1 0 440050586 I 1 1 0 565159104 J 0 5 0 5 0 540973790 I 0 5 0 5 0 587993086 J 0 5 0 5 0 990245881 I 0 5 0 5 1 272389647 Y 1 1 0 781212821 K 1 1 0 601907230 Y 1 2 0 107032431 K 1 2 0 139865882 Y 0 5 0 5 0 990245881 K 0 5 0 5 1 075047604 Y 0 5 0 5 0 540973790 K 0 5 0 5 1 075047604 Error Messages Note that the functions will return a DATA ERROR message
100. ons which use the series definition in the SandMath the Gamma function using the Lanczos formula etc With that said let s delve into the individual functions comprising the High Level Math group First off come those more frequently used so that they have gained their place in the ROM s main FAT Looking at the authorship you ll see the tight collaboration between JM and the author as stated in the opening statements of this manual c Angel M Martin Revision 44_E Page 49 SandMath_44 Manual 3 3 1 Gamma function and associates Let s further separate these by logical groups depending on their similarities and applicability The first one is the GAMMA and related functions 1 GM PSI PSIN LNGM ICGM BETA and ICBT all of them a Quantum leap from the previous functions described in the manual both in terms of the mathematical definition and as it refers to the required programming resources and techniques __ Function Author Description o Z o O ooo Di a GMF JM Baillard _ Reciprocal Gamma Continuous fractions TAg BETA Angel Martin _ Euler s Beta function _ _ o Z o TAJ GAMMA Angel Martin __ Euler s Gamma function Lanczos TAJ ICBT_ JM Baillard Incomplete Beta function __ _ o o Pi IcemM JM Baillard ______ Incomplete Gamma function __ o TA LNGM Angel Martin _ Logarithm Gamma function _ _ _ _
101. ordinary factorial and its value is therefore closer to the square root of the factorial It is denoted by n For an odd positive integer n 2k 1 k gt 1 itis k 2k 1 2i 1 z 1 A common related notation is to use multiple exclamation points to denote a multifactorial the product of integers in steps of two n three n or more The double factorial is the most commonly used variant but one can similarly define the triple factorial n and so on One can define the k th factorial denoted by n recursively for non negative integers as wie 2 if0 lt n lt k nf n k ifn gt k The figures below show the plots for X right a comparison with log abs gamma red versus log abs doublegamma green left Using the Logarithm is helpful to deal with large arguments as these functions go beyond the calculator numeric range very quickly Also ran out of space in the module to have more than one function on this subject thus LGMF was chosen given its more general purpose character The implementation is thru an all MCODE function yet execution times may be large depending on the arguments LGMF may also be used to compute factorials use n 1 and then E X on the result Obviously the accuracy won t be the greatest but it s a reasonable compromise Output pf LGMF x Examples 2 ENTER 100 XEQ LGMF gt Log 100 79 53457468 999 ENTER 123456 XEQ LGMF g
102. pherical Hankel1 Spherical Hankel2 Toronto function Weber and Anger functions Lambert WO Lambert W1 Angel Martin JM Baillard JM Baillard JM Baillard JM Baillard JM Baillard JM Baillard Angel Martin JM Baillard Angel Martin Angel Martin JM Baillard Angel Martin Angel Martin JM Baillard Angel Martin JM Baillard Angel Martin JM Baillard JM Baillard JM Baillard JM Baillard Angel Martin Angel Martin JM Baillard JM Baillard Angel Martin Angel Martin c Angel M Martin Revision 44 E Page 13 SandMath_44 Manual The last section groups the factorial functions circling back from the special functions into the number theory field a timid foray to say the most Name Description Author Section Header Falling Factorial Angel Martin Pochhammer symbol Angel Martin Multi Factorial JM Baillard Logarithm Multi Factorial JM Baillard Poisson Standard Distribution Angel Martin Super Factorial JM Baillard Extended Factorial Angel Martin FCAT Function Catalogue Angel Martin The best way to access FCAT is through the main launcher DAAI then pressing ENTERN CN FCAT and SP FNC are usability enhancements for the admin and housekeeping It invokes the sub function CATALOG with ot keys for individual function launch and general navigation Users of the POWERCL Module will already be familiar with its features as it s exactly the same code which in fact resides in the Library 4 and it
103. reflection formula if you wish The interesting fact about this is how it has been 2 5 implemented if x lt 1 then the MCODE function branches to a a FOCAL program that as part of the calculations calls the MCODE function after doing the change x 1 x which obviously is gt 1 perx gt 2 a gin he Le perz lt 1 c Angel M Martin Revision 44_E Page 64 SandMath_44 Manual Really the direct method isn t very useful at all and it s more of an anecdotal implementation with academic value but not practical The Borwein algorithm provides an iterative alternative to the direct method with a much faster convergence even as a FOCAL program and more comfortable treatment It is implemented in the SandMath as a courtesy of JM Baillard in the function ZETAX For example using ZETAX to calculate Z 1 001 returns the correct solution is a few seconds See the appendices for a FOCAL listing of the program if interested Examples Complete the table below for C x using both the direct method and the Borwein algorithm Use the result in WolframAlfa as reference to also determine their respective errors Borwein 5 0 0039682539682 0 003968254 8 0136E 09 0 003968254 8 0136E 09 5 1 036927755 1 03692775 4 96019E 09 1 036927755 1 38255E 10 1 202056903 1 20205676 1 19096E 07 1 202056903 1 32764E 10 1 6449340668482 1644934066 5 15644E 10 10 58444846 10 58444847 4 77115E 10 We
104. rning X depending on whether the number in Y is even or odd respectively c ngel M Martin Revision 44_E Page 15 SandMath_44 Manual ATAN2 is the two argument variant of arctangent Its expression is given by the following definitions arctan 4 rz gt m arctan y gt 0 r lt 0 m arctan y lt O0 2 lt 0 atan2 y r7 _ _ z y gt 0z y lt 0 2 0 undefined y 0 z E3 E does just what its name implies adds one to the result of dividing the argument in x by one thousand Extensively used throughout this module and in countless matrix programs to prepare the element indexes FLOOR and CEIL The floor and ceiling functions map a real number to the largest previous or the smallest following integer respectively More precisely floor x x is the largest integer not greater than x and ceiling x x is the smallest integer not less than x The SandMath implementation uses the native MOD function through the expressions CEIL x x MOD x 1 and FLOOR x x MOD x 1 GEU is a new constant added to the HP 41 the Euler Mascheroni constant defined as the limiting difference between the harmonic series and the natural logarithm oe y lim In n f Eare 2 i The numerical value of this constant to 10 decimal places is y 0 5772156649 The stack lift is enabled allowing for normal RPN style calculations It appears in formulas to calculate the Psi function
105. routine listing 3 3 3 Lower Incomplete Gamma function In mathematics the upper and the lower incomplete gamma functions are respectively as follow C s x Et e dt ylas T t e dt T E Connection with Kummer s confluent hypergeometric function when the real part of z is positive ya z s tze 7 M 1 8s 1 2 which is the expression used in the SandMath The Upper incomplete Gamma function can be easily obtained from the relationship y s z T s x T 8 c ngel M Martin Revision 44_E Page 53 SandMath_44 Manual 3 3 4 Log Gamma function Many times is easier to calculate the Logarithm of the Gamma function instead of the main Gamma value This could be due to numeric range problems remember that the 41 won t support numbers over E100 or due to the poles and singularities of the main definition The SandMath uses the Stirling approximation to compute LogGamma as given by the following expression directly obtained from the formula in page 27 1 2n T z n 27 Inz z 21n z In zsinh z 2 z 8l z This approximation is also good to more than 8 decimal digits for z with a real part greater than 8 For smaller values we ll use the functional equation to extend it to the region where it s accurate enough and then back calculate the result as appropriate tu The picture on the left shows the LogGamma function for _ positive b arguments Interestingly it has a negati
106. s the usual conventions for the stack registers entering x as the last parameter in register X despite the written form Output 2 ee T Examples PI ENTER 0 6 ENTER 0 7 FL LERCH gt 0 7 7 0 6 5 170601130 3 ENTER 4 6 ENTER 0 8 FL LERCH gt 0 8 3 4 6 3 152827048 Or a1 H R instead c ngel M Martin Revision 44_E Page 87 SandMath_44 Manual Kelvin Functions In applied mathematics the Kelvin functions of the first kind Berv x and Beiv x and of the Second kind Kerv x and Keiv x are the real and imaginary parts respectively of Si 4 P f mifa dy re for the ist Kind Ky ze for the 2nd Kind These functions are named after William Thomson ist Baron Kelvin For integers n Bern x and Bein x have the following series expansion Ber x con 7 2 L kIT n k 1 4 and on k Beno 2 onl Dl 2 2 Z kT n k 1 4 The figure below shows Ber n x and Ker n x for the first 4 integer orders and real arguments Ber n X and Bei n x are available in the SandMath implemented as FOCAL programs written by JM Baillard Both values are calculated simultaneously by KLV and left in X Y registers as follows Stack Output Example 2 SQRT PI FL KLV gt ber sqrt 2 x 0 674095951 X lt gt Y gt bei sqrt 2 n 1 597357210 Or ml H K instead c Angel M Martin Revision 44_E Page 88 San
107. s wasted by eGObEEP and W instead This method is used in several functions of the SandMath module like the RCL math functions just described Appendix 3 A trip down to Memory Lane From the HP 41 User s Handbook Automatic Primary Extended Memory Data Storage Data Storage Stack Registers Registers T Foo Roo Fd ALPHA Ra The standard Raon Y Roo HP 41 0 has a You can add up to four memory modules X LAST X o f 83 Rag Red bringing the total te 100 primary and 219 Pall memeryniodulas whe primary storage extended storage registers allocated to storage registers m PGE each additional module would Rea account for the following register R addresses S The Function Rej US The Indirect Address Register Ros 10 0000 The Desired Register Recalled into the reqister Rio 2 3400 Storage Register Arithmetic Arithmetic can be pertormed upon the contents of all storage registers by executing followed by the artthmetic function followed in tum by the register address For example Opertion Result O1 Number tn X register is added to the contents of register Eoi and the sum is placed into Eoi The display execution form of this is sT 02 Number in A register is subtracted from the contents of register Rios and the difference is placed into Ros The display execution form of this is 5T 1E Number in X resister is multiphed by the contents of register Ros and the product is placed into Ros The display ex
108. scovery was posthumously published in 1712 in his work Katsuyo Sampo Bernoulli s also posthumously in his Ars Conjectandi of 1713 Ada Lovelace s note G on the analytical engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage s machine As a result the Bernoulli numbers have the distinction of being the subject of the first computer program I 01 LBL BN2 i TE ob Barnaulli numi iwib If r3 oz K 1 TA Beare a i 03 GTOO1 1 4 o4 x 0 p N 05 INCX 06 X 1 a2 07 RIN kN ne 08 ODD T a p all w oo CLX ks 10 X 0 7 0 25 11 RTN i 3 G 12 l2 2 il r 13 M lt 14 GTO 00 15 6 There are several or rather many algorithms and approaches to l6 1 X the calculation of Bn In this particular example we ll use the 17 RIN expression based on the Riemann s Zeta function according to l1 amp LBL OO a which the values of the Riemann zeta function satisfy 19 Si X 20 M lt 21 GTOOO n 1 n Bn ae l 23 1 X for all integers n20 The expression n 1 n for n 0 is to be ca lie understood as the limit of x 1 x when x gt 0 aa F 26 XoY Sanaan oo 27 LBLO1 The FOCAL program on the right is a super short application of the 36 STOM ZETA function even if it s used for negative arguments Obviously 39 CHS we ve single cased the troublesome points to avoid execution times 30 INCX unreasonably long but apart from that it s quite generic in its 31 ZETA approach
109. see that not only is the Borwein algorithm faster and more capable in range but also their results are more accurate than the direct approach MCODE or not 13 digit internal subroutines notwithstanding 02 j DRO So oLa X Note The following links to the MAA and the now defunct Zetagrid make fascinating reading on the Zeta zeros current trends and historic perspective make sure you don t miss them http www maa org editorial mathgames mathgames_10_18_04 html http www zetagrid net c Angel M Martin Revision 44_E Page 65 SandMath_44 Manual Appendix 8 Putting Zeta to work Bernoulli numbers In mathematics the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory The values of the first few Bernoulli numbers are BO 1 B1 1 2 B2 1 6 B3 0 B4 1 30 B5 0 B6 1 42 B7 0 B8 1 30 If the convention B1 1 2 is used this sequence is also known as the first Bernoulli numbers with the convention B1 1 2 is known as the second Bernoulli numbers Except for this one difference the first and second Bernoulli numbers agree Since Bn 0 for all odd n gt 1 and many formulas only involve even index Bernoulli numbers some authors write Bn instead of B2n The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli after whom they are named and independently by Japanese mathematician Seki Kowa Seki s di
110. ster R101 and 55555 000 in register R5 This requires some indirect addressing as well say using register Y the sequence would be 101 ENTER 5 STO IND Y and then 55555 STO 5 Then execute RCL IND 101 press RAM SHIFT EEX 0 1 gt to obtain 55555 00 in X Note general purpose prompt lengtheners are a better alternative to the EEX implementation used here Their advantage of course is that they are applicable to all mainframe prompting functions not only to the enhanced RCL Thus for instance you could use it with STO as well removing the need for indirect addressing to store 5 in R101 The AMC_OS X module has a general purpose prompt lengthener activated by pressing the ON key while the function prompt is up c Angel M Martin Revision 44_E Page 28 SandMath_44 Manual Pressing ALPHA at the RCL prompt will invoke function AIRCL _ _ This will in turn prompt for a data register number and once filled it ll append the integer part of the value stored in that register to the ALPHA register thus equivalent to what AINT does with the x register Note that AIRCL _ _ is fully programmable When entered in a program you d ignore the prompts and the program step following it will be used to hold the register number to be used by ARCLI when the program runs This technique is known as non mergea functions to work around the limitation of the OS Too bad we can t use the Byte Table location
111. t Log 123456 578 0564932 And now lets move to something completely unrelated Fourier Series representation of a given function f x c Angel M Martin Revision 44_E Page 45 SandMath_44 Manual Appendix 5 Primorials a primordial view Welcome to the intersection between factorials and prime numbers In number theory primorial is a function from natural numbers to natural numbers similar to the factorial function but rather than multiplying successive positive integers only successive prime numbers are multiplied The name primorial attributed to Harvey Dubner draws an analogy to primes the same way the name factorial relates to factors There are two conflicting definitions that differ in the interpretation of the argument the first interprets the argument as an ndex into the sequence of prime numbers so that the function is strictly increasing while the second interprets the argument as a bound on the prime numbers to be multiplied so that the function value at any composite number is the same as at its predecessor The figures below plot both definitions comparing their shape and slopes Of as a function of n plotted logarithmically fy m as a function of n red dots compared to nl Both plots are logarithmic Natural primorial right plot In general for a positive integer n such a primorial n can also be defined namely as the product of those Prime primorial left p
112. t are not counted separately The formula is nl kl n k The general operation include the following enhanced features Gets the integer part of the input values forcing them to be positive Checks that neither one is Zero and that n gt r Uses the minimum of r n r to expedite the calculation time Checks the Out of Range condition at every multiplication so if it occurs its determined as soon as possible The chain of multiplication proceeds right to left with the largest quotients first e The algorithm works within the numeric range of the 41 Example nCr 335 167 is calculated without problems e It doesn t perform any rounding on the results Partial divisions are done to calculate NCR as opposed to calculating first NPR and dividing it by r Provision is made for those cases where n 0 and r 0 returning zero and one as results respectively This avoids DATA ERROR situations in running programs and is consistent with the functions definitions for those singularities Note as well that there is no final rounding made to the result This was the subject of heated debates in the HP Museum forum with some good arguments for a final rounding to ensure that the result is an integer The SandMath implementation however does not perform such final conditioning as the algorithm used seems to always return an integer already Pls Report examples of non conformance if you run into them Example Calculate the number
113. t the applicability of this method extends to the Integro Differential forms of the Bessel functions and so could be used to calculate second primitives or derivatives as well The expressions used in the SandMath for functions ITJ and ITI are as follows D L x K x Tm 2F3 n 1 2 n 2 2 n 1 2 n 2 p 2 n 1 x 4 D Ja K x T m oF 3 n 1 2 n 2 2 nt1 p 2 n 2 p 2 nt x 4 Where K 2 7 sqrt z and u 1 for the integral primitive Just in case you don t believe it take a look at this WolframAlpha s link http www wolframalpha com input 7i integrate 2 amp 8bessell 28N Y2CX 29 29 1 2fnt l1 3 x 300 dx gmt tty 2S A H l k 5 i 1 2 f n 1 3 x moo ax 2 7 tt AS ha ntl E 1 1 4 Nothing short of magical if you ask me what I d call going out with a bang Examples 1 4 ENTER 3 XFL ITJ gt 0 3 J 1 4 x dx 1 049262785 1 4 ENTER 3 SFL ITI gt 0 3 1 1 4 x dx 2 918753200 1 ENTER 3 FL ITJ gt 0 3 J 1 x dx 1 260051955 0 ENTER 10 XFL ITJ gt 0 10 J 0 x dx 1 067011304 50 ENTER 30 FL ITJ gt 0 30 J 50 x dx 1 478729947 E 8 c Angel M Martin Revision 44_E Page 96 SandMath_44 Manual Appendix 11 Looking for Zeros Once again just connecting the dots a brute force crude implementation of a root finder for Bessel functions made possible once the major task i
114. te it refers to its relation to the Gamma function as follows 2 Uy SDS Tety As a graphical example the picture below shows B X 0 5 for values of x between 4 and 4 As it s expected the same Gamma problem points are inherited by the Beta function Betaly 5 15 10 Biy z The implementation on the SandMath makes no attempt to discover new approaches or utilize any numeric equivalence it simple applies the definition formula using the Gamma subroutine Obvious disadvantages include the reduced numeric range aggravated by the multiplication of gamma values in the numerator Execution time corresponds to three times that of the Gamma function plus the small overhead to perform the Alpha Data checks and the arithmetic operations between the three gamma values 3 3 7 Incomplete Beta Function The incomplete beta function a generalization of the beta function is defined as B x a b 1 1 1 t dt 0 For x 1 the incomplete beta function coincides with the complete beta function The relationship between the two functions is like that between the gamma function and its generalization the incomplete gamma function And it s given in terms of the Hypergeometric function the expression by Bizi a b F a 1 b a 1 z iT c Angel M Martin Revision 44_E Page 56 SandMath_44 Manual 3 3 8 Bessel functions and Modified The next logical group would be the Bessel functions
115. tory to make it all fit in an 8k footprint factor and to allow reuse with other modules This is totally transparent to the end user just make sure it is installed in your system and that the revisions match See the relevant Library 4 documentation if interested Function Launchers and Mass key assignments As any good theme module worth its name the SandMath has its own mass Key assignment routine Use it to assign the most common functions within the ROM to their dedicated keys for a convenient mapping to explore the functions Besides that a distinct feature of this module is the function launchers used to access diverse functions grouped by categories These include the Hyperbolic the Fractions the RCL Math and the Special Function groups This saves memory registers for key assignments whilst maintaining the standard keyboard available also in USER mode for other purposes This is the fourth incarnation of the SandMath project which in turn has had a fair number of revisions and iterations on its own 7he new distinct addition has been a secondary Function address Table FAT to provide access to more functions exceeding the limit per page imposed by the operating system Some other refinements consisted in a rationalization of the backbone architecture as well as a more modular approach to each of pages of the module Gone are the 8k vs 12k distinctions of the past as now the Matrix and Polynomial functions have
116. ulator No rounding is made until the final comparison which is done on 10 digit values From the definition above it s clear that BS coincides with either Jn x or In x depending on the status of the CPU flag 9 and for positive orders The functions JBS and IBS are just MCODE extensions of BS that set up the specific settings prior to invoking it and depending on the signs of the orders and the arguments possibly adjust the result after it s completed c Angel M Martin Revision 44_E Page 60 SandMath_44 Manual The second function BS2 is only used for second kind functions with integer orders It s a finite sum and not an infinite summation Its contribution to the final result grows as the function order increases Its main goal was to reduce execution time as much as possible derived from the speed gains of MCODE versus FOCAL The definition of f n x is as follows f n x x 2 k n k H k H ntk k 0 1 2 The definition of gk n x is as follows g n x x 2 n k 1 k 3 k 0 1 n 1 The calculation of Gamma uses the Lanczos approximation implemented in the GAMMA function of the SandMath Despite being reasonably fast its execution time is noticeably longer than that of the Factorial for integer indexes therefore BS2 will use FACT instead for integer orders The Harmonic Numbers H n are obtained using another SandMath function as subroutine 21 N You see that the inter
117. uperfactorial in green hyperfactorial in purple eee ee ee ee m mxi fo Dont expect quantum leaps in number theory A ee eer erent eens eee here it is after all one of the most difficult branches of math c Angel M Martin Revision 44_E Page 41 SandMath_44 Manual Pochhammer symbol Rising and falling empires In mathematics the Pochhammer symbol introduced by Leo August Pochhammer is the notation x where n is a non negative integer Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below Care needs to be taken to check which interpretation is being used in any particular article The symbol x is used for the rising factorial Sometimes called the Pochhammer function Pochhammer polynomial ascending factorial rising sequential product or upper factorial a o 2 1 2 2 2 n 1 The symbol X n is used to represent the falling factorial sometimes called the descending factorial 2 falling sequential product lower factorial 1 u x 1 a 2 2 n 1 These conventions are used in combinatory However in the theory of special functions in particular the hypergeometric function the Pochhammer symbol x n is used to represent the rising factorial Extreme caution is therefore needed in interpreting the meanings of both notations The figures below show the rising left and falling factoria
118. ur coefficients in the stack registers T Z Y X using a ENTER b ENTER c ENTER d ENTER CROOT uses the well known Cardano Vieta formulas to obtain the roots The highest order coefficient doesn t need to be equal to 1 but errors will occur if the first term is zero for obvious reasons c ngel M Martin Revision 44_E Page 19 SandMath_44 Manual The SandMath implementation does reasonably well with multiple roots but sure enough you can find corner cases that will make it fail yet not more so than an equivalent FOCAL program Appendix 2 lists the code as well as an equivalent FOCAL program to compare the sizes much shorter but surely much slower and with data registers requirements Both cases can return real or complex roots If the roots are complex the functions will flag it in the following manners 1 QROOT will clear the Z register indicating that X and Y contain the real and imaginary parts of the two solutions Conversely if Z 0 then X and amp contain the two real roots 2 CROOT will leave the calculator in RAD mode in CROOT indicating that X and Y contain the real and imaginary parts of the second and third roots The real root will always be placed in the Z register Conversely if the calculator is set in DEG mode then registers Z Y and X have the three real roots QROUT outputs the contents of the X and Y registers to the display interpreted by the value in Z to determine whether there are to
119. used independently from the SandMath but sure it is a powerful complement for these specific subjects VCALC Vera VSTO__ VADST c Angel M Martin Revision 44_E Page 40 SandMath_44 Manual Quick recap a summary table of the different factorial functions available in the SandMath _ Function Author Description _ _ _ _ O Z o O o _ MFCT Multifactorial Logarithm Multifactorial SFCT Superfactorial XFCT Extended FACT POCH Angel Martin Pochhammer symbol FFACT Falling factorial Large numbers in a calculator like the HP 41 represent a challenge Not only the numeric range represents a problem but also the reduced accuracy limits the practical application of the field Nevertheless the few functions that follow contribute to add further examples of the ingenuity and what s possible using this venerable platform This was the last section added to the SandMath in revision E It also required compacting the few gaps available and transferring some code to the last available space in the Library 4 module Make sure you have matching revision of those two The functions in the table above operate only on integers i e no extension to real numbers using GAMMA Below one of such extensions the Hyperfactorial in a 3D visualization from WolframWorld Re H iz Irna Aez A z The figure on the left shows a plot of the four functions on the real line Fibonacci in blue double factorial in red s
120. ve results region between 1 and 2 so it isn t always positive Note also the asymptotic behavior near 3 the origin due to the Gamma function pole The implementation on the SandMath uses the analytical continuation to calculate LogGamma for arguments less than 9 ncluding negative values Obvious problems like the poles at negative integer will yield DATA ERROR messages but outside that the approximation should hold since I z n T z X z i i 1 2 n it follows Ln F z n Ln T z Ln X z i i 1 2 n Notice also that the same error will occur when trying to calculate LogGamma when Gamma is negative which occurs between even negative numbers and their immediately lower inferior one see the plot in page 27 See the following link for a detailed description of another implementation using Lanczos for both cases to calculate Gamma and LogGamma on the 41 by Steven Thomas Smith http www homuseum org cai sys cqiwrap hpmuseunrvarticles cgi read 941 An excellent implementation of Gamma and related functions for the 41 is available on the following link written by Jean Marc Baillard very complete and detailed Attp www hpmuseum org sottware 41 4lgamdgm htm c Angel M Martin Revision 44_E Page 54 SandMath_44 Manual 3 3 5 Digamma function In mathematics the digamma function is defined as the logarithmic derivative of the gamma function P r z log T x rae It is the
121. ve and negative values for real z gt 0 4 and so cannot be used for practical numerical computation The W function may be approximated using Newton s method with successive approximations to w W z so z w e being wje 7 ut TT A __ tae s eva wens The implementation in the SandMath uses this iterative method to solve for W z the roots of its functional equation given the functions argument z An important consideration is the selection of the initial estimations For that the general practice is to start with Ln x as lower limit and 1 Ln x as upper value Another aspect of the W function is the existence of two branches The second branch is defined for arguments between 1 e and 0 with function values between 1 and infinite The lower branch is also available in the SandMath as the function WL1 In fact the MCODE algorithm is the same one with just different initial estimations depending on the branch to calculate c Angel M Martin Revision 44_E Page 67 SandMath_ 44 Manual Example 1 calculate W for x 5 5 WLO gt RUNNING followed by 1 326724665 We can use the inverse Lambert function AWL to check the accuracy of the results simply executing it after WLO and comparing with the original argument Note the AWL will be seen later on in the Secondary FAT Sub functions group This it requires FL to call it not XEQ 5 WLO EFL AWL gt 4 9999999
122. w hp41 org SandMath Overlay 2009 Lujan Garcia Published under the GNU software licence agreement c Angel M Martin Revision 44_E Page 2 SandMath_44 Manual Table of Contents Revision E 1 Introduction Function Launchers and Mass key assignments Used Conventions and disclaimers Getting Started Accessing the functions Main and Dedicated Launchers the Overlay Appendix 1 Launcher Maps Function index at a glance 2 Lower Page Functions in Detail 2 1 SandMath44 Group Elementary Math functions Number Displaying and Coordinate conversions Base Conversions First Second and Third degree Equations Appendix 2 FOCAL program listing Additional Test Functions rounded and otherwise 2 2 Fractions Calculator Fraction Arithmetic and displaying 2 3 Hyperbolic Functions Direct and Indirect Hyperbolics Errors and Examples 2 4 Recall Math Individual RCL Math functions RCL Launcher the Total Rekall Appendix 3 A trip down memory lane 3 Upper Page Functions in Detail 3 1 Statistics Probability Statistical Menu Another type of Launcher Alea jacta est Combinations and Permutations Linear Regression Let s not digress Ratios Sorting and Register Maxima Probability Distribution Function And what about Prime Factorization Appendix 4 Prime Factors decomposition Distance between two Points c Angel M Martin Revision 44_E 23 25 26 27 28
123. w real roots or the Real amp Imaginary parts of the complex roots It will be automatically invoked by QROOT always and by CROOT real roots when they are executed in RUN mode Note that CROOT will not display the first real root which will be located in Z CVIETA is a driver program for CROOT including the prompts for the equation coefficients The results are placed in the stack following the same conventions as for CROOT explained above Example _1 Calculate the roots of the equation AX 2 3X 3x 2 2 ENTER 3 ENTER ENTER 2 CROOT gt Z 0 500 Y 1 000 X 2 000 f x 2x 3 3x2 3x 2 From the final prompt you know all roots are real The value in Z blinks briefly in the display before the final prompt above is presented use RCL Z or RDN RDN to retrieve it No user registers are used X ri tt 1 H1 4 1 13 reor pm i mH N Tm Example 2 Calculate the three solutions of the equation xX x7 x 1 0 1 LENTER ENTER ENTER CROOT gt Z 1 000 Y 1 000 X 1 E 10 d Shown as rounded number for the real part c ngel M Martin Revision 44_E Page 20 SandMath_44 Manual Appendix 2 CVIETA equivalent FOCAL program replaced now with an all MCODE implementation 01 LBL CVIETA r ea N2 02 6 imaginary part F 03 R 66 RCLO01 cbrt x R3 2 04 ST T 0 q 2 a 3 in
124. y of cases and only runs into limiting situations in very few instances if at all remember that exceeding 24 characters will shift off the display the left characters first that is the original number which doesn t result into any data loss Obviously prime numbers don t have any other factors than themselves For instance for x 17777 PFCT will return J which indeed is hardly debatable Note that only the last two prime factors found will be stored in Y and Z and that the original number will remain in X after the execution terminates A more capable prime factorization program is available in the ALGEBRA module using the matrix functions of the Advantage and Advanced Matrix ROMs to save the solutions in a results matrix See the appendices for a listing of the program used in the SandMath and the more comprehensive one ema Dp RAE Shown on the left there s an even simpler version that doesn t 7 INT consolidate the multiple factors which will aggravate the length Fa limitation of the ALPHA registers of 24 chrs max The core of the action FA is performed by PRIME therefore the fast execution due to the res MCODE speed 6 7 ros See the appendix in the next pages with both the actual code for PFCT F g in the SandMath and for PRMF a more capable implementation using F 40 the Matrix functions from the HP Advantage to store the prime factor Fal PRIME and their repetition indexes really the best wa
125. y to present the results 12 SF OO 13 AINT For that second case the function PF gt X restores the original argument F 44 FS2C 00 from the matrix values Also function TOTNT is but a simple extension r 45 GTO 01 using the same approach 16 7 18 F F F F F F F c ngel M Martin Revision 44_E Page 37 SandMath_44 Manual Appendix 4 Prime factor decomposition The FOCAL programs listed below are for PFCT included in the SandMath and PRMF a more capable implementation that uses the Matrix functions from the HP Advantage or the AVD_MATRIX ROM PRMF stores all the different prime factors and their repetition indices in a n x 2 matrix The matrix is re dimensioned dynamically each time a new prime factor is found and the repetition index is incremented each time the same prime factor shows up A matrix is a much better place than the ALPHA register for sure as is done in PFCT The drawback is of course the execution speed much faster in PFCT PF gt X is the reverse function that restores the original number from the values stored in the matrix TOTNT Totient function is but a simple extension also shown in the listings below PRMF PC gt X and TOTNT are included in the Advanced MATRIX ROM Below is the program listing implemented in the SandMath 1 LBL PRIME 32 GTO 03 skip if Prime 2 CF OO 33 ST L 3 INT 34 LASTX reduced number ABS 35 GTO 00 5 CLA 36 Prim

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