Artifact a85e26e172fe888645ca490506f4d7630b74d782edec2ce8de06b9b7b235aa1f:
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r37/packages/rataprx/rataprx.rlg
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2011-09-02 18:13:33
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Wed Feb 3 23:17:05 MET 1999 REDUCE 3.7, 15-Jan-99 ... 1: 1: 2: 2: 2: 2: 2: 2: 2: 2: 2: 3: 3: % Tests of the rataprx package. % Authors: Lisa Temme, Wolfram Koepf (koepf@zib.de) % periodic decimal representations rational2periodic(1/3); _ 0.3 periodic2rational(ws); 1 --- 3 rational2periodic(-1/3); _ -0.3 periodic2rational(ws); - 1 ------ 3 rational2periodic(1.2/3); 0.4 periodic2rational(ws); 2 --- 5 rational2periodic(1/3.4); ________________ 0.2941176470588235 periodic2rational(ws); 5 ---- 17 rational2periodic(1.2/3.4); ________________ 0.3529411764705882 periodic2rational(ws); 6 ---- 17 rational2periodic(352673/3124); periodic({11289,100},{1, 4, 8, 5, 2, 7, 5, 2, 8, 8, 0, 9, 2, 1, 8, 9, 5, 0, 0, 6, 4, 0, 2, 0, 4, 8, 6, 5, 5, 5, 6, 9, 7, 8, 2, 3, 3, 0, 3, 4, 5, 7, 1, 0, 6, 2, 7, 4, 0, 0, 7, 6, 8, 2, 4, 5, 8, 3, 8, 6, 6, 8, 3, 7, 3, 8, 7, 9, 6, 4}) periodic2rational(ws); 352673 -------- 3124 rational2periodic(53765/5216); periodic({1030770,100000},{7, 0, 5, 5, 2, 1, 4, 7, 2, 3, 9, 2, 6, 3, 8, 0, 3, 6, 8, 0, 9, 8, 1, 5, 9, 5, 0, 9, 2, 0, 2, 4, 5, 3, 9, 8, 7, 7, 3, 0, 0, 6, 1, 3, 4, 9, 6, 9, 3, 2, 5, 1, 5, 3, 3, 7, 4, 2, 3, 3, 1, 2, 8, 8, 3, 4, 3, 5, 5, 8, 2, 8, 2, 2, 0, 8, 5, 8, 8, 9, 5}) periodic2rational(ws); 53765 ------- 5216 % continued fractions % of numbers cfrac pi; 1146408 {---------, 364913 1 3 + ------------------------------------------------------} 1 7 + ------------------------------------------------ 1 15 + ----------------------------------------- 1 1 + ----------------------------------- 1 292 + --------------------------- 1 1 + --------------------- 1 1 + --------------- 1 1 + --------- 1 2 + --- 1 cfrac(pi,3); 355 1 {-----,3 + ----------------} 113 1 7 + ---------- 1 15 + --- 1 cfrac(pi,20); 1146408 {---------, 364913 1 3 + ------------------------------------------------------} 1 7 + ------------------------------------------------ 1 15 + ----------------------------------------- 1 1 + ----------------------------------- 1 292 + --------------------------- 1 1 + --------------------- 1 1 + --------------- 1 1 + --------- 1 2 + --- 1 oldprec:=precision 20; oldprec := 12 cfrac pi; 14885392687 contfrac(-------------,{3, 4738167652 {1,7}, {1,15}, {1,1}, {1,292}, {1,1}, {1,1}, {1,1}, {1,2}, {1,1}, {1,3}, {1,1}, {1,14}, {1,2}, {1,1}, {1,1}, {1,2}, {1,2}, {1,2}, {1,2}}) cfrac(pi^2); 27053934029 contfrac(-------------,{9, 2741136618 {1,1}, {1,6}, {1,1}, {1,2}, {1,47}, {1,1}, {1,8}, {1,1}, {1,1}, {1,2}, {1,2}, {1,1}, {1,1}, {1,8}, {1,3}, {1,1}, {1,10}, {1,5}, {1,1}, {1,3}}) cfrac(pi*e*sqrt(2)); 22809995021 contfrac(-------------,{12, 1888712428 {1,12}, {1,1}, {1,68}, {1,1}, {1,3}, {1,1}, {1,2}, {1,1}, {1,6}, {1,1}, {1,15}, {1,1}, {1,5}, {1,1}, {1,1}, {1,2}, {1,1}, {1,8}, {1,2}}) precision oldprec; 20 % of rational functions cfrac((x+2/3)^2/(6*x-5),x); 2 9*x + 12*x + 4 {-----------------, 54*x - 45 6*x + 13 1 ---------- + -------------} 36 24*x - 20 ----------- 9 cfrac((x+2/3)^2/(6*x-5),x,0); 6*x + 13 6*x + 13 {----------,----------} 36 36 cfrac((x+2/3)^2/(6*x-5),x,1); 2 9*x + 12*x + 4 {-----------------, 54*x - 45 6*x + 13 1 ---------- + -------------} 36 24*x - 20 ----------- 9 cfrac((x+2/3)^2/(6*x-5),x,10); 2 9*x + 12*x + 4 {-----------------, 54*x - 45 6*x + 13 1 ---------- + -------------} 36 24*x - 20 ----------- 9 cfrac((x*8-7/2)^4/(x^5-2/3),x); 4 3 2 196608*x - 344064*x + 225792*x - 65856*x + 7203 {----------------------------------------------------, 5 48*x - 32 4*x + 7 524288*x - 458752 144060*x - 574709 1/(--------- + 1/(------------------- + 1/(------------------- + 1/( 16384 245 29503488 1428871832845410168479416320*x + 5604409773009942380402638848 --------------------------------------------------------------- + 1 110370509467032373008840289 /(4638094170692621015213105500565148574652*x - 3747067916403980284808800355871185905903)/365473525943308099\ 0834049614798549893120))))} cfrac((x*8-7/2)^4/(x^5-2/3),x,2); 524288*x - 458752 {---------------------, 2 128*x + 112*x + 49 1 -----------------------------------} 4*x + 7 1 --------- + --------------------- 16384 524288*x - 458752 ------------------- 245 % of analytic functions cfrac(e^x,x,10); 5 4 3 2 - x - 30*x - 420*x - 3360*x - 15120*x - 30240 {----------------------------------------------------, 5 4 3 2 x - 30*x + 420*x - 3360*x + 15120*x - 30240 x 1 + ------------------------------------------------------------} - x 1 + ------------------------------------------------------ x 2 + ------------------------------------------------ - x 3 + ------------------------------------------ x 2 + ------------------------------------ - x 5 + ------------------------------ x 2 + ------------------------ - x 7 + ------------------ x 2 + ------------ - x 9 + ------ 2 % default order is 4 cfrac(e^x,x); 3 2 x + 9*x + 36*x + 60 {-----------------------, 2 3*x - 24*x + 60 x 1 + ---------------------------} - x 1 + --------------------- x 2 + --------------- - x 3 + --------- x 2 + --- 5 cfrac(x^2/(x-1)*e^x,x); 6 4 2 x + 3*x + x {----------------, 4 2 3*x - x - 1 2 - x ----------------------------} 2 - 2*x 1 + ---------------------- 2 x 1 + ---------------- 2 x 1 + ---------- 2 x 1 + ---- 1 cfrac(x^2/(x-1)*e^x,x,2); 2 x {----------, 2 2*x - 1 2 - x ---------------} 2 - 2*x 1 + --------- 1 cfrac(atan(x),x,10); 9 7 5 3 307835*x + 4813380*x + 19801782*x + 29609580*x + 14549535*x {----------------------------------------------------------------------------, 10 8 6 4 2 19845*x + 1091475*x + 9459450*x + 28378350*x + 34459425*x + 14549535 x -----------------------------------------------------------------} 2 x 1 + ----------------------------------------------------------- 2 4*x 3 + ----------------------------------------------------- 2 9*x 5 + ----------------------------------------------- 2 16*x 7 + ----------------------------------------- 2 25*x 9 + ----------------------------------- 2 36*x 11 + ---------------------------- 2 49*x 13 + --------------------- 2 64*x 15 + -------------- 2 81*x 17 + ------- 19 cfrac(asin(x),x,5); 5 3 69049*x - 717780*x + 922320*x {---------------------------------, 4 2 145125*x - 871500*x + 922320 x ----------------------------------------} 2 - x 1 + ---------------------------------- 2 - 17*x 6 + ---------------------------- 2 - 549*x 10 + --------------------- 2 - 69049*x 238 + ------------- 1098 % not implemented cfrac(log(x),x,4); ***** not yet implemented cfrac(asech(x),x,5); ***** not yet implemented cfrac(sin sqrt x,x,4); ***** not yet implemented % wrong input cfrac(1,x); cfrac(1,x) cfrac(x,x,x); cfrac(x,x,x) cfrac(x,x,x,5); cfrac(x,x,x,5) % Pade representations pade(sin(x),x,0,3,3); 2 x*( - 7*x + 60) ------------------ 2 3*(x + 20) pade(tanh(x),x,0,5,5); 4 2 x*(x + 105*x + 945) ----------------------- 4 2 15*(x + 28*x + 63) pade(atan(x),x,0,5,5); 4 2 x*(64*x + 735*x + 945) -------------------------- 4 2 15*(15*x + 70*x + 63) pade(1/(x*sin(x)),x,0,3,2); 2 x + 6 -------- 2 6*x pade(sin(x)/x^2,x,0,10,1); 10 8 6 4 2 - x + 110*x - 7920*x + 332640*x - 6652800*x + 39916800 --------------------------------------------------------------- 39916800*x pade(sin(x)/x^2,x,0,10,2); 10 8 6 4 2 - x + 110*x - 7920*x + 332640*x - 6652800*x + 39916800 --------------------------------------------------------------- 39916800*x pade(sin(x)/x^2,x,0,10,3); 10 8 6 4 2 - 23*x + 4620*x - 451440*x + 22619520*x - 498960000*x + 3113510400 --------------------------------------------------------------------------- 2 19958400*x*(x + 156) pade(exp(x),x,0,10,10); 10 9 8 7 6 5 4 (x + 110*x + 5940*x + 205920*x + 5045040*x + 90810720*x + 1210809600*x 3 2 10 + 11762150400*x + 79394515200*x + 335221286400*x + 670442572800)/(x 9 8 7 6 5 4 - 110*x + 5940*x - 205920*x + 5045040*x - 90810720*x + 1210809600*x 3 2 - 11762150400*x + 79394515200*x - 335221286400*x + 670442572800) pade(sin(x),x,0,20,20); 18 (420*x*( - 1670454796485655407005204111896505853962882149730473*x 16 + 3750723201144687025399857898162770954823924964685652842*x 14 - 3582506072289078352047596683909246967322825072732178695360*x 12 + 1869589702800106132374452333905246037435005346891034203856000*x 10 - 573228615690533362277150976788354558105622438150759712792761600*x 8 + 103954806107838160002753187442833010398502453693126706480803904000*x - 10728835421367893971671793497263049404983813914658053829579862016000 6 *x + 4 575186523659337692549014637994968894181652462542936979129823993856000*x - 13299404753182468245973779380302173780050163417906548608594453340160000 2 *x + 8461166135144631096871438729757765368229020614386275136075764211\ 20 7120000))/(9703514253649496804459335098386178033709237613*x 18 + 26693188916859840432281615685297667671827346329880*x 16 + 40131841713883475746635715860433350742371877282920840*x 14 + 42773075706893663042738071933996865427898294402559424000*x 12 + 35329791607150874190393563954056763271257094118524405024000*x 10 + 23367407211060385995097426988941972393423826920849430345472000*x 8 + 12411237061763347778639777646354605230793727402242210950475520000*x 6 + 5182014156622335713728846799235531962702355938650410017404846080000*x 4 + 1615241584293926563589087600285609358152611856120931412435087564800000*x 2 + 337066298264605104501019771103522770139245794549642179643364545331200000*x + 35536897767607450606860042664982614546561886580422355571518209689190400000 ) % no Pade Approximation exists pade(exp(1/x),x,0,5,5); ***** no Pade Approximation exists % wrong order pade(sin(x)/x^2,x,0,10,0); ***** Pade Approximation of this order does not exist % not implemented pade(factorial(x),x,1,3,3); ***** not yet implemented % extended Pade representations pade(asech(x),x,0,3,3); 2 2 2 - 3*log(x)*x + 8*log(x) + 3*log(2)*x - 8*log(2) + 2*x ----------------------------------------------------------- 2 3*x - 8 taylor(ws-asech(x),x,0,10); 11 13 6 43 8 1611 10 11 log(x)*(0 + O(x )) + (-----*x + ------*x + -------*x + O(x )) 768 2048 81920 pade(sin(sqrt(x)),x,0,3,3); 3 2 x*(56447*x - 4851504*x + 132113520*x - 885487680) ------------------------------------------------------ 3 2 7*sqrt(x)*(179*x - 7200*x - 2209680*x - 126498240) taylor(ws-sin(sqrt(x)),x,0,10); 1 13/2 1399187 15/2 - ------------*x + --------------------*x 6227020800 229747925062656000 1364974757 17/2 12300718539103 19/2 - -------------------------*x + ------------------------------*x 10397012600785434624000 6310321239870308548952064000 21/2 + O(x ) end; 4: 4: 4: 4: 4: 4: 4: 4: 4: Time for test: 12710 ms, plus GC time: 270 ms 5: 5: Quitting Wed Feb 3 23:17:42 MET 1999