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