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Automatic expansion of C-fractionsfor solutions of Riccati equations
via Guessing and Proving.
The examples from Cuyt et alii., Handbook of Continued Fractions for Special Functions.
Maulat & Salvy
This worksheet demonstrates the use of the submodule gfun:-ContFrac, for power series solutions of Riccati differential equations : y'=p y^2 + q y + r with p,q,r rational in the variable z.
The first section gives a few examples, and demonstrates how to print some information along the way.A comprehensive list of all the examples from Cuyt et alii is then given in the following section.
r e s t a r t ;s t a r t t i m e : = t i m e ( ) ;
w i t h ( g f u n ) :
i f g f u n : - v e r s i o n ( ) < 3 . 7 0 t h e n e r ro r "O ld g fun vers ion . P lease download the l a tes t ve rs ion on g f u n ' s w e b s i t e , h t t p : / / p e r s o . e n s - l y o n . f r / b r u n o .salvy /sof tware / the-gfun-package/ ."f i ;
wi th(gfun: -ContFrac) :
First examples.The procedure Riccati_to_Cfrac takes as input :- f: a function to be expanded (or a series)- y: a function name- z: the variable name
and optionally :- a number (the series expansion order)
R i c c a t i _ t o _ C f r a c ( e x p ( z ) , y , z ) ;
(1.2)(1.2)
(1.4)(1.4)
(1.5)(1.5)> >
(1.3)(1.3)
> >
> >
> >
(1.1)(1.1)
(1.6)(1.6)
> >
The infolevel command enables to print details on the computation.The 'demo' field concerns the main computation lines ; it can be set to 0 or 1.
in fo leve l [demo] :=1;
R i c c a t i _ t o _ C f r a c ( t a n ( z ) , y , z , 2 0 ) ;c o m p u t i n g s e r i e s . . .. . . d o n e .c o m p u t i n g a R i c c a t i e q u a t i o n ( u s i n g a g u e s s i n g a p p r o a c h ) . . .. . . d o n e .c o m p u t i n g a C - f r a c t i o n e x p a n s i o n . . .. . . d o n e . ( i n . 6 6 9 s e c o n d s )
in fo leve l [demo] :=0;
More information on how the continued fraction expansion is computed and proved can be printed using the 'gfuncontfrac' information field.
i n f o l e v e l [ g f u n c o n t f r a c ] : = 1 ;
R i c c a t i _ t o _ C f r a c ( t a n ( z ) , y , z , 2 0 ) ;Guess ing a fo rmu lac o n j e c t u r e ( o n 8 c o e f f i c i e n t s )
> > (2.1)(2.1)
(2.2)(2.2)> >
> >
(1.7)(1.7)
(1.1)(1.1)
(1.6)(1.6)
d e f i n i n g a s e q u e n c e H ( n , z ) , w h i c h r e l a t e s t o c o n v e r g e n c e .l e m m a : t h e c o n j e c t u r e h o l d s i f f t h e r e e x i s t s a n u n b o u n d e d i( n ) s . t . , H ( i ( n ) , z ) t e n d s t o 0 . ( i . e . , t h e i r v a l u a t i o n s t e n d t o i n f i n i t y )p r o v i n g L i m i t ( v a l ( H ( n , z ) ) , n = i n f i n i t y ) = i n f i n i t y :- c o m p u t i n g a r e c u r r e n c e f o r H ( n , z ) . ( w h i c h d o e s n o t c o n c l u d e )
- r e d u c i n g t h e r e c u r r e n c e o r d e r f o r H ( n , z ) . . .- . . . d o n e .
QED.
in fo leve l [demo] :=0;i n f o l e v e l [ g f u n c o n t f r a c ] : = 0 ;
C-fractions formulas of Riccati solutions in Cuyt et alii.In this section, formulas from the compendium by Cuyt et alii (2008) are recoveredautomatically. All explicit C-fractions solutions of Riccati equations are recovered.
Formulas are stored in a table gfun:-ContFrac:-Cuyt_Cfrac, which is accessed as follows:
chapter:=11;
l a b e l s ( c h a p t e r ) ;
(2.1.1)(2.1.1)
(2.1.3)(2.1.3)
(2.3)(2.3)
> >
> >
(2.1.2)(2.1.2)
(2.2)(2.2)
> >
> >
> >
(1.1)(1.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
Cuyt_Cfrac[chapter ] [1 .3 ] ;ez
In what follows, only the parameters related to the series exponents are given example values.
Chapter 11
c h a p t e r : = 1 1 ;
l a b e l s ( c h a p t e r ) ;
N:=20;
Generic expansion, with finite series as input.The order of the series is doubled when the C-fraction coefficients are in z^2.
f o r f o r m u l a i n l a b e l s ( c h a p t e r ) d o
p r i n t ( s p r i n t f ( " % a . % a " , c h a p t e r , f o r m u l a ) ) ; f : = C u y t _ C f r a c [ c h a p t e r ] [ f o r m u l a ] :
i f f o r m u l a i n { 1 . 3 , 2 . 2 , 7 . 1 , 7 . 2 } t h e n S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) : e l i f f o r m u l a = 7 . 4 t h e n S : = s e r i e s ( f , z , 2 * N ) ; e l s e S : = M u l t i S e r i e s : - s e r i e s ( f , z , 2 * N ) f i ; R i c c a t i _ t o _ C f r a c ( S , y , z ) ;
od;"11.1.2"
(2.2)(2.2)
(1.1)(1.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
"11.1.3"
"11.2.2"
"11.2.4"
"11.3.7"
(2.2)(2.2)
(1.1)(1.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
"11.4.5"
"11.4.8"
"11.5.5"
"11.6.4"
(2.2)(2.2)
(1.1)(1.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
"11.6.9"
"11.7.1"
"11.7.2"
> >
(2.1.5)(2.1.5)
> >
> >
(2.2)(2.2)
(1.6)(1.6)
> >
(2.1.4)(2.1.4)
(2.2.3)(2.2.3)
(2.2.4)(2.2.4)
(2.1.6)(2.1.6)
> >
(2.2.2)(2.2.2)
> >
> >
(1.1)(1.1)
(2.2.1)(2.2.1)
"11.7.4"
f : = - ( - 2 * z + ( 1 + z ) ^ 2 - 1 ) / ( ( 1 + z ) ^ 2 - 1 ) ;
f a c t o r ( f ) ;
Chapter 12
c h a p t e r : = 1 2 ;
N:=25:The parameter l in formulas (12.6.21) and (12.6.22) is set to a consistent value.
v a l l : = 3 ;
i n f o l e v e l [ g f u n c o n t f r a c ] : = 0 ;
f o r f o r m u l a i n l a b e l s ( c h a p t e r ) d o
p r i n t ( s p r i n t f ( " % a . % a " , c h a p t e r , f o r m u l a ) ) ; f : = C u y t _ C f r a c [ c h a p t e r ] [ f o r m u l a ] :
i f f o r m u l a i n { 6 . 2 1 , 6 . 2 2 } t h e n
> >
(2.2)(2.2)
(1.1)(1.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.2.4)(2.2.4)
p r i n t ( f , l = v a l l ) ; f : = s u b s ( l = v a l l , f ) ; f i ;
R i c c a t i _ t o _ C f r a c ( M u l t i S e r i e s : - s e r i e s ( f , z , N ) , y , z )
od:"12.6.17"
"12.6.21"
"12.6.22"
> >
(2.2)(2.2)
(1.1)(1.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.2.4)(2.2.4)
"12.6.23"
Chapter 13
(2.3.4)(2.3.4)
> >
(2.2)(2.2)
> >
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.3.2)(2.3.2)
> >
(2.2.4)(2.2.4)
> >
> > (2.3.1)(2.3.1)
(2.3.3)(2.3.3)
(1.1)(1.1)
> >
In this chapter, one parameter (k) is given a value, and some variable changes are performed for simplicity.
chapter:=13;
l a b e l s ( c h a p t e r ) ;
v a l k : = 3 ;
N:=34:f o r f o r m u l a i n l a b e l s ( c h a p t e r ) d o
p r i n t ( s p r i n t f ( " % a . % a " , c h a p t e r , f o r m u l a ) ) ;
f : = C u y t _ C f r a c [ c h a p t e r ] [ f o r m u l a ] ; i f f o r m u l a = 3 . 5 t h e n f : = s u b s ( k = v a l k , f ) ; e l i f f o r m u l a = 4 . 9 t h e n f : = s u b s ( z = z / s q r t ( I * P i ) , f ) e n d i f ;
p r i n t ( f ) ;
S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) ;
R i c c a t i _ t o _ C f r a c ( S , y , z )
od:"13.1.11"
"13.2.20"
(2.3.4)(2.3.4)
> >
(2.2)(2.2)
(1.1)(1.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.2.4)(2.2.4)
"13.3.5"
"13.4.9"
Chapter 14
(2.3.4)(2.3.4)
(2.4.3)(2.4.3)
> >
> >
(2.2)(2.2)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
> >
(2.4.4)(2.4.4)
(2.4.2)(2.4.2)
> >
(2.4.5)(2.4.5)
> >
(1.1)(1.1)
c h a p t e r : = 1 4 ;
N:=20;
l a b e l s ( c h a p t e r ) ;
Parameters n and nu are given example values.v a l n : = 4 ;v a l n u : = 3 / 2 ;
f o r f o r m u l a i n l a b e l s ( c h a p t e r ) d o
p r i n t ( s p r i n t f ( " % a . % a " , c h a p t e r , f o r m u l a ) ) ;
f : = C u y t _ C f r a c [ c h a p t e r ] [ f o r m u l a ] ; p r i n t ( f ) ;
i f f o r m u l a = 1 . 1 6 t h e n S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) a s s u m i n g n u : : r e a l ; e l i f f o r m u l a = 1 . 2 0 t h e n S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) a s s u m i n g 1 - n u > 0 ; e l s e S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) ; f i ;
R i c c a t i _ t o _ C f r a c ( S , y , z )
od:"14.1.16"
"14.1.20"
(2.3.4)(2.3.4)
> >
(2.5.1)(2.5.1)
> >
(2.2)(2.2)
> >
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
(2.4.5)(2.4.5)
(1.1)(1.1)
"14.2.24"
Chapter 15Chapter 15 contains parametrized hypergeometric ratios, which are long to compute with.For this reason, information is printed along the computation, and additional arguments are used, as detailed below.
c h a p t e r : = 1 5 ;
N:=20;
(2.5.3)(2.5.3)
(2.3.4)(2.3.4)
> >
(2.5.4)(2.5.4)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
(2.4.5)(2.4.5)
> >
(1.1)(1.1)
> >
l a b e l s ( c h a p t e r ) ;
in fo leve l [demo] :=1;
f o r f o r m u l a i n l a b e l s ( c h a p t e r ) d o
p r i n t ( s p r i n t f ( " % a . % a " , c h a p t e r , f o r m u l a ) ) ;
f : = C u y t _ C f r a c [ c h a p t e r ] [ f o r m u l a ] ;
p r i n t ( f ) ;
S : = s e r i e s ( f , z , N ) ;
R i c c a t i _ t o _ C f r a c ( S , y , z , 1 5 ) ;
od:"15.3.3"
c o m p u t i n g s e r i e s . . .. . . d o n e .c o m p u t i n g a R i c c a t i e q u a t i o n ( u s i n g a g u e s s i n g a p p r o a c h ) . . .. . . d o n e .c o m p u t i n g a C - f r a c t i o n e x p a n s i o n . . .. . . d o n e . ( i n 1 0 6 . 2 5 2 s e c o n d s )
"15.3.4"
c o m p u t i n g s e r i e s . . .. . . d o n e .c o m p u t i n g a R i c c a t i e q u a t i o n ( u s i n g a g u e s s i n g a p p r o a c h ) . . .
(2.3.4)(2.3.4)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.6.2)(2.6.2)
(2.5.2)(2.5.2)
> >
> >
> > (2.5.6)(2.5.6)
(2.6.3)(2.6.3)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
> >
> > (2.6.1)(2.6.1)
(1.1)(1.1)
. . . d o n e .c o m p u t i n g a C - f r a c t i o n e x p a n s i o n . . .. . . d o n e . ( i n 7 . 4 5 4 s e c o n d s )
in fo leve l [demo] :=0;
Chapter 16c h a p t e r : = 1 6 ;
N:=20;
l a b e l s ( c h a p t e r ) ;
f o r f o r m u l a i n l a b e l s ( c h a p t e r ) d o
p r i n t ( s p r i n t f ( " % a . % a " , c h a p t e r , f o r m u l a ) ) ;
f : = C u y t _ C f r a c [ c h a p t e r ] [ f o r m u l a ] ;
p r i n t ( f ) ;
S : = s e r i e s ( f , z , N ) ;
R i c c a t i _ t o _ C f r a c ( S , y , z ) ;
od:"16.1.13"
(2.3.4)(2.3.4)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
"16.1.14"
"16.2.4"
"16.3.4"
"16.5.7"
(2.3.4)(2.3.4)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
> >
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)
(2.7.1)(2.7.1)
(2.7.2)(2.7.2)
> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
> >
> >
(2.7.3)(2.7.3)
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
ez
Chapter 17In this chapter, some functions require to compute the series by hand, as well as the Riccati equation.The formulas are still produced and proved automatically.
c h a p t e r : = 1 7 ;
l a b e l s ( c h a p t e r ) ;
N:=20;
f o r f o r m u l a i n l a b e l s ( c h a p t e r ) d o
p r i n t ( s p r i n t f ( " % a . % a " , c h a p t e r , f o r m u l a ) ) ;
f := subs(n=nn , Cuyt_Cf rac [chapter ] [ fo rmula ] ) ;
i f f o r m u l a = 1 . 4 4 t h e n
l p r i n t ( " A d - h o c s e r i e s c o m p u t a t i o n , a n d R i c c a t i e q u a t i o n computat ion.") ; f : = s u b s ( z = z / I , f / e x p ( - I * P i ) ) ; p r i n t ( f ) ; f : = e v a l ( f , H a n k e l H 1 = C o n t F r a c : - m y H a n k e l H 1 ) ; f : = s i m p l i f y ( f , p o w e r , s y m b o l i c , e x p ) ; S : = m a p ( s i m p l i f y , M u l t i S e r i e s : - s e r i e s ( f , z , N ) ) ;
d e q : = d i f f ( y ( z ) , z ) - ( z ^ 2 + ( 2 * n u + 2 ) * z ^ 2 * y ( z ) - y ( z ) ^ 2 )/ z ^ 3 : R i c c a t i _ t o _ C f r a c ( S , y , z , i n p u t _ r i c c a t i = d e q ) ;
(2.3.4)(2.3.4)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
e l i f f o r m u l a = 1 . 4 6 t h e n
l p r i n t ( " A d - h o c s e r i e s c o m p u t a t i o n . " ) ; f : = s u b s ( z = I * z , f ) ; p r i n t ( f ) ; f : = e v a l ( f , H a n k e l H 2 = C o n t F r a c : - m y H a n k e l H 2 ) ; f : = s i m p l i f y ( f , p o w e r , s y m b o l i c , e x p ) ; S : = m a p ( s i m p l i f y , M u l t i S e r i e s : - s e r i e s ( f , z , N ) ) ;
R i c c a t i _ t o _ C f r a c ( S , y , z ) ;
e l i f f o r m u l a = 2 . 3 5 t h e n p r i n t ( f ) ; S : = m a p ( f a c t o r , s e r i e s ( f , z , N ) ) ; R i c c a t i _ t o _ C f r a c ( S , y , z ) ;
e l s e p r i n t ( f ) ; S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) ; R i c c a t i _ t o _ C f r a c ( S , y , z ) ;
f i ;
od:"17.1.37"
"17.1.39"
(2.3.4)(2.3.4)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
"17.1.44"
" A d - h o c s e r i e s c o m p u t a t i o n , a n d R i c c a t i e q u a t i o n c o m p u t a t i o n . "
"17.1.46"
" A d - h o c s e r i e s c o m p u t a t i o n . "
(2.3.4)(2.3.4)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
"17.2.32"
"17.2.35"
Chapter 18
> >
(2.3.4)(2.3.4)
(2.8.4)(2.8.4)
(2.8.5)(2.8.5)
> >
> >
(2.2)(2.2)
(2.8.2)(2.8.2)
(2.5.5)(2.5.5)
> > (2.8.1)(2.8.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
> >
(2.8.3)(2.8.3)
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
> >
c h a p t e r : = 1 8 ;
l a b e l s ( c h a p t e r ) ;
N:=40;
a l p h a : = 2 ;
f o r f o r m u l a i n l a b e l s ( c h a p t e r ) d o
p r i n t ( s p r i n t f ( " % a . % a " , c h a p t e r , f o r m u l a ) ) ;
f : = C u y t _ C f r a c [ c h a p t e r ] [ f o r m u l a ] ; p r i n t ( f ) ; S : = M u l t i S e r i e s : - s e r i e s ( f , z , N ) ; R i c c a t i _ t o _ C f r a c ( S , y , z ) ;
od:"18.2.16"
"18.2.17"
(2.3.4)(2.3.4)
(2.8.5)(2.8.5)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
> > (2.8.1)(2.8.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
"18.2.22"
"18.3.4"
> >
(2.3.4)(2.3.4)
> >
> >
(2.8.5)(2.8.5)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
> > (2.8.1)(2.8.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
> >
> >
(2.5.2)(2.5.2)
(3.1.1)(3.1.1)
> >
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
> >
"18.4.20"
Other examples.
logarithmic derivative of Airy function, at infinityAn indication that this function is of interest appears in the Chudnovsky brothers article of 1991.We perform a slight linear fractional transformation to remove the first terms, and prove the formula.
f : = A i r y A i ( x ) ;
S :=Mul t iSer ies : -asympt ( f ,x ,100 ) :d S : = d i f f ( S , x ) :dlog:=asympt(dS/S,x,100):S t0 := subs (x=1 / t^2 ,d log ) :s e r i e s ( t * S t 0 , t , 9 0 ) :S t := eva l (%,csgn=1) :
(2.3.4)(2.3.4)
(3.2.2)(3.2.2)
(3.2.1)(3.2.1)
(3.2.3)(3.2.3)
> >
(2.8.5)(2.8.5)
(3.2.5)(3.2.5)
> >
> >
(2.2)(2.2)
> >
(2.5.5)(2.5.5)
> > (2.8.1)(2.8.1)
> >
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
> >
> > > >
(2.5.2)(2.5.2)
> >
(3.1.3)(3.1.3)
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(3.2.4)(3.2.4)
> >
(1.1)(1.1)
> >
(3.1.2)(3.1.2)
Formulanumtheory [c f rac ] (S t , t , 10 ,s imregular ) :numtheory[cfrac] (%, t ,s imregular ,quot ients) ;
Ad hoc transformation to remove the first terms.S : = e v a l ( s e r i e s ( 1 + t ^ 3 / 4 / ( 1 + S t ) , t , 9 0 ) , t = z ^ ( 1 / 3 ) ) :g fun: -ContFrac : -R icca t i_ to_Cf rac (subs( t=z ,S ) ,y , z ,10 ) ;
dlmf 4.25.5 f : = e x p ( 2 * A * a r c t a n ( 1 / z ) ) ;
A : = 2 / 3 ;
g : = ' g ' ;
g : = s u b s ( z = 1 / z , s o l v e ( f = 1 + 2 * A / ( z - A + g ( z ) ) , g ( z ) ) ) ;
G := map(normal ,ser ies (g ,z ,40 ) ) :g fun: -ContFrac: -R iccat i_ to_Cfrac(G,y ,z ,20 ) ;
(2.3.4)(2.3.4)
(2.8.5)(2.8.5)
(3.2.5)(3.2.5)
> >
(2.2)(2.2)
(2.5.5)(2.5.5)
> > (2.8.1)(2.8.1)
(1.6)(1.6)
(2.1.4)(2.1.4)
(2.4.1)(2.4.1)> >
(2.2.4)(2.2.4)
(2.5.2)(2.5.2)
(2)(2)
> >
(2.7.4)(2.7.4)
(2.6.4)(2.6.4)
(2.4.5)(2.4.5)
(1.1)(1.1)
> >
> > t i m e ( ) - s t a r t t i m e ;193.940
