d-ai37 776 research in algebraic ?rnipulation(u ... · d-ai37 776 research in algebraic...
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D-AI37 776 RESEARCH IN ALGEBRAIC ?RNIPULATION(U) MASSACHUSETTS IINST OF TECH CAMBRIDGE LAB FOR COMPUTER SCIENCEJ MOSES 14 DEC 8B. AFOSR-TR-84 88 7 AFOSR-8-0250
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AFOSR'.TR.
SECURITY CLASSIFICATION OF THIS PAGE
REPORT DOCUMENTATION PAGEis REPORT SECURITY CLASSIFICA1 ION lb. RESTRICTIVE MARKINGS
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4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S)
AFOSR.TR, 76& NAME OF PERFORMING ORGANIZATION b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION
Massachusetts Institute of (If .pplicabiu) Air Force Office of Scientific ResearchTechnology ,,
- 6c. ADDRESS (Cit). Slate and ZIP Codu) 7b. ADDRESS (City. State and ZIP Code)
Laboratory for Computer Science, 545 Tech- Directorate of Mathematical & Information0 nology Square,; Cambridge MA 02139 Sciences, Bolling AFB DC 20332
"2Sm. NAME OF FUNDINGlSPONSORING lab. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER" 4. ORGANIZATION Of appUcable)
AFOSR NM AFOSR-80-0250&c. ADDRESS (City. State and ZIP Code) 10. SOURCE OF FUNDING NO.
PROGRAM PROJECT TASK WORK UNIT
Bolling AFB DC 20332 ELEMENT NO. NO. NO. NO.
11. TITLE lnceae Security Ciaaafication, I '*U % " "
RESEARCH IN ALGEBRAIC MANIPULATION; 3n up 9 61102F 2304 A412. PERSONAL AUTHOR(S)
Joel Moses13& TYPE OF REPORT 13b. TIME COVERED 114. DATE OF REPORT (Yr.. %lo. Day) 15. PAGE COUNTInterim FROMI/7/82 T030/6/83 14 DEC 83 9
1. SUPPLEMENTARY NOTATION
17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse If neceuary aid identify b
FIELD GROUP SUB. GR.
19, ABSTRACT 'Continue on reverse if necesary and identify by block nurber"
-"-': L Professor Shunro Watanabe of Tapan has been visiting the group since March 1982. In the.'. past year he has been able, with the support of the group, to develop a major new pr'ogramLA- for solving in closed form a wide variety of ordinary differential equations.
""" The MACSYMA system uses two large packages for solving ODEs. The first package was writtenby Jeffrey Golden of the group. It solves equations by using a few well known techniques(e.g., separation of variables). It is quite weak in dealing with second order equations,
Lsuch as hypergeometric functions. The second package, due to Ed Lafferty of the MITRECorporation, incorporates many more techniques, but is still not a very general solver ofODEs.
Professor Watanabe's approach is to convert most second order ODEs into (CONTINUED)
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'P_' -" UNCLASSIFIEOUNLIMITEO 7- SAME AS RPT :9 OTIC USERS - UIJCL,:J1I'
22. NAME OF RESPONSIBLE INDIVIDUAL 22L, TELEPHONE NUMBER 22L OFFICE SYMBOL.nude . rra Code o
Dr. Robert N. Buchal '0. 767- 4939EDT DIFORM 1473, 83 APR OF I JAN 73 I OBSOLEtE 1"' ''-* ---!'T,
1 7N'L !,l IC.ATIN 0I, THIS PAC.
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UNCLASSIFI ED
SECURITY CLASSIFICATION OF THIS PAGE
ITEM #19, ABSTRACT, CONTINUED: instances of the so-called P-function. These functions,originally due to Riemann, have been extensively studied by the Japanese mathematicianFukuhara (also spelled Hukuhara). For the past year, Watanabe has implemented a packagethat transforms a large class of equations into P-functions. He has used Kaxe's tableas a test bed.
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AFOSR-TR.PROGRESS REPORT 1982/83
Professor Shunro Watanabe of Japan has been visiting our group since March,
1982. In the past year he has been able, with the support of our group, to
develop a major new program for solving in closed form a wide variety of
ordinary differential eqat .
The MACSYMA system uses two large packages for solving ODEs, The first
package was written by Jeffrey Golden of our group. It solves equations
- by using a few well known techniques (e.g., separation of variables). It
is quite weak in dealing with second order equations, such as hypergeometric
functions. The second package, due to Ed Lafferty of the MITRE Corporation,
incorporates many more techniques, but is still not a very general solver of
ODEs.
Professor Watanabe's approach is to convert most second order ODEs into
instances of the so-called P-function. These functions, originally due to
Riemann, have been extensively studied by the Japanese mathematician
Fukuhara (also spelled Hukuhara). For the past year, Watanabe has implemented
a package that transforms a large class of equations into P-functions. He has
used Kamke's table as a test bed. Several examples from Kamke using this
package, are given in Appendix I.
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-P APPENDIX I
(D2) [DSK, SWATAN]
(C3) showtime:trueSTime: 5 msec.
(C4) loadfile(pmain,fasl);
PMAIN FASL DSK SWATAN being loaded Kamke examole 378aLoading doneTime= 411 msec.(D4) DONE
(C5) batch(exampl,2);
(C6) k378a:x,(x-l),(x+l)-2*'diff(y.x.2)+2*x-(x+l)*(x-3)*'diff(y,x)-2*(x-1).y=0;Time- 57 msec.
22 d Y dY
(D6) (X - 1) X (X + 1) --- + 2 (X - 3) X (X + 1) -- - 2 (X - 1) Y = 02 dX
dX
(C7) lode2(k378a);dY
2 (2 X - 6) --d Y dX Z Y
we solve --- + ---------------- = 02 2 3 2
dX X - 1 X + 2 X + X
SOLVE FASL DSK MACSYM being loadedLoading done
PREMAP FASL DSK SWATAN being loadedLoading done
U...j we use Y -
2(X + 1)
dU2- 2
2 U dX d Uthe result is ------------ -- 0
2 X - 1 2x - x dX
dY2 2 --dY dX 2Y
we solve --- ------- ------- 0
2 X- 1 2dX X - x
PEQPI FASL DSK SWATAN being loadedLoading done
PHYPGN FASL DSK SWATAN being loadedLoading done ....
_e ' J'
* " 'i " % " " % % " -"" "' ' "" . . . . " - •" " ". . . .. • . - ,
.4o 4
" (D2) [DSK SWATAN] a e-%a-re 344
*, (C3) showtime:trueSTime- 4 msec.
* .:(C4) batch(exampl,7);
- - (C5) K344:X-4*'DIFF(Y,X.2) (EXP(2/X)-V-2).Y=O;Time- 65 msec.
24 d Y 2/X 2
(D5) X --- + (%E - V ) Y 02
dX
(C6) loadfile(pmain,fasl);
-,'. . PIAIN FASL DSK SWATAN being loadedLoading doneTime- 413 msec.
(D6) DONE
d (C7) lode2(k344);
*,2 2/X 2d Y (%E -V )Y
we solve -- ---------------- 0
2 4dX X
PIVTR FAS DSK SWATAN being loadedLoading done
4.
PTRINV FASL DSK SWATAN being loadedLoading done
we useT -
x".-. dY
2 2 --dY dT 2T 2
the result is --- + + (%E -V )Y 02 T
- dT
2 2--d Y dX 2 X 2
we solve --- + ---- + (%E -V )Y 0
2 XdX
PIVTR2 FASL DSK SWATAN being loadedLoading done
PTREXP FASL DSK SWATAN being loadedLoading done
PDVTR FASL DSK SWATAN being loaded
4..,v ,..,- ,• " " . . . . . . . . .. .. . .,9.'' - - , , , " ' " ,", "4 , -" . "-""" '" ,,. " • ,,--" ."." ." "•"•"-" "" - " - "
the type is hypergeometricthe solution may be written by Riemann's P-functions as follows
,0 1 INF)
y:p[0 0 2 (x)
i,'..T [ 1 3 1
PNOPM FASL DSK SWATAN being loaded*: Loading done
we solve[0 1 INF J[ J
y:P [ 1 0 - 1 ) (x)[ )[0 3 -2
/ 2C (X- 1)
y- (K2 I -------- dX + KI) X'I J 2
] xcontinue? type y or n
, Y;
SIN FASL DSK MACSYM being loadedLoading done
SININT FASt DSK NACSYM being loadedLoading done
SCHATC FASL DSK MACSYM being loadedLoading doneTime= 21253 msec.
2
(D7) - 2 K2 X LOG(X) + K2 X + Ki X - K2
Time= 23672 msec.(D8) BATCH DONE
(C9) closefile(buffersave);
.5 e
.oI'5 ,- ,- ,.. . .
Loading done
Loading done" SOLVE FASL DSK MACSYM being 1: ded
"= U
we use Yx
22X 2 dU
the result is U (%E - V ) + ... 0
2dX
2d Y 2 X 2
we solve --- + (%E - V ) Y - 02
dXx
we use T - %EdY
2 -- 2 2d Y dT (V - T ) Y
the result is --- +- -- - ----------- 02 T 2
dT TdY
2 -- 2 2dY dX (X -V)Y
we solve --- + ---------- 0
2 X 2dX X
PCONFL FASL DSK SWATAN being loadedLoading donethe equation is confluent type
PCNEQO FASL DSK SWATAN being loadedLoading donethe solution may be representable by Fukuhara's P-function.
[ INF INF 0 J[ V
y-P [ 2 (x)
I)% ]( XI - v ]( 2J
PCFPTM FASL DSK SWATAN being loadedLoading doney- Y (X)
-, B, ABS(V)Times 15604 msec.
q.(D7) Y MXB. ABS(V)
%- - . - -° - - J.. ..
W .,
(D2) [DSK. SWATANI
(C3) showtime trueSTimez 5 msec.
(C4) batchiexamIl.12); Kamke exam.'!e .06
(C5) k406: 16 (X'3-1)2-' DIFF(YX.2)-2? XY=O;Time= 41 mrsec.
23 2 d Y
(D5) 16 (X - 1) + 27 X Y 0
dX..
(C6) k406t:48*x-2,(x-1)-2 'diff(y,x,2)+32,x*(x-I)^2*'diff(yx)+9*x~y=O;Time- 50 msec.
22 2 dY 2 dY
(D6) 48 (X 1) X --- + 32 (X - 1) X -- 9 X Y 02 dX
dX
"* (C7) loadfile(pmain.,fasl);
PMAIN FASL DSK SWATAN being loaded
5..Loading doneTime- 426 msec.
. (D7) DONE
(C8) lode2(k406t);dY
2 2--dY dX 3 Y
we solve --- - + - .......-------------- 0
2 3 X 3 2dX 1C X - 32 X + 16 X
SOLVE FASL DSK MACSYM being loaded
Loading done
PHYPGN FASL DSK SWATAN being loaded
Loading donethe type is hypergeometricthe solution may be written by Riemann's P-functions as follows
[ 0 1 INF )[ J( 13 1)
y-P 3 4 3 J (x)(1 ]
.(0 - 03[ 4 ]
PHGHP FASL DSK SWATAN being leaded
Loading done
Li z" -. ""
W -jr -. 1 -0' CZ -w- 0!w
PALGS F-,SL CSK SAATAN being loadedLoadi ig done
we solve,0 1 INF ]
l[ )•.[1 1 1 )
1/4 - - ... ]y= (X - 1) P [ 3 2 12 ] (x)
-[ ]
0 0 -
(4)
PALG4 FASL DSK SWATAN being loaded
Loading done
P0H - '.SZL I.SK SWATAN being loadedLoading done
1/4 2 1/4 1/4KI T (2 SQRT(T + T + 1) + SQRT(3) (T + 1)) (X - 1)
y3 - - - - - - - - -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -3 1/12
(T - 1)
1/4 2 1/4 1/4K2 T (SQRT(3) (T + 1) - 2 SQRT(T + T + 1)) (X - 1)
-------------------------------------------------------------------------------------
3 1/12(T - 1)
" where t=x-(1/3)Time= 13449 msec.
1/4 2 1/4 1/4KI T (2 SQRT(T + T + 1) + SQRT(3) (T 1 1)) (X - 1)
(D8) ------------------------------------------------------------3 1/12
(T-1
1/4 2 1/4 1/4K2 7 (SQRT(3) (T + 1) - 2 SQRT(T + T + 1)) (X - 1)
---- --------------------------------------------------------------3 1/12
(T - 1)
Time- 15278 msec.(D9) BATCH DONE
(ClO) closefile(buffer,save);
-.
* -- . . . -. . . . . , o O .- ,o - --. o'. - *,
I,, :(D2) [DSK. SWATAN]
(C3) showtie:true$Time= 5 msec.
Kamke exam~le 'SO
(C4) batch(exampl,13);
(C5) K180:X-2*'DIFF(YX.2)+3,X,'DIFF(Y,X.I)+(X-2+1-V-2)*Y=F(X):Time= 47 msec.
22 d Y dY 2 2
(D5) X --- + 3 X -- + (X - V + 1) Y = F(X)2 dX
dX
(C6) loadfile(pmain,fasl);
PMAIN FASL DSK SWATAN being loadedLoading doneTime= 413 msec.(D6) DONE
(C7) lode2(k180);dY
2 3d 2 2
d Y dX (X - V + 1) Y F(X)we solve --- + -----------------
2 X 2 2dX X x
dY2 3 -- 2 2d Y dX (X - V + 1) Y
first we solve --- +---- + -- ---------- 02 X 2
dX X
SOLVE FASL DSK MACSYM being loadedLoading done
PCONFL FASL DSK SWATAN being loadedLoading done
the equation is confluent type
PCNEQO FASL DSK SWATAN being loadedLoading done
the solution may be representable by Fukuhara's P-function.[ INF INF 0 J[ ][ 3 ]-I - v I)
ylP[ 2 ](x)
C% V* - -, IE 2*F [ 2 b
PCFPTM FASL DSK SWATAN bigloaded
• ~~~~~~...,o.... ... %....................... .......... .. ... ...-.......... . *... . . .,.
Loading doneY (X)B. ABS(V)
yz------------------
xY (X)
B. ABS(V)the solution of the homog. eq. is-------------
x
PNONH FASL DSK SWATAN being loaded
Loading donea special sol. for the nonhom. eq. is
x/
I (%PI F(T) J (T) Y (X) - %PI F(T) Y (T) J (X)) dT) ABS(V) ABS(V) ABS(V) ABS(V)
4-. /
0
2Xcontinue? type y or nn;Time: 9943 msec.
x/[I (%PI F(T) J (T) Y (X) %PI F(T) Y (T) J (X)) dTJ ABS(V) ABS(V) ABS(V) ABS(V)I
0(07) -----------------------------------------------------------------------
2X
Y (x)B, ABS(V)
Time= 12131 msec.(D8) BATCH DONE
(C9) closefile(buffer,save):
.,
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