111 L3, L 111 "- MICROCOPY RESOLUTION TEST CHART NATINAL BUREAU OF STANOARDS1963-A

Transcription

1 HD-R137 IEEEEEE EAERCH IN RLGEBRFIIC MANIPULATION(U) MASSACHUSETTS I'll INST OF TECH CAMBRIDGE LAB FOR COMPUTER SCIENCE J MOSES 26 JUN 8-1 RFOSR-TR RFOSR UNCLRSSIFIED F/G 12/1 NL

2 111 L3, L 111 "- MICROCOPY RESOLUTION TEST CHART NATINAL BUREAU OF STANOARDS1963-A

3 SE U-IT Vr( I PAGE Is. REPORT Its-wril I ~ CTIVE MARKINGS UNCLASSIFIED 2&. SECURITY CLASSIFICATION AUTHORITY 3. DiSTRIBUTION/AVAI LABILITY OF REPORT Approved for public release; distribution 2b. OECLASSIFICATION/DOWNGRAOING SCHEDULE unlimited. 4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5.1MONITORING ORGANIZATION REPORT NUMBER(S) 6.. NAME OF PERFORMING ORGANIZATION b. OFFICE SYMBOL 7. NAFORITRN ORGANIZA 036 Massachusetts Institute of (Itappicalil) Air Force Office of Scientific TRcs.arch 6c. ADDRESS (City. State and ZIP Code) 7b. ADDRESS (City, State and ZIP Code) Laboratory for Computer Science, 545 Directorate of Mathematical & Information *Technology Square, Cambridge MA Sciences, Boiling AFB DC G. NAME OF FUNOING/SPONSORING Sti. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATIONI NUMBER ORGANIZATION (fapciie AORNM AFOSR O. ADDRESS 101ly. State &vd ZIP Code) 10. SOURCE OF FUNDING NOS. PROGRAM PROJECT TASK WORK UNIT *- Bolling AFB DO ELIEME NT NO. NO. NO. NO. 11. TITLE (incltude Security C~aiuatieetionj SEE REMARKS 61102F 2304 A4 * 12. PERSONAL AUTHORI 13.. TYPS Of REPORT 13.TIME COVERED 114. DATE OF REPORT (yr.. Moa.. Day) 15. PAGE COUN Interim FROM/7/81 TD30/6/ JUN SUPPLEMENTARY NOTATION Ne 1?. COSATI CODES 18. SUBJECT TERMS (Continue on reverse it neeuary and identifyb by) R FIELD OUSU. 19. ABSTRACT (Contianue on reverse it necesary and identify by block number; This was a transitional year. One in which the investigators phased dow research on the integration of algebraic functions (described in last year's report), and one in which they began a new line of research on the solution of ordinary differential equations in closed form. Barry Trager continued his doctoral thesis research on the integration of algebraic function while working at IBM Research in Yorktown Heights. The report on this work * previously submitted still gives a very good account of this line of research. The approach he has taken is to determine the minimal extension field in which the integral can be expressed. This can lead to a tremendous improvement in both the running time and the size and complexity of the expression that is produced. (CMNIMME) TRISOUT ION/AVAILABILIT Y OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION UNCLASSIIEDIUMLIMITED 52SAME AS RPT. OTIC USERS 0 UNCLAZZ"IF1ED 22. NAME Of RESPONSIBLE INDIVIDUAL 22b TELEPHONE NUMBER 22c OFFICE SYMOA0L #include Area Code) Dr. Robert N. Duchal (,?0.1) 7G : 00 FORM 1473,83 APR EDITION OF I JAN 72IS OBSOLETE - *_ L I, T~ N'AASSIF ICATION or THIS PAGE

4 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE ITEM #11, TITLE: RESEARCH IN ALGEBRAIC MANIPULATION 1 JULY JUNE 82. ITEM #19, ABSTRACT, CONTINUED: The major effort in the last few months of the period was on the solution of ordinary differential equations (ODEs) by a Japanese visitor, Professor Shunro Watanabe. Professor Watanabe has begun to develop a subsystem of MACSYMA that solves ODEs by converting them to P-functions, originally studied by Riemann. One example of this approach to solving ODEs is given within. One goal of this work is to solve a large percentage of Kamke's equations using this general approach. iacces35iof For - NTIS GRA.:I DTIC Ti-B ; un~nn ou/ By ibuti! / Avilability Codes 1"ist Special IN-I. A!.,I r t,ptv CLA. IFrA,(1 r mr1 THl.% P. -,F

5 o. -.. AFOSR-TR-..- PROGRESS REPORT FOR THE PERIOD u - - ' -- it"r - " - This was a transitional year. One in which we phased down research on 1 i- C,,,-, r, '. % the integration of algebraic functions (dee-t-bed in last year's repzr), and -aav.-is "whieh"- bag--a new line of research on the solution of ordinary differential equations in closed form. Barry Trager continued his doctoral thesis research on the integration of algebraic function while working at IMB Research in Yorktown Heights. The report on this work previously submitted still gives a very good account of this line of research. The approach he has taken is to determine the minimal extension field in which the integral can be expressed. This can lead to a tremendous improvement in both the running time and the size and complexity of the expression that is produced. The major effort in the last few months of the period was on the solution.of ordinary differential equations (ODEs) by a Japanese visitor, Prof. Shunro Watanabe. Professor Watanabe has begun to develop a subsystem of MACSYMA that solves ODEs by converting them to P-functions, originally studied by Riemann. One example of this approach to solving ODEs ia given below. One goal of this work is to solve a large percentage of Kamke's equations using this general approach. Y. 1ir

6 PALGS FASL DSK SWATAN being loaded we Solve (0 1 INF ) -V 1/4 [ ye (x 1) P [ ) (x) ( 1 J ( 0 - ] 4) PALG4 FASL DSK SWATAN being loaded POIASZ FASL DSK SWATAN being loaded 1/4 2 1/4 1/4 KI T (2 SQRT(T + T + 1) + SQRT(3) (T * 1)) (X - 1) 3 1/12 (1-1) 1/ /4 K2 T (SQRT(3) (T + 1) - 2 SQRT(T + T + 1)) (X - 1) /12 (T - 1) where tax^(1/3) Time *sec. 1/4 2 1/4 1/4 KI T (2 SQRT(T * T * 1) * SQRT(3) (T * 1)) (X - 1) (DO) ww w /12 (T - 1) 1/4 2 1/4 1/4 K2 T (SQRT(3) (T + 1) - 2 SQRT(T + T + 1)) (X - 1) /12 (T - 1) Times asec. (DO) BATCH DONE (CIO) closefile(bufer.save); A I R r: U, 7 ' Sic Wt -1,..i Chier. Technical Information Division

7 (MZ) [DSK. SWATAN- (C3) showi*me.t!ue$ Time. 5 esec. (C4) batchfexampl.12); Lmauke exa =pe 4D6 (C5) k406:16-(x-3-1)-2-'diff(yx.2)z?,xy.0; Time. 41 sec dY (D5) 16 (X - 1) --- * 27 X Y 0 ". 2 dx (CS) k46t:48x-2(x-)^2. 'diff(yx,2)32.xe(x-1)-z.' dff(y.x -gox.y-0 ; Time, 50 msec dY 2 dy (DO) 48 (X - 1) X (X - 1) X -- + g X Y 0 o 2 dx dx (C7) loadl ile(pmain.fasl); PMAIN FASL DSK SWATAN being loaded Tim msec. (07) DONE (CS) lodez(k406t); dy dy dx 3Y we solve X 3 2 ail 16 X - 32 X + 16 X SOLVE FASL DSK 14ACSYN being loaded PHYPGN FASL DSK SWATAN being loaded the type is hypergeometric the solution may be written by Riosann's P-functions as follows, 0' 11I1 sap 3 4 PIM9P FASL CS1 SWATAN being leaded -, Ledial dloes

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