Contains reports on two research projects.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E

McLoud, Veronica E.

Nacamuli, R. M.

Pennell, Martha M.

DSpace@MIT

XXIV. COMPUTATION RESEARCHResearch StaffMartha M. Pennell Gail M. Fratar Veronica E. McLoudT. H. Brooks R. M. NacamuliA. A COMPUTER INDEXING PROGRAMToday there are several indexes (e. g. , KWIC, Chemical Titles) which have beenprepared essentially by computer techniques. For the most part the computer pro-grams involved are too sophisticated to be used for indexing a small number of titlessuch as would be found in one's personal library or that of a small research group.We have written a computer program that permutes and alphabetizes the key wordsin a title of, at most, 132 characters for a maximum of 70 titles. The index obtainedconsists of entries 132 characters wide with the key word made to lie in column 67 andthe rest of the title shifted and folded over if it extends beyond the allotted length.The most difficult obstacle to overcome in such a program is to define what is meantby a key word. In our program a key word was said to have at least four letters. Afteralphabetization those entries in the index whose key words were not of interest weremanually discarded. Such a method is possible only if the number of titles is small.Some other indexing programs have used a table of non-key words which the programreferred to each time. Such a method has the disadvantage that titles consisting of non-key words are ignored. Moreover, if several groups with diverse interests are to usesuch a program, the table of non-key words changes radically and must be reconstructedeach time.Thus far, we have indexed 212 titles with the following results:Output 1555 title sAcceptable titles 1275Unacceptable titles 286Number of acceptable key words 75Average occurrence 3. 73Percentage of unacceptable titles 8 %The program is now being used to help update "Basic Data."'Martha M. Pennell, R. M. NacamuliThis work was supported in part by the Joint Services Electronics Programs(U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E).QPR No. 82 293(XXIV. COMPUTATION RESEARCH)References1. S. C. Brown, Quarterly Progress Report No. 80, Research Laboratory of Elec-tronics, M.I.T., January 15, 1966, pp. 83-85.B. EXAMPLE OF SYMBOLIC MANIPULATION OF POLYNOMIALS IN MADThe ease of solution of numerical problems by means of a digital computer is oftenlessened by necessary prior tedious mathematical manipulations such as the examplediscussed here, the multiplication of polynomials. The problem from which this examplearose is the following.Let x = sin 0 cos c, y = sin 0 sin 4, z = cos 0 (1)w1 = x 4 +y4 + - 3/5w2 = +6 + - 15/11[x4 +y4+ ]+ 30/77 (2)3 = x8 y8 + 8 - 28/15[x6 +y6+z6 ] + 154/143[x4 +y4+z 4 ]  7/39.Let f(x, y, z) be a product of 2 to 4 factors each of which is wl, w2 or w 3 . We wantto evaluate integrals of the formd f(x, y, z) sin 0 dO. (3)We can approach the problem in two ways, either by numerical approximation or for-mal integration. The latter method, which is preferable, would involve so much tediousalgebra to expand the product f(x, y, z) that it would be futile to attempt it unlessthe process could be mechanized. Investigation proved that this could be donequite easily.Our problem is to find an algorithm that will construct symbolically the productP(x,y, z) - Q(x,y,z), where P,Q are polynomials in x,y, z.If n is the highest power of x,y, z occurring in P, then P can be representedby the (n+1) X (n+1)X(n+1) array P1, where PI(I,J,K) is the value of the coef-ficient of the term P(x,y, z) in which the powers of x,y, z are I, J, K, respectively.If m is the highest power of x,y,z occurring in Q, then Q can be defined simi-larly by an (m+l) X (m+1) X (m+1) array Q1. If we define a new array ANS, whereANS(I',J',K') is the coefficient of the term of the product P(x,y,z) - Q(x,y,z) inwhich the powers of x, y, z are I', J', K', thenQPR No. 82 294(XXIV. COMPUTATION RESEARCH)ANS(I', J', K') = Pl(I,J,K) - Q (II, JJ, KK).I+II=I'J+JJ=J'K+KK=K'This algorithm was programmed in the MAD 1 language as listed below. Here X1,Y1,Z1 are the maximum powers of x,y, z in P(x,y, z), and X2,Y2, Z2 are the maximumpowers of x, y, z occurring in Q(x, y, z).:7 77t L FUNtCTIi (P ,I ,t , 1, 1,9Z I, XC, -,2 )E 'C [ULT.I TEG E I ,J ,K ,I I ,JJ ,KK,X1 ,YI 1 , 1 ,X ,YZT'H ZERO, FOR I=, I.0. (Xl+X+1)*(Y 1 +Ye+1)*(ZI+L +)ZELO PNS (I) C.T'T LCCOP, TCF I-C,!,I.G.X-T' LOCP,FO J= C, 1,J.G.Y 1T'. LCCP, FCF K-C, ,I.0.Z11T L FC, % II=C,1,I.C.X'il LCCP, FOF jJ, 1,J.C.Y2T'l. LOOP, FOE KK:OC,1, a..Z2LCOP / ~S (T+1I ,J+JJ ,i i) -A (1+ 1I ,.1 + , K)+P 1 ( , )I I (II ,JJ ,KK)F'lUCTICE FTUF<The routine may be called repeatedly if the product of more than two polynomials(2) is desired. From (1) the powers of sin 0, cos 0, sin 4, cos ) in the term whosecoefficient is ANS(I', J', K') are easily obtained. The desired integration (3) can now beperformed analytically term by term with the use of the recurrence relationSm+l n-1sin x cos x n - . m n-2sin x cos x dx x + n sin x cos x dx,S incs xd m+n m + nm,n > 0.The calling program and the subroutine were debugged and run on the time-sharingsystem at the Computation Center, M. I. T.Veronica E. McLoud, Martha M. PennellReferences1. B. Arden, B. Galler and R. Graham, "The Michigan Algorithm Decoder," Univer-sity of Michigan, April 1965.QPR No. 82 295

Computation Research

http://dspace.mit.edu/bitstream/1721.1/55489/1/RLE_QPR_082_XXIV.pdf

Computation Research

Abstract

Similar works

Full text

Available Versions

DSpace@MIT