Bijlsma, S.J.

NARCIS 

Algebraic operations in Algol 60 : (the saddle-point method)

CWI's Institutional Repository

• STICHTING 2e BOERHAAVESTRAAT 49 AMSTERDAM AFDELING TOEGEPASTE WISKUNDE TN 45 Algebraic operations in ALGOL 60 (The saddlepoint method) by S.J .. Bijlsma May 1966 The Mathematical Centre at Amsterdam, founded the J~lth of February, 1946, is a non-profit i11Stitution aiming at the promotion of pure mathematics and its applications, and is sponsored by the Netherlands Government through the Nether lands Organization for the 1\dvancement of Pure Research ( Z. W. 0. ) and the Central Organization for 1\pplied Scientific Research in the Nether lands (T. N. O. ), by the Municipality of An1sterdam and by several industries. In the following we want to study the asymptotic behaviour of ~(s) defined by~ ( 1 ) cp(s) = C for· large positive values o'f s, assu1ning that f(z) en g(z) (f(z) = R(x 9y) + iI(x,y)) are analytic ~unctions of a complex variable z and R (x ,Y) - +00 at both ends or the conto11r co We consider a complex transformation :frarn the z to the w, .. ,, (-· u + iv) plane g w = f(z) ori u +iv= R(x,y) + iI(x,y) • • • This transfo11nat1.on is at a Let c' be the tre.nsf'or1ned conto1..ir c as is illustrated in the figure~ C '' 'W CCil-7[7 ' '. .. ;:'" , · .. · ·: •:: _ 0 ~> ,ur m : ; r ••*:..• •• <· ... 7 ... then we haveg (2) f e C ,sw g ( z ( w) ) dz dw O dWo c' The idea of the saddlepoint method is to reduce ce into such a path that .... sw = °"'SU e ·· e becomes as small as possible on tl'.iis path,. i 0 eo • c 1 is reduced into c'' o :tin 2 Clld) In the following .. we asst11ne that z 0 1.s a .. . the local substitution 00 (3) e first order 1 -dw .. , , "'" dw 0 . ' ... ,, saddle:point P 1 (2) changes . . . -· In order to calculate the asymptotic series I we first dete2·In1ne by • into i means of' program SJB 040166/1 the zero of the derivative of' the complex :function f(z)o Then program SJB 040166/2 is used to invert the series 2 . . . dz to calculate the series in Pe after which one can integrate formula (3) • in a elementary wayo To be more specific, letg Zo 2 + 000 + ..,, ....... ·.-•.~·.•·. n! (z 2 p 0 Suppose we putg \ t.i i=O i+1 a.p J.. k and then we can find the coefficients a. by comparison of coefficientso l. .. .. This g1vesi 00 + 000 + b ( + 000 =· 1 0 i · ·O As a result we findg .. let the coefficient of pJ in ( moreover n . 0 i= . • 1 0 b • 2 . l. a.p1 ) besom (l.j) then we have 1 c;:a 3 OIIID n+2 n-1 • ►ill { - k + 2) + a. o a o} 1. nc :J. where if l. Ifg a. --n k=3 i=1 - a. 1 else if 1 > 1 theng J a. o som (l J.. - z ) + a • Q • then the coe~fici.ent eking g(z(p)) •- e + 0 is given by .. Assum=Lng that g e -k k d. o som (i,k l. i) and n n P + eco we havei k g = o (i + 1 )a. J. so that formula ( 3) changes intog 00 e + 000 + pn + ooo)dp _oo a.nd as a result we havei 1T -s e 0 0 0 + 5 C, ooo (2n 1 ) ) n n s O 2 The coe:fficients are given by prograzn SJB 040166/20 + ooa) 4 -The fonnulas f ( z) and g ( z) 1 written in Polish notation are punched in the input paper tapeo The operation symbols and function symbols are punched as integers according to the following table 1 + 2 I 3 4 0 sin 5 cos 6 exp 7 ln 8 sqrt 9 arctg 10 The va.riabl.e z is punched as the integer 11 o A complex nurnber c is punched as the integer 12 11 f'oJ.lowed by two real n1.J111bers respect i veJ.y • 0 equal to the real. and imaginary part of co Exa,tnple g Z X Z .. • .. • ., • , < presented on the tape by the :following sequence o:f ntJrnbers ~ 2 3 1 1 1 1 1 12 2 0 8 1 1 • mentioned accuracyo Input of SJB 040166/2 2) • • 0 • • is the index of the last of' the coefficients c Output of this progra,m are the calculated coefficients respectively equal to the real and represented as • • imaginary part a pair of real nu1nbers of gkc The following MoCo Standardprocedures are used~ COMPLEX ZERO (AP 217)~ A procedure giving the zero o:r a :function of a XEEN(n) .. complex vax-iable c g An integer procedure assigning to its identifier a number which can be brought into the machine . ~ by the console and which is used here to deter1nine the array, -lenghts H and HC • in which ' • • • ,1 formulas respectively complex n111nbers are storede The idea of manipulating formu1as, as used in this note 9 is due to Rill Po ·v·o do Riet (ref o 2 and 3 c) o ?~g;!!?, ccm:n,::ient Opdrachtnr R1261 / TW160. Codenr SJB 040166 / le Bepaling reele e11 irnaginaire delen; !v.!!.!~g~r. krnax, kcmax; kmax:=XEEN(1023 ); kcmax~:::XEEN(1023X1024): 1024 ; ~~,~~g~l:-.. in~eger i ,k~kc ,x,y fz ,R, I; real xO ,yO ,XO, YO; ~P!~ge1~ ~r;r~y H[O:kmax,1:3]; real ~ryay HC[O:kcmax,1:2], e[l:3]; in1~~g~r, procedure STORE (i 9l,j); value i,.j; !!1:~~g~~,, i,l!lj~ 12,~gi,!:w STORE~=~::=k+l; !f k>kmftX t.hen ~egin PUTEXT1(~ k too large );stop end; I end~ • '. T ll-ffild -H[k,1 ]:=1;H[ksr2 ]:=l;H[k,3 ]:=j ~!~l._~g~!-", p:::·,9cedure S(i 1 j); y~~~~=- i ,j ~ ~1!~ege~. i ,j; S := 1f i=O t_her1 ~l else if j-0 then i else STORE(i,1.,j); 'W'tt: fl ik 'Trv« 7 • _.,_ - -- • l!,l~~ge,!·. pr~.cedure D(i 1 j); ~.!l~_eg~:r;, itj; D:-S(i,P(NU ......... ER(STORECN(-lJO)),j)); lP;~~~g.e:: .. P,r9:3~¢l~r~. P(i ,j); ~sa.~~:e 1 ,j ~ ~n~~g~r i ~j; P:~-if i=OV j=O then O else if i==-1 then j else if j=l then i else STORE (i~21)j), - _,___ ---- ;n;t Jlilllll,JII I FT 7 P li r:n I =• - "1 I ill IC & 10 CT ~,E.~~g~,r,, r9£~dure Q(iij); ~nteg:311" i rj; Q:= STORE (i,3 ~j), !r1t_eg~-~I pr9,9~??;re SIN(i)t i~t~g~r i; SIN:-STORE(l,O&i); i11t~g~,! P,1:9~~¢l_ur,~_ COS(i); ~p.t~g~,r i, COS:==STORE(2 ,O,l), ! .. 1~t~$er E!'P~.~,9.!l:~- EXP (j); -~Il-~f.!K~r i; EXP:=STORE (3 ,o ~i) 3 i!!!·~-~er P,.r~c,~~~1re LN(i); ~p~eKe1~ i; LN != STORE (4,0 "i) ~ . inte er I?~r:.?cedure SQRT(i)~ ~.!!t.,~g~r. i; SQRT~-STORE (5,0,i); ~~~~g~: E.~?~~du!~. ARCTAN(i); !P:t,eger i, ARCTAN:=STORE (6!)0,i); L~t~$~}~_ E,~:.?£f}.?~re NUMBER(i); !!!~!3S~E,, i; NUMBER:=STORE(-1 ~O ,i), integ~~- P!?C~.?~!'!=:. STORECN(al,a2)§r:_e.~~- al,a23 a brl~gi!l STORECN :=kc :=kc+ 1; if' kc>kcma.x then !Jfg~n PUTEXTI(---'kc HC [kc ~Jw ):-al; HC [kc,2 ]:=a2 encl; P,!.,o.,cedure RI(i,R,I) ;yal?~- i, .i!1tege:r, i,R,I; ~:.~, in ~!1!~g~,t. rl ,r2 ~il ,i2; if i=z then beg~~ R:=-=x;I:.:::y er1g else ) ;stop end; · : n1::a: n • ~f_ H[islJ~-1 then P.~g,~n R:=NU ER(STORECN(HC[H[i,3],1)1:0)); I:-•- NU ER(STORECN(O~HC[H[i,3]1 2])) end else !f.r H[i,2]=1 then !?-.~S~rl;. RI(H[i,l],rl,il);RI(H[i,3]fr2,i2)t R:=S(r1$r2); I:= S(il ti2) _, 6 ow,, end else if H(i,2 ]=2 then r,,egin RI(H[i,1 ],r:1 ,il) ;RI(H[i~:0 ],r;·2 ,i2); . else R:-•D(P(rJl ,1ro2} ,P(il ,i2}}; I:~S('P(rl ,·12) ,P(r·2 ,il)) end if. H[i,2 }=3 then P,~~~-1?:. RI(H[i,1 ],rJl ,il) ;RI(H[i.3 ],r~2 ,i2); - ' R: ·-Q(S(P(rl ,r2) ,P(]l ,i2)) ,S(P(r~2 ,r2) t)P(i2 ,i2) )) ; ~ - - ' ' ' . I:=Q(D(P(il ,r2) ,P(rl ti2)) ,S(P(r2 ,r2) ,P(i2,i2))) end else ' if H[i,1 ]:=3 /\ H[i,2 }=o then pr,~~·?, RI(H[i,3 ],rl ,il); end else R:- ·P(EXP{rl) ,COS(il}); • • ' I:=P( EXP(rl) 9SIN (il)) if H[i,1}::2 /\ H[i,2]=0 then JJ,eg~n, RI(H[i,3],rl,il); R:=P(P(NU ER(STORECN(. 5,0)),,COS(rl)), S( EXP(il) ,Q(l ,EXP(il)))); I:==P(P(NU. ER(STORECN(. 5,0)),SIN"(rl)), D( Q(l ,EXP{il )) ,EXP(il) )) end else • if H[i,1 ]=1 /\ H[i,2 J=O then p~gi~, RI(H(i,3]~·r·l,il),; end else R:-P(P(NV ER(ST·ORECN(., 5,0)),SIN(rl)), S(EXP(i1.),Q(1,EXP(il )))),; I:=P(P(N·u-MBER(STORECN(-. 5,0)),COS(rl)), D(Q(l .EXP(il )),EXP(ll))) if H[i,1 ]=4 /\ H[i 92 }=O then !!~g~P,-, RI(H[i,3 J,rl ,i.1); • else end R::==LN (SQR;T(S(P(1~1. ,rl) ,P(1.l ,il )) )) ; I::..=ARCTAN(Q(il ,rl)) = 1 cc, !,f. H[i 5l }=5 /\ H[i,2 ]=O then 7P.ezl!:·,., RI(H[.i 93 ]grJ. ,il.}; els~ ' R: •RP(SQR~T(S(.P(i0l ,rl ),P(il ,i,l) )) , COS(P(ARC 'i .. AN(Q(il ,rJl)), N1JMBER(S·TORECN(. 5 510)}))); I:--P(SQRT(S(P(rll,1·1),P(il ,11 ))), SIN ( P(.A.RC TAN ( Q(il ,r·l)), Nl~·MBER(S'TORECN(. 5 ,0) )))) er1d if H[i,1 ]=6 /\ H[i,,2 ]=O then '!?,~f5!1:1. RI(H[i,3 ],rl ,11.); R:-•P(Nl.T .., ..... ER(STORECN(. 5,0)) 9 D(ARCTAN(Q(P(NT; ER(STORECN(-1,0)),rl), S(il 9Nl.IMBER(STORECN (-1,0))))), ~A.RCT ~L\.N(Q(P(NUMBER(STORECN (-1,0 )) ,rl), S(il,1))})); I:~-P(NU ........ ER(STORECN(-~ 5,0)), LN (SQRT(Q(S{P.(rl 1rl) ,P(D(l ,il) ,D(l ,il))), S(P(r]_ ,r·1) ,P(S(l ,il) ,S(l ,il)) )) )) ) ,.._, C ti end; real Et'?.~~~~~,e. VALUECN(i); val·ue i; ~·nt,~&~E. i; ~p.f!3g~r Pt~.~,1~~;!-lr:971 DI(i); value i; ~r:t~~f?i~:, i; DI:= if H[i,l l=-1 then O else if i~- z the:n 1 Flse • I 1 -- we,,, '. · 1 I ¥· Pl M TX f tr 7 ',11 if H[i,2 }-=1 then S(DI(H[i,J~ ]} 11DI(H[i,3 ])} else if H[i 512 }=2 then S(P(H[i ,1 ],DI(H[i,3 ])) ,P(H[:i~3 ],DI(H[i91 ]})) else ':I tr ., • if H[i,2 }=3 then Q(D(P(H[i,3 ],DI(H[i 91 ])) ,P(H[i,1 )9 DI(H[i,3 ]))), P(H(i,3 ],H[i,3 ])) else if H[i,1 ]::::J_ /\ H[i,2]=0 then P(COS(H[:i,3]),DI(H[i,3]}) else ..... 1•· if H[i,1}=2 A. H[i,2J=O then P(NU .J...,ER(S1'10RECN'(-l~O)), P(SIN (H[i,:3 ]},DI(H[i,3 ])) ) else if H[i,1}=3 A H[i,2]=0 then P(EXP(H[i,3]),DI(H[i,3])) else -- ' - -if H[i,1}=4 /\ H[i,2]=0 then P(Q(l,H[i~3]},DI(H[i.3])) else _____ .. .,. ·-· --..,- - .._ ...... if H[i,1}=5 /\ H[i,2]=:0 then P(Q(l,SQRT(H[i,3])),DI(H[i,3]}) else 11 1 J.rat s:r :n: :au r P(Q(l,S(l,P(H[i,3 ],H[i,3 ]))),DI(H[i,3 ])); ' • - 8 -if H[i,1 J=-1 the11. VAL t:·EcN (1) else if j_:.:-::x ·t.hen ... 7 17 xO else if i.:.:y then yO else ---if H[i,2 }=1. t:he.~ (R.IF(H[i,1 ])-+RIF(H(i,3 ])) else. • "&4rn'■ 1 if H[i,2 }=2 then (RIF(H[1.,l ]}xRIF(H[i,3 ])) else • • • ·t··• H[· 2 1--·~ 1. 1, rQ..I then (illF(H[i,1 ]) /RIF(H[i.3 ]}) c)lse WT? n f 1f H[i,1 ]=1 /\ H[i,2}=0 then si:r1(RIF(H[i,3])) else if H[i,1}=2 A H[i,2}=0 then cos(RIF(H[i,3])) else . . if H[i,1}=3 /\ H[i,2}=0 then exp(RIF(H[i,3])) else !~. H[i,1]=4 A H[i,2]=0 then ln(RIF(H[i,3])) else I if H[i,1]=5 /\ H[i,2}=0 then sqrt(RIF(H[i,3])) else arctan(RIF(H[i,3])); pr,o,c,e9-;1,re, CZERO(xO,yO,r,s,e); real r,s,xO,yO; ,arr,aYs e; pep;~n real a,b ,c ,d,el ,e2 te3 ,g,h,rO ,rl ,r2 ,s0 ,s1 ,s2 ,t,u; I el:·-e[l ; e2:=e[2 ; e3:=e[3 ; a:-xO; g:==sqrt(axa + yO )x.1 + .1; h:=O; c:=-. 5; d:=O; xO:==a+g; rO:==r; sO:=s; xO:=a-g; rl:=r; s1:=s; ro: .. -rO-rl; s0:--s0-sl; xO: a; r2 :=r; s2 :==s, LL: rl:=rl-r2; sl:-s1-s2; t: rOxc - sOxd -rl; u:=rOXd +sOxc -s1; a:=(t.-rl)Xc-(u-sl)Xd-rl; b:==(t.-rl)Xd +(u-sl)Xc-s1; rO:=tXc:-UXd; s0:-t><d+uxc; t:-- 2. Ox((l. O+c)Xr2-dxs2); u:=-2. Ox((l. O+c)Xs2+dXr2); CSQRT(axa-bxb+2. ox(txrO uxsO) ,2. Ox(axb+txsO+uxr~o) ,c,d); if axc+bxd<O then !J~g:i~ c:--c; d:---d end; a:=a+c; b:==b+d; c: ·--(axt+bxu)/(axa+bxb); d::=(a><u-bxt)/(axa+·bxb); a:· ·sqrt(cxc+dxd),110_; a:·-gxc-hXd; h:=gxd+•h>c:c.;; g:--a,; xO :=xO+g; yO :=j,rO+h; rO:--rJ_; sO:---s1; rl:==~1·2; sl:=s2; r2:==rz s2:=s; XO: ·-xO; YO:=-=yO end CZERO; ip :=b :=b/(2. Oxt·p); • • 9 =-if a,<0 then p,eg~n ip::--=if b>O then I"P else -rp; rp:--abs(t,) end; -CSQRT:·--rp end CSQRT; ,.._ . - , - . i11teg:er P~,o~e9:ur,e f; 7 j r~g~.n ~nt':3~.e,~~, r";re[il 1:·1,r2; switch F:-prod,som,quot,verech,sj_n,cos"exp, ln,sqrt,arctan, var ,getal;r· :--read;g,o~~. F[r ]; prod: f:=P(f,f); g9~9 END; som: f:-=S{f,f); ~ot9. END; c:1.·uot: f:;--=Q(f ,f); go~o END; versch: f:---D(f ,f); 8:?!~ END; sin: f:---SIN(f); -~?to END; cos: f:=-COS(f); g~~9 END; exp: f:=-=EXP(f); go~9 END; In: f: LN(f); goto. END; sqrt: f:-•SQRT(f); go~tp END; 1 arctan: f:=ARCTAN(f); g~:~9. END; var: getal: END: f :=z; g~~10, END; rl :~-read;r2 :--read;-f :--NlTMBER(STORECN (rl ,r2 )) ; end; • '• BEGIN OF CALCN: k:•-kc:-- -1.; NUMBER(STORECN(O,O)); Nl1' ER(STORECN(l,O)); end (1nd z:==STORE(-2,0,0); x:=STORE(-2,0,0); y: -STORE(-2,0PO); xO:=.read; yO:::=rE.ad, for i:-1,2.,3 do e[i]:-read; RI(DI(f),R,I); CZERO(xO,yO,RIF(R),RIF(I)~e); P. LCR; FLOP(3,1,XO); PUTEXTl( ., ); FLOP(3,1,YO) • • 1 10 -t?,EJg~n commEJn.t Inverte:r"en van reeks met c·omplexe coeff. Rl261 SJB 040166/2; ~-n~~g~r,~, k·m.ax&kc~max,rnO; kma:1{:-= XEEN(102~3); kcmax:== XEEN(1023 x 1024): 1024;mO:-•read; d[l:mO], a[O:m0-2], H[0:1. .. max,l:3],h[O:m0-2], b[O:m0-2], e[O:mO- -2 ]. g[O :m0-2 ]; ~,:l"!',a.y, HC [O:kcmax,1 :2 ],fac[l :mO ]; -i,1,j; ~~~iE~. STORE:- k:- k + 1; -~,f- k > kmax then -~p,g~1a PUTEXTl ( ..... , H[k,1 ]:-• i; H[k,2 ]:-- l; H[.k,3 ]:= j too large ) ; stop end; end; • ~p.~ege:3_ P,r,9,9,~.d!1r~., S(i,j); ,ra.lue i,j; ~nt~g~!, i,j; S:= ~ -~ .. - -· !fr i - 0 then j else if j ~:.! 0 then i else STORE(i,1,j); ~P.,t_e~.Af-, p;r9c~~u.r~, D(i,j); !ntefiet· i,j; D:=-= S(i,P(NUMBER(STORECN(-1,0)),j)); ~n~eg~-;r EF,?,9~?~:~- P(i,j); v:ilue i,j; if_ i = 0 V j -- 0 then O else if i . .::: 1 then j else if j = 1 then i else·-STORE(i,2,j); ~.nt~ger~ p·r99!39-:U!~ .. Q(i,j); ~~tr,ge:rJ. i,j; Q:-- STORE(i,3~j); ~·~~-~.eger0 2r,?1~1~1d;J}~~. SIN(i); ~l}~.eger i; SIN:-- STORE(l,O,i); ~·~~ee;=:r: P,:r?,c,~~:~r-~, COS(i); inte~er, i; COS:- STORE(2,0.i); i~t-~Sze:, pro~~.?:--1Fe. EXP{i); ~ntege! i; EXP:= STORE(3,0,i); ip.~.eger, procdled~~-r~IT LN(i); !P~~g~;1;:. i; LN:= STORE(4,0,i); !n~eg';r. P!9Ce_dur_e SQRT(i); -~n.t~g~!, i; SQRT:= STORE(5,0,i); ~nt~ger pi:9ce~!:re. ARCTAN(i); ~;n~~g,erp i; ARCTAN:= STORE(6,0,i); ~~t~ger, P!?cerl~.1F~- STORECN(al,a2); real al,a2; ~~~g~l?-- STORECN :=kc:=k.c+ 1; HC[kc,1]:=al; HC[kc,2]:--a2 or1d• ~ J ~- , 'Tl tall!? q •· ' ~!:t,eg~:r~~. p.1~0,9i~?:1r~, pr(i,j); v·a,l,J.e i .j; ~~t~ge,~, i ,j; ~-·· - , ... ., b2 :~~HC (i,1 }<HC[j,2 ]+ HC [i,2 }<HC[j,1. ]; pr :·:-:STORECN (bl ~b2) . t:-"!!. .. • l .. v~. ue 1,J, 'llteef •• ,,,,, • ,..., 11 --p~gin real bl,b2; bl:=HC[i,l]t-HC[],1]; b2:=HC[i,2]+HC[j,2]; • so:- STORECN(bl ,b2) end; !~:~~.~gE~.~- l?r?~~?1~!~, di(i,j); v~:1l·u.e i,j; ~!!ttf-g~~ i.j; _b~g!;n real bl,b2; bl:-=HC[i,IJ-HC[j,1]; b2:--HC[i,2]-HC[j,2]; di:- STORECN(bl ,b2) .o.nd e, ., ..... , ~r,.~t~g~-r:. Eroce~tlre qu(i ,j}; ·\1·alue- i,j; !n~~g!=rq i,j; ~eglp real b,1.b2; bl :=(HC [i,1. ]><HC[j,1 ]-:-HC [i,2 ]xHC [j,2 ])/ (HC [J~l ]xHC [j ,1 ]+HC[j,2 }><HC[j,2 ]); b2:=(HC[i,2 )><HC[ .. j 171 ]-HC [i,1 ]xHC[j 12 ])/ (HC [j,1 }><HC[j,1 ]+HC[j,2 }<HC[j~2 ]) ; q.u:- STORECN(bl,b2) end; !~~eg~:. pr9~-e~~r~1 ex(i); °'\79.lue i; ~p.~1eg~~;,. i; p1~gip.. real bl,b2; bl:··exp(HC[i~l])xcos(HC[i,.2 ]); end; b2 :- -exp(HC [i ,1 ])Xsin(HC [i,2 ]) ; ex:=STORECN (bl ,b2) :ipt~g1e!. P,;r!=}9E:-d~;r~, lg(i); value i; ,~~y~er, i; p~g~p, r~eal bl ,b2.; bl !·-ln(sqrt(HC[i,1 }xHC[i,1 ]+HC[i,2 ]xHC [i92 ])) ; . .. b2:=arctan( HC[i,2VHC[i,1]); lg:=STORECN(bl ,b2) end; i!ltreger. pr,op~1d~!!3. sq(i); value i; ~nteg~r, i; a b.ef;i~ real bl ,b2; bl :r•~sq.rt(HC[i 9l }<HC[i,1 ]+HC[i,2 ]xHC[i,2 ])x ' end; " t:. 'LZi ;z; c~os(arcta!1(HC [i,2 ]/HC [i,1 ]) /2); b2: .-sqrt(HC [i,1 }<HC [i,1 ]+HC [i,2 }<HC [i112 ])>< sin (a.rcta.n ( HC [i ,21'HC [i ,1 ]) /2) j sq :=STORECN (bl ,b2) ' ~?~gip,1 real bl,b2; bl:--sin(HC[i,l])x(exp{HC[i,2])+exp(-HC[i,2]))/2 ; . . ' lJ2 : .. -cos ( HC [i 91 ])x ( exp(-HC [! ,2 ])-exp( HC [i ,2 ]) ) /2; . . . si:=-=STORECN (bl ,b2) end; • _, 12 = ~?t<:~~,r. --~ :("O(·e1iure co(i); y'"a,lue i; in~;e.ge~r 1.; ~ • ' M .. ·~ b_~r;~i~~- rea.l b 1 ,b2; bl :-=c~os ( HC [1,1 ])x ( exp( HC [i 02 ])+exp(-HC [i ,2 ]) ) /2 ; b2 :- -sin ( HC [i ,1 ])x ( e:-i.:.p(-HC [:i ,2 ])-exp(HC [i ,2 ]) ) /2 ; Cli:=STORECN (bl. ~b2) • end; i;nteg:e1'> p~~o~e?~~:~ aI"(i); ·va.lue i;tr1t~g~£) .. i; . ··-··. ~· ?~zip "f.f:,?-1 UI 1ol ~b2, bl :-(:1-·rctan(-HC [i,1 ]/(HC [i ,2 J-:1) )-ar.ictan(-HC [i,1 ]/ (HC [:i,2 }r 1))) /2; b2 :=-lr1(sc~1:r·t( ((1-HC [i,2 ])x(l-HC[!,2 ])+HC [i,l ]xHC[i,1 ])/ ( (l+HC[i,2 ]}X(l~-HC [i 92 ])+HC[i,1. ]xHC [i,1 ])))/2 ; • . ~.rt :==STORECN (~Jl ,b2) ..... A:>, w Lrit;:,g~:!. E_:rc,ced'U.re sum(i?g,i'ittE:i); Yi?-~1?:~- g,b,; inte~~!. i.g.b,ei; ,..__ . tl,efr~.n-, real s; s:=O; f9,.r.:,. i:=g st~p 1 !;!!!,:,}~& b ~:? s:=ao(s,ei); sum:=s end; !Y?-~~p,;er~ o·r.iccedure som (l,j}; v:~.l~.].e 1; }nt.eg:E?:r:. 1,j; !:'esl~. lntt=: f:r~ i; som :-ii 1-1 then a[.j] else sum(i,0 0 j,pr·(s.[i],sinm(l-l,j-i))) ~• 111 -- re r -,,c • .r~f..~al Pr.o~~~?·u.r~, F AC (i); "'"valtlf~ ],; integer i; • •• ,•••""• -•~c• !?~g,~~t int.er,:~17,. j; A: .~! 1 i<kf theri F AC:.:-; f'ac[i] else heg~!·1. for j::-.:2 _s~ep; 1 until mo do factkf+j]:= fac(kf+j-1 ]x(kf+j); --~ ' ·-- - -·- .. kf :-= kf·+mO; p;.r.•t:?_ A en(i ·--encl; ,i!1~_eg~r pr~:>_r:;~d~.r~e DI(i); value i; ~1~1;i:?~e:, i; DI:= if H[i,1 J=:-1 the:n O else if i=z then 1 else m,t Ill -- - \I I ■ kCII: •r 11 .. . ... " . ~" . if H[i,2]=1 then S(DI(H[i,1]).DI(H[i,3])) else ............ «-· vs t•r•·r: if H[i 512}=2 then S(P(H[i,1],DI(H[i,3])).P(II[i,3],DI(H[i,1]))) else 'C 1 r -zw; I F/U. ; , • -if H[i,2]=:3 then Q(D(P(H[i!)3],DI(H[i,1])),P(H[i,1],DI(H[i,3]))). P(H[i,3 ],H[i,3 ]}} else if' H[i,J_ J=l. A H[i,2 }=O ·then P(COS(H[i,3 ]),DI(H[i,3 ])) else if H[i,l ]=2 /\ H[i92}=0 then P(NlTMBER(s=roRECN(-1,0)), ':lz • •• P(SIN{H[i,3 ]),DI(H[i,3 ]))) else if Hf~i,1 ]=3 /\ H[i,2 }=O then P(E.XP(H[i,3 ]) ,DI(H[i,3 ]) ) else ,, ", •= 13 1ft J.f., H[i,1 }=4 /\ H[i!)2]=0 thE·n P(Q{l,H[i,3]),DI(H[i,3 ])) else . . if H[i,1.];:-.:5 1\ H[i,2}=0. then P(Q(l,SQRT(H[i,3]}):>DI(H[i,3])) else P(Q(l,S(l,P(H[i,3 ],H[i,3 ]))) ~DI(H[i,3 ])) ; ·-~ ' . . -"' intep;~,r P:O~·~?~!~~. v,..LA.Lt;·E(:1); V"j.lue i; j~~~g~r. i; VALlTE:~~ if H[i .• 1 J::-1 then H[i ,3] else if i=z then zO else if H[i,2}=1 ·then so(VALlIE(H[l,1 ]),VALUE(H[i,3])) else if H["lt,2}=2 then pr(VALUE(H[i,1 ]),VALUE(H[i,3])) else • if H[i,2]==3 then q~1(VAL{JE(H[i,1]),VALLTE(H[i,3])) else if H[i"l ]--1 /\ H[i,2 }=O then si(\7 AL l.TE(H[i,3 ])) else -· . ... .. - . if H[i,1]=2 /\ H[i,2]=0 then co(VALUE(H[i,3])) else . if H[i,1}=3 A H[i,2}::0 then ex(VALlJE(H[i,3])) else if H[i,1]=4 /\ H[i,2}-:0 then lg(VALl.TE(H[i,3])) else if H[i,1}=5 /\ H[:i,2}=:0 then sq(VALUE(H[i,3])) else ar(VALU.E(H[i,3])); -· , ' ,... . . . ~!1 te~err: P:-:9?,ed~I'~ _ f; b~e;ip, !.1:1~~~~1~. r;r;leal rl ,r2; s'Witch F: -prod,som,quot,versch,sin,cos,exp, ln,sqrt,arctan,va.r 1getal;r:- ,,,read;got?,, F[r ]; prod: f:=P(f ,f); g~~10 END; som: f:· -S(f,f); go~g END; quot: f:0-::Q(f ,f); g?t~1 END; versch: f:- D(f.f); g9:r9. END; sin: f:--SIN(f); g9t9, END; cos: f:·-COS(f); g~1r,,~. END; exp: f:=EXP(f); goto END; ~- ' . -ln: f:--LN(f); g;?t(? .. END; sqr~t: f:-SQRT(f); g:9,t~-s END; arctan: f::=ARCTAN(f); goto END; var: f:-*z; goto END; getal: rl :-r·ead;r2 :• -read;f :=NlJ ER(STORECN(rl,r·2)); • • END: end; SJ I \W - -- - --- -~=- --- - • -. . . for n:- -1 _s~~P 1 until m0-2 do • stop end; - 14 :::; -i-;.-2))), pr ( c [2 ],sum ( i ,1. ,n-1,pr (a[i ],a[n-i ]) ) ) ) ,pr· ( c [2 ],pr (ST ORECN (-2, 0) ,a[O ]) ) ) ; . . HC [K+n,1. ]:·-HC [a[n],1 ]; HC [K+n,2 ]:--HC [a[n],2 ]; kc:·-K+n; a[n]:==K+n end; Kl:·-K+m0-2; for~ n:• -1 ~~~E- 1 until m0-2 do HC[Kl·+n,1 ]:==HC(e[n],1 ]; HC [Kl+n,2 ]:--HC [e[n],2 ]; kc:=Kl+n; e[n]:=Kl+n end; f9r" n:=O _s~ep 1 until m0-2 do ' be~in. g[n]:=sum (i. ,0 ,n,pr(pr(STORECN (i + 1,0) ,a[i]) ,e[n-i ]) ) ; 1 1 ' S .. k PlfNLCR; PUTEXTl( g[ -..); SFIXP(2,0,n); Pl,TTEXTl( . }= ); • • FLOP(3,1,HC [n],1]); PVTEXTl( , ); FLOP(3,1,HC[g[n],2]) • end er1d; BEGIN OF CALCN:k:·-kc::=-1 • :=O; f·a.c[.1]:--l;NUMBER(STORECN(O,O)); NU ER(STORECN(l,O)); z:-STORE(-2,0,0); xO:=read; yO :=read;zO :· ··STORECN (xO ,yO); for n:==l ~~-'yE. 1 ;!}'ltA~. mo do pe~~-P= if n---1 t.hen d[l ]:-DI(f) else d[n]:=DI(d[n-1 ]) end; for n:=O .~~.!3-P 1 until m0-2 ,jo Jl,e~i.p ~f- n--0 then h[O ]:·-f else h[n]:=DI(h[n-1 ]) end; SP end end References 1o HoAo Lauwerier 2o RoPo Vodo Riet 3o RoPc Vodo Riet 15 IIPI' Asymptotic Expansions Math r Centre tract 13 1'-1a.tho Centruzn. (1966) Algebraic operations in ALGOL 60 The Cauchy-problem I Report TW 97 1 Matho Centrurn ( 1965) application o:r a method for a1gebraic manipulation in ALGOL 60 Report TW 99 • Math" Centrtull { 1966) 

Algebraic operations in Algol 60 : (the saddle-point method)

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