share/contrib/format/format.ps

   1 %!PS-Adobe-2.0
   2 %%Creator: dvips 5.518 Copyright 1986, 1993 Radical Eye Software
   3 %%Title: format.dvi
   4 %%CreationDate: Thu Sep  7 13:50:06 1995
   5 %%Pages: 12
   6 %%PageOrder: Ascend
   7 %%BoundingBox: 0 0 612 792
   8 %%EndComments
   9 %DVIPSCommandLine: dvips -o format.ps format.dvi
  10 %DVIPSSource:  TeX output 1995.07.06:1710
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  12 /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
  13 /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
  14 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
  15 ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
  16 isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
  17 hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
  18 TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get
  19 round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{
  20 statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0
  21 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn
  22 begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X
  23 array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo
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  25 FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{
  26 pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}
  27 B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup
  28 length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B
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  30 /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp
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  32 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff
  33 ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice
  34 ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{
  35 ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn
  36 /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1
  37 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D
  38 }B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0
  39 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add
  40 .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage
  41 userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook
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  48 ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley
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  50 scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave
  51 transform round exch round exch itransform moveto rulex 0 rlineto 0
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  53 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}
  54 B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{
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  56 -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{
  57 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end
  58 %%EndProcSet
  59 %%BeginProcSet: special.pro
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  62 false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B
  63 /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit
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  68 true def end /@MacSetUp{userdict /md known{userdict /md get type
  69 /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup
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  71 N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath
  72 clippath mark{transform{itransform moveto}}{transform{itransform lineto}
  73 }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{
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  75 closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39
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  91 maxlength dict begin /magscale false def normalscale currentpoint TR
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  93 /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx
  94 psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy
  95 scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR
  96 /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{
  97 psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
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  99 moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict
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1354 FFE007FFC0221F7F9E23>120 D E end
1355 %%EndProlog
1356 %%BeginSetup
1357 %%Feature: *Resolution 300dpi
1358 TeXDict begin
1359
1360 %%EndSetup
1361 %%Page: 1 1
1362 1 0 bop 404 482 a FC(An)21 b(expression)g(formatter)h(for)g
1363 FB(Ma)n(csyma)812 602 y FA(Bruce)15 b(R.)h(Miller)1146
1364 584 y Fz(\003)848 700 y FA(July)g(6,)g(1995)893 891 y
1365 Fy(Abstract)423 956 y Fx(A)g(pac)o(k)n(age)i(for)e(formatting)i
1366 (algebraic)h(expressions)g(in)f Fw(Ma)o(csyma)1463 940
1367 y Fv(1)1496 956 y Fx(is)g(de-)365 1001 y(scrib)q(ed.)k(It)14
1368 b(pro)o(vides)i(facilities)g(for)e(user-directed)i(hierarc)o(hical)h
1369 (structuring)f(of)365 1047 y(expressions,)g(as)d(w)o(ell)h(as)g(for)f
1370 (directing)i(simpli\014cati)q(ons)i(to)c(selected)h(sub)q(expres-)365
1371 1093 y(sions.)24 b(It)15 b(emphasizes)h(a)f(seman)o(tic)h(rather)g
1372 (than)f(syn)o(tactic)h(description)i(of)d(the)365 1138
1373 y(desired)h(form.)j(The)14 b(pac)o(k)n(age)g(also)h(pro)o(vides)h
1374 (utilities)h(for)c(obtaining)k(e\016cien)o(tly)365 1184
1375 y(the)11 b(co)q(e\016cien)o(ts)h(of)e(p)q(olynomials)q(,)j
1376 (trigonometric)g(sums)d(and)i(p)q(o)o(w)o(er)e(series.)17
1377 b(Sim-)365 1230 y(ilar)e(capabilities)h(w)o(ould)e(b)q(e)g(useful)g(in)
1378 g(other)f(computer)h(algebra)g(systems.)365 1275 y Fy(Keyw)o(ords:)23
1379 b Fx(algebraic)18 b(structure,)f(computer)f(algebra,)i
1380 Fw(Ma)o(csyma)p Fx(,)d(simpli\014-)365 1321 y(cation,)f(soft)o(w)o
1381 (are,)f(transformation.)262 1458 y Fu(1)69 b(In)n(tro)r(duction)262
1382 1549 y Ft(In)15 b(a)f(general)i(purp)q(ose)g(Computer)e(Algebra)h
1383 (System)f(\(CAS\),)h(an)o(y)g(particular)g(mathe-)262
1384 1599 y(matical)9 b(expression)j(can)g(tak)o(e)f(on)g(a)g(v)n(ariet)o(y)
1385 g(of)g(forms:)k(expanded)d(form,)e(factored)h(form)262
1386 1649 y(or)i(an)o(ything)g(in)g(b)q(et)o(w)o(een.)20 b(Eac)o(h)14
1387 b(form)e(ma)o(y)g(ha)o(v)o(e)h(adv)n(an)o(tages;)g(a)h(giv)o(en)f(form)
1388 f(ma)o(y)g(b)q(e)262 1699 y(more)f(compact)h(than)h(another,)g(or)f
1389 (allo)o(w)f(clear)i(expression)h(of)e(certain)i(algorithms.)h(Or)262
1390 1748 y(it)e(ma)o(y)f(simply)g(b)q(e)i(more)f(informativ)o(e,)e
1391 (particularly)i(if)g(it)h(has)g(ph)o(ysical)f(signi\014cance.)324
1392 1798 y(A)k(CAS)h(con)o(tains)g(man)o(y)d(to)q(ols)j(for)f(transforming)
1393 f(expressions.)30 b(Ho)o(w)o(ev)o(er,)19 b(most)262 1848
1394 y(are)f(lik)o(e)g Fs(Ma)o(csyma)p Ft('s[1])g Fr(factor)g
1395 Ft(and)g Fr(expand)p Ft(,)g(op)q(erating)g(only)g(on)g(the)h(en)o(tire)
1396 g(ex-)262 1898 y(pression)e(or)f(its)h(top)f(lev)o(el.)26
1397 b(A)o(t)17 b(the)g(other)g(extreme)g(are)g(op)q(erations)g(lik)o(e)f
1398 Fr(substpart)262 1948 y Ft(whic)o(h)h(extract)h(a)f(sp)q(eci\014c)h
1399 (part)g(of)e(an)h(expression,)i(then)e(transform)f(and)h(replace)h(it.)
1400 262 1997 y(Unfortunately)m(,)13 b(the)h(means)g(of)f(sp)q(ecifying)h
1401 (the)h(piece)g(of)e(in)o(terest)i(is)f(purely)g(syn)o(tactic,)262
1402 2047 y(requiring)i(the)h(user)h(to)e(k)o(eep)h(close)g(w)o(atc)o(h)g
1403 (on)f(the)h(form)e(of)h(the)h(argumen)o(ts)f(to)g(a)o(v)o(oid)262
1404 2097 y(error.)324 2147 y(The)f(pac)o(k)n(age)f(describ)q(ed)i(here)g
1405 (giv)o(es)f(users)h(of)e Fs(Ma)o(csyma)h Ft(more)f(con)o(trol)g(o)o(v)o
1406 (er)h(the)262 2197 y(structure)e(of)e(expressions,)j(and)d(it)g(do)q
1407 (es)i(so)f(using)f(a)g(more)g(seman)o(tic,)g(almost)f(algebraic,)262
1408 2246 y(language)i(describing)i(the)g(desired)h(structure.)20
1409 b(It)14 b(also)f(pro)o(vides)g(a)h(seman)o(tic)e(means)h(of)262
1410 2296 y(addressing)g(parts)g(of)f(an)g(expression)i(for)e(particular)g
1411 (simpli\014cations.)k(F)m(or)c(example,)f(to)p 262 2331
1412 573 2 v 307 2358 a Fq(\003)325 2369 y Fp(Email:)p Fo(miller)o(@ca)o
1413 (m.n)o(is)o(t.g)o(ov)308 2397 y Fv(1)325 2409 y Fp(The)k(use)g(of)g
1414 (commercial)d(pro)q(ducts)h(or)i(their)f(names)g(implies)g(no)g
1415 (endorsemen)o(t)e(b)o(y)j(NIST,)h(the)262 2448 y(Dept.)10
1416 b(of)h(Commerce)f(or)h(the)g(U.S.)g(Go)o(v)o(ernmen)o(t)o(.)967
1417 2574 y Ft(1)p eop
1418 %%Page: 2 2
1419 2 1 bop 262 307 a Ft(rearrange)16 b(an)f(expression)h(in)o(to)f(a)f
1420 (series)j(in)e Fr(eps)f Ft(through)i(order)g(5,)e(whose)i(terms)f(will)
1421 262 357 y(b)q(e)e(p)q(olynomial)o(s)e(in)h(the)h Fr(x)g
1422 Ft(and)f Fr(y)p Ft(,)g(whose)i(co)q(e\016cien)o(ts,)f(in)f(turn,)h
1423 (will)f(b)q(e)h(trigonometric)262 407 y(sums)g(in)g Fr(l)h
1424 Ft(and)g Fr(g)f Ft(with)h(factored)g(co)q(e\016cien)o(ts)h(one)f(uses)h
1425 (the)g(command:)370 488 y Fr(format)p Ft(\()p Fr(foo)p
1426 Fn(;)7 b Ft(\045)p Fr(series)o Ft(\()p Fr(eps)p Fn(;)f
1427 Fr(5)p Ft(\))p Fn(;)h Ft(\045)o Fr(poly)p Ft(\()p Fr(x)o
1428 Fn(;)g Fr(y)p Ft(\))p Fn(;)f Ft(\045)p Fr(tr)o(ig)p Ft(\()p
1429 Fr(l)p Fn(;)g Fr(g)p Ft(\))p Fn(;)h Ft(\045)o Fr(facto)o(r)p
1430 Ft(\);)262 570 y(|)13 b(more)g(easily)g(in)o(v)o(ok)o(ed)g(than)h
1431 (describ)q(ed.)324 620 y(An)i(expression)h(`formatting')12
1432 b(to)q(ol)k(for)f(a)h(general)g(purp)q(ose)h(system)e(w)o(as)h(rep)q
1433 (orted)262 669 y(in)f([4)o(])g(for)h(Scratc)o(hpad,)h(predating)f(the)g
1434 (user-sp)q(eci\014ed)i(canonical)d(represen)o(tations)j(of)262
1435 719 y Fs(Axiom)g Ft(and)h(the)g(author's)g(system)f Fs(MA)o(O)p
1436 Ft(.)g(Je\013rey)i(Golden[3)o(])e(prop)q(osed)i(a)e(similar)262
1437 769 y(system)11 b(for)h Fs(Ma)o(csyma)p Ft(|)g(although)f(nev)o(er)h
1438 (implemen)o(ted,)e(his)i(design)g(pro)o(vided)f(inspi-)262
1439 819 y(ration)f(and)g(a)g(go)q(o)q(d)h(naming)d(con)o(v)o(en)o(tion)i
1440 (for)h(the)g(pac)o(k)n(age)f(describ)q(ed)j(here.)18
1441 b(A)11 b(di\013eren)o(t)262 869 y(addressing)i(sc)o(heme)h(for)f
1442 (directing)g(simpli\014cations)e(in)i(Mathematica)f(w)o(as)h(rep)q
1443 (orted)i(in)262 918 y([5)o(].)h(Th)o(us)11 b(the)h(general)f(idea)f(b)q
1444 (ehind)h(these)h(to)q(ols)f(is)g(not)f(new,)i(y)o(et)f(the)g(to)q(ols)f
1445 (themselv)o(es)262 968 y(are)j(not)h(commonl)o(y)d(a)o(v)n(ailable)g
1446 (in)i(most)f(CAS.)h(F)m(urther,)h(w)o(e)f(feel)h(that)f(our)g(syn)o
1447 (thesis)i(is)262 1018 y(unique.)i(And)c(while)f(our)h(syn)o(tax)g(ma)o
1448 (y)e(b)q(e)i(a)g(bit)f(baro)q(que,)h(including)f(man)o(y)f(k)o(eyw)o
1449 (ords,)262 1068 y(w)o(e)j(ha)o(v)o(e)f(found)h(the)g(pac)o(k)n(age)g
1450 (to)g(b)q(e)g(an)g(indisp)q(ensable)g(to)q(ol)f(in)h(practice.)324
1451 1139 y(Tw)o(o)f(mo)q(dules)g(are)i(do)q(cumen)o(ted)f(in)f(this)h(rep)q
1452 (ort.)20 b(The)14 b(principal)g(to)q(ol,)e Fr(format)p
1453 Ft(,)h(is)262 1188 y(describ)q(ed)g(in)f(Section)g(2.)18
1454 b(It)12 b(uses)h(pro)q(cedures)h(in)e Fr(coeflist)e Ft(whic)o(h)i
1455 (obtain)f(co)q(e\016cien)o(ts)262 1238 y(of)g(p)q(olynomials,)f
1456 (trigonometric)h(sums)h(and)g(p)q(o)o(w)o(er)h(series.)19
1457 b(The)13 b(latter)g(mo)q(dule)e(can)i(b)q(e)262 1288
1458 y(useful)g(alone;)f(it)h(is)g(do)q(cumen)o(ted)g(in)g(section)h(3.)j
1459 (An)d(app)q(endix)f(discusses)i(implemen)o(ta-)262 1338
1460 y(tion)e(issues.)19 b(The)14 b Fs(Lisp)g Ft(source)h(co)q(de)g(ma)o(y)d
1461 (b)q(e)j(obtained)e(from)f(the)j(author.)262 1473 y Fu(2)69
1462 b Fm(FORMAT)p Fu(;)25 b(F)-6 b(ormatting)23 b(expressions)262
1463 1588 y Fr(format)p Ft(\()p Fl(expr)p Ft(,)p Fl(template)652
1464 1594 y Fk(1)668 1588 y Ft(,)p Fn(:)7 b(:)g(:)e Ft(\))783
1465 b Fl(F)m(unction)365 1663 y Ft(Recursiv)o(ely)10 b(arranges)g
1466 Fl(expr)f Ft(according)g(to)g(the)h Fl(chain)g Ft(of)f(templates,)g
1467 Fl(template)1625 1669 y Fj(i)1639 1663 y Ft(.)324 1738
1468 y(Eac)o(h)17 b Fl(template)h Ft(indicates)f(the)h(desired)h(form)c(for)
1469 i(an)g(expression;)j(either)e(the)g(ex-)262 1788 y(p)q(ected)f(form)d
1470 (or)i(that)g(in)o(to)g(whic)o(h)f(it)h(will)e(b)q(e)j(transformed.)24
1471 b(A)o(t)16 b(the)g(same)f(time,)g(the)262 1838 y(indicated)e(form)f
1472 (implies)f(a)i(set)i(of)e Fl(pie)n(c)n(es)p Ft(;)g(the)h(next)g
1473 (template)e(in)h(the)h(c)o(hain)g(applies)f(to)262 1888
1474 y(those)j(pieces.)25 b(F)m(or)16 b(example,)e Fr(\045poly\(x\))h
1475 Ft(sp)q(eci\014es)i(the)g(transformation)d(in)o(to)h(a)g(p)q(oly-)262
1476 1938 y(nomial)9 b(in)i Fn(x)p Ft(,)h(with)g(the)g(pieces)i(b)q(eing)e
1477 (the)g(co)q(e\016cien)o(ts.)19 b(The)13 b(passiv)o(e)f
1478 Fr(\045frac)f Ft(treats)i(the)262 1987 y(expression)i(as)f(a)f
1479 (fraction;)g(the)i(pieces)g(are)f(the)h(n)o(umerator)e(and)g
1480 (denominator.)324 2037 y(Whereas)e(the)g(next)g(template)e(formats)g
1481 (all)g(pieces)i(of)f(the)g(previous)h(la)o(y)o(er,)f(p)q(ositional)262
1482 2087 y Fl(subtemplates)i Ft(ma)o(y)f(b)q(e)i(used)g(to)g(sp)q(ecify)g
1483 (formats)e(for)h(eac)o(h)h(piece)h(individually)m(.)g(This)f(is)262
1484 2137 y(most)c(useful)j(when)f(the)h(pieces)g(ha)o(v)o(e)f(unique)h
1485 (roles)f(and)g(need)h(to)f(b)q(e)h(treated)g(di\013eren)o(tly)m(,)262
1486 2187 y(suc)o(h)i(as)g(a)g(fraction's)f(n)o(umerator)g(and)h
1487 (denominator.)324 2236 y(The)g(full)f(syn)o(tax)h(of)f(a)g(template)g
1488 (is)604 2318 y Fl(keywor)n(d)t Ft(\()p Fl(p)n(ar)n(ameter)t
1489 Fn(;)7 b(:)g(:)g(:)n Ft(\)[)p Fl(subtemplate)r Fn(;)g(:)g(:)g(:)n
1490 Ft(])p Fn(:)262 2399 y Ft(The)20 b(recognized)i(k)o(eyw)o(ords)e(are)h
1491 (describ)q(ed)h(in)e(T)m(able)f(1.)37 b(The)21 b(parameters)f(\(if)g
1492 (not)262 2449 y(needed\))15 b(and)f(subtemplates)g(\(along)e(with)i
1493 (paren)o(theses)i(and)e(brac)o(k)o(ets\))h(are)f(optional.)967
1494 2574 y(2)p eop
1495 %%Page: 3 3
1496 3 2 bop 324 307 a Ft(In)17 b(addition)f(to)i(the)g(k)o(eyw)o(ord)f
1497 (templates,)g(arithmetic)g(patterns)h(are)g(recognized.)262
1498 357 y(This)11 b(is)g(an)h(expression)g(in)o(v)o(olving)e(addition,)g(m)
1499 o(ultiplicatio)o(n)f(and)i(exp)q(onen)o(tiation)h(con-)262
1500 407 y(taining)d(a)i(single)f(instance)i(of)e(a)h(k)o(eyw)o(ord)g
1501 (template.)16 b(In)11 b(e\013ect,)i(the)e(system)g(`solv)o(es')f(the)
1502 262 457 y(expression)17 b(to)g(b)q(e)g(formatted)f(for)g(the)i(corresp)
1503 q(onding)f(part,)h(formats)d(it)h(accordingly)262 506
1504 y(and)d(reinserts)j(it.)i(Eg.)13 b(,)g Fr(format\(X,a+\045factor\))d
1505 Ft(is)k(equiv)n(alen)o(t)f(to)h Fr(a+factor\(X-a\))p
1506 Ft(.)324 556 y(An)o(y)e(other)i(template)e(is)g(assumed)h(to)f(b)q(e)h
1507 (a)g(function)f(to)h(b)q(e)g(applied)f(to)h(the)g(expres-)262
1508 606 y(sion;)g(the)h(result)h(is)e(then)i(formatted)e(according)h(to)f
1509 (the)i(rest)g(of)e(the)i(template)e(c)o(hain.)262 714
1510 y Fi(Examples)262 797 y Fr(\(c1\))20 b(format\(\(a+b*x\)*\(c-x\))o
1511 (^2,\045p)o(oly\()o(x\),\045f)o(actor)o(\);)436 847 y(3)348
1512 b(2)523 b(2)262 897 y(\(d1\))20 b(b)i(x)43 b(-)22 b(\(2)f(b)h(c)g(-)f
1513 (a\))h(x)43 b(+)22 b(c)f(\(b)g(c)h(-)g(2)f(a\))h(x)f(+)h(a)f(c)262
1514 996 y(\(c2\))f(format\(\(1+2*e*\(q+r*c)o(os\(g\))o(\)^2\))o(^4,\045s)o
1515 (eries)o(\(e,2\))o(,\045tri)o(g\(g\),)o(\045f\);)610
1516 1046 y(2)131 b(2)348 b(2)262 1096 y(\(d2\))20 b(1)i(+)g(e)f(\(4)h(\(r)
1517 43 b(+)21 b(2)h(q)g(\))f(+)h(4)f(cos\(2)g(g\))g(r)44
1518 b(+)21 b(16)h(cos\(g\))e(q)i(r\))349 1195 y(2)174 b(4)152
1519 b(2)44 b(2)130 b(4)349 b(4)283 1245 y(+)22 b(e)43 b(\(3)22
1520 b(\(3)f(r)43 b(+)22 b(24)f(q)44 b(r)f(+)22 b(8)f(q)h(\))f(+)h(3)g
1521 (cos\(4)e(g\))i(r)697 1345 y(3)458 b(2)130 b(2)327 1395
1522 y(+)21 b(24)h(cos\(3)f(g\))g(q)h(r)43 b(+)22 b(24)f(cos\(g\))f(q)i(r)g
1523 (\(3)f(r)43 b(+)22 b(4)g(q)f(\))654 1494 y(2)65 b(2)131
1524 b(2)327 1544 y(+)21 b(12)h(cos\(2)f(g\))g(r)43 b(\(r)h(+)21
1525 b(6)h(q)f(\)\))h(+)f(...)262 1644 y(\(c3\))f(format\(\(1+2*a+a^2\)*b)e
1526 (+)k(a*\(1+2*b+b^2\),\045s)o(um,\045p)o(roduc)o(t,\045fa)o(ctor)o(\);)
1527 567 1694 y(2)217 b(2)262 1743 y(\(d3\))20 b(a)i(\(b)f(+)h(1\))43
1528 b(+)22 b(\(a)f(+)h(1\))43 b(b)262 1843 y(\(c4\))20 b
1529 (format\(expand\(\(1+x^4)o(\)*y^2)o(+\(1+)o(x^8\)*)o(y^4\),)o
1530 (\045p\(y\))o(,\045f\(a)o(^2-2\))o(\);)414 1893 y(4)131
1531 b(2)174 b(4)130 b(2)153 b(4)262 1943 y(\(d4\))20 b(\(x)44
1532 b(-)21 b(a)h(x)43 b(+)22 b(1\))f(\(x)43 b(+)22 b(a)g(x)43
1533 b(+)21 b(1\))h(y)414 2042 y(2)305 b(2)283 b(2)327 2092
1534 y(+)21 b(\(x)44 b(-)21 b(a)h(x)f(+)h(1\))f(\(x)44 b(+)21
1535 b(a)h(x)f(+)h(1\))f(y)262 2192 y(\(c5\))f(format\(expand\(\(a+x\)^)o
1536 (3-a^3)o(\),\045f)o(-a^3\))o(;)545 2242 y(3)87 b(3)262
1537 2291 y(\(d5\))20 b(\(x)i(+)f(a\))44 b(-)21 b(a)262 2374
1538 y(format)13 b Ft(can)i(also)f(b)q(e)h(used)g(to)g(fo)q(cus)g
1539 (simpli\014cations)d(on)i(manageable)f(pieces)j(of)e(large)262
1540 2424 y(expressions.)967 2574 y(3)p eop
1541 %%Page: 4 4
1542 4 3 bop 262 2454 a
1543 gsave currentpoint currentpoint translate 270 rotate neg exch neg exch
1544 translate
1545  262 2454 a 300 2489 a Fl(Class)p 511
1546 2504 2 50 v 140 w(T)m(emplate)15 b(\(w/abbr)n(ev.\))122
1547 b(Co)n(er)n(cion)14 b(to)p 1648 2504 V 417 w(Pie)n(c)n(es)g(&)h(or)n
1548 (dering)p 275 2506 2076 2 v 300 2541 a(A)o(lgebr)n(aic)467
1549 2526 y Fj(a)p 511 2556 2 50 v 536 2541 a Fr(\045poly\()p
1550 Fn(x)692 2547 y Fk(1)709 2541 y Fn(;)7 b(:)g(:)g(:)n
1551 Fr(\))p Ft(,)13 b Fr(\045p)181 b Ft(P)o(olynomial)10
1552 b(in)j Fn(x)1337 2547 y Fj(i)p 1648 2556 V 1674 2541
1553 a Ft(co)q(e\016cien)o(ts)i(\(ascending)f(exps.\))p 511
1554 2606 V 536 2591 a Fr(\045series\()p Fn(\017;)7 b(n)p
1555 Fr(\))p Ft(,)k Fr(\045s)186 b Ft(Series)15 b(in)e Fn(\017)h
1556 Ft(through)g(order)g Fn(n)p 1648 2606 V 454 w Ft(")p
1557 511 2655 V 536 2640 a Fr(\045Taylor\()p Fn(\017;)7 b(n)p
1558 Fr(\))253 b Ft(T)m(a)o(ylor)12 b(in)h Fn(\017)h Ft(through)g(order)h
1559 Fn(n)p 1648 2655 V 441 w Ft(")p 511 2705 V 536 2690 a
1560 Fr(\045monicpoly\()p Fn(x)802 2696 y Fk(1)818 2690 y
1561 Fn(;)7 b(:)g(:)g(:)n Fr(\))p Ft(,)13 b Fr(\045mp)50 b
1562 Ft(Monic)13 b(p)q(olynomial)e(in)i Fn(x)1460 2696 y Fj(i)p
1563 1648 2705 V 1674 2690 a Ft(leading)g(co)q(ef.)18 b(then)d(co)q
1564 (e\016cien)o(ts)p 511 2755 V 536 2740 a Fr(\045trig\()p
1565 Fn(x)692 2746 y Fk(1)709 2740 y Fn(;)7 b(:)g(:)g(:)n
1566 Fr(\))p Ft(,)13 b Fr(\045t)181 b Ft(T)m(rigonometric)11
1567 b(sum)i(in)h Fn(x)1477 2746 y Fj(i)p 1648 2755 V 1674
1568 2740 a Ft(sin)f(co)q(efs)i(\(ascending)g(m)o(ult\),)c(then)k(cos)p
1569 511 2805 V 536 2790 a Fr(\045coeff\()p Fn(v)q(;)7 b(n)p
1570 Fr(\))271 b Ft(P)o(olynomial)10 b(in)j Fn(v)p 1648 2805
1571 V 341 w Ft(Co)q(e\016cien)o(t)h(of)f Fn(v)1947 2775 y
1572 Fj(n)1984 2790 y Ft(and)h(remainder.)p 275 2807 2076
1573 2 v 300 2842 a Fl(Sums)p 511 2857 2 51 v 140 w Fr(\045sum)424
1574 b Fl(p)n(assive)1173 2827 y Fj(b)p 1648 2857 V 1674 2842
1575 a Ft(terms)13 b(\()p Fr(inpart)g Ft(order)2044 2827 y
1576 Fj(c)2062 2842 y Ft(\))p 511 2907 2 50 v 536 2892 a Fr(\045partfrac\()p
1577 Fn(x)p Fr(\))p Ft(,)e Fr(\045pf)157 b Ft(P)o(artial)12
1578 b(fraction)i(decomp.)j(in)d Fn(x)p 1648 2907 V 409 w
1579 Ft(")p 275 2908 2076 2 v 300 2943 a Fl(Pr)n(o)n(ducts)p
1580 511 2958 2 50 v 80 w Fr(\045product)p Ft(,)e Fr(\045prod)202
1581 b Fl(p)n(assive)p 1648 2958 V 501 w Ft(factors)14 b(\()p
1582 Fr(inpart)f Ft(order\))p 511 3008 V 536 2993 a Fr(\045factor)p
1583 Ft(,)f Fr(\045f)290 b Ft(factored)14 b(form)p 1648 3008
1584 V 695 w(")p 511 3058 V 536 3043 a Fr(\045factor\()p Fl(minp)n(oly)p
1585 Fr(\))p Ft(,)e Fr(\045f)105 b Ft(factored)14 b(with)g(elemen)o(t)f
1586 (adjoined)p 1648 3058 V 379 w(")p 511 3107 V 536 3092
1587 a Fr(\045sqfr)p Ft(,)g Fr(\045sf)311 b Ft(square-free)15
1588 b(factored)f(form)p 1648 3107 V 485 w(")p 275 3109 2076
1589 2 v 300 3144 a Fl(F)m(r)n(actions)p 511 3159 2 50 v 68
1590 w Fr(\045frac)402 b Fl(p)n(assive)p 1648 3159 V 501 w
1591 Ft(n)o(umerator)13 b(&)h(denominator)p 511 3209 V 536
1592 3194 a Fr(\045ratsimp)p Ft(,)e Fr(\045r)268 b Ft(rationally)12
1593 b(simpli\014ed)p 1648 3209 V 578 w(")p 275 3210 2076
1594 2 v 300 3245 a Fl(Complex)p 511 3260 2 50 v 81 w Fr(\045rectform)p
1595 Ft(,)g Fr(\045g)246 b Ft(Gaussian)13 b(form)p 1648 3260
1596 V 362 w(real)g(&)i(imaginary)c(parts)p 511 3310 V 536
1597 3295 a Fr(\045polarform)292 b Ft(P)o(olar)13 b(form)p
1598 1648 3310 V 430 w(magnitude)f(&)i(phase)p 275 3312 2076
1599 2 v 300 3347 a Fl(`Bags')412 3332 y Fj(d)p 511 3362 2
1600 51 v 536 3347 a Fr(\045equation)p Ft(,)e Fr(\045eq)224
1601 b Ft(equation)p 1648 3362 V 470 w(left-hand)13 b(side)h(&)g(righ)o
1602 (t-hand)g(side)p 511 3412 2 50 v 536 3397 a Fr(\045relation\()p
1603 Fn(r)q Fr(\))p Ft(,)d Fr(\045rel)139 b Ft(Relation;)12
1604 b Fn(r)g Fh(2)f(f)p Ft(=)p Fn(;)c(>;)g Fh(\025)p Fn(;)g(<;)g
1605 Fh(\024)p Fn(;)g Fh(6)p Ft(=)p Fh(g)p 1648 3412 V 364
1606 w Ft(")p 511 3461 V 536 3447 a Fr(\045list)402 b Ft(list)p
1607 1648 3461 V 570 w(elemen)o(ts)p 511 3511 V 536 3496 a
1608 Fr(\045matrix)358 b Ft(matrix)p 1648 3511 V 503 w(ro)o(ws)14
1609 b(\(use)h Fr(\045list)d Ft(to)i(target)g(elemen)o(ts\))p
1610 275 3513 2076 2 v 300 3548 a Fl(Gener)n(al)p 511 3563
1611 2 50 v 95 w Fr(\045expression,)20 b(\045expr)118 b Fl(p)n(assive)p
1612 1648 3563 V 501 w Ft(the)14 b(op)q(erands)h(\()p Fr(inpart)e
1613 Ft(order\))p 511 3613 V 536 3598 a Fr(\045preformat\()p
1614 Fn(T)802 3604 y Fk(1)819 3598 y Fn(;)7 b(:)g(:)g(:)n
1615 Fr(\))140 b Ft(F)m(ormat)12 b(according)h(to)h(c)o(hain)g
1616 Fn(T)1565 3604 y Fj(i)p 1648 3613 V 1674 3598 a Ft(the)g(result,)g(\()p
1617 Fi(not!)j Ft(the)e(parts\))p 275 3639 870 2 v 320 3666
1618 a Fg(a)339 3678 y Fp(See)c(section)e(3)j(for)f(the)g(in)o(terpreta)o
1619 (tion)d(of)j(p)q(olynomials,)d(trigonometri)o(c)h(sums)h(and)h(series)g
1620 (used)f(b)o(y)h(this)g(pac)o(k)n(age.)323 3707 y Fg(b)339
1621 3718 y Fp(A)i(passiv)o(e)e(k)o(eyw)o(ord)g(do)q(es)g(not)h(transform)e
1622 (the)i(expression)d(but)j(treats)f(it)h(as)g(a)h(sum,)e(fraction)g(or)h
1623 (whatev)o(er.)k(Note)c(that)g Fo(\045sum)f Fp(\()p Fo(\045product)p
1624 Fp(\))e(treats)i(an)275 3758 y(expression)e(whic)o(h)i(do)q(es)g(not)g
1625 (ha)o(v)o(e)f Fo("+")h Fp(\(resp.)f Fo("*")p Fp(\))g(as)i(its)f(main)f
1626 (op)q(erator)f(as)j(a)f(single)f(term)g(\(factor\).)323
1627 3786 y Fg(c)339 3797 y Fp(The)h(order)g(of)g(the)g(pieces)f(corresp)q
1628 (onds)e(to)j(the)g(in)o(ternal)e(ordering;)h(subtemplat)o(e)f(usage)h
1629 (ma)o(y)g(b)q(e)h(a)o(wkw)o(ard.)321 3826 y Fg(d)339
1630 3838 y Fp(See)g(the)f(do)q(cumen)o(tati)o(on)e(of)k Fo(coerce)p
1631 854 3838 11 2 v 10 w(bag)f Fp(for)g(a)g(description)e(of)i(the)g(co)q
1632 (ercions)e(used.)262 2454 y
1633 currentpoint grestore moveto
1634  262 2454 a 715 2546 a Ft(T)m(able)k(1:)k(T)
1635 m(emplate)c(k)o(eyw)o(ords.)967 2574 y(4)p eop
1636 %%Page: 5 5
1637 5 4 bop 575 2441 a
1638 gsave currentpoint currentpoint translate 270 rotate neg exch neg exch
1639 translate
1640  575 2441 a 600 2476 a Fl(Class)p 851
1641 2490 2 50 v 181 w(T)m(emplate)14 b(\(w/abbr)n(ev.\))p
1642 1378 2490 V 138 w(F)m(unction)p 575 2492 2117 2 v 600
1643 2527 a(T)m(ar)n(geting)771 2512 y Fj(a)p 851 2542 2 50
1644 v 877 2527 a Fr(\045arg\(n\))p 1378 2542 V 373 w Ft(F)m(ormats)e(the)j
1645 (n-th)f(argumen)o(t)p 851 2592 V 877 2577 a Fr(\045lhs)p
1646 Ft(\()p Fl(r)p Ft(\))p 1378 2592 V 389 w(F)m(ormats)e(the)j(lefthand)f
1647 (side)g(of)f(an)h(equation)f(or)h(relation)f(\(default)h
1648 Fr("=")p Ft(\).)p 851 2642 V 877 2627 a Fr(\045rhs)p
1649 Ft(\()p Fl(r)p Ft(\))p 1378 2642 V 389 w(F)m(ormats)e(the)j(righ)o
1650 (thand)e(side.)p 851 2691 V 877 2676 a Fr(\045element\(i,..\))p
1651 Ft(,)p Fr(\045e)o(l)p 1378 2691 V 142 w Ft(F)m(ormats)f(an)i(elemen)o
1652 (t)f(of)h(a)f(matrix.)p 851 2741 V 877 2726 a Fr(\045num)p
1653 Ft(,)p Fr(\045denom)p 1378 2741 V 295 w Ft(F)m(ormats)f(the)j(n)o
1654 (umerator)e(or)h(denominator)e(of)h(a)h(fraction.)p 851
1655 2791 V 877 2776 a Fr(\045match\()p Fn(P)6 b Fr(\))p 1378
1656 2791 V 318 w Ft(F)m(ormats)12 b(all)h(sub)q(expressions)j(for)e(whic)o
1657 (h)f Fn(P)6 b Ft(\()p Fn(expr)q Ft(\))14 b(returns)h(T)m(rue.)p
1658 575 2793 2117 2 v 600 2828 a Fl(Contr)n(ol)p 851 2842
1659 2 50 v 141 w Fr(\045if\()p Fn(P)992 2834 y Fk(1)1010
1660 2828 y Fn(;)7 b(:)g(:)g(:)m Fr(\)[)p Fn(T)1144 2834 y
1661 Fk(1)1170 2828 y Fn(:)g(:)g(:)e(;)i(T)1268 2834 y Fj(n)p
1662 Fk(+1)1332 2828 y Fr(])p 1378 2842 V 50 w Ft(Find)14
1663 b(\014rst)g Fn(P)1615 2834 y Fj(i)1629 2828 y Ft(\()p
1664 Fn(expr)q Ft(\))d Fh(!)p Fr(True)p Ft(,)i(then)h(format)e(expr)j(using)
1665 e Fn(T)2366 2834 y Fj(i)2380 2828 y Ft(,)h(else)g Fn(T)2508
1666 2834 y Fj(n)p Fk(+1)2573 2828 y Ft(.)p 575 2844 2117
1667 2 v 600 2879 a Fl(Subtemplate)p 851 2894 2 50 v 60 w
1668 Fr(\045noop)p 1378 2894 V 417 w Ft(Do)q(es)g(nothing;)f(used)i(to)f
1669 (\014ll)f(a)g(subtemplate)h(slot.)600 2929 y Fl(A)o(ids)p
1670 851 2944 V 196 w Fr([)p Fn(T)923 2935 y Fk(1)942 2929
1671 y Fn(;)7 b(T)985 2935 y Fk(2)1003 2929 y Fn(;)g(:)g(:)g(:)n
1672 Fr(])p 1378 2944 V 312 w Ft(Creates)15 b(a)f(template)f(c)o(hain)g
1673 (where)i(an)f(individual)e(template)h(w)o(as)h(exp)q(ected.)p
1674 851 2994 V 877 2979 a Fr(\045ditto\()p Fn(T)6 b Fr(\))p
1675 1378 2994 V 321 w Ft(Rep)q(eats)15 b(the)f(template)f(so)h(that)g(it)g
1676 (applies)f(to)h(follo)o(wing)d(pieces.)p 575 2995 2117
1677 2 v 600 3030 a Fl(Convenienc)n(e)p 851 3045 2 50 v 51
1678 w Fr(\045subst\()p Fl(e)n(qns,)p Fn(:)c(:)g(:)t Fr(\))p
1679 1378 3045 V 208 w Ft(Substitutes)15 b Fl(e)n(qns)g Ft(in)o(to)e
1680 (expression;)h(result)h(is)f(formatted)e(at)i(next)h(la)o(y)o(er)p
1681 851 3095 V 877 3080 a Fr(\045ratsubst\()p Fl(e)n(qns,)p
1682 Fn(:)7 b(:)g(:)s Fr(\))p 1378 3095 V 143 w(lratsubst)p
1683 Ft('s)12 b Fl(e)n(qns)i Ft(in)o(to)f(expression;)i(result)g(is)e
1684 (formatted)g(at)h(next)g(la)o(y)o(er)p 575 3121 870 2
1685 v 620 3148 a Fg(a)639 3160 y Fp(T)m(argetting)9 b(templates)g(are)i
1686 (basically)e(shorthand)g(equiv)n(alen)o(ts)g(of)i(structuring)e
1687 (templates)f(using)j(subtemplate)o(s.)1306 2441 y
1688 currentpoint grestore moveto
1689  1306
1690 2441 a 620 2532 a Ft(T)m(able)i(1:)18 b(T)m(emplate)12
1691 b(k)o(eyw)o(ords)i(con)o(tin)o(ued.)967 2574 y(5)p eop
1692 %%Page: 6 6
1693 6 5 bop 262 307 a Fr(\(c6\))20 b(foo:X^2*SIN\(Y\)^4-2*X)o(^2*SI)o
1694 (N\(Y\))o(^2+X^)o(4*COS)o(\(Y\)^4)501 357 y(-2*X^4*COS\(Y\)^2+X^4)o
1695 (+X^2)o(+1$)262 407 y(\(c7\))g(trigsimp\(foo\);)436 457
1696 y(4)87 b(2)109 b(4)196 b(4)86 b(2)153 b(4)262 506 y(\(d7\))20
1697 b(\(x)44 b(+)21 b(x)h(\))f(cos)h(\(y\))f(-)g(2)h(x)43
1698 b(cos)21 b(\(y\))h(+)f(x)44 b(+)21 b(1)262 606 y(\(c8\))f
1699 (format\(foo,\045p\(x\),tri)o(gsimp)o(\);)414 656 y(4)87
1700 b(4)152 b(2)87 b(4)262 706 y(\(d8\))20 b(x)44 b(sin)21
1701 b(\(y\))g(+)h(x)43 b(cos)21 b(\(y\))h(+)f(1)262 789 y
1702 Ft(The)14 b(follo)o(wing)d(examples)i(illustrate)h(the)g(use)h(of)e
1703 (subtemplates)262 872 y Fr(\(c9\))20 b(l1:[1+2*a+a^2,1+2*b+)o(b^2,1)o
1704 (+2*c)o(+c^2])o($)262 922 y(\(c10\))g(format\(l1,\045list,\045f\))o(;)
1705 567 971 y(2)196 b(2)f(2)262 1021 y(\(d10\))20 b([\(a)h(+)h(1\))f(,)h
1706 (\(b)f(+)h(1\))f(,)h(\(c)f(+)h(1\))f(])262 1121 y(\(c11\))f
1707 (format\(l1,\045list[\045no)o(op,\045f)o(]\);)436 1171
1708 y(2)414 b(2)65 b(2)262 1220 y(\(d11\))20 b([a)43 b(+)22
1709 b(2)g(a)f(+)h(1,)f(\(b)h(+)f(1\))h(,)f(c)43 b(+)22 b(2)g(c)f(+)h(1])262
1710 1320 y(\(c12\))e(format\(l1,\045list[\045no)o(op,\045d)o(itto)o
1711 (\(\045f\)])o(\);)436 1370 y(2)414 b(2)196 b(2)262 1420
1712 y(\(d12\))20 b([a)43 b(+)22 b(2)g(a)f(+)h(1,)f(\(b)h(+)f(1\))h(,)f(\(c)
1713 g(+)h(1\))f(])262 1503 y Ft(The)14 b(follo)o(wing)d(examples)i
1714 (illustrate)h(the)g(usage)g(with)g(`bags.')262 1586 y
1715 Fr(\(c13\))20 b(format\([a=b,c=d,e=f)o(],\045eq)o(uati)o(on\);)262
1716 1636 y(\(d13\))g([a,)h(c,)h(e])f(=)h([b,)f(d,)g(f])262
1717 1735 y(\(c14\))f(format\(\045,\045list\);)262 1785 y(\(d14\))g([a)i(=)f
1718 (b,)h(c)f(=)h(d,)f(e)h(=)f(f])262 1885 y(\(c15\))f
1719 (m1:matrix\([a^2+2*a+)o(1=q,b)o(^2+2)o(*b+1=)o(r],)610
1720 1934 y([c^2+2*c+1=s,d^2+2)o(*d+1=)o(t]\)$)262 1984 y(\(c16\))g
1721 (format\(m1,\045equation)o(,\045mat)o(rix[)o(\045noop)o(,\045lis)o
1722 (t[\045no)o(op,\045f)o(actor)o(]]\);)414 2034 y([)43
1723 b(2)284 b(2)239 b(])414 2084 y([)22 b(a)43 b(+)22 b(2)f(a)h(+)f(1)44
1724 b(b)f(+)22 b(2)f(b)h(+)f(1)h(])65 b([)22 b(q)43 b(r)22
1725 b(])262 2134 y(\(d16\))42 b([)610 b(])22 b(=)f([)131
1726 b(])414 2183 y([)43 b(2)458 b(2)65 b(])g([)22 b(s)43
1727 b(t)22 b(])414 2233 y([)g(c)43 b(+)22 b(2)f(c)h(+)f(1)87
1728 b(\(d)22 b(+)f(1\))87 b(])262 2316 y Ft(The)14 b(more)f(concise)i
1729 Fr(format\(m1,\045eq,\045e)o(l\(2,2)o(\),\045f)o(\);)c
1730 Ft(obtains)i(the)i(same)e(result.)324 2366 y(And)h(a)f(more)g(in)o(v)o
1731 (olv)o(ed)g(example:)262 2449 y Fr(\(c17\))20 b
1732 (sqrtp\(f\):=not\(atom\()o(f\)\))f(and)i(op\(f\)='sqrt$)967
1733 2574 y Ft(6)p eop
1734 %%Page: 7 7
1735 7 6 bop 262 307 a Fr(\(c18\))20 b(first\(solve\(a*x^2+b)o(*x-\(b)o
1736 (-2*a)o(\)/4,x)o(\)\);)654 357 y(2)261 b(2)523 407 y(sqrt\(b)42
1737 b(+)22 b(a)g(b)f(-)h(2)f(a)h(\))f(+)h(b)262 457 y(\(d18\))e(x)i(=)f(-)h
1738 (------------------)o(----)o(---)741 506 y(2)g(a)262
1739 606 y(\(c19\))e(format\(\045,\045rhs,\045pref)o(ormat)o(\(\045p\()o
1740 (match)o(\(sqrt)o(p\)\)\),)697 656 y(\045match\(sqrtp\),\045arg\()o
1741 (1\),\045f)o(\);)523 706 y(sqrt\(\(b)g(-)i(a\))f(\(b)h(+)f(2)h(a\)\))86
1742 b(b)262 756 y(\(d19\))20 b(x)i(=)f(-)h(------------------)o(----)o(-)d
1743 (-)j(---)741 805 y(2)g(a)283 b(2)21 b(a)262 921 y Ff(2.1)55
1744 b(User)18 b(de\014ned)h(templates)262 997 y Ft(New)g(templates)g(can)g
1745 (b)q(e)h(de\014ned)g(b)o(y)f(giving)e(the)j(template)e(k)o(eyw)o(ord)h
1746 (the)h(prop)q(ert)o(y)262 1047 y Fr(formatter)p Ft(;)c(the)i(v)n(alue)f
1747 (should)g(b)q(e)h(a)f(function)g(\(or)g(lam)o(b)q(da)e(expression\))k
1748 (of)e(the)g(ex-)262 1097 y(pression)d(to)g(b)q(e)g(formatted)f(and)h
1749 (an)o(y)f(parameters)h(for)g(the)g(template.)324 1147
1750 y(F)m(or)f(example,)f Fr(\045rectform)g Ft(and)i Fr(\045if)f
1751 Ft(could)h(b)q(e)g(de\014ned)h(as)262 1227 y Fr(put\(\045rectform,l)o
1752 (ambda)o(\([c],)632 1277 y(block\([r:rectform)o(list\()o(c\)],)697
1753 1327 y(format_piece\(r[1]\))j(+\045I*)j(format_piece\(r[2]\)\))o(\),)
1754 349 1376 y(formatter\)$)262 1426 y(put\(\045if,)f(lambda\([x,test],)501
1755 1476 y(if)i(test\(x\))e(then)h(format_piece\(x,1\))501
1756 1526 y(else)261 b(format_piece\(x,2\))o(\),)370 1576
1757 y(formatter\)$)324 1656 y Ft(T)m(o)q(ols)13 b(useful)h(for)f
1758 (de\014ning)h(templates)f(are)i(the)f(follo)o(wing.)262
1759 1729 y Fr(format)p 397 1729 14 2 v 14 w(piece)p Ft(\()p
1760 Fl(pie)n(c)n(e)p Ft(,)p Fh(f)p Fl(nth)p Fh(g)p Ft(\))785
1761 b Fl(F)m(unction)365 1809 y Ft(F)m(ormat)12 b(a)h(giv)o(en)g(piece)h
1762 (of)f(an)g(expression,)h(automatically)c(accoun)o(ting)j(for)365
1763 1859 y(subtemplates)g(and)g(the)h(remaining)c(template)i(c)o(hain.)18
1764 b(A)13 b(sp)q(eci\014c)h(subtem-)365 1909 y(plate,)g(rather)g(than)g
1765 (the)h(next)f(one,)g(can)g(b)q(e)g(selected)i(b)o(y)e(sp)q(ecifying)g
1766 Fl(nth)p Ft(.)262 2012 y Fr(coerce)p 397 2012 V 14 w(bag)p
1767 Ft(\()p Fl(op)p Ft(,)p Fl(expr)p Ft(\))897 b Fl(F)m(unction)365
1768 2092 y Ft(A)o(ttempts)19 b(to)f(co)q(erce)j Fl(expr)e
1769 Ft(in)o(to)f(an)g(expression)i(with)e Fl(op)i Ft(\(one)f(of)f
1770 Fr("=",)365 2142 y("#",)j("<",)g("<=",)g(">",)g(">=",)g("[")12
1771 b Ft(or)i Fr(matrix)p Ft(\))e(as)h(the)h(top-lev)o(el)f(op-)365
1772 2192 y(erator.)38 b(It)21 b(co)q(erces)i(the)e(expression)g(b)o(y)f(sw)
1773 o(apping)g(op)q(erands)h(b)q(et)o(w)o(een)365 2242 y(la)o(y)o(ers)c({)g
1774 (but)g(only)f(if)g(adjacen)o(t)h(la)o(y)o(ers)g(are)g(also)g(lists,)g
1775 (matrices)f(or)h(rela-)365 2292 y(tions.)h(This)13 b(mo)q(del)e
1776 (assumes)i(that)g(a)f(list)h(of)f(equations,)h(for)f(example,)g(can)365
1777 2341 y(b)q(e)19 b(view)o(ed)e(as)h(an)g(equation)f(whose)h(sides)h(are)
1778 f(lists.)29 b(Certain)18 b(com)o(bina-)365 2391 y(tions,)13
1779 b(particularly)g(those)h(in)o(v)o(olving)d(inequalities)h(ma)o(y)g(not)
1780 h(b)q(e)h(meaning-)365 2441 y(ful,)f(ho)o(w)o(ev)o(er,)h(so)g(some)f
1781 (caution)g(is)h(advised.)967 2574 y(7)p eop
1782 %%Page: 8 8
1783 8 7 bop 262 307 a Fu(3)69 b Fm(COEFLIST)p Fu(;)26 b(Determini)o(ng)21
1784 b(co)r(e\016cien)n(ts)262 398 y Ft(W)m(e)11 b(de\014ne)j(the)e
1785 (`algebras')g(of)f(p)q(olynomials,)e(trigonometric)i(sums)h(and)g(p)q
1786 (o)o(w)o(er)g(series)i(to)262 448 y(b)q(e)g(those)h(expressions)g(that)
1787 f(can)g(b)q(e)h(cast)f(in)o(to)f(the)i(follo)o(wing)c(forms.)270
1788 573 y Fh(P)s Ft(\()p Fn(v)338 579 y Fk(1)357 573 y Fn(;)c(:)g(:)g(:)n
1789 Ft(\))41 b(=)555 502 y Fe(\()589 573 y Fn(P)644 512 y
1790 Fe(\014)644 537 y(\014)644 562 y(\014)644 587 y(\014)681
1791 573 y Fn(P)17 b Ft(=)769 533 y Fe(X)793 622 y Fj(i)836
1792 573 y Fn(c)854 579 y Fj(i)867 573 y Fn(v)888 550 y Fj(p)905
1793 554 y Fd(1)p Fc(;i)887 584 y Fk(1)945 573 y Fn(v)966
1794 550 y Fj(p)983 554 y Fd(2)p Fc(;i)965 584 y Fk(2)1029
1795 573 y Fh(\001)7 b(\001)g(\001)1078 502 y Fe(\))1118 573
1796 y Fn(;)269 715 y Fh(T)j Ft(\()p Fn(v)338 721 y Fk(1)357
1797 715 y Fn(;)d(:)g(:)g(:)n Ft(\))41 b(=)555 644 y Fe(\()589
1798 715 y Fn(T)642 655 y Fe(\014)642 680 y(\014)642 705 y(\014)642
1799 729 y(\014)679 715 y Fn(T)17 b Ft(=)764 676 y Fe(X)788
1800 764 y Fj(i)831 715 y Fn(c)849 721 y Fj(i)870 715 y Ft(cos\()p
1801 Fn(m)977 721 y Fk(1)p Fj(;i)1018 715 y Fn(v)1038 721
1802 y Fk(1)1066 715 y Ft(+)9 b Fh(\001)e(\001)g(\001)o Ft(\))i(+)1223
1803 676 y Fe(X)1247 764 y Fj(i)1290 715 y Fn(s)1309 721 y
1804 Fj(i)1330 715 y Ft(sin\()p Fn(m)1433 698 y Fb(0)1433
1805 725 y Fk(1)p Fj(;i)1474 715 y Fn(v)1494 721 y Fk(1)1522
1806 715 y Ft(+)g Fh(\001)e(\001)g(\001)n Ft(\))1627 644 y
1807 Fe(\))1668 715 y Fn(;)305 857 y Fh(S)s Ft(\()p Fn(v)q(;)g
1808 Fh(O)q Ft(\))42 b(=)555 786 y Fe(\()589 857 y Fn(S)639
1809 797 y Fe(\014)639 822 y(\014)639 847 y(\014)639 872 y(\014)676
1810 857 y Fn(S)15 b Ft(=)779 805 y Fj(n)760 818 y Fe(X)784
1811 906 y Fj(i)826 857 y Fn(c)844 863 y Fj(i)858 857 y Fn(v)879
1812 840 y Fj(p)896 844 y Fc(i)912 857 y Ft(;)7 b Fn(p)952
1813 863 y Fj(n)986 857 y Fh(\024)k(O)1063 786 y Fe(\))1104
1814 857 y Fn(:)262 1017 y Ft(The)i(v)n(ariables)g Fn(v)538
1815 1023 y Fj(i)565 1017 y Ft(ma)o(y)f(b)q(e)i(an)o(y)f(atomic)e
1816 (expression)k(in)e(the)h(sense)h(of)e Fr(ratvars)p Ft([1)n(].)k(The)262
1817 1066 y(shorthands)g Fr(operator\(op\))e Ft(and)i Fr(match\(predicate)o
1818 (\))d Ft(ma)o(y)h(b)q(e)j(used)g(to)e(sp)q(ecify)i(all)262
1819 1116 y(sub)q(expressions)12 b(ha)o(ving)d Fr(op)g Ft(as)h(an)g(op)q
1820 (erator,)h(or)f(that)g(pass)g(the)h(predicate,)g(resp)q(ectiv)o(ely)m
1821 (.)324 1166 y(The)18 b(co)q(e\016cien)o(ts)g Fn(c)644
1822 1172 y Fj(i)675 1166 y Ft(and)g Fn(s)779 1172 y Fj(i)810
1823 1166 y Ft(are)g(general)g Fs(Ma)o(csyma)g Ft(expressions.)30
1824 b(In)18 b(principle)262 1216 y(they)12 b(w)o(ould)g(b)q(e)g(indep)q
1825 (enden)o(t)i(of)e(the)g(v)n(ariables)g Fn(v)1067 1222
1826 y Fj(i)1081 1216 y Ft(,)g(but)g(in)g(practice)h(they)g(ma)o(y)d(con)o
1827 (tain)262 1266 y(non-p)q(olynomia)o(l)j(dep)q(endence)19
1828 b(\(or)d(non-trigonometric,)f(in)g(the)i(trigonometric)e(case\).)262
1829 1316 y(These)f(non-p)q(olynomial)c(cases)15 b(w)o(ould)d(include)i
1830 (expressions)h(lik)o(e)e(\(1)8 b(+)g Fn(x)p Ft(\))1460
1831 1300 y Fj(n)1483 1316 y Ft(,)13 b(where)h Fn(n)g Ft(is)262
1832 1365 y(sym)o(b)q(olic.)20 b(Lik)o(ewise,)15 b(\()p Fn(x)674
1833 1350 y Fj(a)694 1365 y Ft(\))710 1350 y Fj(b)742 1365
1834 y Ft(is,)g(in)g(general,)g(m)o(ultiv)n(alued;)e(unless)j
1835 Fn(a)e Ft(=)g(1)h(or)g Fn(b)f Fh(2)f Fn(Z)19 b Ft(or)262
1836 1415 y Fr(radexpand=all)p Ft(,)10 b(it)k(will)f(not)h(b)q(e)g(in)o
1837 (terpreted)i(as)e Fn(x)1110 1400 y Fj(ab)1156 1415 y
1838 Fh(2)e(P)s Ft(.)19 b(F)m(urthermore,)13 b(w)o(e)i(extend)262
1839 1465 y(the)f(algebras)h(to)f(include)g(lists,)g(v)o(ectors,)h(matrices)
1840 f(and)g(equations,)g(b)o(y)g(in)o(terpreting)h(a)262
1841 1515 y(list)e(of)g(p)q(olynomials,)e(sa)o(y)m(,)h(as)i(a)g(p)q
1842 (olynomial)c(with)k(lists)g(as)g(co)q(e\016cien)o(ts.)324
1843 1565 y(The)20 b(exp)q(onen)o(ts)h Fn(p)637 1571 y Fj(i)671
1844 1565 y Ft(in)e(series)j(are)e(restricted)i(to)e(n)o(um)o(b)q(ers,)h
1845 (but)f(the)g(exp)q(onen)o(ts)262 1614 y Fn(p)283 1620
1846 y Fj(j;i)333 1614 y Ft(and)14 b(m)o(ultiples)e Fn(m)629
1847 1620 y Fj(j;i)681 1614 y Ft(for)h(p)q(olynomials)e(and)i(trigonometric)
1848 g(sums)g(ma)o(y)f(b)q(e)i(general)262 1664 y(expressions)h(\(excluding)
1849 f(bags\).)324 1714 y(The)j(follo)o(wing)d(functions)j(construct)h(a)f
1850 (list)f(of)g(the)i(co)q(e\016cien)o(ts)f(and)g(`k)o(eys',)g(that)262
1851 1764 y(is,)12 b(the)h(exp)q(onen)o(ts)h(or)f(m)o(ultiples.)j(Note)d
1852 (that)g(these)h(are)g(sparse)g(represen)o(tations)h(|)d(no)262
1853 1814 y(co)q(e\016cien)o(ts)j(are)f(zero.)392 1903 y Fr(coeffs)p
1854 Ft(\()p Fn(P)q(;)7 b(v)607 1909 y Fk(1)624 1903 y Fn(;)g(:)g(:)g(:)n
1855 Ft(\))41 b Fh(!)g Ft([[\045)p Fr(poly)p Fn(;)7 b(v)1017
1856 1909 y Fk(1)1033 1903 y Fn(;)g(:)g(:)g(:)n Ft(])p Fn(;)g
1857 Ft([)p Fn(c)1161 1909 y Fk(1)1178 1903 y Fn(;)g(p)1218
1858 1909 y Fk(1)p Fj(;)p Fk(1)1263 1903 y Fn(;)g(:)g(:)g(:)m
1859 Ft(])p Fn(;)g(:)g(:)g(:)n Ft(])289 1966 y Fr(trig)p 380
1860 1966 14 2 v 15 w(coeffs)p Ft(\()p Fn(T)s(;)g(v)606 1972
1861 y Fk(1)624 1966 y Fn(;)g(:)g(:)g(:)n Ft(\))41 b Fh(!)713
1862 2028 y Ft([[\045)p Fr(trig)p Fn(;)7 b(v)899 2034 y Fk(1)915
1863 2028 y Fn(;)g(:)g(:)g(:)n Ft(])p Fn(;)g Ft([[)p Fn(c)1055
1864 2034 y Fk(1)1072 2028 y Fn(;)g(m)1127 2034 y Fk(1)p Fj(;)p
1865 Fk(1)1172 2028 y Fn(;)g(:)g(:)g(:)m Ft(])p Fn(;)g(:)g(:)g(:)n
1866 Ft(])p Fn(;)g Ft([[)p Fn(s)1391 2034 y Fk(1)1408 2028
1867 y Fn(;)g(m)1463 2011 y Fb(0)1463 2038 y Fk(1)p Fj(;)p
1868 Fk(1)1508 2028 y Fn(;)g(:)g(:)g(:)n Ft(])p Fn(;)g(:)g(:)g(:)m
1869 Ft(]])279 2090 y Fr(series)p 414 2090 V 14 w(coeffs)p
1870 Ft(\()p Fn(S;)g(v)q(;)g Fh(O)q Ft(\))41 b Fh(!)g Ft([[\045)p
1871 Fr(series)p Fn(;)7 b(v)q(;)g Fh(O)q Ft(])p Fn(;)f Ft([)p
1872 Fn(c)1175 2096 y Fk(1)1190 2090 y Fn(;)h(p)1230 2096
1873 y Fk(1)1248 2090 y Ft(])p Fn(;)g(:)g(:)g(:)e(;)i Ft([)p
1874 Fn(c)1383 2096 y Fj(n)1404 2090 y Fn(;)g(p)1444 2096
1875 y Fj(n)1466 2090 y Ft(]])279 2152 y Fr(Taylor)p 414 2152
1876 V 14 w(coeffs)p Ft(\()p Fn(S;)g(v)q(;)g Fh(O)q Ft(\))41
1877 b Fh(!)g Ft([[\045)p Fr(Taylor)p Fn(;)7 b(v)q(;)g Fh(O)q
1878 Ft(])p Fn(;)f Ft([)p Fn(c)1175 2158 y Fk(1)1190 2152
1879 y Fn(;)h(p)1230 2158 y Fk(1)1248 2152 y Ft(])p Fn(;)g(:)g(:)g(:)e(;)i
1880 Ft([)p Fn(c)1383 2158 y Fj(n)1404 2152 y Fn(;)g(p)1444
1881 2158 y Fj(n)1466 2152 y Ft(]])262 2242 y(The)j(latter)h(t)o(w)o(o)f
1882 (functions)g(b)q(oth)h(expand)f(an)h(expression)g(through)g(order)g
1883 Fh(O)q Ft(,)f(but)h(the)g(se-)262 2292 y(ries)h(v)o(ersion)f(only)g
1884 (carries)h(expands)g(arithmetic)f(op)q(erations)g(and)h(is)f(often)g
1885 (considerably)262 2342 y(faster)j(than)g Fr(Taylor_coeffs)p
1886 Ft(.)262 2449 y Fi(Examples:)967 2574 y Ft(8)p eop
1887 %%Page: 9 9
1888 9 8 bop 262 307 a Fr(\(c20\))20 b(cl1:coeffs\(\(a+b*x\)*)o(\(c-x\))o
1889 (^2,x)o(\);)719 357 y(2)174 b(2)262 407 y(\(d20\))20
1890 b([[\045poly,x],[a)f(c)j(,0],[b)f(c)43 b(-)21 b(2)h(a)g(c,1],[a)e(-)i
1891 (2)f(b)h(c,2],[b,3]])262 506 y(\(c21\))e(map\('first,rest\(coe)o(ffs\()
1892 741 556 y(\(a+b*x\)*\(c-x\)^2=q)o(0+q1*)o(x+q2*)o(x^2+q)o(3*x^3)o
1893 (,x\)\))o(\);)479 606 y(2)218 b(2)262 656 y(\(d21\))20
1894 b([a)i(c)43 b(=)22 b(q0,)f(b)g(c)44 b(-)21 b(2)h(a)g(c)f(=)h(q1,)f(a)g
1895 (-)h(2)g(b)f(c)h(=)f(q2,)h(b)f(=)h(q3])262 756 y(\(c22\))e
1896 (trig_coeffs\(2*\(a+co)o(s\(x\)\))o(*cos)o(\(x+3*)o(y\),x,)o(y\);)262
1897 805 y(\(d22\))g([[\045trig,x,y],[],[[1)o(,0,3])o(,[2)e
1898 (a,1,3],[1,2,3]]])262 905 y(\(c23\))i(series_coeffs\(\(a+b*)o(x\)*\(c)o
1899 (-x\)^)o(2,x,2)o(\);)806 955 y(2)174 b(2)262 1005 y(\(d23\))20
1900 b([[\045series,x,2],[a)f(c)i(,0],[b)g(c)43 b(-)22 b(2)f(a)h(c,1],[a)e
1901 (-)i(2)f(b)h(c,2]])262 1104 y(\(c24\))e(coeffs\(\(a+b*x\)*sin\()o
1902 (x\),x\))o(;)262 1154 y(\(d24\))g([[\045poly,x],[a)f(sin\(x\),0],[b)h
1903 (sin\(x\),1]])262 1254 y(\(c25\))g(coeffs\(\(a+log\(b\)*x\))o(*\(c-l)o
1904 (og\(x)o(\)\)^2,)o(opera)o(tor\(l)o(og\)\);)981 1303
1905 y(2)174 b(2)262 1353 y(\(d25\))20 b([[\045poly,log\(x\),log\()o(b\)],[)
1906 o(a)f(c)i(,0,0],[c)42 b(x,0,1],[-)20 b(2)i(a)g(c,1,0],)894
1907 1403 y([-)f(2)g(c)h(x,1,1],[a,2,0],[x,)o(2,1]])262 1518
1908 y Ff(3.1)55 b(Related)19 b(functions)262 1618 y Fr(get)p
1909 331 1618 14 2 v 15 w(coef)p Ft(\()p Fl(clist)p Ft(,)p
1910 Fn(k)555 1624 y Fk(1)571 1618 y Fn(;)7 b(:)g(:)g(:)n
1911 Ft(\))880 b Fl(F)m(unction)365 1695 y Ft(Gets)19 b(the)g(co)q
1912 (e\016cien)o(t)g(from)d(the)j(co)q(e\016cien)o(t)g(list)f
1913 Fl(clist)f Ft(corresp)q(onding)i(to)365 1745 y(the)13
1914 b(k)o(eys)g Fn(k)547 1751 y Fj(i)560 1745 y Ft(.)18 b(The)13
1915 b(k)o(eys)g(are)f(matc)o(hed)g(to)g(v)n(ariable)f(p)q(o)o(w)o(ers)j
1916 (when)e Fl(clist)g Ft(is)g(a)365 1795 y Fr(\045poly)p
1917 Ft(,)h Fr(\045series)f Ft(or)h Fr(\045Taylor)g Ft(form.)j(If)d
1918 Fl(clist)g Ft(is)h(a)f Fr(\045trig)g Ft(then)h Fn(k)1440
1919 1801 y Fk(1)1472 1795 y Ft(should)365 1845 y(b)q(e)h
1920 Fr(sin)e Ft(or)h Fr(cos)f Ft(and)h(the)g(remaining)e(k)o(eys)i(are)h
1921 (matc)o(hed)e(to)h(m)o(ultipliers.)262 1945 y Fr(uncoef)p
1922 Ft(\()p Fl(clist)p Ft(\))1034 b Fl(F)m(unction)365 2022
1923 y Ft(Reconstructs)17 b(the)f(expression)g(from)e(a)h(co)q(e\016cien)o
1924 (t)h(list)e Fl(clist)p Ft(.)21 b(The)16 b(co)q(e\016-)365
1925 2072 y(cien)o(t)f(list)e(can)h(b)q(e)h(an)o(y)e(of)g(the)i(co)q
1926 (e\016cien)o(t)g(list)e(forms.)262 2172 y Fr(partition)p
1927 463 2172 V 13 w(poly)p Ft(\()p Fl(expr)p Ft(,)p Fl(test)p
1928 Ft(,)p Fn(v)762 2178 y Fk(1)779 2172 y Fn(;)7 b(:)g(:)g(:)n
1929 Ft(\))672 b Fl(F)m(unction)365 2249 y Ft(P)o(artitions)10
1930 b Fl(expr)g Ft(in)o(to)f(t)o(w)o(o)g(p)q(olynomials;)f(the)i(\014rst)h
1931 (is)e(made)g(of)g(those)i(mono-)365 2299 y(mials)g(for)i(whic)o(h)g
1932 (the)h(function)f Fl(test)g Ft(returns)i Fr(true)d Ft(and)h(the)h
1933 (second)g(is)f(the)365 2349 y(remainder.)18 b(The)c(test)h(function)f
1934 (is)f(called)h(on)g(the)g(p)q(o)o(w)o(ers)h(of)e(the)h
1935 Fn(v)1471 2355 y Fj(i)1485 2349 y Ft(.)262 2449 y Fr(partition)p
1936 463 2449 V 13 w(trig)p Ft(\()p Fl(expr)p Ft(,)p Fl(sintest)p
1937 Ft(,)p Fl(c)n(ostest)p Ft(,)p Fn(v)946 2455 y Fk(1)963
1938 2449 y Fn(;)7 b(:)g(:)g(:)m Ft(\))489 b Fl(F)m(unction)967
1939 2574 y Ft(9)p eop
1940 %%Page: 10 10
1941 10 9 bop 365 307 a Ft(T)m(rigonometric)10 b(analog)h(to)h
1942 Fr(partition)p 1012 307 14 2 v 14 w(poly)p Ft(;)f(The)h(functions)g
1943 Fl(sintest)g Ft(and)365 357 y Fl(c)n(ostest)h Ft(select)h(sine)g(and)f
1944 (co<sine)g(terms,)f(resp)q(ectiv)o(ely;)i(eac)o(h)g(are)f(called)g(on)
1945 365 407 y(the)i(m)o(ultipliers)d(of)h(the)h Fn(v)782
1946 413 y Fj(i)796 407 y Ft(.)262 510 y Fr(partition_serie)o(s)p
1947 Ft(\()p Fl(expr)p Ft(,)p Fl(test)p Ft(,)o Fn(v)q Ft(,)p
1948 Fh(O)q Ft(\))658 b Fl(F)m(unction)262 583 y Fr(partition)p
1949 463 583 V 13 w(Taylor)p Ft(\()p Fl(expr)p Ft(,)p Fl(test)p
1950 Ft(,)p Fn(v)q Ft(,)p Fh(O)q Ft(\))665 b Fl(F)m(unction)365
1951 662 y Ft(Analog)13 b(to)h Fr(partition)p 759 662 V 14
1952 w(poly)f Ft(for)g(series.)262 769 y Fi(Examples:)262
1953 848 y Fr(\(c26\))20 b(get_coef\(CL1,2\);)262 898 y(\(d26\))g(a)i(-)f(2)
1954 h(b)g(c)262 998 y(\(c27\))e(uncoef\(cl1\);)458 1047 y(3)348
1955 b(2)152 b(2)371 b(2)262 1097 y(\(d27\))20 b(b)i(x)43
1956 b(+)22 b(\(a)f(-)h(2)f(b)h(c\))f(x)44 b(+)21 b(\(b)h(c)43
1957 b(-)22 b(2)f(a)h(c\))f(x)h(+)f(a)h(c)262 1197 y(\(c28\))e
1958 (partition_poly\(\(a+b)o(*x\)*\()o(c-x\))o(^2,'e)o(venp,)o(x\);)697
1959 1247 y(2)131 b(2)109 b(3)152 b(2)262 1297 y(\(d28\))20
1960 b([\(a)h(-)h(2)g(b)f(c\))h(x)43 b(+)22 b(a)f(c)h(,)f(b)h(x)43
1961 b(+)22 b(\(b)f(c)44 b(-)21 b(2)h(a)f(c\))h(x])262 1412
1962 y Ff(3.2)55 b(Supp)r(ort)19 b(functions)262 1512 y Fr(matching)p
1963 441 1512 V 14 w(parts)p Ft(\()p Fl(expr,)13 b(pr)n(e)n(dic)n(ate,)h(ar)
1964 n(gs)p Fn(:)7 b(:)g(:)f Ft(\))522 b Fl(F)m(unction)365
1965 1591 y Ft(Returns)14 b(a)e(list)g(of)g(all)f(sub)q(expressions)k(of)d
1966 Fl(expr)g Ft(for)h(whic)o(h)f(the)h(application)365 1641
1967 y Fr(predicate\(piece,arg)o(s)p Fn(:)7 b(:)f(:)t Fr(\))13
1968 b Ft(returns)j Fr(True)p Ft(.)262 1744 y Fr(function)p
1969 441 1744 V 14 w(calls)p Ft(\()p Fl(expr,functions)p Fn(:)7
1970 b(:)g(:)t Ft(\))631 b Fl(F)m(unction)365 1823 y Ft(Returns)15
1971 b(a)f(list)f(of)g(all)g(calls)g(in)h Fl(expr)g Ft(in)o(v)o(olving)d(an)
1972 o(y)j(of)f Fl(functions)p Ft(.)262 1926 y Fr(function)p
1973 441 1926 V 14 w(arguments)p Ft(\()p Fl(expr,functions)p
1974 Fn(:)7 b(:)g(:)s Ft(\))544 b Fl(F)m(unction)365 2005
1975 y Ft(Returns)15 b(a)f(list)f(of)g(all)g(argumen)o(t)g(lists)h(for)f
1976 (calls)h(to)f Fl(functions)i Ft(in)e Fl(expr)p Ft(.)262
1977 2112 y Fi(Examples:)262 2191 y Fr(\(c29\))20 b
1978 (t2:\(a+log\(b\)*x\)*\(c-)o(log\(x)o(\)\)^2)o($)262 2241
1979 y(\(c30\))g(matching_parts\(t2,c)o(onsta)o(ntp\))o(;)262
1980 2290 y(\(d30\))g([2,)h(-)h(1])262 2390 y(\(c31\))e
1981 (function_calls\(t2,l)o(og\);)262 2440 y(\(d31\))g([log\(x\),)g
1982 (log\(b\)])957 2574 y Ft(10)p eop
1983 %%Page: 11 11
1984 11 10 bop 262 307 a Fu(4)69 b(Av)l(ailabilit)m(y)262
1985 398 y Ft(This)13 b(pac)o(k)n(age)h(has)g(b)q(een)g(tested)i(in)d
1986 (Macsyma)g(Inc.'s)h(v)o(ersions)g(418.85)e(for)h(Genera)i(8.3)262
1987 448 y(and)c(418.1)e(for)i(Sparc)h(computers)f(under)h(SunOS)g(4.1.3,)d
1988 (as)j(w)o(ell)e(as)h(the)h(DOE)f(`maxim)o(a')262 498
1989 y(v)o(ersion)j(4.155.)i(The)e Fs(Lisp)g Ft(source)h(co)q(de)g(is)f(a)o
1990 (v)n(ailable)e(from)g(the)i(author.)262 635 y Fu(Ac)n(kno)n(wledgmen)n
1991 (ts)262 726 y Ft(The)c(author)g(wishes)h(to)f(thank)f(Je\013rey)j
1992 (Golden)d(\(Macsyma,)h(Inc.\))17 b(for)10 b(sharing)g(his)f(ideas)262
1993 776 y(whic)o(h)15 b(led)g(to)g(expanding)g(the)h(scop)q(e)g(of)f(the)h
1994 (pac)o(k)n(age.)22 b(W)m(e)14 b(thank)i(Ric)o(hard)e(F)m(ateman)262
1995 826 y(\(U.)f(C.)h(Berk)o(eley\))h(for)e(teac)o(hing)h(us)g(m)o(uc)o(h)f
1996 (ab)q(out)h(the)g(in)o(ternals)g(of)f Fs(Ma)o(csyma)p
1997 Ft(.)262 963 y Fu(References)282 1054 y Ft([1])20 b(Sym)o(b)q(olics)d
1998 (Inc.,)j Fa(Macsyma)e(Reference)i(Man)o(ual)p Ft(,)f(Sym)o(b)q(olics)e
1999 (Inc.,)j(Burlington)347 1104 y(MA.,)13 b(1988.)282 1187
2000 y([2])20 b(Kn)o(uth,)13 b(Donald,)e Fa(Semin)o(umerical)f(Algorithms,)h
2001 (The)i(art)g(of)g(computer)f(program-)347 1236 y(ming)p
2002 Ft(,)f(V)m(ol.)h(2.)i(Addison-W)m(esley)m(,)e(1969.)282
2003 1319 y([3])20 b(Golden,)13 b(Je\013rey)i(P)m(.,)e(priv)n(ate)g(comm)o
2004 (unication.)282 1402 y([4])20 b(Griesmer,)13 b(J.)h(H.,)g(Jenks,)h(R.)f
2005 (D.)f(and)h(Y)m(un,)g(Y.)g(Y.,)g(\\A)g(F)o(ORMA)m(T)g(statemen)o(t)g
2006 (in)347 1452 y(SCRA)m(TCHP)m(AD,")e(SIGSAM)i(Bulletin)f(\(9\),)h(1975,)
2007 e(pp.)i(24{25.)282 1535 y([5])20 b(Barnett,)e(M.)e(P)m(.)32
2008 b(and)16 b(P)o(erry)m(,)h(K.)g(R.)e(,)i(\\Hierarc)o(hical)f(Addressing)
2009 i(in)e(Sym)o(b)q(olic)347 1585 y(Computation,")11 b(Computers)i(Math.)h
2010 (Applic.)f(,)g(to)h(app)q(ear.)262 1722 y Fu(A)68 b(Implemen)n(tati)o
2011 (on)262 1813 y Ft(In)15 b(this)g(app)q(endix,)g(w)o(e)g(describ)q(e)i
2012 (some)e(of)f(the)i(most)e(imp)q(ortan)o(t)f(elemen)o(ts)i(of)g(the)h
2013 (im-)262 1863 y(plemen)o(tation.)24 b(It)16 b(is)h(not)f(our)g(in)o
2014 (ten)o(tion)g(to)h(describ)q(e)h(ev)o(ery)f(facet)g(in)f(detail,)g
2015 (rather,)262 1913 y(w)o(e)d(o\013er)i(it)e(as)h(an)f(o)o(v)o(erview)h
2016 (to)f(the)i(lisp)e(co)q(de,)h(and)f(as)h(a)f(guide)h(to)g(an)o(y)o(one)
2017 f(wishing)g(to)262 1963 y(implem)o(en)o(t)f(similar)f(facilities)i(for)
2018 g(another)i(CAS.)262 2079 y Ff(A.1)55 b(Co)r(e\016cien)n(t)19
2019 b(Lists)262 2156 y Ft(The)11 b(fundamen)o(tal)e(algorithm)f(for)i(con)o
2020 (v)o(erting)h(p)q(olynomials,)d(p)q(oisson)j(series,)h(etc.)29
2021 b(in)o(to)262 2205 y(canonical)11 b(represen)o(tations,)j(suc)o(h)f(as)
2022 g(the)g(co)q(e\016cien)o(t)g(lists)f(de\014ned)i(here,)f(is)f(as)h
2023 (follo)o(ws.)262 2255 y(First,)j(an)g(`arithmetic')d(is)j(implemen)o
2024 (ted)e(for)i(the)g(new)h(represen)o(tation.)25 b(That)16
2025 b(is,)g(the)262 2305 y(co)q(de)i(to)f(add,)h(m)o(ultiply)d(and)i(exp)q
2026 (onen)o(tiate)h(\(at)g(least\))f(ob)r(jects)i(in)e(the)h(new)g(form)e
2027 (is)262 2355 y(written)11 b(\(See)h([2)o(])f(for)g(algorithms\).)k(An)c
2028 (expression)h(is)f(then)h(con)o(v)o(erted)g(recursiv)o(ely;)g(de-)262
2029 2405 y(p)q(ending)e(on)f(the)i(main)d(op)q(erator)i(of)g(the)g
2030 (expression,)i(its)e(argumen)o(ts)f(are)i(\014rst)f(con)o(v)o(erted)957
2031 2574 y(11)p eop
2032 %%Page: 12 12
2033 12 11 bop 262 307 a Ft(and)16 b(then)i(they)f(are)h(com)o(bined)e
2034 (appropriately)m(.)26 b(A)o(toms)16 b(are)h(con)o(v)o(erted)h(in)f
2035 (whatev)o(er)262 357 y(w)o(a)o(y)c(is)g(appropriate)h(for)g(the)g
2036 (represen)o(tation.)324 407 y(This)19 b(is)h(the)g(metho)q(d)f(used)i
2037 (in)o(ternally)e(b)o(y)g(the)i(CRE)e(and)h(P)o(oisson)g(facilities)e
2038 (of)262 457 y Fs(Ma)o(csyma)p Ft(.)31 b(An)18 b(issue)h(for)f(us)g(w)o
2039 (as)g(whether)i(it)d(w)o(as)h(b)q(est)i(to)d(lev)o(erage)i(these)g
2040 (exist-)262 506 y(ing)13 b(facilities)g(b)o(y)h(transforming)f(\014rst)
2041 i(to)f(CRE)g(or)g(P)o(oisson)g(represen)o(tations)i(and)e(from)262
2042 556 y(there)j(in)o(to)e(co)q(e\016cien)o(t)h(lists,)g(or)f(whether)i(w)
2043 o(e)f(should)g(reimplemen)o(t)d(the)k(metho)q(ds)e(for)262
2044 606 y(con)o(v)o(ersions)f(directly)g(in)o(to)f(co)q(e\016cien)o(t)i
2045 (list)f(form.)324 656 y(In)e(the)g(end,)h(w)o(e)f(decided)h(to)f
2046 (reimplemen)o(t)e(the)i(metho)q(d)f(for)h(p)q(olynomial)c(and)k(series)
2047 262 706 y(arithmetic.)23 b(The)16 b(primary)f(reason)h(is)g(that)h(the)
2048 f(CRE)g(\(and)g(T)m(a)o(ylor\))f(transforms)g(the)262
2049 756 y(en)o(tire)i(expression)h(in)o(to)e(CRE)g(form,)f(including)h
2050 (what)h(will)e(b)q(ecome)i(the)g(co)q(e\016cien)o(ts.)262
2051 805 y(This)f(is)h(unnecessary)i(w)o(ork)d(for)g(our)h(purp)q(oses,)h
2052 (and)f(in)f(the)i(application)d(to)h Fr(format)p Ft(,)262
2053 855 y(the)d(w)o(ork)f(ma)o(y)e(immediately)f(b)q(e)k(undone)g(at)g(the)
2054 g(next)g(step.)18 b(Indeed,)c(if)d(an)h(expression)262
2055 905 y(had)j(already)h(b)q(een)h Fr(format)p Ft('d,)d(the)i(curren)o(t)i
2056 (co)q(de)e(ma)o(y)e(lea)o(v)o(e)i(the)h(co)q(e\016cien)o(ts)g(in)e(the)
2057 262 955 y(correct)g(form.)324 1005 y(The)k(P)o(oisson)f(pac)o(k)n(age)g
2058 (do)q(es)h(not)f(carry)h(out)g(an)o(y)f(transformations)f(of)g(the)i
2059 (co)q(ef-)262 1054 y(\014cien)o(ts)e(and,)f(so,)h(w)o(as)f(suitable)g
2060 (for)g(use)h(in)f(con)o(v)o(ersion)h(to)f(trigonometric)f(co)q
2061 (e\016cien)o(t)262 1104 y(lists.)i(Ultimately)m(,)8 b(w)o(e)k(rewrote)h
2062 (m)o(uc)o(h)d(of)h(the)h(existing)f(p)q(oisson)g(pac)o(k)n(age)g(an)o
2063 (yw)o(a)o(y)m(.)16 b(This)262 1154 y(w)o(as)e(b)q(oth)h(to)g(add)g
2064 (\015exibilit)o(y)e(\(particularly)h(to)h(allo)o(w)e(non-in)o(tegral)h
2065 (m)o(ultipliers\))f(that)262 1204 y(w)o(ould)e(b)q(e)i(useful)g(b)q
2066 (oth)f(here)i(and)e(to)h(users)h(of)e(the)h(P)o(oisson)f(pac)o(k)n
2067 (age,)g(and)g(also)g(to)h(rem-)262 1254 y(edy)e(a)f(long)g(standing)h
2068 (limitatio)o(n)d(of)j(the)g(pac)o(k)n(age)g(|)f(it)h(failed)e(to)i
2069 (detect)i(when)e(enco)q(ded)262 1303 y(trigonometric)g(argumen)o(ts)h
2070 (exceeded)j(the)e(predeclared)h(b)q(ounds)f(resulting)g(in)f(spurious)
2071 262 1353 y(computations.)k(Con)o(tact)d(the)h(author)f(for)g
2072 (information)e(ab)q(out)i(this)g(alternate)h(P)o(oisson)262
2073 1403 y(pac)o(k)n(age.)i(Ho)o(w)o(ev)o(er,)10 b(w)o(e)g(ha)o(v)o(e)f(an)
2074 h(implem)o(en)o(tation)d(of)h Fr(trig_coeffs)g Ft(that)h(a)o(v)o(oids)g
2075 (using)262 1453 y(P)o(oisson,)16 b(should)f(our)i(alternativ)o(e)e(P)o
2076 (oisson)h(pac)o(k)n(age)g(b)q(e)h(unacceptable)g(for)f(whatev)o(er)262
2077 1503 y(reason.)324 1553 y Fr(Taylor)p 459 1553 14 2 v
2078 14 w(coeffs)p Ft(,)11 b(the)h(alternativ)o(e)f(con)o(v)o(ersion)h(to)g
2079 (series)h(co)q(e\016cien)o(t)g(lists,)e(do)q(es)i(use)262
2080 1602 y Fr(Taylor)f Ft(as)h(describ)q(ed)j(ab)q(o)o(v)o(e;)c(it)i(is)f
2081 (useful)h(when)g(full)e(T)m(a)o(ylor)g(expansions)i(are)g(needed.)262
2082 1719 y Ff(A.2)55 b(F)-5 b(ormat)262 1795 y Ft(The)16
2083 b(basic)g(op)q(eration)f(of)h(the)g(formatting)d(program)i(is)g
2084 (relativ)o(ely)g(simple;)g(it)g(is)h(data-)262 1845 y(driv)o(en)i(b)o
2085 (y)g(the)h(templates.)30 b(The)19 b(\014rst)g(template)e(in)h(the)h(c)o
2086 (hain)e(is)h(examined)f(and)h(if)262 1895 y(it)c(is)h(a)g(kno)o(wn)f
2087 (formatting)f(template,)g(format)h(binds)g(the)i(remaining)d(template)h
2088 (c)o(hain)262 1945 y(and)h(the)h(subtemplates.)24 b(It)15
2089 b(then)i(calls)e(the)h(function)g(asso)q(ciated)g(with)f(the)i
2090 (template)262 1994 y(on)g(the)h(expression)h(and)e(an)o(y)g(parameters)
2091 g(giv)o(en)g(to)h(the)g(template.)28 b(Eac)o(h)18 b(template)262
2092 2044 y(function)c(transforms)h(the)h(expression)g(appropriately)f(and)g
2093 (then)g(calls)g Fr(format)p 1573 2044 V 15 w(piece)262
2094 2094 y Ft(on)e(the)i(appropriate)f(pieces.)324 2144 y(The)e(function)f
2095 Fr(format)p 702 2144 V 15 w(piece)g Ft(determines)h(if)f(there)i(is)e
2096 (a)h(subtemplate)f(that)h(should)262 2194 y(b)q(e)i(applied)f(to)g(a)h
2097 (giv)o(en)f(piece)h(or)g(if)f(the)h(next)g(template)f(in)g(the)h(c)o
2098 (hain)f(should)h(b)q(e)g(used.)262 2243 y(It)h(then)h(recursiv)o(ely)h
2099 (in)o(v)o(ok)o(es)e Fr(format)f Ft(to)i(format)e(the)i(giv)o(en)f
2100 (piece)h(with)g(the)g(selected)262 2293 y(template.)957
2101 2574 y(12)p eop
2102 %%Trailer
2103 end
2104 userdict /end-hook known{end-hook}if
2105 %%EOF