Artifact 094c02c02254f1eebfc392e97e33ae83606640f2ddc35d07a1bf87d9fe63948f:
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r37/packages/support/0patches.red
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2011-09-02 18:13:33
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module patches; % Patches to correct problems in REDUCE 3.7. % Author: Anthony C. Hearn. % Copyright (c) 2001 Anthony C. Hearn. All Rights Reserved. global '(patch!-date!*); patch!-date!* := "7-Sep-2001"; % Bugs fixed by these patches. % 28 Jun 99. Gnuplot handling on the Macintosh was not correct. % 7 Aug 99. The evaluation of df(tan((sqrt(1-x^2)*asin acos x % + 2*sqrt(1-x^2)*x)/x),x) did not terminate. % 20 Oct 99. The sequence a1:=12x^2-16x+3; a2:=3x-4; off mcd; % on combineexpt; e^(a1/a2); gave the wrong answer. % 8 Nov 99. factorize(2*c*s*u^3*v^5-2*c*s*u^3*v +2*c*s*u*v^5-2*c*s*u*v % -s^2*u^4*v^4+s^2*u^4+s^2*u^2*v^6-s^2*u^2*v^4-s^2*u^2*v^2 % +s^2*u^2 +s^2*v^6-s^2*v^2+u^4*v^4-u^4*v^2 -v^4+v^2) gave % a catastrophic error. % 9 Nov 99. Patched procedures generated a "redefined" message. % 16 Nov 99. Some EXCALC calculations could cause a catastrophic error. % 18 Dec 99. Integrations could give catastrophic errors because some % kernels were not unique. % 31 Jan 00. The sequence weight x=1,y=1; wtlevel 10; factor x; led to % the error that x was invalid as a kernel. % 5 Feb 00. The sequence x := mat((1,2)); sign sqrt 42; led to a % spurious error. % 6 Feb 00. The sequence on complex; sqrt(i*sqrt(3)-1); gave a wrong % result. % 10 Feb 00. Some root evaluations could lead to an error like % <equation> invalid as scalar. % 18 Feb 00. A sequence like m := mat((a,b),(c,d)); det sub(a=1,m); % would cause a type mismatch error. % 18 Apr 00. Complaints about the pattern matching limit of 5 terms % are resolved by the addition of a variable matchlength!*, % whose initial value of 5 can be changed as needed. % 22 Apr 00. The RULE mechanism left spurious expressions in various % non-local variables. % 28 Jul 00. A sum index within a derivative was treated as an identifier % (e.g., sum(x^n/factorial n*sub(x=0,df(cos x,x,n)),n,0,5); % 2 Aug 00. With complex on, some factorizations seemed to run forever % (e.g., factorize (400y^12+400y^10*z+40y^9*z^2+100y^8*z^2 % +20y^7*z^5+120y^7*z^4+20y^7*z^3+41y^6*z^4+60y^5*z^7 % +60y^5*z^5+20y^4*z^7+6y^4*z^6+20y^4*z^5 % +2y^3*z^6+9y^2*z^8+6y*z^8+z^8)) % 29 Aug 00. The sequence load_package gentran,scope; matrix a(10,10); % on gentranopt; gentran a(1,1) ::=: a(1,1); caused a % segmentation violation or similar error. % 19 Sep 00. Clearing some sqrt rules could lead to a spurious % "not found" message. % 20 Sep 00. The sequence load_package algint; % int(1/sqrt((2*e^c-y)/(e^c*y)),y); % caused a catastrophic error. % 8 Nov 00. Some sequences did not optimize completely when the SCOPE % command "optimize" was used. % 20 Nov 00. The sum operator did not always preserve a noncom order % (e.g., noncom u,v; sum(u(m)*v(1-m),m,0,1);) % 12 Dec 00. int with four arguments did not automatically load the % defint package. % 13 Dec 00. Some gcd calculations could produce an endless loop. E.g., % in on numval,rounded; y:=x^4+x3*x^3+x2*x^2+x1*x+x0; % on fullroots; solve(y,x); % 9 Jan 01. SOLVE did not return results in same order as the given % variables (e.g., solve({y=x+t^2,x=y+u^2},{x,y,u,t}); % 14 Jan 01. Some resultants (e.g. resultant(p^3-3p^2-a,3p*(p-2),p)) % caused an error. % 19 Jan 01. Some algebraic integrals could produce a catastrophic % error when the algint package was loaded. % 22 Jan 01. Some algebraic integrals could produce a spurious zero % divisor message when the algint package was loaded (e.g., % int((sqrt((-sqrt(a^4*x^2+4)+a^2*x)/(2*x)) % *(-sqrt(a^4*x^2+4)*a^2*x-a^4*x^2-4))/(2*(a^4*x^2+4)),x)) % 23 Jan 01. Inverses of matrices containing non-commuting objects % could be incorrect (e.g. noncom q; 1/mat((1,0,0), % (x/p*q 1,1,0),(x*y/(2p*(p-1))*q 1*q 1,y/(p-2)*q 1,1))). % 2 Feb 01. Some calls of SOLVE could produce a "zero divisor" error % error (e.g., solve(sqrt x*sqrt((4x^2*x+1)/x)-1=0,x)). % 9 Feb 01. The patched version of combine!-logs included an undefined % macro. % 20 Feb 01. Even with combineexpt on, expressions like a*a^x and % e*e^(2/(2-x)) did not simplify adequately. % 6 Mar 01. With algint loaded, some integrals would abort before % completion (e.g., int((x^(2/3)*sqrt(sqrt(y)*sqrt(pi) + 2pi % *y*x)*sqrt(- sqrt(y)*sqrt(pi)+2pi*y*x))/(4pi*y*x^3 - x),x)). % 7 Apr 01. Factorizations with rounded and complex rounded were not % supported. % 23 Apr 01. A term could be dropped from the solution of a set of % linear equations involving surds. % 7 May 01. Patches of 19 Sep 00 updated to correct bug. % 1 Jun 01. With precise on, sqrt(x^2) returned x rather than abs(x). % 11 Jun 01. Expressions involving noncommuting arguments of a % commuting operator were not handled correctly. % E.g., operator p,q; noncom q; p q a*p q b -p q b*p q a; % 15 Jun 01. Patch of 9 Jan 01 gave a catastrophic error with depend % statements, e.g., depend u,z; solve(u=y/x,z). % 7 Aug 01. Resultants could return an error or the wrong sign % (replaces patch of 14 Jan 01). % 9 Aug 01. Rules for tan, like let tan(~x)=>sin x/cos x, made some % integrals not return a closed form (e.g., int(1/sin(y)^2,y)). % 24 Aug 01. Some factorizations with complex on could have the wrong % sign. % 7 Sep 01. Some solve problems could cause a catastrophic error in % SQFRF. % Alg declarations. fluid '(matchlength!*); matchlength!* := 5; flag('(matchlength!*),'share); fluid '(!*sqrtrulep); patch alg; % 20 Oct 99, 20 Feb 01. symbolic procedure exptunwind(u,v); begin scalar x,x1,x2,y,z,z2; a: if null v then return u; x := caar v; x1 := cadr x; x2 := caddr x; y := cdar v; v := cdr v; if !*combineexpt and length u=1 and null cdr(z2 := kernels u) then u := {(({'expt,car z2,ldeg u} . 1) . lc u)}; while (z := assocp1(x1,v)) and (z2 := simp {'plus,{'times,x2,y},{'times,caddar z,cdr z}}) and (!*combineexpt or (fixp numr z2 and fixp denr z2)) do <<if fixp numr z2 and fixp denr z2 then <<x2 := divide(numr z2,denr z2); if car x2>0 then <<if fixp x1 then u := multf(x1**car x2,u) else u := multpf(mksp(x1,car x2),u); z2 := cdr x2 ./ denr z2>>; y := numr z2>> else y := 1; x2 := prepsq(quotf(numr z2,y) ./ denr z2); v := delete(z,v)>>; if !*combineexpt and y=1 and fixp x1 then <<while (z := assocp2(x2,v)) and cdr z=1 and fixp cadar z do <<x1 := cadar z * x1; v := delete(z,v)>>; if eqcar(x2,'quotient) and fixp cadr x2 and fixp caddr x2 and cadr x2<caddr x2 then <<z := nrootn(x1**cadr x2,caddr x2); if cdr z = 1 then u := multd(car z,u) else if car z = 1 then u := multf(formsf(x1,x2,1),u) else <<u := multd(car z,u); v := (list('expt,cdr z,x2) . 1) . v>>>> else u := multf(formsf(x1,x2,y),u)>> else u := multf(formsf(x1,x2,y),u); go to a end; % 31 Jan 00. symbolic procedure factor1(u,v,w); begin scalar x,y,z,r; y := lispeval w; for each j in u do if (x := getrtype j) and (z := get(x,'factor1fn)) then apply2(z,u,v) else <<while eqcar(j:=reval j,'list) and cdr j do <<r:=append(r,cddr j); j:=cadr j>>; x := !*a2kwoweight j; if v then y := aconc!*(delete(x,y),x) else if not(x member y) then msgpri(nil,j,"not found",nil,nil) else y := delete(x,y)>>; set(w,y); if r then return factor1(r,v,w) end; % 5 Feb 00. algebraic (let sign(sqrt ~a) => 1 when sign a=1); % 18 Feb 00. symbolic procedure getrtype u; begin scalar x,y; return if null u then nil else if atom u then if not idp u then not numberp u and getrtype1 u else if flagp(u,'share) then if (x := eval u) eq u then nil else getrtype x else if (x := get(u,'avalue)) and not(car x memq '(scalar generic)) or (x := get(u,'rtype)) and (x := list x) then if y := get(car x,'rtypefn) then apply1(y,nil) else car x else nil else if not idp car u then nil else if (x := get(car u,'avalue)) and (x := get(car x,'rtypefn)) then apply1(x,cdr u) else if car u eq 'sub then 'yetunknowntype else getrtype2 u end; symbolic procedure let3(u,v,w,b,flgg); begin scalar x,y1,y2,z; x := u; if null x then <<u := 0; return errpri1 u>> else if numberp x then return errpri1 u; y2 := getrtype v; if b and idp x then <<remprop(x,'rtype); remprop(x,'avalue)>>; if (y1 := getrtype x) then return if z := get(y1,'typeletfn) then lispapply(z,list(x,v,y1,b,getrtype v)) else typelet(x,v,y1,b,getrtype v) else if y2 and not(y2 eq 'yetunknowntype) then return if z := get(y2,'typeletfn) then lispapply(z,list(x,v,nil,b,y2)) else typelet(x,v,nil,b,y2) else letscalar(u,v,w,x,b,flgg) end; % 18 Apr 00. symbolic procedure mcharg1(u,v,w); if null u and null v then list nil else begin integer m,n; m := length u; n := length v; if flagp(w,'nary) and m>2 then if m<=matchlength!* and flagp(w,'symmetric) then return mchcomb(u,v,w) else if n=2 then <<u := cdr mkbin(w,u); m := 2>> else return nil; return if m neq n then nil else if flagp(w,'symmetric) then mchsarg(u,v,w) else if mtp v then list pair(v,u) else mcharg2(u,v,list nil,w) end; % 19 Sep 00, 7 May 01, 15 Jun 01. symbolic procedure letscalar(u,v,w,x,b,flgg); begin if not atom x then if not idp car x then return errpri2(u,'hold) else if car x eq 'df then if null letdf(u,v,w,x,b) then nil else return nil else if getrtype car x then return let2(reval x,v,w,b) else if not get(car x,'simpfn) then <<redmsg(car x,"operator"); mkop car x; return let3(u,v,w,b,flgg)>> else nil else if null b and null w then <<remprop(x,'avalue); remprop(x,'rtype); remflag(list x,'antisymmetric); remprop(x,'infix); remprop(x,'kvalue); remflag(list x,'linear); remflag(list x,'noncom); remprop(x,'op); remprop(x,'opmtch); remprop(x,'simpfn); remflag(list x,'symmetric); wtl!* := delasc(x,wtl!*); if flagp(x,'opfn) then <<remflag(list x,'opfn); remd x>>; rmsubs(); return nil>>; if eqcar(x,'expt) and caddr x memq frlis!* then letexprn(u,v,w,!*k2q x,b,flgg) else if eqcar(x,'sqrt) then <<!*sqrtrulep := t; let2({'expt,cadr x,'(quotient 1 2)},v,w,b)>>; x := simp0 x where !*precise = t; return if not domainp numr x then letexprn(u,v,w,x,b,flgg) else errpri1 u end; symbolic procedure setk1(u,v,b); begin scalar x,y,z,!*uncached; !*uncached := t; if atom u then <<if null b then <<if not get(u,'avalue) then msgpri(nil,u,"not found",nil,nil) else remprop(u,'avalue); return nil>> else if (x:= get(u,'avalue)) then put!-avalue(u,car x,v) else put!-avalue(u,'scalar,v); return v>> else if not atom car u then rerror(alg,25,"Invalid syntax: improper assignment"); u := car u . revlis cdr u; if null b then <<z:=assoc(u,wtl!*); if not(y := get(car u,'kvalue)) or not (x := assoc(u,y)) then <<if null z and null !*sqrtrulep then msgpri(nil,u,"not found",nil,nil)>> else put(car u,'kvalue,delete(x,y)); if z then wtl!*:=delasc(u,wtl!*); return nil>> else if not (y := get(car u,'kvalue)) then put!-kvalue(car u,nil,u,v) else <<if x := assoc(u,y) then <<updoldrules(u,v); y := delasc(car x,y)>>; put!-kvalue(car u,y,u,v)>>; return v end; % 7 May 01. symbolic procedure clearrules u; begin scalar !*sqrtrulep; return rule!-list(u,nil) end; % 2 Feb 01. symbolic procedure simprad(u,n); if !*reduced then multsq(radfa(numr u,n),invsq radfa(denr u,n)) else begin scalar iflag,x,y,z; if !*rationalize then << y:=list(denr u,1); u:=multf(numr u, exptf(denr u,n-1)) ./ 1 >> else y := radf(denr u,n); if n=2 and minusf numr u then <<iflag := t; x := radf(negf numr u,n)>> else x := radf(numr u,n); z := simp list('quotient,retimes cdr x, retimes cdr y); if domainp numr z and domainp denr z then z := multsq(mkrootsq(prepf numr z,n), invsq mkrootsq(prepf denr z,n)) else <<if iflag then <<iflag := nil; z := negsq z>>; z := mkrootsq(prepsq z,n)>>; z := multsq(multsq(if !*precise and evenp n then car x ./ 1 else car x ./ 1, 1 ./ car y), z); if iflag then z := multsq(z,mkrootsq(-1,2)); return z end; symbolic procedure radfa(u,n); begin scalar x,y; x := fctrf u; if numberp car x then x := append(zfactor car x,cdr x) else x := (car x ./ 1) . cdr x; y := 1 ./ 1; for each j in x do y := multsq(y,radfb(car j,cdr j,n)); return y end; symbolic procedure radfb(u,m,n); begin scalar x,y; x := radf(u,n); y := exptf(car x,m) ./ 1; return multsq(exptsq(mkrootlsq(cdr x,n),m),y) end; % 20 Feb 01. symbolic procedure reval2(u,v); if v or null !*combineexpt or dmode!* then !*q2a1(simp!* u,v) else !*q2a1((simp!* u where !*mcd = nil),v); % 1 Jun 01. symbolic procedure simpexpt2(u,n,flg); begin scalar m,n,x,y; if u=1 then return 1 ./ 1; m:=numr n; if pairp u then << if car u eq 'expt then <<n:=multsq(m:=simp caddr u,n); if !*precise then u := list('abs,cadr u) else u := cadr u; return simpexpt1(u,n,flg)>> else if car u eq 'sqrt and not !*keepsqrts then return simpexpt2(cadr u, multsq(1 ./ 2,n),flg) else if car u eq 'times and not !*precise then <<x := 1 ./ 1; for each z in cdr u do x := multsq(simpexpt1(z,n,flg),x); return x>> else if car u eq 'times and (y:=split!-sign cdr u) and car y then <<x := simpexpt1(retimes append(cadr y,cddr y),n,flg); for each z in car y do x := multsq(simpexpt1(z,n,flg),x); return x>> else if car u eq 'quotient and (not !*precise or posnump caddr u and posnump prepsq n ) then <<if not flg and !*mcd then return simpexpt1(prepsq simp!* u,n,t); n := prepsq n; return quotsq(simpexpt{cadr u,n},simpexpt{caddr u,n})>> else if car u eq 'minus and not !*precise and not(cadr u = 1) then return (multsq(simpexpt list(-1,expon), simpexpt list(cadr u,expon))) where expon=prepsq n>>; if null flg then <<if null(dmode!* and idp u and get(u,dmode!*)) then u := prepsq simp!* u; return simpexpt1(u,n,t)>> else if numberp u and zerop u then return nil ./ 1 else if not numberp m then m := prepf m; n := prepf denr n; if m memq frlis!* and n=1 then return list ((u . m) . 1) . 1; if !*mcd or not numberp m or n neq 1 or atom u or denr simp!* u neq 1 then return simpx1(u,m,n) else return mksq(u,m) end; endpatch; % Algint declarations. fluid '(!*noacn !*structure !*tra !*trmin gaussiani intvar sqrtflag); fluid '(!*pvar listofallsqrts listofnewsqrts); global '(modevalcount); patch algint; % 20 Sep 00. symbolic procedure algebraiccase(expression,zlist,varlist); begin scalar rischpart,deriv,w,firstterm; scalar sqrtflag,!*structure; sqrtflag:=t; sqrtsave(listofallsqrts,listofnewsqrts,list(intvar . intvar)); rischpart:= errorset!*(list('doalggeom,mkquote expression), !*backtrace); newplace list (intvar.intvar); if atom rischpart then << if !*tra then prin2t "Inner integration failed"; deriv:=nil ./ 1; rischpart:=deriv >> else if atom car rischpart then << if !*tra or !*trmin then prin2t "The 'logarithmic part' is not elementary"; return (nil ./ 1) . expression >> else << rischpart:=car rischpart; deriv:=!*diffsq(rischpart,intvar) where sqrtflag=nil; if !*tra or !*trmin then << prin2t "Inner working yields"; printsq rischpart; prin2t "with derivative"; printsq deriv >> >>; deriv:=!*addsq(expression,negsq deriv); if null numr deriv then return rischpart . (nil ./ 1); if null involvesq(deriv,intvar) then return !*addsq(rischpart, !*multsq(deriv,((mksp(intvar,1) .* 1) .+ nil) ./ 1)) . (nil ./ 1); varlist:=getvariables deriv; zlist:=findzvars(varlist,list intvar,intvar,nil); varlist:=setdiff(varlist,zlist); firstterm:=simp!* car zlist; w:=sqrt2top !*multsq(deriv,invsq !*diffsq(firstterm,intvar)); if null involvesq(w,intvar) then return !*addsq(rischpart,!*multsq(w,firstterm)) . (nil ./ 1); if !*noacn then interr "Testing only logarithmic code"; deriv:=transcendentalcase(deriv,intvar,nil,zlist,varlist); return !*addsq(car deriv, rischpart) . cdr deriv end; % 22 Jan 01, 9 Feb 01. symbolic procedure combine!-logs(coef,logarg); begin scalar ans,dencoef,parts,logs,lparts,!*rationalize,trueimag; !*rationalize:=t; coef:=simp!* coef; if null numr logarg then return coef; parts:=split!-real!-imag numr coef; if null numr cdr parts then return multsq(coef,logarg); dencoef:=multf(denr coef,denr logarg); if !*tra then << prin2t "attempting to find 'real' form for"; mathprint list('times,list('plus,prepsq car parts, list('times,prepsq cdr parts,'i)), prepsq logarg) >>; logarg:=numr logarg; logs:= 1 ./ 1; while pairp logarg do << if ldeg logarg neq 1 then interr "what a log"; if atom mvar logarg then interr "what a log"; if car mvar logarg neq 'log then interr "what a log"; logs:=!*multsq(logs, !*exptsq(simp!* cadr mvar logarg,lc logarg)); logarg:=red logarg >>; logs:=rationalizesq logs; ans:=multsq(!*multsq(car parts,logs),1 ./ dencoef); lparts:=split!-real!-imag numr logs; if numr difff(denr cdr lparts,intvar) then interr "unexpected denominator"; lparts:=!*multsq(denr cdr lparts ./ 1,car lparts) . cdr lparts; if not onep denr car lparts then interr "unexpected denominator"; trueimag:=quotsq(addf(!*exptf(numr car lparts,2), !*exptf(numr cdr lparts,2)) ./ 1, !*exptf(denr logs,2) ./ 1); if numr diffsq(trueimag,intvar) then ans:=!*addsq(ans, !*multsq(gaussiani ./ multf(2,dencoef), !*multsq(simplogsq trueimag,cdr parts))); trueimag:=!*multsq(car lparts,!*invsq(numr cdr lparts ./ 1)); if numr diffsq(trueimag,intvar) then ans:=!*addsq(ans,!*multsq(!*multsq(cdr parts,1 ./ dencoef), !*k2q list('atan,prepsq!* trueimag))); return ans; end; % 6 Mar 01. symbolic procedure modevalvar v; begin scalar w; if atom v then <<if (w := get(v,'modvalue)) then return w; put(v,'modvalue,modevalcount); modevalcount := modevalcount+1; return modevalcount-1>> else if car v neq 'sqrt then <<if !*tra then <<princ "Unexpected algebraic:"; print v>>; error1()>> else if numberp cadr v then return (mksp(v,1) .* 1) .+ nil; w := modeval(!*q2f simp cadr v,!*pvar); w := assoc(w,listofallsqrts); if w then return cdr w else return 'failed end; endpatch; % Excalc declarations. global '(basisforml!* detm!* indxl!* metricd!* metricu!*); smacro procedure ldpf u; caar u; smacro procedure lowerind u; list('minus,u); patch excalc; % 16 Nov 99. symbolic procedure mkmetric u; begin scalar x,y,z,okord; putform(list(cadr u,nil,nil),0); put(cadr u,'indxsymmetries, '((lambda (indl) (tot!-sym!-indp (evlis '((nth indl 1) (nth indl 2))))))); put(cadr u,'indxsymmetrize, '((lambda (indl) (symmetrize!-inds '(1 2) indl)))); flag(list cadr u,'covariant); okord := kord!*; kord!* := basisforml!*; x := simp!* caddr u; y := indxl!*; metricu!* := t; for each j in indxl!* do <<for each k in y do setk(list(cadr u,lowerind j,lowerind k),0); y := cdr y>>; for each j on partitsq(x,'basep) do if ldeg ldpf j = 2 then setk(list(cadr u,lowerind cadr mvar ldpf j, lowerind cadr mvar ldpf j), mk!*sq lc j) else setk(list(cadr u,lowerind cadr mvar ldpf j, lowerind cadr mvar lc ldpf j), mk!*sq multsq(lc j,1 ./ 2)); kord!* := okord; x := for each j in indxl!* collect for each k in indxl!* collect simpindexvar list(cadr u,lowerind j,lowerind k); z := subfg!*; subfg!* := nil; y := lnrsolve(x,generateident length indxl!*); subfg!* := z; metricd!* := mkasmetric x; metricu!* := mkasmetric y; detm!* := mk!*sq detq x end; endpatch; % Ezgcd declarations. fluid '(image!-set reduced!-degree!-lclst unlucky!-case); symbolic smacro procedure polyzerop u; null u; patch ezgcd; % 8 Nov 99. symbolic procedure ezgcdf(u,v); begin scalar kordx,x; kordx := kord!*; x := errorset2{'ezgcdf1,mkquote u,mkquote v}; if null errorp x then return first x; setkorder kordx; return gcdf1(u,v) end; symbolic procedure poly!-gcd(u,v); begin scalar !*exp,z; if polyzerop u then return poly!-abs v else if polyzerop v then return poly!-abs u else if u=1 or v=1 then return 1; !*exp := t; if quotf1(u,v) then z := v else if quotf1(v,u) then z := u else if !*gcd then z := gcdlist list(u,v) else z := 1; return poly!-abs z end; symbolic procedure gcdlist3(l,onestep,vlist); begin scalar unlucky!-case,image!-set,gg,gcont,l1,w,w1,w2, reduced!-degree!-lclst,p1,p2; l1:=for each p in l collect p . ezgcd!-comfac p; l:=for each c in l1 collect quotfail1(car c,comfac!-to!-poly cdr c, "Content divison in GCDLIST3 failed"); gcont:=gcdlist for each c in l1 collect cddr c; if domainp gcont then if not(gcont=1) then errorf "GCONT has numeric part"; l := sort(for each p in l collect poly!-abs p,function ordp); w := nil; while l do << w := car l . w; repeat l := cdr l until null l or not(car w = car l)>>; l := reversip w; w := nil; if null cdr l then return multf(gcont,car l); if domainp (gg:=car (l:=sort(l,function degree!-order))) then return gcont; if ldeg gg=1 then << if division!-test(gg,l) then return multf(poly!-abs gg,gcont) else return gcont >>; if onestep then << p1 := poly!-abs car l; p2 := poly!-abs cadr l; if p1=p2 then << if division!-test(p1,cddr l) then return multf(p1,gcont) >> else << gg := poly!-gcd(lc p1,lc p2); w1 := multf(red p1, quotfail1(lc p2, gg, "Division failure when just one pseudoremainder step needed")); w2 := multf(red p2,negf quotfail1(lc p1, gg, "Division failure when just one pseudoremainder step needed")); w := ldeg p1 - ldeg p2; if w > 0 then w2 := multf(w2, (mksp(mvar p2, w) .* 1) .+ nil) else if w < 0 then w1 := multf(w1, (mksp(mvar p1, -w) .* 1) .+ nil); gg := ezgcd!-pp addf(w1, w2); if division!-test(gg,l) then return multf(gg,gcont) >>>>; return gcdlist31(l,vlist,gcont,gg,l1) end; endpatch; % Int declarations. fluid '(!*failhard !*purerisch !*reverse !*trdint sqrt!-places!-alist badpart ccount cmap cmatrix content cval denbad denominator!* lhs!* loglist orderofelim rhs!* tanlist !*intflag!* indexlist intvar listofnewsqrts listofallsqrts lorder power!-list!* sqrtflag sqrtlist sqrt!-intvar basic!-listofnewsqrts basic!-listofallsqrts sqfr sillieslist varlist); global '(!*number!* !*seplogs !*spsize!* !*statistics gensymcount); patch int; % 18 Dec 99. symbolic procedure findtrialdivs zl; begin scalar dlists1,args1; for each z in zl do if exportan z then <<if car z eq 'tan then <<args1 := (mksp(z,2) .* 1) .+ 1; tanlist := (args1 ./ 1) . tanlist>> else args1 := !*kk2f z; dlists1 := (z . args1) . dlists1>>; return dlists1 end; % 12 Dec 00. symbolic procedure simpdint u; begin scalar low,upp,fn,var,x,y; if length u neq 4 then rerror(int,2,"Improper number of arguments to INT"); load!-package 'defint; fn := car u; var := cadr u; low := caddr u; upp := cadddr u; low := reval low; upp := reval upp; if low = upp then return nil ./ 1 else if null getd 'new_defint then nil else if upp = 'infinity then if low = 0 then if not smemql('(infinity unknown), x := defint!* {fn,var}) then return simp!* x else nil else if low = '(minus infinity) then return mkinfint(fn,var) else if freeof(var,low) then if not smemql('(infinity unknown), x := defint!* {fn,var}) and not smemql('(infinity unknown), y := indefint!* {fn,var,low}) then return simp!* {'difference,x,y} else nil else nil else if upp = '(minus infinity) or low = 'infinity then return negsq simpdint {fn,var,upp,low} else if low = '(minus infinity) then return simpdint{prepsq simp{'sub,{'equal,var,{'minus,var}},fn}, var,{'minus,upp},'infinity} else if low = 0 then if freeof(var,upp) and not smemql('(infinity unknown), x := indefint!* {fn,var,upp}) then return simp!* x else nil else if freeof(var,upp) and freeof(var,low) and not smemq('(infinity unknown), x := indefint!* {fn,var,upp}) and not smemql('(infinity unknown), y := indefint!* {fn,var,low}) then return simp!* {'difference,x,y}; return mkdint(fn,var,low,upp) end; % 1 Jun 01. symbolic procedure simpint u; if atom u or null cdr u or cddr u and (null cdddr u or cddddr u) then rerror(int,1,"Improper number of arguments to INT") else if cddr u then simpdint u else begin scalar ans,dmod,expression,variable,loglist,oldvarstack, !*intflag!*,!*purerisch,cflag,intvar,listofnewsqrts, listofallsqrts,sqrtfn,sqrt!-intvar,sqrt!-places!-alist, basic!-listofallsqrts,basic!-listofnewsqrts,coefft, varchange,w,!*precise; !*intflag!* := t; variable := !*a2k cadr u; if not(idp variable or pairp variable and numlistp cdr variable) then <<varchange := variable . intern gensym(); if !*trint then prin2t {"Integration kernel", variable, "replaced by simple variable", cdr varchange}; variable := cdr varchange>>; intvar := variable; w := cddr u; if w then rerror(int,3,"Too many arguments to INT"); listofnewsqrts:= list mvar gaussiani; listofallsqrts:= list (cadr mvar gaussiani . gaussiani); sqrtfn := get('sqrt,'simpfn); put('sqrt,'simpfn,'proper!-simpsqrt); if dmode!* then << if (cflag:=get(dmode!*, 'cmpxfn)) then onoff('complex, nil); if (dmod := get(dmode!*,'dname)) then onoff(dmod,nil)>> where !*msg := nil; begin scalar dmode!*,!*exp,!*gcd,!*keepsqrts,!*limitedfactors,!*mcd, !*rationalize,!*structure,!*uncached,kord!*, ans1,denexp,badbit,nexp,oneterm; !*keepsqrts := !*limitedfactors := t; !*exp := !*gcd := !*mcd := !*structure := !*uncached := t; dmode!* := nil; if !*algint then << sqrt!-intvar:=!*q2f simpsqrti variable; if (red sqrt!-intvar) or (lc sqrt!-intvar neq 1) or (ldeg sqrt!-intvar neq 1) then interr "Sqrt(x) not properly formed" else sqrt!-intvar:=mvar sqrt!-intvar; basic!-listofallsqrts:=listofallsqrts; basic!-listofnewsqrts:=listofnewsqrts; sqrtsave(basic!-listofallsqrts,basic!-listofnewsqrts, list(variable . variable))>>; coefft := (1 ./ 1); expression := int!-simp car u; if varchange then <<depend1(car varchange,cdr varchange,t); expression := int!-subsq(expression,{varchange})>>; denexp := 1 ./ denr expression; nexp := numr expression; while not atom nexp and null cdr nexp and not depends(mvar nexp,variable) do <<coefft := multsq(coefft,(((caar nexp) . 1) . nil) ./ 1); nexp := lc nexp>>; ans1 := nil; while nexp do begin scalar x,zv,tmp; if atom nexp then <<x := !*f2q nexp; nexp := nil>> else <<x := !*t2q car nexp; nexp := cdr nexp>>; x := multsq(x,denexp); zv := findzvars(getvariables x,list variable,variable,nil); begin scalar oldzlist; while oldzlist neq zv do << oldzlist := zv; foreach zz in oldzlist do zv:=findzvars(distexp(pseudodiff(zz,variable)), zv,variable,t)>>; % The following line was added to make, for example, % int(df(sin(x)/x),x) return the expected result. zv := sort(zv, function ordp) end; tmp := ans1; while tmp do <<if zv=caar tmp then <<rplacd(car tmp,addsq(cdar tmp,x)); tmp := nil; zv := nil>> else tmp := cdr tmp>>; if zv then ans1 := (zv . x) . ans1 end; if length ans1 = 1 then oneterm := t; nexp := ans1; ans := nil ./ 1; badbit:=nil ./ 1; while nexp do <<u := cdar nexp; if !*trdint then <<princ "Integrate"; printsq u; princ "with Zvars "; print caar nexp>>; ans1 := errorset!*(list('integratesq,mkquote u, mkquote variable,mkquote loglist, mkquote caar nexp), !*backtrace); nexp := cdr nexp; if errorp ans1 then badbit := addsq(badbit,u) else <<ans := addsq(caar ans1, ans); badbit:=addsq(cdar ans1,badbit)>>>>; if !*trdint then <<prin2 "Partial answer="; printsq ans; prin2 "To do="; printsq badbit>>; if badbit neq '(nil . 1) then <<setkorder nil; badbit := reordsq badbit; ans := reordsq ans; coefft := reordsq coefft; if !*trdint then <<princ "Retrying..."; printsq badbit>>; if oneterm and ans = '(nil . 1) then ans1 := nil else ans1 := errorset!*(list('integratesq,mkquote badbit, mkquote variable,mkquote loglist,nil), !*backtrace); if null ans1 or errorp ans1 then ans := addsq(ans,simpint1(badbit . variable . w)) else <<ans := addsq(ans,caar ans1); if not smemq(variable, ans) then ans := nil ./ 1; if cdar ans1 neq '(nil . 1) then ans := addsq(ans, simpint1(cdar ans1 . variable . w)) >>>>; end; ans := multsq(coefft,ans); if !*trdint then << prin2t "Resimp and all that"; printsq ans >>; put('int,'simpfn,'simpiden); put('sqrt,'simpfn,sqrtfn); << if dmod then onoff(dmod,t); if cflag then onoff('complex,t)>> where !*msg := nil; oldvarstack := varstack!*; varstack!* := nil; ans := errorset!*(list('int!-resub,mkquote ans,mkquote varchange),t); put('int,'simpfn,'simpint); varstack!* := oldvarstack; return if errorp ans then error1() else car ans end; symbolic procedure mkdint(fn,var,low,upp); begin scalar x,!*precise; if getd 'defint0 and not((x := defint0 {fn,var,low,upp}) eq 'failed) then return simp x else if not smemq('infinity,low) and not smemq('infinity,upp) then <<x := prepsq!* simpint {fn,var}; if not eqcar(x,'int) then return simp!* {'difference, subeval {{'equal,var,upp},x}, subeval {{'equal,var,low},x}}>>; return mksq({'int,fn,var,low,upp},1) end; % 9 Aug 01. symbolic procedure transcendentalcase(integrand,svar,xlogs,zlist,varlist); begin scalar divlist,jhd!-content,content,prim,sqfr,dfu,indexlist, sillieslist,originalorder,wrongway,power!-list!*, sqrtlist,tanlist,loglist,dflogs,eprim,dfun,unintegrand, sqrtflag,badpart,rhs!*,lhs!*,gcdq,cmap,cval,orderofelim,cmatrix; scalar ccount,denominator!*,result,denbad,temp; gensymcount:=0; integrand:=sqrt2top integrand; if !*trint then << printc "Extension variables z<i> are"; print zlist>>; begin scalar w,gg; gg:=1; foreach z in zlist do <<w := subs2 diffsq(simp z,svar); gg := !*multf(gg,quotf(denr w,gcdf(denr w,gg)))>>; gg := quotf(gg,gcdf(gg,denr integrand)); unintegrand := (!*multf(gg,numr integrand) ./ !*multf(gg,denr integrand)); if !*trint then << printc "After unnormalization the integrand is "; printsq unintegrand >> end; divlist := findtrialdivs zlist; sqrtlist := findsqrts zlist; divlist := trialdiv(denr unintegrand,divlist); prim := sqfree(cdr divlist,zlist); jhd!-content := content; printfactors(prim,nil); eprim := sqmerge(countz car divlist,prim,nil); printfactors(eprim,t); sqfr := for each u in eprim collect multup u; if !*reverse then zlist := reverse zlist; indexlist := createindices zlist; dfu:=dfnumr(svar,car divlist); loglist := append(loglist,factorlistlist prim); loglist := mergein(xlogs,loglist); loglist := mergein(tanlist,loglist); cmap := createcmap(); ccount := length cmap; if !*trint then <<printc "Loglist "; print loglist>>; dflogs := difflogs(loglist,denr unintegrand,svar); if !*trint then <<printc "************ 'Derivative' of logs is:"; printsq dflogs>>; dflogs := addsq((numr unintegrand) ./ 1,negsq dflogs); gcdq := gcdf(denr dflogs,denr dfu); dfun := !*multf(numr dfu,denbad:=quotf(denr dflogs,gcdq)); denbad := !*multf(denr dfu,denbad); denbad := !*multf(denr unintegrand,denbad); dflogs := !*multf(numr dflogs,quotf(denr dfu,gcdq)); dfu := dfun; rhs!* := multbyarbpowers f2df dfu; if checkdffail(rhs!*,svar) then <<if !*trint then printsq checkdffail(rhs!*,svar); interr "Simplification fails on above expression">>; if !*trint then << printc "Distributed Form of Numerator is:"; printdf rhs!*>>; lhs!* := f2df dflogs; if !*trint then << printc "Distributed Form of integrand is:"; printdf lhs!*; terpri()>>; cval := mkvect(ccount); for i := 0:ccount do putv(cval,i,nil ./ 1); power!-list!* := tansfrom(rhs!*,zlist,indexlist,0); lorder:=maxorder(power!-list!*,zlist,0); originalorder := for each x in lorder collect x; if !*trint then << printc "Maximum order for variables determined as "; print lorder >>; if !*statistics then << !*number!*:=0; !*spsize!*:=1; foreach xx in lorder do !*spsize!*:=!*spsize!* * (xx+1) >>; dfun:=solve!-for!-u(rhs!*,lhs!*,nil); backsubst4cs(nil,orderofelim,cmatrix); if !*statistics then << prin2 !*number!*; prin2 " used out of "; printc !*spsize!* >>; badpart:=substinulist badpart; dfun:=df2q substinulist dfun; result:= !*multsq(dfun,!*invsq(denominator!* ./ 1)); result:= !*multsq(result,!*invsq(jhd!-content ./ 1)); dflogs:=logstosq(); if not null numr dflogs then << if !*seplogs and (not domainp numr result) then << result:=mk!*sq result; result:=(mksp(result,1) .* 1) .+ nil; result:=result ./ 1 >>; result:=addsq(result,dflogs)>>; if !*trint then << terpri(); printc "*****************************************************"; printc "************ THE INTEGRAL IS : **********************"; printc "*****************************************************"; terpri(); printsq result; terpri()>>; if badpart then begin scalar n,oorder; if !*trint then printc "plus a part which has not been integrated"; lhs!*:=badpart; lorder:=maxorder(power!-list!*,zlist,0); oorder:=originalorder; n:=length lorder; while lorder do << if car lorder > car originalorder then wrongway:=t; if car lorder=car originalorder then n:= n-1; lorder:=cdr lorder; originalorder:=cdr originalorder >>; if !*trint and wrongway then printc "Went wrong way"; dfun:=df2q badpart; dfun:= !*multsq(dfun,invsq(denbad ./ 1)); badpart := dfun; if wrongway then << if !*trint then printc "Resetting...."; result:=nil ./ 1; dfun := integrand; badpart:=dfun >>; if rootcheckp(unintegrand,svar) then return simpint1(integrand . svar.nil) . (nil ./ 1) else if !*purerisch or allowedfns zlist then << badpart := dfun; dfun := nil ./ 1 >> else << !*purerisch:=t; if !*trint then <<printc " Applying transformations ..."; printsq dfun>>; temp := get('tan,'opmtch); remprop('tan,'opmtch); denbad:=transform(dfun,svar); if denbad=dfun then <<dfun:=nil ./ 1; badpart:=denbad; put('tan,'opmtch,temp)>> else <<denbad:=errorset!*(list('integratesq,mkquote denbad, mkquote svar,mkquote xlogs, nil), !*backtrace); put('tan,'opmtch,temp); if not atom denbad then << denbad:=car denbad; dfun:=untan car denbad; if (dfun neq '(nil . 1)) then badpart:=untan cdr denbad; if car badpart and not(badpart=denbad) then << wrongway:=nil; lhs!*:=f2df car badpart; lorder:=maxorder(power!-list!*,zlist,0); n:=length lorder; while lorder do << if car lorder > car oorder then wrongway:=t; if car lorder=car oorder then n:= n-1; lorder:=cdr lorder; oorder:=cdr oorder >>; if wrongway or (n=0) then << if !*trint then printc "Still backwards"; dfun := nil ./ 1; badpart := integrand>>>>>> else <<badpart := dfun; dfun := nil ./ 1 >>>>>>; if !*failhard then rerror(int,9,"FAILHARD switch set"); if !*seplogs and not domainp result then << result:=mk!*sq result; if not numberp result then result:=(mksp(result,1) .* 1) .+ nil; result:=result ./ 1>>; result:=addsq(result,dfun) end else badpart:=nil ./ 1; return (sqrt2top result . badpart) end; endpatch; unfluid '(indexlist); patch limits; % 1 Jun 01. symbolic procedure simplimit u; begin scalar fn,exprn,var,val,old,v,!*precise,!*protfg; if length u neq 4 then rerror(limit,1, "Improper number of arguments to limit operator"); fn:= car u; exprn := cadr u; var := !*a2k caddr u; val := cadddr u; !*protfg := t; old := get('cot,'opmtch); put('cot,'opmtch, '(((!~x) (nil . t) (quotient (cos !~x) (sin !~x)) nil))); v := errorset!*({'apply,mkquote fn,mkquote {exprn,var,val}},nil); put('cot,'opmtch,old); !*protfg := nil; return if errorp v or (v := car v) = aeval 'failed then mksq(u,1) else simp!* v end; endpatch; % Matrix declarations. fluid '(!*bezout); patch matrix; % 7 Aug 99, 23 Apr 01. symbolic procedure quotfexf!*1(u,v); if null u then nil else (if x then x else (if denr y = 1 then numr y else if denr (y := (rationalizesq y where !*rationalize = t))=1 then numr y else rerror(matrix,11, "Catastrophic division failure")) where y=rationalizesq(u ./ v)) where x=quotf(u,v); % 23 Jan 01. symbolic procedure lnrsolve(u,v); begin scalar temp,vlhs,vrhs,ok, !*exp,!*solvesingular; if !*ncmp then return clnrsolve(u,v); !*exp := t; if asymplis!* or wtl!* then <<temp := asymplis!* . wtl!*; asymplis!* := wtl!* := nil>>; vlhs := for i:=1:length car u collect intern gensym(); vrhs := for i:=1:length car v collect intern gensym(); u := car normmat augment(u,v); v := append(vlhs,vrhs); ok := setkorder v; u := foreach r in u collect prsum(v,r); v := errorset!*({function solvebareiss, mkquote u,mkquote vlhs},t); if caar v memq {'singular,'inconsistent} then <<setkorder ok; rerror(matrix,13,"Singular matrix")>>; v := pair(cadr s,car s) where s = cadar v; u := foreach j in vlhs collect coeffrow(negf numr q,vrhs,denr q) where q = cdr atsoc(j,v); setkorder ok; if temp then <<asymplis!* := car temp; wtl!* := cdr temp>>; return for each j in u collect for each k in j collect if temp then resimp k else cancel k; end; % 23 Apr 01. symbolic procedure extmult(u,v); if null u or null v then nil else (if x then cdr x .* (if car x then negf c!:subs2multf(lc u,lc v) else c!:subs2multf(lc u,lc v)) .+ extadd(extmult(!*t2f lt u,red v), extmult(red u,v)) else extadd(extmult(red u,v),extmult(!*t2f lt u,red v))) where x = ordexn(car lpow u,lpow v); % 14 Jan 01, 7 Aug 01. symbolic procedure resultant(u,v,var); if domainp u and domainp v then 1 else begin scalar x; kord!* := var . kord!*; if null domainp u and null(mvar u eq var) then u := reorder u; if null domainp v and null(mvar v eq var) then v := reorder v; x := if !*bezout then bezout_resultant(u,v,var) else polyresultantf(u,v,var); setkorder cdr kord!*; return x end; symbolic procedure resultantsq(u,v,var); if domainp numr u and domainp numr v and denr u = 1 and denr v = 1 then 1 ./ 1 else begin scalar x; kord!* := var . kord!*; if null domainp numr u and null(mvar numr u eq var) then u := reordsq u; if null domainp numr v and null(mvar numr v eq var) then v := reordsq v; x := if !*bezout then bezout_resultant(!*q2f u,!*q2f v,var) else polyresultantf(!*q2f u,!*q2f v,var); setkorder cdr kord!*; return !*f2q x end; symbolic procedure polyresultantf(u,v,var); begin scalar beta,cd,cn,delta,gam,r,s,temp,x; cd := cn := r := s := 1; gam := -1; if domainp u or domainp v then return 1 else if ldeg u<ldeg v then <<s := (-1)^(ldeg u*ldeg v); temp := u; u := v; v := temp>>; while v do <<delta := ldeg u-ldegr(v,var); beta := negf(multf(r,exptf(gam,delta))); r := lcr(v,var); gam := multf(exptf(negf r,delta),exptf(gam,1-delta)); temp := u; u := v; if not evenp ldeg temp and not evenp ldegr(u,var) then s := -s; v := quotf(pseudo_remf(temp,v,var),beta); if v then <<cn := multf(cn,exptf(beta,ldeg u)); cd := multf(cd, exptf(r,(1+delta)*ldeg u-ldeg temp+ldegr(v,var))); if (x := quotf(cd,cn)) then <<cn := 1; cd := x>>>>>>; return if not domainp u and mvar u eq var then nil else if ldeg temp neq 1 then quotf(multf(s,multf(cn,exptf(u,ldeg temp))),cd) else u end; symbolic procedure lcr(u,var); if domainp u or mvar u neq var then u else lc u; symbolic procedure ldegr(u,var); if domainp u or mvar u neq var then 0 else ldeg u; symbolic procedure pseudo_remf(u,v,var); !*q2f simp pseudo!-remainder {mk!*sq(u ./ 1),mk!*sq(v ./ 1),var}; symbolic procedure bezout_resultant(u,v,w); begin integer n,nm; scalar ap,ep,uh,ut,vh,vt; if domainp u or null(mvar u eq w) then return if not domainp v and mvar v eq w then exptf(u,ldeg v) else 1 else if domainp v or null(mvar v eq w) then return if mvar u eq w then exptf(v,ldeg u) else 1; n := ldeg v - ldeg u; if n < 0 then return multd((-1)**(ldeg u*ldeg v), bezout_resultant(v,u,w)); ep := 1; nm := ldeg v; uh := lc u; vh := lc v; ut := if n neq 0 then multpf(w to n,red u) else red u; vt := red v; ap := addf(multf(uh,vt),negf multf(vh,ut)); ep := b!:extmult(!*sf2exb(ap,w),ep); for j := (nm - 1) step -1 until (n + 1) do <<if degr(ut,w) = j then <<uh := addf(lc ut,multf(!*k2f w,uh)); ut := red ut>> else uh := multf(!*k2f w,uh); if degr(vt,w) = j then <<vh := addf(lc vt,multf(!*k2f w,vh)); vt := red vt>> else vh := multf(!*k2f w,vh); ep := b!:extmult(!*sf2exb(addf(multf(uh,vt), negf multf(vh,ut)),w),ep)>>; if n neq 0 then <<ep := b!:extmult(!*sf2exb(u,w),ep); for j := 1:(n-1) do ep := b!:extmult(!*sf2exb(multpf(w to j,u),w),ep)>>; return if null ep then nil else lc ep end; endpatch; % Ncpoly declarations. fluid '(!*complex !*trnc dipvars!*); patch ncpoly; % 9 Jan 01. symbolic procedure nc_factsolve(s,vl,all); begin scalar v,sb,ns,so,soa,sol,nz,w,q,z,r,abort; v:= numr simp car vl; ns:=for each e in s collect numr simp e; r:=t; while r do <<r:=nil; s:=ns; ns:=nil; for each e in s do if not abort then <<e:=absf numr subf(e,sb); while(q:=quotf(e,v)) do e:=q; if null e then nil else if domainp e or not(mvar e member vl) then abort:=t else if null red e and domainp lc e then <<w:=mvar e; sb:=(w . 0).sb; r:=t; vl:=delete(w,vl)>> else if not member(e,ns) then ns:=e.ns >>; >>; if abort or null vl then return nil; nc_factorize_timecheck(); if null ns and vl then <<sol:={for each x in vl collect x.1}; goto done>>; s:=for each e in ns collect prepf e; if !*trnc then <<prin2 "solving "; prin2 length s; prin2 " polynomial equations for "; prin2 length vl; prin2t "variables"; for each e in s do writepri(mkquote e,'only);>>; w:=(cdr solveeval{'list.s,'list.vl} where dipvars!*=nil); loop: nc_factorize_timecheck(); if null w then goto done; so:= cdr car w; w:=cdr w; soa:=nil; if smemq('i,so) and null !*complex then go to loop; for each y in vl do if not smember(y,so) then <<soa:=(y . 1) . soa; nz:=t>>; for each y in so do <<z:=nc_factorize_unwrap(reval caddr y,soa); nz:=nz or z neq 0; soa:=(cadr y . z).soa; >>; if not nz then goto loop; q:=assoc(car vl,soa); if null q or cdr q=0 then go to loop; soa := for each j in soa collect (car j . sublis(soa,cdr j)); sol := soa . sol; if all then go to loop; done: sol:=for each s in sol collect append(sb,s); if !*trnc then <<prin2t "solutions:"; for each w in sol do writepri(mkquote('list. for each s in w collect {'equal,car s,cdr s}),'only); prin2t "-------------------------"; >>; return sol end; endpatch; % Plot declarations. global '(!*plotpause !*plotusepipe dirchar!* opsys!* plotcleanup!* plotcmds!* plotcommand!* plotdir!* plotdta!* plotheader!* tempdir!*); patch plot; % 28 Jun 99. symbolic procedure init_gnuplot(); << !*plotpause := -1; plotcleanup!* := {}; tempdir!* := getenv 'tmp; if null tempdir!* then tempdir!* := getenv 'temp; dirchar!* := "/"; plotcommand!* := "gnuplot"; opsys!* := assoc('opsys, lispsystem!*); if null opsys!* then opsys!* := 'unknown else opsys!* := cdr opsys!*; if getenv "gnuplot" then plotdir!* := getenv "gnuplot" else if null plotdir!* and not (opsys!* = 'unix) then plotdir!* := get!-lisp!-directory(); if opsys!* = 'win32 then << plotcommand!* := "wgnuplot"; plotheader!* := ""; dirchar!* := "\"; plotdta!* := for each n in {"gnutmp.tm1", "gnutmp.tm2", "gnutmp.tm3", "gnutmp.tm4", "gnutmp.tm5", "gnutmp.tm6", "gnutmp.tm7", "gnutmp.tm8"} collect gtmpnam n; plotcleanup!* := if null tempdir!* then {"erase gnutmp.tm*"} else {bldmsg("erase %w\gnutmp.tm*", tempdir!*)} >> else if opsys!* = 'msdos then << plotheader!* := ""; % ?? "set term vga"; dirchar!* := "\"; plotdta!* := for each n in {"gnutmp.tm1", "gnutmp.tm2", "gnutmp.tm3", "gnutmp.tm4", "gnutmp.tm5", "gnutmp.tm6", "gnutmp.tm7", "gnutmp.tm8"} collect gtmpnam n; plotcmds!*:= gtmpnam "gnutmp.tm0"; plotcleanup!* := if null tempdir!* then {"erase gnutmp.tm*"} else {bldmsg("erase %w\gnutmp.tm*", tempdir!*)} >> else if opsys!* = 'riscos then << plotheader!* := ""; dirchar!* := "."; plotdta!* := for i:=1:10 collect tmpnam(); plotcmds!*:= tmpnam(); plotcleanup!* := bldmsg("remove %w", plotcmds!*) . for each f in plotdta!* collect bldmsg("remove %w", f) >> else if opsys!* = 'unix then << plotheader!* := "set term x11"; plotdta!* := for i:=1:10 collect tmpnam(); plotcmds!*:= tmpnam(); plotcleanup!* := bldmsg("rm %w", plotcmds!*) . for each f in plotdta!* collect bldmsg("rm %w", f) >> else if opsys!* = 'finder then << plotcommand!* := "gnuplot"; plotcmds!*:= "::::gnuplot:reduce.plt"; plotheader!* := ""; dirchar!* := ":"; plotdta!* := for each n in {"::::gnuplot:gnutmp.tm1", "::::gnuplot:gnutmp.tm2", "::::gnuplot:gnutmp.tm3", "::::gnuplot:gnutmp.tm4", "::::gnuplot:gnutmp.tm5", "::::gnuplot:gnutmp.tm6", "::::gnuplot:gnutmp.tm7", "::::gnuplot:gnutmp.tm8"} collect gtmpnam n; plotcleanup!* := nil >> else << rederr bldmsg("gnuplot for %w not available yet", opsys!*); plotdta!* := for i:=1:10 collect tmpnam(); plotcmds!*:= tmpnam(); plotheader!* := "set term dumb" >>; if 'pipes member lispsystem!* then !*plotusepipe:=t else plotcommand!* := bldmsg("%w %w", plotcommand!*, plotcmds!*); if plotdir!* then plotcommand!* := bldmsg("%w%w%w", plotdir!*, dirchar!*, plotcommand!*); nil >>; endpatch; patch poly; % 7 Aug 99. symbolic procedure rationalizesq u; begin scalar !*structure,!*sub2,v,x; if x := get(dmode!*,'rationalizefn) then u := apply1(x,u); powlis!* := '(i 2 (nil . t) -1 nil) . powlis!*; v := subs2q u; powlis!* := cdr powlis!*; return if domainp denr v then v else if (x := rationalizef denr v) neq 1 then <<v := multf(numr v,x) ./ multf(denr v,x); if null !*algint and null !*rationalize then v := gcdchk v; subs2q v>> else u end; % 6 Feb 00, 7 Sep 01. symbolic procedure sqfrf u; begin integer n; scalar !*gcd,units,v,w,x,y,z,!*msg,r; !*gcd := t; if (r := !*rounded) then <<on rational; u := numr resimp !*f2q u>>; n := 1; x := mvar u; v := gcdf(u,diff(u,x)); u := quotf(u,v); if flagp(dmode!*,'field) and ((y := lnc u) neq 1) then <<u := multd(!:recip y,u); v := multd(y,v)>>; while degr(v,x)>0 do <<w := gcdf(v,u); if u neq w then z := (quotf(u,w) . n) . z; v := quotf(v,w); u := w; n := n + 1>>; if r then <<on rounded; u := numr resimp !*f2q u; z := for each j in z collect numr resimp !*f2q car j . cdr j>>; if v neq 1 and assoc(v,units) then v := 1; if v neq 1 then if n=1 then u := multf(v,u) else if (w := rassoc(1,z)) then rplaca(w,multf(v,car w)) else if null z and ((w := rootxf(v,n)) neq 'failed) then u := multf(w,u) else if not domainp v then z := aconc(z,v . 1) else errach {"sqfrf failure",u,n,z}; return (u . n) . z end; % 2 Aug 00. symbolic procedure sqfrp u; begin scalar !*ezgcd, dmode!*; if null getd 'ezgcdf1 then load_package ezgcd; !*ezgcd := t; return domainp gcdf!*(u,diff(u,mvar u)) end; % 13 Dec 00. symbolic procedure gcdk(u,v); begin scalar lclst,var,w,x; if u=v then return u else if domainp u or degr(v,(var := mvar u))=0 then return 1 else if ldeg u<ldeg v then <<w := u; u := v; v := w>>; if quotf1(u,v) then return v else if !*heugcd and (x := heu!-gcd(u,v)) then return x else if ldeg v=1 or getd 'modular!-multicheck and modular!-multicheck(u,v,var) or not !*mcd then return 1; a: w := remk(u,v); if null w then return v else if degr(w,var)=0 then return 1; lclst := addlc(v,lclst); if x := quotf1(w,lc w) then w := x else for each y in lclst do if atom y and not flagp(dmode!*,'field) or not (domainp y and (flagp(dmode!*,'field) or ((x := get(car y,'units)) and y member (for each z in x collect car z)))) then while (x := quotf1(w,y)) do w := x; u := v; v := prim!-part w; if degr(v,var)=0 then return 1 else go to a end; % 19 Jan 01. symbolic procedure quarticf pol; begin scalar !*sub2,a,a2,a0,b,dsc,p,p1,p2,q,shift,var; var := mvar pol; p := shift!-pol pol; a := coeffs car p; shift := caddr p; if cadr a then rerror(poly,16,list(pol,"not correctly shifted")) else if cadddr a then return list(1,pol); a2 := cddr a; a0 := caddr a2; a2 := car a2; a := car a; q := quadraticf1(a,a2,a0); if not(q eq 'failed) then <<a2 := car q; q := cdr q; a := exptsq(addsq(!*k2q mvar pol,shift),2); b := numr subs2q quotsq(addsq(multsq(!*f2q car q,a), !*f2q cadr q), !*f2q cadr p); a := numr subs2q quotsq(addsq(multsq(!*f2q caddr q,a), !*f2q cadddr q), !*f2q cadr p); a := quadraticf!*(a,var); b := quadraticf!*(b,var); return multf(a2,multf(car a,car b)) . nconc!*(cdr a,cdr b)>> else if null !*surds or denr shift neq 1 then return list(1,pol); shift := numr shift; if knowndiscrimsign eq 'negative then go to complex; dsc := powsubsf addf(exptf(a2,2),multd(-4,multf(a,a0))); p2 := minusf a0; if not p2 and minusf dsc then go to complex; p1 := not a2 or minusf a2; if not p1 then if p2 then p1 := t else p2 := t; p1 := if p1 then 'positive else 'negative; p2 := if p2 then 'negative else 'positive; a := rootxf(a,2); if a eq 'failed then return list(1,pol); dsc := rootxf(dsc,2); if dsc eq 'failed then return list(1,pol); p := invsq !*f2q addf(a,a); q := multsq(!*f2q addf(a2,negf dsc),p); p := multsq(!*f2q addf(a2,dsc),p); b := multf(a,exptf(addf(!*k2f mvar pol,shift),2)); a := powsubsf addf(b,q); b := powsubsf addf(b,p); knowndiscrimsign := p1; a := quadraticf!*(a,var); knowndiscrimsign := p2; b := quadraticf!*(b,var); knowndiscrimsign := nil; return multf(car a,car b) . nconc!*(cdr a,cdr b); complex: a := rootxf(a,2); if a eq 'failed then return list(1,pol); a0 := rootxf(a0,2); if a0 eq 'failed then return list(1,pol); a2 := powsubsf addf(multf(2,multf(a,a0)),negf a2); a2 := rootxf(a2,2); if a2 eq 'failed then return list(1,pol); p := addf(!*k2f mvar pol,shift); q := addf(multf(a,exptf(p,2)),a0); p := multf(a2,p); a := powsubsf addf(q,p); b := powsubsf addf(q,negf p); knowndiscrimsign := 'negative; a := quadraticf!*(a,var); b := quadraticf!*(b,var); knowndiscrimsign := nil; return multf(car a,car b) . nconc!*(cdr a,cdr b) end; % 7 Apr 01. symbolic procedure factorize u; (begin scalar x,y; x := simp!* u; y := denr x; if not domainp y then typerr(u,"polynomial"); u := numr x; if u = 1 then return {'list, if !*nopowers then 1 else {'list,1,1}} else if fixp u then !*ifactor := t; if !*force!-prime and not primep !*force!-prime then typerr(!*force!-prime,"prime"); u := if dmode!* and not(dmode!* memq '(!:rd!: !:cr!:)) then if get(dmode!*,'factorfn) then begin scalar !*factor; !*factor := t; return fctrf u end else rerror(poly,14, list("Factorization not supported over domain", get(dmode!*,'dname))) else fctrf u; return facform2list(u,y) end) where !*ifactor = !*ifactor; % 7 Apr 01, 24 Aug 01. symbolic procedure factor!-prim!-sqfree!-f u; begin scalar x,y,!*msg,r; r := !*rounded; if r and univariatep numr u and lc numr u=1 and denr u=1 then return unifactor u else if r or !*complex or !*rational then <<if r then on rational; u := numr resimp !*f2q car u . cdr u>>; if null !*limitedfactors then <<if null dmode!* then y := 'factorf else <<x := get(dmode!*,'sqfrfactorfn); y := get(dmode!*,'factorfn); if x and not(x eq y) then y := 'factorf>>; if y then <<y := apply1(y,car u); u := (exptf(car y,cdr u) . for each j in cdr y collect(car j . cdr u)); go to ret>>>>; u := factor!-prim!-sqfree!-f!-1(car u,cdr u); ret: if r then <<on rounded; u := car u . for each j in cdr u collect (numr resimp !*f2q car j . cdr j)>>; return u end; % 7 Apr 01. symbolic procedure unifactor u; if not eqcar(u := root_val list mk!*sq u,'list) then errach {"unifactor1",u} else 1 . for each j in cdr u collect if not eqcar(j,'equal) then errach{"unifactor2",u} else addsq(!*k2q cadr j,negsq simp caddr j); % 11 Jun 01. symbolic procedure noncomp u; !*ncmp and noncomp1 u; symbolic procedure noncomp1 u; if null pairp u then nil else if pairp car u then noncomfp u else flagp(car u,'noncom) or noncomlistp cdr u; symbolic procedure noncomlistp u; pairp u and (noncomp1 car u or noncomlistp cdr u); endpatch; % Rlisp declarations. fluid '(newrules!*); patch rlisp; % 9 Nov 99. symbolic procedure load!-package u; begin scalar x,y; if stringp u then return load!-package intern compress explode2 u else if null idp u then rederr list(u,"is not a package name") else if memq(u,loaded!-packages!*) then return u else if or(atom(x:= errorset(list('evload,list('quote,list u)), nil,!*backtrace)), cdr x) then rederr list("error in loading package",u,"or package not found"); loaded!-packages!* := u . loaded!-packages!*; x := get(u,'package); if x then x := cdr x; a: if null x then go to b else if null atom get(car x,'package) then load!-package car x else if or(atom(y := errorset(list('evload, list('quote,list car x)), nil,!*backtrace)), cdr y) then rederr list("module",car x,"of package",u, "cannot be loaded"); x := cdr x; go to a; b: if (x := get(u,'patchfn)) then begin scalar !*usermode,!*redefmsg; eval list x end end; % 22 April 00. symbolic procedure begin11 x; begin scalar mode,result,newrule!*; if cursym!* eq 'end then if terminalp() and null !*lisp!_hook then progn(cursym!* := '!*semicol!*, !*nosave!* := t, return nil) else progn(comm1 'end, return 'end) else if eqcar((if !*reduce4 then x else cadr x),'retry) then if programl!* then x := programl!* else progn(lprim "No previous expression",return nil); if null !*reduce4 then progn(mode := car x,x := cadr x); program!* := x; if eofcheck() then return 'c else eof!* := 0; add2inputbuf(x,if !*reduce4 then nil else mode); if null atom x and car x memq '(bye quit) then if getd 'bye then progn(lispeval x, !*nosave!* := t, return nil) else progn(!*byeflag!* := t, return nil) else if null !*reduce4 and eqcar(x,'ed) then progn((if getd 'cedit and terminalp() then cedit cdr x else lprim "ED not supported"), !*nosave!* := t, return nil) else if !*defn then if erfg!* then return nil else if null flagp(key!*,'ignore) and null eqcar(x,'quote) then progn((if x then dfprint x else nil), if null flagp(key!*,'eval) then return nil); if !*output and ifl!* and !*echo and null !*lessspace then terpri(); result := errorset!*(x,t); if errorp result or erfg!* then progn(programl!* := list(mode,x),return 'err2) else if !*defn then return nil; if null !*reduce4 then if null(mode eq 'symbolic) then x := getsetvars x else nil else progn(result := car result, (if null result then result := mkobject(nil,'noval)), mode := type result, result := value result); add2resultbuf((if null !*reduce4 then car result else result), mode); if null !*output then return nil else if null(semic!* eq '!$) then if !*reduce4 then (begin terpri(); if mode eq 'noval then return nil else if !*debug then prin2t "Value:"; rapply1('print,list list(mode,result)) end) else if mode eq 'symbolic then if null car result and null(!*mode eq 'symbolic) then nil else begin terpri(); result:= errorset!*(list('print,mkquote car result),t) end else if car result then result := errorset!*(list('assgnpri,mkquote car result, (if x then 'list . x else nil), mkquote 'only), t); if null !*reduce4 then return if errorp result then 'err3 else nil else if null(!*mode eq 'noval) then progn(terpri(), prin2 "of type: ", print mode); return nil end; endpatch; % Roots declarations. % fluid '(rootacc!#!# rootacc!#!# !*noeqns); % patch roots; % 10 Feb 00. % Commented out since now solved another way (7 Apr 01). % symbolic procedure root_val x; % roots x % where rootacc!#!#=p, iniprec!#=p where p=precision 0, !*msg=nil, % !*noeqns=t; % endpatch; % Scope declarations. global '(kvarlst prevlst varlst!*); patch scope; % 29 Aug 00. symbolic procedure maxtype type; if atom type then type else if pairp cdr type then cadr type else car type; % 8 Nov 00. symbolic procedure prepmultmat(preprefixlist); begin scalar tlcm,var,varexp,kvl,kfound,pvl,pfound,tel,ratval,ratlst, newvarlst,hvarlst; hvarlst:= nil; while not null (varlst!*) do <<var := car varlst!*; varlst!* := cdr varlst!*; if flagp(var,'ratexp) then <<tlcm:=1; remflag(list var,'ratexp); foreach elem in get(var,'varlst!*) do if pairp cdr elem then tlcm := lcm2(tlcm,cddr elem); varexp:=fnewsym(); tel:=(varexp.(if tlcm = 2 then list('sqrt,var) else list('expt,var, if onep cdr(tel:=simpquot list(1,tlcm)) then car tel else list('quotient,car tel,cdr tel)))); if assoc(var,kvarlst) then <<kvl:=kfound:=nil; while kvarlst and not(kfound) do if caar(kvarlst) eq var then << kvl:=tel.kvl; kfound:=t; pvl:=pfound:=nil; prevlst:=reverse(prevlst); while prevlst and not(pfound) do if cdar(prevlst) eq var then << pvl:=cons(caar prevlst,varexp).pvl; pfound:=t >> else << pvl:=car(prevlst).pvl; prevlst:=cdr(prevlst) >>; if pvl then if prevlst then prevlst:=append(reverse prevlst,pvl) else prevlst:=pvl >> else << kvl:=car(kvarlst).kvl; kvarlst:=cdr kvarlst>>; if kvl then if kvarlst then kvarlst:=append(reverse kvl,kvarlst) else kvarlst:=reverse kvl >> else preprefixlist:=tel.preprefixlist; ratlst:=newvarlst:=nil; foreach elem in get(var,'varlst!*) do if pairp cdr elem then << ratval:=divide((tlcm * cadr elem)/(cddr elem),tlcm); ratlst:=cons(car elem,cdr ratval).ratlst; if car(ratval)>0 then newvarlst:=cons(car elem,car ratval).newvarlst >> else newvarlst:=elem.newvarlst; if ratlst then << put(varexp,'varlst!*,reverse ratlst); hvarlst:=varexp.hvarlst >>; if newvarlst then << put(var,'varlst!*,reverse newvarlst); hvarlst:=var.hvarlst >> else remprop(var,'varlst!*) >> else hvarlst:=var.hvarlst >>; varlst!* := hvarlst; return preprefixlist end; endpatch; % Solve declarations. fluid '(!*multiplicities vars!*); global '(multiplicities!*); patch solve; % 9 Jan 01. symbolic procedure !*solvelist2solveeqlist u; begin scalar x,y,z; u := for each j in u collect solveorder j; for each j in u do <<if caddr j=0 then rerror(solve,2,"zero multiplicity") else if null cadr j then x := for each k in car j collect list('equal,!*q2a k,0) else x := for each k in pair(cadr j,car j) collect list('equal,car k,!*q2a cdr k); if length vars!* > 1 then x := 'list . x else x := car x; z := (caddr j . x) . z>>; z := sort(z,function ordp); x := nil; if !*multiplicities then <<for each k in z do for i := 1:car k do x := cdr k . x; multiplicities!* := nil>> else <<for each k in z do << x := cdr k . x; y := car k . y>>; multiplicities!* := 'list . reversip y>>; return 'list . reversip x end; % 9 Jan 01, 15 Jun 01. symbolic procedure solveorder u; begin scalar v,w,x,y,z; v := vars!*; x := cadr u; if length x<length v then v := setdiff(v,setdiff(v,x)); if null x or x=v then return u; y := car u; while x do <<z := (car x . car y) . z; x := cdr x; y := cdr y>>; w := v; a: if null w then return reversip x . v . cddr u else if null(y := depassoc(car w,z)) then return u else x := cdr y . x; w := cdr w; go to a end; symbolic procedure depassoc(u,v); if null v then nil else if u = caar v then car v else if depends(caar v,u) then nil else depassoc(u,cdr v); % 2 Feb 01. symbolic procedure check!-solns(z,ex,var); begin scalar x; if errorp (x := errorset2 {'check!-solns1,mkquote z,mkquote ex,mkquote var}) then return check!-solns1(z,(numr simp!* prepf ex where !*reduced=t),var) else return car x end; symbolic procedure check!-solns1(z,ex,var); begin scalar x,y,fv,sx,vs; fv := freevarl(ex,var); for each z1 in z do fv := union(fv,union(freevarl(numr caar z1,var), freevarl(denr caar z1,var))); fv := delete('i,fv); if fv then for each v in fv do if not flagp(v,'constant) then vs := (v . list('quotient,1+random 999,1000)) . vs; sx := if vs then numr subf(ex,vs) else ex; while z do if null cadar z then <<z := nil; x := 'unsolved>> else if <<y := numr subf(ex,list(caadar z . mk!*sq caaar z)); null y or fv and null(y := numr subf(sx,list(caadar z . mk!*sq subsq(caaar z,vs)))) or null numvalue y>> then <<x := car z . x; z := cdr z>> else z := cdr z; return if null x then 'unsolved else x end; % 7 Apr 01. symbolic procedure solvequadratic(a2,a1,a0); if !*rounded and numcoef a0 and numcoef a1 and numcoef a2 then for each z in cdr root_val list mkpolyexp2(a2,a1,a0) collect simp!* (if eqcar(z,'equal) then caddr z else errach {"Quadratic confusion",z}) else begin scalar d; d := sqrtq subtrsq(quotsqf(exptsq(a1,2),4),multsq(a2,a0)); a1 := quotsqf(negsq a1,2); return list(subs2!* quotsq(addsq(a1,d),a2), subs2!* quotsq(subtrsq(a1,d),a2)) end; endpatch; % Sum declarations. fluid '(sum_last_attempt_rules!* !*zeilberg); patch sum; % 28 Jul 00. symbolic procedure freeof!-df(u, v); if atom u then t else if car(u) eq 'df then freeof!-df(cadr u, v) and not smember(v,cddr u) else freeof!-dfl(cdr u, v); symbolic procedure freeof!-dfl(u, v); if null u then t else freeof!-df(car u,v) and freeof!-dfl(cdr u,v); symbolic procedure simp!-sum u; begin scalar y; y := cdr u; u := car u; if not atom y and not freeof!-df(u, car y) then if atom y then return !*p2f(car fkern(list('sum,u)) .* 1) ./ 1 else return sum!-df(u, y); u := simp!* u; return if null numr u then u else if atom y then !*p2f(car fkern(list('sum,prepsq u)) .* 1) ./ 1 else if !*zeilberg then gosper!*(mk!*sq u,y) else simp!-sum0(u,y) end; symbolic procedure sum!-subst(u,x,a); if u = x then a else if atom u then u else sum!-subst(car u, x,a) . sum!-subst(cdr u,x,a); symbolic procedure sum!-df(u,y); begin scalar w,z,upper,lower,dif; if length(y) = 3 then << lower := cadr y; upper := caddr y; dif := addsq(simp!* upper, negsq simp!* lower); if denr dif = 1 then if null numr dif then return simp!* sum!-subst(u, car y, upper) else if fixp numr dif then dif := numr dif else dif := nil else dif := nil; if dif and dif <= 0 then return nil ./ 1 >>; if null dif then << z := 'sum . (u . y); let sum_last_attempt_rules!*; w:= opmtch z; rule!-list (list sum_last_attempt_rules!*,nil); return if w then simp w else mksq(z,1)>>; z := nil ./ 1; a: if dif < 0 then return z; z := addsq(z,simp!* sum!-subst(u, car y, list('plus,lower,dif))); dif := dif - 1; go to a end; % 20 Nov 00. symbolic procedure termlst(u,v,klst); begin scalar x,kern,lst; if null u then return nil else if null klst or domainp u then return list multsq(v,!*f2q u); kern := car klst; klst := cdr klst; x := setkorder list kern; u := reorder u; v := reorder(numr v) ./ reorder(denr v); while not domainp u and mvar u eq kern do << lst := nconc(termlst(lc u, multsq(!*p2q lpow u, v),klst),lst); u := red u>>; if u then lst := nconc(termlst(u,v,klst),lst); setkorder x; return lst end; endpatch; % Taylor declarations. fluid '(!*taylorautocombine); patch taylor; % 1 Jun 01. symbolic procedure simptaylor u; if remainder(length u,3) neq 1 then Taylor!-error('wrong!-no!-args,'taylor) else if null subfg!* then mksq('taylor . u,1) else begin scalar !*precise,arglist,degree,f,ll,result,var,var0; if !*taylorautocombine and not ('taysimpsq memq mul!*) then mul!* := aconc!*(mul!*,'taysimpsq); f := simp!* car u; u := revlis cdr u; arglist := u; while not null arglist do << var := car arglist; var := if eqcar(var,'list) then cdr var else {var}; for each el in var collect begin el := simp!* el; if kernp el then return mvar numr el else typerr(prepsq el,'kernel) end; var0 := cadr arglist; degree := caddr arglist; if not fixp degree then typerr(degree,"order of Taylor expansion"); arglist := cdddr arglist; ll := {var,var0,degree,degree + 1} . ll>>; result := taylorexpand(f,reversip ll); return if smember('Taylor!*,result) then result else mksq('taylor . prepsq f . u,1) end; endpatch; endmodule; end;