module solve; % Solve one or more algebraic equations. % Author: David R. Stoutemyer. % Major modifications by: Anthony C. Hearn, Herbert Melenk and % Donald R. Morrison. create!-package('(solve ppsoln glsolve solvealg solvetab quartic),nil); % Other packages needed by solve package. load!-package 'matrix; fluid '(!*allbranch !*exp !*ezgcd !*fullroots !*limitedfactors !*multiplicities !*nonlnr !*notseparate !*numval !*numval!* !*rounded !*solvealgp !*solvesingular !!gcd !:prec!: asymplis!* dmode!* kord!* vars!* !*!*norootvarrenamep!*!*); % NB: !*!*norootvarrenamep!*!* is internal to this module, and should % *never* be changed by a user. fluid '(inside!-solveeval solve!-gensymcounter); solve!-gensymcounter := 1; global '(!!arbint multiplicities!*); switch allbranch,fullroots,multiplicities,nonlnr,solvesingular; put('fullroots,'simpfg,'((t (rmsubs)))); !*nonlnr := t; % Put it on for now. flag('(!*allbranch multiplicities!*),'share); % ***** Some Non-local variables ***** !*allbranch := t; % Returns all branches of solutions if T. % !*multiplicities Lists all roots with multiplicities if on. % !*fullroots := t; % Computes full roots of cubics and quartics. !*solvesingular := t; % Default value. % !!gcd SOLVECOEFF returns GCD of powers of its arg in % this. With the decompose code, this should % only occur with expressions of form x^n + c. algebraic operator one_of; put('arbint,'simpfn,'simpiden); % algebraic operator arbreal; symbolic operator expand_cases; symbolic procedure simp!-arbcomplex u; simpiden('arbcomplex . u) where dmode!*=nil; deflist('((arbcomplex simp!-arbcomplex)),'simpfn); % ***** Utility Functions ***** symbolic procedure freeofl(u,v); null v or freeof(u,car v) and freeofl(u,cdr v); symbolic procedure allkern elst; % Returns list of all top-level kernels in the list of standard % quotients elst. Corrected 5 Feb 92 by Francis Wright. if null elst then nil else union(kernels numr car elst, allkern cdr elst); symbolic procedure topkern(u,x); % Returns list of top level kernels in the standard form u that % contain the kernel x; for each j in kernels u conc if not freeof(j,x) then list j else nil; symbolic procedure coeflis ex; % Ex is a standard form. Returns a list of the coefficients of the % main variable in ex in the form ((expon . coeff) (expon . coeff) % ... ), where the expon's occur in increasing order, and entries do % not occur of zero coefficients. We need to reorder coefficients % since kernel order can change in the calling function. begin scalar ans,var; if domainp ex then return (0 . ex); var := mvar ex; while not domainp ex and mvar ex=var do <>; if ex then ans := (0 . reorder ex) . ans; return ans end; % ***** Evaluation Interface ***** put('solve,'psopfn,'solveeval); symbolic procedure solveeval u; begin scalar !*ezgcd,!!gcd,vars!*; integer nargs; if atom u then rerror(solve,1,"SOLVE called with no equations") else if null dmode!* then !*ezgcd := t; if not inside!-solveeval then solve!-gensymcounter := 1; nargs := length u; u := (if nargs=1 then solve0(car u,nil) else if nargs=2 then solve0(car u, cadr u) else <>) where inside!-solveeval = t; return !*solvelist2solveeqlist u end; symbolic procedure solve!-gensym(); begin scalar w; w := explode solve!-gensymcounter; solve!-gensymcounter := solve!-gensymcounter+1; while length w < 4 do w := '!0 . w; % If users have things to solve with names like G0001 in them, there % could be confusion. return compress ('g . w) end; symbolic procedure !*solvelist2solveeqlist u; begin scalar x,y,z; for each j in u do < 1 then x := 'list . x else x := car x; z := (caddr j . x) . z>>; z := sort(z,function ordp); x := nil; if !*multiplicities then <> else <>; multiplicities!* := 'list . reversip y>>; % Now check for redundant solutions. % if length vars!*>1 then z := check_solve_redundancy z; return 'list . reversip x end; % symbolic procedure check_solve_redundancy u; % % We assume all solutions are prefixed by LIST. % begin scalar x,y; % x := for each j in u collect cdr j; % Remove the LIST. % for each j in u do if not supersetlist(cdr j,x) then y:= j . y; % return reversip!* y % end; symbolic procedure supersetlist(u,v); % Returns true if u is a non-equal superset of any element of v. v and (u neq car v and null setdiff(car v,u) or supersetlist(u,cdr v)); % ***** Fundamental SOLVE Procedures ***** Comment most of these procedures return a list of "solve solutions". A solve solution is a list with three fields: the list of solutions, the corresponding variables (or NIL if the equations could not be solved --- in which case there is only one solution in the first field) and the multiplicity; symbolic procedure solve0(elst,xlst); % This is the driving function for the solve package. % Elst is any prefix expression, including a list prefixed by LIST. % Xlst is a kernel or list of kernels. Solves eqns in elst for % vars in xlst, returning either a list of solutions, or a single % solution. begin scalar !*exp,!*notseparate,w; integer neqn; !*exp := !*notseparate := t; % Form a list of equations as expressions. elst := for each j in solveargchk elst collect simp!* !*eqn2a j; neqn := length elst; % There must be at least one. % Determine variables. if null xlst then <> else <>; terpri!* nil>> else <>; if length vars!* = 0 then rerror(solve,3,"SOLVE called with no variables") else if neqn = 1 then if null numr car elst then return if !*solvesingular then list list(for each j in vars!* collect !*f2q makearbcomplex(), vars!*,1) else nil else if length vars!*=1 then if solutionp(w := solvesq(car elst,car vars!*,1)) or null !*solvealgp or univariatep numr car elst then return w; % More than one equation or variable, or single eqn has no solution. elst := for each j in elst collect numr j; w := solvesys(elst,vars!*); if car w eq 't or car w eq 'inconsistent then return cdr w else if car w eq 'failed or null car w then return for each j in elst collect list(list(j ./ 1),nil,1) else errach list("Improper solve solution tag",car w) end; symbolic procedure solutionp u; null u or cadar u and not root_of_soln_p caar u; symbolic procedure root_of_soln_p u; null cdr u and kernp (u := car u) and eqcar(mvar numr u,'root_of); symbolic procedure solveargchk u; if getrtype (u := reval u) eq 'list then cdr reval u else if atom u or not(car u eq 'lst) then list u else cdr u; % ***** Procedures for solving a single eqn ***** symbolic procedure solvesq (ex,var,mul); % Attempts to find solutions for standard quotient ex with respect to % top level occurrences of var and kernels containing variable var. % Solutions containing more than one such kernel are returned % unsolved, and solve1 is applied to the other solutions. Integer % mul is the multiplicity passed from any previous factorizations. % Returns a list of triplets consisting of solutions, variables and % multiplicity. begin scalar e1,oldkorder,x1,y,z; integer mu; ex := numr ex; if null(x1 := topkern(ex,var)) then return nil; oldkorder := setkorder list var; % The following section should be extended for other cases. e1 := reorder ex; setkorder oldkorder; if mvar e1 = var and null cdr x1 and ldeg e1 =1 then return {{{quotsq(negf reorder red e1 ./1, reorder lc e1 ./ 1)}, {var},mul}}; % don't call fctrf here in rounded mode, so polynomial won't get % rounded (since factoring isn't going to succeed) ex := if !*rounded then {1,ex . 1} else fctrf ex; % Now process monomial. if domainp car ex then ex := cdr ex else ex := (car ex . 1) . cdr ex; for each j in ex do <1 and (y := solvedecomp(reverse y,car x1,mu)) then z := append(y,z) else if (degr(y := reorder e1,var) where kord!*={var}) = 1 and not smemq(var,delete(var,x1)) then <> else if x1 then z := append( if null cdr x1 then solve1(e1,car x1,var,mu) % Inhibit use of principle-of-powers-soln, since solvealg % code is more powerful. % else if (y := principle!-of!-powers!-soln(e1,x1,var,mu)) % neq 'unsolved % then y else if not smemq('sol, (x1:=simp!* list('sol,mk!*sq(e1 ./ 1), var))) then solvesq(x1,var,mu) % else list list(list(e1 ./ 1),nil,mu), else mkrootsof(e1 ./ 1,var,mu), z)>>; return z end; symbolic procedure solvedecomp(u,var,mu); % Solve for decomposed expression. At the moment, only one % level of decomposition is considered. begin scalar failed,x; if length(x := solve0(car u,cadadr u))=1 then return nil; u := cdr u; while u do <> else solve0(list('difference,prepsq caar j,caddar u), if cdr u then cadadr u else var); if failed then u := nil else u := cdr u>>; return if failed then nil else adjustmul(x,mu) end; symbolic procedure adjustmul(u,n); % Multiply the multiplicities of the solutions in u by n. if n=1 then u else for each x in u collect list(car x,cadr x,n*caddr x); symbolic procedure solve1(e1,x1,var,mu); Comment e1 is a standard form, non-trivial in the kernel x1, which is itself a function of var, mu is an integer. Uses roots of unity, known solutions, inverses, together with quadratic, cubic and quartic formulas, treating other cases as unsolvable. Returns a list of solve solutions; begin scalar !*numval!*; !*numval!* := !*numval; % Keep value for use in solve11. return solve11(e1,x1,var,mu) end; symbolic procedure solve11(e1,x1,var,mu); begin scalar !*numval,b,coefs,hipow; integer n; % The next test should check for true degree in var. if null !*fullroots and null !*rounded and numrdeg(e1,var)>2 then return mkrootsof(e1 ./ 1,var,mu); !*numval := t; % Assume that actual numerical values wanted. coefs:= errorset!*(list('solvecoeff,mkquote e1,mkquote x1),nil); % if atom coefs then return list list(list(e1 . 1),nil,mu); if atom coefs or atom x1 and x1 neq var then return mkrootsof(e1 ./ 1,var,mu); % solvecoeff problem - no soln. coefs := car coefs; n:= !!gcd; % numerical gcd of powers. hipow := caar reverse coefs; if hipow = 1 then return begin scalar lincoeff,y,z; if null cdr coefs then b := 0 else b := prepsq quotsq(negsq cdar coefs,cdadr coefs); if n neq 1 then b := list('expt,b,list('quotient,1,n)); % We may need to merge more solutions in the following if % there are repeated roots. for k := 0:n-1 do % equation in power of var. <>; return y end else if hipow=2 then return <> else return solvehipow(e1,x1,var,mu,coefs,hipow) end; symbolic procedure getcoeff(u,n); % Get the nth coefficient in the list u as a standard quotient. if null u then nil ./ 1 else if n=caar u then cdar u else if n> else <>; for each j in solvesq(c,'!!x,mu) conc solvesq(addsq(1 ./ 1,multsq(d,subtrsq(d,caar j))), var,caddr j)>> else if solve1test2(coefs,rcoeffs,f) % coefficients antisymmetric then <>; append(solvesq(e1,var,mu),b)>> % equation has no symmetry % now look for real roots before cubics or quartics. We must % reverse the answer from solveroots so that roots come out % in same order from SOLVE. % else if !*numval!* and dmode!* memq '(!:rd!: !:cr!:) % this forces solveroots independent of numval. else if !*rounded and univariatep e1 then reversip solveroots(e1,var,mu) else if null !*fullroots then mkrootsof(e1 ./ 1,var,mu) else if hipow=3 then for each j in solvecubic(getcoeff(coefs,3), getcoeff(coefs,2), getcoeff(coefs,1), getcoeff(coefs,0)) conc solvesq(subtrsq(d,j),var,mu) else if hipow=4 then for each j in solvequartic(getcoeff(coefs,4), getcoeff(coefs,3), getcoeff(coefs,2), getcoeff(coefs,1), getcoeff(coefs,0)) conc solvesq(subtrsq(d,j),var,mu) % else list list(list(e1 ./ 1),nil,mu) else mkrootsof(e1 ./ 1,var,mu) % We can't solve quintic and higher. end; symbolic procedure solnmerge(u,varlist,mu,y); % Merge solutions in case of multiplicities. It may be that this is % only needed for the trivial solution x=0. if null y then list list(u,varlist,mu) else if u = caar y and varlist = cadar y then list(caar y,cadar y,mu+caddar y) . cdr y else car y . solnmerge(u,varlist,mu,cdr y); symbolic procedure nilchk u; if null u then !*f2q u else u; symbolic procedure solve1test1(coefs,rcoeffs,f); % True if equation is symmetric in its coefficients. f is midpoint. begin integer j,p; if null coefs or caar coefs neq 0 or null !*fullroots then return nil; p := caar coefs + caar rcoeffs; a: if j>f then return t else if (caar coefs + caar rcoeffs) neq p or cdar coefs neq cdar rcoeffs then return nil; coefs := cdr coefs; rcoeffs := cdr rcoeffs; j := j+1; go to a end; symbolic procedure solve1test2(coefs,rcoeffs,f); % True if equation is antisymmetric in its coefficients. f is % midpoint. begin integer j,p; if null coefs or caar coefs neq 0 or null !*fullroots then return nil; p := caar coefs + caar rcoeffs; a: if j>f then return t else if (caar coefs + caar rcoeffs) neq p or numr addsq(cdar coefs,cdar rcoeffs) then return nil; coefs := cdr coefs; rcoeffs := cdr rcoeffs; j := j+1; go to a end; symbolic procedure solveabs u; begin scalar mu,var,lincoeff; var := cadr u; mu := caddr u; lincoeff := cadddr u; u := simp!* caar u; return append(solvesq(addsq(u,lincoeff),var,mu), solvesq(subtrsq(u,lincoeff),var,mu)) end; put('abs,'solvefn,'solveabs); symbolic procedure solveexpt u; begin scalar c,mu,var,lincoeff; var := cadr u; mu := caddr u; lincoeff := cadddr u; if null numr lincoeff then return nil; u := car u; return if freeof(car u,var) % c**(...) = b. then <> else c:=0; solvesq(subtrsq(simp!* cadr u, quotsq(addsq(solveexpt!-logterm lincoeff, simp!* c), simp!* list('log,car u))),var,mu)>> else if freeof(cadr u,var) % (...)**(m/n) = b; then if ratnump cadr u then solve!-fractional!-power(u,lincoeff,var,mu) else << % (...)**c = b. if !*allbranch then <> % c := mkexp list('times, % list('arbreal,!!arbint))>> else c:=1; solvesq(subtrsq(simp!* car u, multsq(simp!* list('expt, mk!*sq lincoeff, mk!*sq invsq simp!* cadr u), simp!* c)),var,mu)>> % (...)**(...) = b : transcendental. % else list list(list subtrsq(simp!*('expt . u),lincoeff),nil,mu) else mkrootsof(subtrsq(simp!*('expt . u),lincoeff),var,mu) end; put('expt,'solvefn,'solveexpt); symbolic procedure solveexpt!-logterm lincoeff; % compute log(lincoeff), ignoring multiplicity and converting % log(-a) to log(a) + i pi. if not !*allbranch or not minusf numr lincoeff then simp!* list('log,mk!*sq lincoeff) else addsq(simp!*'(times i pi), simp!* {'log,mk!*sq(negf numr lincoeff ./ denr lincoeff)}); symbolic procedure solvelog u; solvesq(subtrsq(simp!* caar u,simp!* list('expt,'e,mk!*sq cadddr u)), cadr u,caddr u); put('log,'solvefn,'solvelog); symbolic procedure solveinvpat(u,op); begin scalar c,d,z,f; f:=get(op,'solveinvpat); if smemq('arbint,f) then f:=subst( if !*allbranch then list('arbint,!!arbint:=!!arbint+1) else 0, 'arbint,f); if not !*allbranch then f:={car f}; return for each c in reverse f join solvesq(simp!* subst(caar u,'(~v),subst(mk!*sq cadddr u,'(~r),c)), cadr u,caddr u) end; put('cos,'solveinvpat, { quote (- ~v + acos(~r) + 2*arbint*pi), quote (- ~v - acos(~r) + 2*arbint*pi) }); put('cos,'solvefn, '(lambda(u) (solveinvpat u 'cos))); put('sin,'solveinvpat, { quote (- ~v + asin(~r) + 2*arbint*pi), quote (- ~v - asin(~r) + 2*arbint*pi + pi) }); put('sin,'solvefn, '(lambda(u) (solveinvpat u 'sin))); put('sec,'solveinvpat, { quote (- ~v + asec(~r) + 2*arbint*pi), quote (- ~v - asec(~r) + 2*arbint*pi) }); put('sec,'solvefn, '(lambda(u) (solveinvpat u 'sec))); put('csc,'solveinvpat, { quote (- ~v + acsc(~r) + 2*arbint*pi), quote (- ~v - acsc(~r) + 2*arbint*pi + pi) }); put('csc,'solvefn, '(lambda(u) (solveinvpat u 'csc))); put('tan,'solveinvpat, { quote (- ~v + atan(~r) + arbint*pi)}); put('tan,'solvefn, '(lambda(u) (solveinvpat u 'tan))); put('cot,'solveinvpat, { quote (- ~v + acot(~r) + arbint*pi)}); put('cot,'solvefn, '(lambda(u) (solveinvpat u 'cot))); symbolic procedure mkexp u; list('plus,list('cos,x),list('times,'i,list('sin,x))) where x = reval u; symbolic procedure solvecoeff(ex,var); % Ex is a standard form and var a kernel. Returns a list of % dotted pairs of exponents and coefficients (as standard quotients) % of var in ex, lowest power first, with exponents divided by their % gcd. This gcd is stored in !!GCD. begin scalar clist,oldkord; oldkord := updkorder var; clist := reorder ex; setkorder oldkord; clist := coeflis clist; !!gcd := caar clist; for each x in cdr clist do !!gcd := gcdn(car x,!!gcd); for each x in clist do <>; return clist end; symbolic procedure solveroots(ex,var,mu); % Ex is a square and content free univariate standard form, var the % relevant variable and mu the root multiplicity. Finds insoluble, % complex roots of EX, returning a list of solve solutions. begin scalar y; y := reval list('root_val,mkquote mk!*sq(ex ./ 1)); if not(car y eq 'list) then errach list("incorrect root format",ex); return for each z in cdr y collect % if not(car z eq 'equal) or cadr z neq var % then errach list("incorrect root format",ex) % else list(list simp caddr z,list var,mu) list(list simp z,list var,mu) end; % ***** Procedures for solving a system of eqns ***** symbolic procedure solvesys(exlist,varlis); % Exlist is a list of standard forms, varlis a list of kernels. If % the elements of varlis are linear in the elements of exlist, and % further the system of linear eqns so defined is non-singular, then % SOLVESYS returns a list of a list of standard quotients which are % solutions of the system, ordered as in varlis. begin scalar eqtype,oldkord; oldkord := setkorder varlis; exlist := for each j in exlist collect reorder j; % See if equations are linear or non-linear. if errorp errorset!*(list('solvenonlnrchk,mkquote exlist, mkquote varlis),nil) then eqtype := 'solvenonlnrsys else eqtype := 'solvelnrsys; % Solve for appropriate equation type. if eqtype eq 'solvenonlnrsys and null !*nonlnr then <>; exlist:=errorset!*(list(eqtype,mkquote exlist,mkquote varlis),t); setkorder oldkord; if errorp exlist then error1() else return if eqtype eq 'solvelnrsys then t . car exlist else car exlist end; symbolic procedure solvenonlnrchk(exlist,varlis); % Returns error if equations are nonlinear. (Error used to prevent % unnecessary computation.) for each ex in exlist do for each var in varlis do solvenonlnrchk1(ex,var,varlis); symbolic procedure solvenonlnrchk1(ex,var,varlis); if domainp ex then nil else <1 or not freeofl(lc ex,varlis) then error1()) else if not freeofl(mvar ex,varlis) and not(mvar ex member varlis) then error1() else solvenonlnrchk1(lc ex,var,varlis); solvenonlnrchk1(red ex,var,varlis)>>; % ***** Support for one_of and root_of *****. symbolic procedure mkrootsof(e1,var,mu); begin scalar x; x := if idp var then var else 'q_; if not !*!*norootvarrenamep!*!* then while smember(x,e1) do x := intern compress append(explode x,explode '!_); e1 := prepsq!* e1; if x neq var then e1 := subst(x,var,e1); return list list(list !*k2q list('root_of,e1,x),list var,mu) end; put('root_of,'psopfn,'root_of_eval); symbolic procedure root_of_eval u; begin scalar !*!*norootvarrenamep!*!*,x; !*!*norootvarrenamep!*!* := t; x := solveeval u; if eqcar(x,'list) then x := cdr x else typerr(x,"list"); x := foreach j in x collect if eqcar(j,'equal) then caddr j else typerr(j,"equation"); if null x then rederr "solve confusion in root_of_eval" else if null cdr x then return car x else return('one_of . x) end; put('root_of,'subfunc,'subrootof); symbolic procedure subrootof(l,expn); % Sets up a formal SUB expression when necessary. begin scalar x,y; for each j in cddr expn do if (x := assoc(j,l)) then <>; expn := sublis(l,car expn) . for each j in cdr expn collect subsublis(l,j); %to ensure only opr and individual args are transformed; if null y then return expn; expn := aconc!*(for each j in reversip!* y collect list('equal,car j,aeval cdr j),expn); return if l then subeval expn else mk!*sq !*p2q mksp('sub . expn,1) end; symbolic procedure one_of_p u; % The use of reval is rather heavy handed, but it works. (if eqcar(x,'one_of) then x else nil) where x=reval u; symbolic procedure expand_cases u; makelist expandcase0 cdr u; symbolic procedure expandcase0 u; if null u then nil else if eqcar(car u,'list) then expandcase2 u else expandcase1 u; symbolic procedure expandcase1 u; % One variable case. if null u then nil else if not eqcar(car u,'equal) then typerr(car u,"equation") else (if x then append(for each j in cdr x collect list('equal,cadar u,j), y) else car u . y) where x=one_of_p caddar u,y=expandcase1 cdr u; symbolic procedure expandcase2 u; % Multiple variable case. if null u then nil else if not eqcar(car u,'list) then typerr(car u,"list") else append(for each j in expandcase22 cdar u collect makelist j, expandcase2 cdr u); symbolic procedure expandcase22 u; if null u then list nil else if not eqcar(car u,'equal) then typerr(car u,"equation") else (if x then for each j in cdr x join for each k in y collect list('equal,cadar u,j) . subst(j,cadar u,k) else for each k in y collect car u . k) where x=one_of_p caddar u,y=expandcase22 cdr u; endmodule; module ppsoln; % Solve surd eqns, mainly by principle of powers method. % Authors: Anthony C. Hearn and Stanley L. Kameny. fluid '(!*complex !*msg !*numval !*ppsoln); global '(bfone!*); !*ppsoln := t; % Keep this as internal switch. symbolic procedure solve!-fractional!-power(u,x,var,mu); % Attempts solution of equation car u**cadr u=x with respect to % kernel var and with multiplicity mu, where cadr u is a rational % number. begin scalar v,w,z; v := simp!* car u; w := simp!* cadr u; z := solvesq(subs2 subtrsq(exptsq(v,numr w),exptsq(x,denr w)), var,mu); w := subtrsq(simp('expt . u),x); z := check!-solns(z,numr w,var); % return if z eq 'unsolved then list list(list w,nil,mu) else z return if z eq 'unsolved then mkrootsof(w,var,mu) else z end; symbolic procedure principle!-of!-powers!-soln(ex,x1,var,mu); % Finds solutions of ex=0 by the principle of powers method. Return % 'unsolved if solutions can't be found. begin scalar z; a: if null x1 then return 'unsolved else if suitable!-expt car x1 and not((z := pr!-pow!-soln1(ex,car x1,var,mu)) eq 'unsolved) then return z; x1 := cdr x1; go to a end; symbolic procedure pr!-pow!-soln1(ex,y,var,mu); begin scalar oldkord,z; return 'unsolved; oldkord := updkorder y; z := reorder ex; setkorder oldkord; if ldeg z neq 1 then return 'unsolved; z := coeflis z; if length z neq 2 or caar z neq 0 then errach list("solve confused",ex,z); z := exptsq(quotsq(negsq(cdar z ./ 1),cdadr z ./ 1), caddr caddr y); z := solvesq(subs2 addsq(simp!* cadr y,negsq z),var,mu); z := check!-solns(z,ex,var); return z end; symbolic procedure check!-solns(z,ex,var); begin scalar x,y,fv,sx,vs; fv := delete('i,freevarl(ex,var)); % this does only one random setting!! if fv then for each v in fv do 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 <> where l=if var then list var else nil; symbolic procedure varsift(a,var); if atom a then if not(null a or numberp a or a eq var) then list a else nil else for each c in cdr a join varsift(c,var); symbolic procedure numvalue u; % Find floating point value of sf u. begin scalar !*numval,x,c,cp,p,m; m := !*msg; !*msg := nil; !*numval := t; c := ('i memq freevarl(u,nil)); if (cp := !*complex) then off complex; x := setdmode('rounded,t); p := precision 10; if x eq '!:rd!: then x := 'rounded; % <==== to avoid error later if c then on complex; !*msg := m; u := numr simp prepf u; !*msg := nil; if c then off complex; if x then setdmode(x,t) else setdmode('rounded,nil); if cp then on complex; precision p; !*msg := m; return if eqcar(u,'!:rd!:) and (numvchk(100,z) where z=round!* u) or eqcar(u,'!:cr!:) and (numvchk(10,z) where z=retag crrl u) and (numvchk(10,z) where z=retag crim u) then nil else u end; symbolic procedure numvchk(fact,z); if atom z then fact*abs z<1 else lessp!:(timbf(bfloat fact,abs!: z),bfone!*); endmodule; module glsolve; % Routines for solving a general system of linear eqns. % Author: Eberhard Schruefer. % Modifications by: D. Hartley and R.W. Tucker. %********************************************************************** %*** The number of equations and the number of unknowns are *** %*** arbitrary i.e. the system can be under- or overdetermined. *** %*** Method used is Cramer's rule, realized through exterior *** %*** multiplication. *** %********************************************************************** fluid '(!*solvesingular !*solveinconsistent vars!*); global '(!!arbint); global '(inconsistent!*); % Flag to show if equations were inconsistent. flag('(inconsistent!*),'share); switch solveinconsistent; % !*solveinconsistent := t; % Default value. symbolic procedure solvelnrsys(u,v); % This is hook to general solve package. u is a list of polynomials % (s.f.'s) linear in the kernels of list v. Result is a tagged % standard form for the solutions. % If the system is inconsistent, then the second element of % the list returned is NIL, so that !*solvelist2solveeqlist % gives expr = 0, rather than var = expr. list list(car w,cadr w and v,1) where w = glnrsolve(u,v); symbolic procedure glnrsolve(u,v); % glnrsolve(u: list of sf's, v: list of kernels) % -> (xprs: list of sq's, flg: boolean) % Adapted by D. Hartley and R.W. Tucker from E. Schruefer's glnrsolve. % The equations u must be ordered with respect to the kernels v % If flg is t then xprs is an ordered list of solutions for v. % If flg is nil then xprs is a list of consistency conditions. begin scalar arbvars,sgn,x,y,cnds; inconsistent!* := nil; while u and null x do <>; >>; if null u then % no consistent non-zero equations if cnds then < 1 then vars!* := t . vars!*; return list(for each j in cnds collect !*f2q j,nil)>>>> else % all equations were zero return list(for each j in v collect !*f2q makearbcomplex(),t); u := cdr u; for each j in u do if y := extmult(!*sf2ex(j,v),x) then if inconsistency!-chk y then cnds := numr cancel(lc y ./ lc x) . cnds else x := y; if cnds then < 1 then vars!* := t . vars!*; return list(for each j in cnds collect !*f2q j,nil)>>>>; arbvars := for each j in setdiff(v,lpow x) collect j . makearbcomplex(); if arbvars and null !*solvesingular then rerror(solve,6,"SOLVE given singular equations"); if null red x then return list(for each j in v collect if y := atsoc(j,arbvars) then !*f2q cdr y else nil ./ 1, t); sgn := evenp length lpow x; return list(for each j in v collect if y := atsoc(j,arbvars) then !*f2q cdr y else solve!-mkglsol(j,x,sgn := not sgn,arbvars), t); end; symbolic procedure inconsistency!-chk u; null u or ((nil memq lpow u) and inconsistency!-chk red u); symbolic procedure solve!-mkglsol(u,v,sgn,arbvars); % u is the kernel to be solved for, x the exterior product of all % independent equations, sgn is the current sgn indicator, arbvars % is an a-list (var . arbvar). begin scalar s,x,y; x := nil ./ 1; y := lpow v; for each j on red v do if s := solve!-glsolterm(u,y,j,arbvars) then x := addsq(cancel(s ./ lc v),x); return if sgn then negsq x else x end; symbolic procedure solve!-glsolterm(u,v,w,arbvars); begin scalar x,y,sgn; x := lpow w; a: if null x then return if null car y then lc w else multf(cdr atsoc(car y,arbvars), if sgn then negf lc w else lc w); if car x eq u then return nil else if car x memq v then <> else if y then return nil else <>; go to a end; %**** Support for exterior multiplication **** % Data structure is lpow ::= list of variables in exterior product % lc ::= standard form symbolic procedure !*sf2ex(u,v); %Converts standardform u into a form distributed w.r.t. v %*** Should we check here if lc is free of v? if null u then nil else if domainp u or null(mvar u memq v) then list nil .* u .+ nil else list mvar u .* lc u .+ !*sf2ex(red u,v); symbolic procedure extmult(u,v); % Special exterior multiplication routine. Degree of form v is % arbitrary, u is a one-form. if null u or null v then nil else (if x then cdr x .* (if car x then negf subs2multf(lc u,lc v) else 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); symbolic procedure subs2multf(u,v); % This definition allows for a power substitution that can lead to % a denominator in subs2. There is a possibility that the power % substitution won't be made, but there is no example of this yet. (if denr y neq 1 then x else numr y) where y=subs2(x ./ 1) where x = multf(u,v); symbolic procedure extadd(u,v); if null u then v else if null v then u else if lpow u = lpow v then (lambda x,y; if null x then y else lpow u .* x .+ y) (addf(lc u,lc v),extadd(red u,red v)) else if ordexp(lpow u,lpow v) then lt u .+ extadd(red u,v) else lt v .+ extadd(u,red v); symbolic procedure ordexp(u,v); if null u then t else if car u eq car v then ordexp(cdr u,cdr v) else if null car u then nil else if null car v then t else ordop(car u,car v); symbolic procedure ordexn(u,v); %u is a single variable, v a list. Returns nil if u is a member %of v or a dotted pair of a permutation indicator and the ordered %list of u merged into v. begin scalar s,x; a: if null v then return(s . reverse(u . x)) else if u eq car v then return nil else if u and ordop(u,car v) then return(s . append(reverse(u . x),v)) else <>; go to a end; endmodule; module solvealg; % Solution of equations and systems which can % be lifted to algebraic (polynomial) systems. % Author: Herbert Melenk. % Copyright (c) 1992 The RAND Corporation and Konrad-Zuse-Zentrum. % All rights reserved. % August 1992: added material for % rule set for reduction of trig. polynomial terms to % elementary expressions in sin and cos, % constant expressions in sin, cos and constant roots, % closed form results for trigonometric systems. % general exponentials. % avoiding false soltutions with surds. % % May 1993: better handling of products of exponentials % with common base, % additional computation branch for linear parts of % nonlinear systems. fluid '(!*exptexpand); % from simp.red fluid '( system!* % system to be solved osystem!* % original system on input uv!* % user supplied variables iv!* % internal variables fv!* % restricted variables kl!* % kernels to be investigated sub!* % global substitutions inv!* % global inverse substitutions depl!* % REDUCE dependency list !*solvealgp % true if using this module solvealgdb!* % collecting some data last!-vars!* % collection of innermost aux variables const!-vars!* % variables representing constants !*expli % local switch: explicit solution groebroots!* % predefined roots from input surds !*test_solvealg % debugging support ); fluid '(!*trnonlnr); % If set on, the modified system and the Groebner result % or the reason for the failure are printed. global '(loaded!-packages!*); switch trnonlnr; !*solvealgp := t; % Solvenonlnrsys receives a system of standard forms and % a list of variables from SOLVE. The system is lifted to % a polynomial system (if possible) in substituting the % non-atomic kernels by new variables and appending additonal % relations, e.g. % replace add % sin u,cos u -> su,cu su^2+cu^2-1 % u^(1/3) -> v v^3 - u % ... % in a recursive style. If completely successful, the % system definitely can be treated by Groebner or any % other polynomial system solver. % % Return value is a pair % (tag . res) % where "res" is nil or a structure for !*solvelist2solveeqlist % and "tag" is one of the following: % % T a satisfactory solution was generated, % % FAILED the algorithm cannot be applied (res=nil) % % INCONSISTENT the algorithm could prove that the % the system has no solution (res=nil) % % NIL the complexity of the system could % be reduced, but some (or all) relations % remain still implicit. % rules to be applied locally for converting composite transcendental % function forms into simpler ones algebraic << solvealg!-rules1:= { sin(~alpha + ~beta) => sin(alpha)*cos(beta) + cos(alpha)*sin(beta), cos(~alpha + ~beta) => cos(alpha)*cos(beta) - sin(alpha)*sin(beta), sin(~n*~alpha) => sin(alpha)*cos((n-1)*alpha) + cos(alpha)*sin((n-1)*alpha) when fixp n, cos(~n*~alpha) => cos(alpha)*cos((n-1)*alpha) - sin(alpha)*sin((n-1)*alpha) when fixp n, sin(~alpha)**2 => 1 - cos(alpha)**2}; solvealg!-rules2:= { tan(~alpha) => sin(alpha)/cos(alpha), cot(~alpha) => cos(alpha)/sin(alpha) } ; solvealg!-rules3:= { sin(~alpha)**2 => 1 - cos(alpha)**2 }; % Artificial operator for matching powers in a % product. operator my!-expt; solvealg!-rules4:= { my!-expt(~a,~b)*my!-expt(a,~c) => my!-expt(a,b+c), my!-expt(~a,~b)*a => my!-expt(a,b+1), my!-expt(~a,~b)*a => my!-expt(a,b+1) % my!-expt(~a,~b)/my!-expt(a,~c) => my!-expt(a,b-c) }; >>; symbolic procedure solvenonlnrsys(sys,uv); % interface to algebraic system solver. % factorize the system and collect solutions. % After factoring we resimplify with *exptexpand off % in order to have exponentials to one basis % collected. begin scalar q,r,s,tag,!*exptexpand; s:='(nil); if solve!-psysp(sys,uv) then s:={sys} else for each p in sys do <>; tag:='failed; r:=nil; for each u in s do <<% collect exponentials with same base. u := solvenonlnrcollectexpt u; q:=solvenonlnrsys1(u,uv); if eqcar(q,'failed) then q:=solvenonlnrsyssep(u,uv); if eqcar(q,'failed) then q:=solvenonlnrsyslin(u,uv,nil); if eqcar(q,'not) then q:=solvenonlnrsyslin(u,uv,t); if eqcar(q,'not) then q:='(failed); if car q and car q neq 'failed then tag:=car q; q:= if car q neq 'failed then cdr q else for each j in u collect {{j ./ 1},nil,1}; r:=union(q,r); >>; return tag.r end; symbolic procedure solvenonlnrcollectexpt u; % u is a list of standard forms. Reform these % such that products of exponentials with same basis % are collected. if not smemq('expt,u) then u else <>; symbolic procedure solvenonlnrsyslin(eqs,vars,mode); % Eqs is a system of equations (standard forms, % implicitly equated to zero); this routine tries % to reduce the system recursively by separation, % if one variable occurs in one equation only linearly. % Mode=NIL: simple version: only pure linear variables % are substituted. % T: extended version: replacing variables with % degree 1 and potentially complicated % coefficients. % Returns solution or % '(not) if not applicable % '(failed) if applicable but solution failed. begin scalar d,e,e1,n,s,q,x,v,w,w1,neqs,nvars; v:=vars; var_loop: if null v then return '(not); x:=car v; v:=cdr v; w:=eqs; eqn_loop: if null w then goto var_loop; e:=car w; w:=cdr w; if null e then goto eqn_loop; if domainp e then return '(inconsistent); e1:= reorder e where kord!*={x}; if not mvar e1 =x or ldeg e1>1 or smemq(x,d:=lc e1) or smemq(x,n:=red e1) then goto eqn_loop; if not mode then <>; % linear form found: n*x+d=0. This is basis for a solution % x=-n/d. In a second branch the case {n=0,d=0} has to % be considered if n and d are not constants. n := reorder n; d:=reorder d; % Step 1: substitute in remaining equations, solve % and add linear formula to result. s:= quotsq(negf n ./ 1, d ./ 1); neqs := for each eqn in delete(e,eqs) join <>; nvars:=for each y in delete(x,vars) join if smemq(y,neqs) then {y}; w:= if null neqs then '(t (nil nil 1)) else if null nvars then '(inconsistent) else if cdr neqs then solvenonlnrsys(neqs,nvars) else solvenonlnrsysone(car neqs,car nvars); if car w eq 'failed then return w; w:=add!-variable!-to!-tagged!-solutions(x,s,w); % Step 2: add an eventual solution for n=0,d=0; if domainp d or not mode then return w; w1:=solvenonlnrsys(n.d.eqs,vars); return merge!-two!-tagged!-solutions(w,w1); end; symbolic procedure solvenonlnrsysone(f,x); % equation system has been reduced to one. Solve % it using solvesq. begin scalar q,w; w:=solvesq(f ./ 1,x,1); if null w then return '(inconsistent); % if not smemq('root_of,w) then goto ret; % % here we try to find out whether a root_of % % expression is a useful information or whether % % it is simply an echo of the input. % if cdr w then goto ret; % multiple branches: good. % q := prepsq caar car w; % if not eqcar(q,'root_of) % not on top level: good. % then goto ret; % q:=subst(x,caddr q,cadr q); % if f = numr simp q then return '(failed); %ret: return t.w; end; symbolic procedure add!-variable!-to!-tagged!-solutions(x,s,y); % Y is a tagged solution. Add equation x=s to all members. if eqcar(y,'inconsistent) then y else if null y or null cdr y then {t,{{s},{x},1}} else car y . for each q in cdr y collect {s . car q,x . cadr q,caddr q}; symbolic procedure merge!-two!-tagged!-solutions(w1,w2); % w1 and w2 are tagged solution sets. Merge these and % eliminated inconsistent cases. if car w1='failed or car w2='failed then '(failed) else if car w1='inconsistent then w2 else if car w2='inconsistent then w1 else car w1 . append(cdr w1,cdr w2); symbolic procedure solvenonlnrsyssep(eqs,vars); % Eqs is a system of equations (standard forms, % implicitly equated to zero); this routine tries % to reduce the system recursively by separation, % if one variable occurs only in one equation. begin scalar y,r,s,r0,u,w; if null vars then return '(failed) else if null cdr eqs then <>; for each x in vars do if null y then <>; if null y then return '(failed); r:=car r; s:=solvenonlnrsys(delete(r,eqs),delete(y,vars)); if car s='failed then return s else s:=cdr s; u:=for each s0 in s join << w:=for each q in pair(cadr s0,car s0) join if not smemq('root_of,cdr q) then {car q.prepsq cdr q}; r0:=subf(r,w); r0:=solvesq(r0,y,caddr s0); for each r1 in r0 collect {caar r1. car s0,y.cadr s0,caddr r1} >>; return t.u; end; symbolic procedure solve!-psysp(s,uv); % T if s is a pure polynomial system. null s or (solve!-psysp1(car s,uv) and solve!-psysp(cdr s,uv)); symbolic procedure solve!-psysp1(f,uv); domainp f or ((member(mvar f,uv) or solve!-psysp2(mvar f,uv)) and solve!-psysp1(lc f,uv) and solve!-psysp1(red f,uv)); symbolic procedure solve!-psysp2(v,uv); % t if there is no interaction between v and uv. null uv or (not smember(car uv,v) and solve!-psysp2(v,cdr uv)); symbolic procedure solvenonlnrsys1(system!*,uv!*); % solve one system. begin scalar r,rules; osystem!* := system!*; if solvealgtrig0 system!* then rules:='(solvealg!-rules1); if smemq('tan,system!*) or smemq('cot,system!*) then rules:='solvealg!-rules2.rules; r := evalletsub2({rules,'(solvenonlnrsyspre)},nil); if errorp r then return '(failed) else system!* := car r; r := solvenonlnrsys2(); return r; end; symbolic procedure solvenonlnrsyspre(); (for each p in system!* collect numr simp prepf p) where dmode!* = nil; symbolic procedure solvenlnrsimp(u); % a prepsq including resimplification with additional rules. % begin scalar r; % r := evalletsub2({'(solvealg!-rules3), % {'simp!* ,mkquote u}},nil); % if errorp r then error(99,list("error during postprocessing simp")); % return car r; % end; simp!* u; symbolic procedure solvenonlnrsys2(); % Main driver. We need non-local exits here % because of possibly hidden non algebraic variable % dependencies. if null !*solvealgp then system!*:='(failed) else % against recursion. (begin scalar iv!*,kl!*,inv!*,fv!*,r,!*solvealgp,solvealgdb!*; scalar last!-vars!*,groebroots!*,const!-vars!*,sub!*; % hiding dmode because exponentials need integers. for each f in system!* do solvealgk0 (if dmode!* then numr subf(f,nil) where dmode!*=nil else f); if !*trnonlnr then print list("original kernels:",kl!*); if null cdr system!* and (smemq('sin,system!*)or smemq('cos,system!*)) and (r:=solvenonlnrtansub(prepf car system!*,car uv!*)) then return solvenonlnrtansolve(r,car uv!*); if atom (errorset('(solvealgk1),!*trnonlnr,nil)) where dmode!*=nil then return (system!*:='(failed)); system!*:='list.for each p in system!* collect prepf p; if not('groebner memq loaded!-packages!*) then load!-package 'groebner; for each x in iv!* do if not member(x,last!-vars!*) then for each y in last!-vars!* do depend1(x,y,t); iv!* := sort(iv!*,function (lambda(a,b);depends(a,b))); if !*trnonlnr then << prin2t "Entering Groebner for system"; writepri(mkquote system!*,'only); writepri(mkquote('list.iv!*), 'only); >>; r := list(system!*,'list.iv!*); r := groesolveeval r; if !*trnonlnr then << prin2t "leaving Groebner with intermediate result"; writepri(mkquote r,'only); terpri(); terpri(); >>; if 'sin memq solvealgdb!* then r:=solvealgtrig2 r; r:= if r='(list) then '(inconsistent) else solvealginv r; system!* := r; % set value aside return r; end) where depl!*=depl!* ; symbolic procedure solvealgk0(p); % Extract new top level kernels from form p. if domainp p then nil else <>; symbolic procedure solvealgk1(); % Process all kernels in kl!*. Note that kl!* might % change during processing. begin scalar k,kl0,kl1; k := car kl!*; while k do <>; end; symbolic procedure solvealgk2(k); % Process one kernel. (if member(k,uv!*) then solvealgvb0 k and (iv!*:= k.iv!*) else if atom k then t else if eq(car k,'expt) then solvealgexpt(k,x) else if memq(car k,'(sin cos tan cot)) then solvealgtrig(k,x) else if null x then t else solvealggen(k,x) ) where x=solvealgtest(k); symbolic procedure solvealgtest(k); % Test if the arguments of a composite kernel interact with % the variables known so far. if atom k then nil else solvealgtest0(k); symbolic procedure solvealgtest0(k); % Test if kernel k interacts with the known variables. solvealgtest1(k,iv!*) or solvealgtest1(k,uv!*); symbolic procedure solvealgtest1(k,kl); % list of those kernels in list kl, which occur somewhere % in the composite kernel k. if null kl then nil else if member(k,kl) then list k else if atom k then nil else union(if smember(car kl,cdr k) then list car kl else nil, solvealgtest1(k,cdr kl)); symbolic procedure solvealgvb k; % Restricted variables are those which might establish % non-algebraic relations like e.g. x + e**x. Test k % and add it to the list. fv!* := append(solvealgvb0 k,fv!*); symbolic procedure solvealgvb0 k; % test for restricted variables. begin scalar ak; ak := allkernels(k,nil); if intersection(ak,iv!*) or intersection(ak,fv!*) then error(99,list("transcendental variable dependency from",k)); return ak; end; symbolic procedure allkernels(a,kl); % a is an algebraic expression. Extract all possible inner % kernels of a and collect them in kl. if numberp a then kl else if atom a or a member uv!* then if not member(a,kl) then a.kl else kl else <>; symbolic procedure allkernels1(f,kl); if domainp f then kl else <>; symbolic procedure solvealgexpt(k,x); % kernel k is an exponential form. ( if eqcar(m,'quotient) and fixp caddr m then if cadr m=1 then solvealgrad(cadr k,caddr m,x) else solvealgradx(cadr k,cadr m,caddr m,x) else if null x then solvealgid k else if ((null intersection(w,uv!*) and null intersection(w,iv!*) and null intersection(w,fv!*)) where w=allkernels(m,nil)) then solvealggen(k,x) else solvealgexptgen(k,x) ) where m = caddr k; symbolic procedure solvealgexptgen(k,x); % Kernel k is a general exponentiation u ** v. begin scalar bas,xp,nv; bas := cadr k; xp := caddr k; if solvealgtest1(xp,uv!*) then return solvealgexptgen1(k,x) else if solvealgtest1(bas,uv!*) then return solvealggen(k,x); % remaining case: "constant" exponential expression to % replaced by an id for syntatical reasons nv := '( % old kernel ( (expt !&alpha n)) % new variable ( !&beta) % substitution ( ((expt !&alpha n) . !&beta) ) % inverse ( (!&beta (expt !&alpha n) !& )) % new equations nil ); nv:=subst(bas,'!&alpha,nv); nv:=subst(solve!-gensym(),'!&beta,nv); nv:=subst(xp,'n,nv); return solvealgupd(nv,nil); end; symbolic procedure solvealgexptgen1(k,x); % Kernel k is a general exponentiation u ** v. % where v is an expression in a solution variable, u % is constant. Transform all kernels with same basis % and compatible exponent to common exponent denominator % form. begin scalar bas,xp,xpl,nv,q,r,nk,sub; bas := cadr k; xp := caddr k; % collect all exponentials with this basis. xpl:={(1 ./ 1).xp}; for each k in kl!* do if eqcar(k,'expt) and cadr k=bas and <> then <>; % compute common denominator. q:=1; for each e in xpl do q:=lcm(q,denr car e); % the new artificial kernel. nk:=reval{'expt,bas,{'quotient,xp,q}}; sub := for each e in xpl collect {'expt,bas,cdr e}. {'expt,nk,numr car e * q/denr car e}; system!*:=sublis(sub,system!*); return solvealggen(nk,x); end; symbolic procedure solvealgradx(x,m,n,y); % error(99,"forms e**(x/2) not yet implemented"); solvealgexptgen1({'expt,x,{'quotient,m,n}},y); symbolic procedure solvealgrad(x,n,y); % k is a radical exponentiation expression x**1/n. begin scalar nv,m,!β !&beta := solve!-gensym(); nv:= '( % old kernel ( (expt !&alpha (quotient 1 !&n))) % new variable ( !&beta) % substitution ( ((expt !&alpha (quotient 1 !&n)) . !&beta) ) % inverse % ( (!&beta !&alpha (expt !& !&n)) ) nil % new equation ( (difference (expt !&beta !&n) !&alpha) ) ); m := list('!&alpha.x,'!&beta.!&beta,'!&n.n); nv := subla(m,nv); % prepare roots for simple surds. if null y or y={x} then groebroots!* := ({'plus,{'expt,!&beta,n},reval{'minus,x}} .{{{'equal,!&beta,{'expt,x,{'quotient,1,n}}}}}).groebroots!*; return solvealgupd(nv,y); end; symbolic procedure solvealgtrig0(f); % examine if sin/cos identies must be applied. begin scalar args,r,c; args :=for each a in solvealgtrig01(f,nil) collect (union(kernels numr q,kernels denr q) where q=simp a); while args do <>; return r; end; symbolic procedure solvealgtrig01(f,args); if atom f then args else if memq(car f,'(sin cos tan cot)) then if constant_exprp cadr f then args else union({cadr f},args) else solvealgtrig01(cdr f,solvealgtrig01(car f,args)); algebraic << operator p_sign; let p_sign(~x) = if sign(x)=0 then 1 else sign(x); >>; symbolic procedure solvealgtrig(k,x); % k is a trigonometric function call. begin scalar nv,m,s,!&alpha,!β solvealgdb!* := union('(sin),solvealgdb!*); if x then if cdr x then error(99,"too many variables in trig. function") else x := car x; solvealgvb k; nv := '( % old kernels ( (sin !&alpha)(cos !&alpha)(tan !&alpha)(cot !&alpha) ) % new variables ( (sin !&beta) (cos !&beta) ) % substitutions ( ((sin !&alpha) . (sin !&beta)) ((cos !&alpha) . (cos !&beta)) %%% these should be handled now by the ruleset. %%% ((tan !&alpha) . (quotient (sin !&beta) (cos !&beta))) %%% ((cot !&alpha) . (quotient (cos !&beta) (sin !&beta))) ) % inverses ( ((sin !&beta) (cond ((and !*expli (test_trig)) '(!&loc (p_sign (!&!& !&)))) (t '(!&x (!&!& (root_of (equal (sin !&alpha) !&) !&x)))))) ((cos !&beta) (cond ((and !*expli (test_trig)) '(!&x (plus (!&!& (times !&loc (acos !&))) (times 2 pi !&arb)))) (t '(!&x (!&!& (root_of (equal (cos !&alpha) !&) !&x)))))) ) % new equation ( (plus (expt (sin !&beta) 2)(expt (cos !&beta) 2) -1) ) ); % invert the inner expression. s := if x then solvealginner(cadr k,x) else 1; !&beta := solve!-gensym(); m := list('!&alpha . (!&alpha:=cadr k), '!&beta . !&beta, '!&loc . solve!-gensym(), '!&arb . {'arbint,!!arbint:=!!arbint+1}, '!&x . x, '!&!& . s); nv:=sublis!-pat(m , nv); if x then last!-vars!*:= append(last!-vars!*,{{'sin,!&beta},{'cos,!&beta}}) else const!-vars!* := append(const!-vars!*,{{'sin,!&beta}.{'sin,!&alpha}, {'cos,!&beta}.{'cos,!&alpha}}); return solvealgupd(nv,nil); end; symbolic procedure solvealgtrig2 u; % r is a result from goesolve; remove trivial relations % like sin^2 + cos^2 = 1. begin scalar r,w,op,v,rh; for each s in cdr u do <>; return 'list . reverse r; end; symbolic procedure solvealggen(k,x); % k is a general function call; processable if SOLVE % can invert the function. begin scalar nv,m,s; if cdr x then error(99,"too many variables in function expression"); x := car x; solvealgvb k; nv := '( % old kernels ( !&alpha ) % new variables ( !&beta ) % substitutions ( ( !&alpha . !&beta) ) % inverses (( !&beta '(!&x (!&!& !&)))) % new equation nil); % invert the kernel expression. s := solvealginner(k,x); m := list('!&alpha . k, '!&beta . solve!-gensym(), '!&x . x, '!&!& . s); nv:=sublis!-pat(m , nv); return solvealgupd(nv,nil); end; symbolic procedure solvealgid k; % k is a "constant" kernel, however in a syntax unprocessable % for Groebner (e.g. expt(a/2)); replace temporarily begin scalar nv,m,s; nv := '( % old kernels ( !&alpha ) % new variables ( ) % substitutions ( ( !&alpha . !&beta) ) % inverses (( !&beta nil . !&alpha)) % new equation nil); % invert the kernel expression. m := list('!&alpha . k, '!&beta . solve!-gensym()); nv:=sublis(m , nv); return solvealgupd(nv,nil); end; symbolic procedure solvealginner(s,x); <>; symbolic procedure solvealgupd(u,innervars); % Update the system and the structures. begin scalar ov,nv,sub,inv,neqs; ov := car u; u := cdr u; nv := car u; u := cdr u; sub:= car u; u := cdr u; inv:= car u; u := cdr u; neqs:=car u; u := cdr u; for each x in ov do kl!*:=delete(x,kl!*); for each x in innervars do for each y in nv do depend1(y,x,t); sub!* := append(sub,sub!*); iv!* := append(nv,iv!*); inv!* := append(inv,inv!*); system!* := append( for each u in neqs collect <>, for each u in system!* collect numr subf(u,sub) ); return t; end; symbolic procedure solvealginv u; % Reestablish the original variables, produce inverse % mapping and do complete value propagation. begin scalar v,r,s,m,lh,rh,y,z,tag,sub0,sub,!*expli,noarb,arbs; scalar abort; integer n; sub0 := for each p in sub!* collect (cdr p.car p); tag := t; r := for each sol in cdr u join <>; for each eqn in reverse cdr sol do <>; if not smemq('root_of,rh) then s:=(lh.prepsq rh).s else fv!*:=lh.fv!*; >>; if (m:=assoc(lh,inv!*))then <>; % if local variable, append to substitution. if not member(lh,uv!*) and !*expli then << sub:=append(sub,{lh .(z:=prepsq subsq(rh,sub))}); if smember(lh,r) then r:=subst(z,lh,r) >>; % append to the final output. if (member(lh,uv!*) or not !*expli) % inhibit repeated same values. and not<< z:=subsq(rh,sub); n:=length member(z,r); n>0 and lh=nth(v,length v + 1 - n)>> then <>; >>; % Classify result. % for each x in uv!* do % if tag and not member(x,v) and smember(x,r) then tag:=nil; if !*test_solvealg then if abort then yesp "ABORTED" else < "; writepri(mkquote ('list .for each u in pair(v,r) collect {'equal,car u,prepsq cdr u}) ,'only); prin2t "================================"; yesp "continue?"; >>; if not abort then {reverse r . reverse v} >>; return solvealg!-verify(tag,r); end; symbolic procedure solvealgdepends(u,x); % inspect u for explicit dependency of x, being careful for % root_of subexpressions. if u=x then t else if atom u then nil else if eqcar(u,'root_of) then if x=caddr u then nil else solvealgdepends(cadr u,x) else solvealgdepends(car u,x) or solvealgdepends(cdr u,x); symbolic procedure test_trig(); begin scalar lh,rh,r; lh := cadr kl!*; rh:= caddr kl!*; if member(lh . nil, solvealgdb!*) then return nil; r := not !*complex and not smemq('i,kl!*) and not smemq('!:gi!:,kl!*) and not smemq('!:cr!:,kl!*) and not smemq('root_of,kl!*); if not r then solvealgdb!* := append(solvealgdb!*,{('sin.cdr lh).nil,('cos.cdr lh).nil}); return r; end; fluid '(!*solvealg_verify); % the idea of the following procedure is to exclude isolated % solutions which give a substantial residue when subsituted % into the equation system under "on rounded"; as long as no % good criterion for a residue to be small has been found, this % step is disabled. symbolic procedure solvealg!-verify(tag,r); <>; end; tag . for each q in r collect car q . cdr q . list 1 >>; symbolic procedure solvealg!-verify1 s; % verify solution s for the current equation system. begin scalar sub,nexpli,x,y,sys,sum,fail; sub:= for each u in pair(cdr s,car s) collect if not nexpli then <>; % a non explicit solution cannot be tested. if nexpli then return nil; sum := nil ./ 1; for each u in osystem!* do if not fail then <>; return if fail then nil else sum; end; symbolic procedure sublis!-pat(a,u); % like sublis, but replace lambda expressions by matching their % actual arguments. begin scalar v; if atom u then return <>; v:=assoc(car u,a); if v and (v:=cdr v) and eqcar(v,'lambda) then return sublis!-pat((caadr v.cadr u).a,caddr v); return sublis!-pat1(a,u); end; symbolic procedure sublis!-pat1(a,l); if null l then nil else if atom l then sublis!-pat(a,l) else sublis!-pat(a,car l) . sublis!-pat1(a,cdr l); %---------------------------------------------------------------- % section for single trigonometric polynomials %---------------------------------------------------------------- symbolic procedure solvenonlnrtansub(p,x); % Perform tangent substitution. if not smemq('sin,p) and not smemq('cos,p) then if smemq(x,p) then nil else nil.p else if car p='cos then if smemq(x,cdr p) then (cdr p). '(quotient (difference 1(expt tg!- 2)) (plus 1(expt tg!- 2))) else nil.p else if car p='sin then if smemq(x,cdr p) then (cdr p). '(quotient (times 2 tg!-) (plus 1(expt tg!- 2))) else nil.p else (if ca and cd and (car ca = car cd or null car ca or null car cd) then (car ca or car cd).(cdr ca.cdr cd)) where ca=solvenonlnrtansub(car p,x), cd=solvenonlnrtansub(cdr p,x); symbolic procedure solvenonlnrtansolve(u,x); begin scalar p,v,s,z,r,si,y; v:=caar u; u:= prepf numr simp cdr u; s:=solveeval {u,'tg!-}; si:=solveeval {{'equal,v,'!&!&},x}; for each q in cdr s do <>; >>; return t.r; end; symbolic procedure solvenonlnrtansolve1 u; % Find all cos**2. if atom u then nil else if car u='expt and eqcar(cadr u,'cos) and caddr u=2 then {u . {'difference,1,{'expt,{'sin,cadr cadr u},2}}} else union(solvenonlnrtansolve1 car u,solvenonlnrtansolve1 cdr u); endmodule; module solvetab; % Simplification rules for SOLVE. % Author: David R. Stoutemyer. % Modifications by: Anthony C. Hearn and Donald R. Morrison. put('asin, 'inverse, 'sin); put('acos, 'inverse, 'cos); put('atan,'inverse,'tan); put('acot,'inverse,'cot); put('asec,'inverse,'sec); put('acsc,'inverse,'csc); algebraic; Comment Rules for reducing the number of distinct kernels in an equation; operator sol; % for all a,b,c,d,x such that ratnump c and ratnump d let % sol(a**c-b**d, x) = a**(c*lcm(c,d)) - b**(d*lcm(c,d)); for all a,b,c,d,x such that not fixp c and ratnump c and not fixp d and ratnump d let sol(a**c-b**d, x) = a**(c*lcm(den c,den d)) - b**(d*lcm(den c,den d)); for all a,b,c,d,x such that a freeof x and c freeof x let sol(a**b-c**d, x) = e**(b*log a - d*log c); for all a,b,c,d,x such that a freeof x and c freeof x let sol(a*log b + c*log d, x) = b**a*d**c - 1, sol(a*log b - c*log d, x) = b**a - d**c; for all a,b,c,d,f,x such that a freeof x and c freeof x let sol(a*log b + c*log d + f, x) = sol(log(b**a*d**c) + f, x), sol(a*log b + c*log d - f, x) = sol(log(b**a*d**c) - f, x), sol(a*log b - c*log d + f, x) = sol(log(b**a/d**c) + f, x), sol(a*log b - c*log d - f, x) = sol(log(b**a/d**c) - f, x); for all a,b,d,f,x such that a freeof x let sol(a*log b + log d + f, x) = sol(log(b**a*d) + f, x), sol(a*log b + log d - f, x) = sol(log(b**a*d) - f, x), sol(a*log b - log d + f, x) = sol(log(b**a/d) + f, x), sol(a*log b - log d - f, x) = sol(log(b**a/d) - f, x), sol(log d - a*log b + f, x) = sol(log(d/b**a) + f, x), sol(log d - a*log b - f, x) = sol(log(d/b**a) - f, x); for all a,b,c,d,x such that a freeof x and c freeof x let sol(a*log b + c*log d, x) = b**a*d**c - 1, sol(a*log b - c*log d, x) = b**a - d**c; for all a,b,d,x such that a freeof x let sol(a*log b + log d, x) = b**a*d - 1, sol(a*log b - log d, x) = b**a - d, sol(log d - a*log b, x) = d - b**a; for all a,b,c,x let sol(log a + log b + c, x) = sol(log(a*b) + c, x), sol(log a - log b + c, x) = sol(log(a/b) + c, x), sol(log a + log b - c, x) = sol(log(a*b) - c, x), sol(log a - log b - c, x) = sol(log(a/b) - c, x); for all a,c,x such that c freeof x let sol(log a + c, x) = a - e**(-c), sol(log a - c, x) = a - e**c; for all a,b,x let sol(log a + log b, x) = a*b - 1, sol(log a - log b, x) = a - b, sol(cos a - sin b, x) = sol(cos a - cos(pi/2-b), x), sol(sin a + cos b, x) = sol(sin a - sin(b-pi/2), x), sol(sin a - cos b, x) = sol(sin a - sin(pi/2-b), x), sol(sin a + sin b, x) = if !*allbranch then sin((a+b)/2)* cos((a-b)/2) else a+b, sol(sin a - sin b, x) = if !*allbranch then sin((a-b)/2)* cos((a+b)/2) else a-b, sol(cos a + cos b, x) = cos((a+b)/2)*cos((a-b)/2), sol(cos a - cos b, x) = if !*allbranch then sin((a+b)/2)* sin((a-b)/2) else a-b, sol(asin a - asin b, x) = a-b, sol(asin a + asin b, x) = a+b, sol(acos a - acos b, x) = a-b, sol(acos a + acos b, x) = a-b; endmodule; module quartic; % Procedures for solving cubic, quadratic and quartic % eqns. % Author: Anthony C. Hearn. % Modifications by: Stanley L. Kameny. fluid '(!*sub2 !*rounded !*trigform dmode!*); !*trigform := t; % Default value. switch trigform; symbolic procedure multfq(u,v); % Multiplies standard form U by standard quotient V. begin scalar x; x := gcdf(u,denr v); return multf(quotf(u,x),numr v) ./ quotf(denr v,x) end; symbolic procedure quotsqf(u,v); % Forms quotient of standard quotient U and standard form V. begin scalar x; x := gcdf(numr u,v); return quotf(numr u,x) ./ multf(quotf(v,x),denr u) end; symbolic procedure cubertq u; % Rationalizing the value in this and the following function leads % usually to neater results. % rationalizesq simpexpt list(mk!*sq subs2!* u,'(quotient 1 3)); % simprad(u,3); symbolic procedure sqrtq u; % rationalizesq simpexpt list(mk!*sq subs2!* u,'(quotient 1 2)); % simprad(u,2); symbolic procedure subs2!* u; <>; symbolic procedure solvequadratic(a2,a1,a0); % A2, a1 and a0 are standard quotients. % Solves a2*x**2+a1*x+a0=0 for x. % Returns a list of standard quotient solutions. % Modified to use root_val to compute numeric roots. SLK. if !*rounded and numcoef a0 and numcoef a1 and numcoef a2 then for each z in cdr root_val mkpolyexp2(a2,a1,a0) collect simp!* 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; symbolic procedure numcoef a; denr a = 1 and domainp numr a; symbolic procedure mkpolyexp2(a2,a1,a0); % The use of 'x is arbitrary here, since it is not used by root_val. <>; symbolic procedure solvecubic(a3,a2,a1,a0); % A3, a2, a1 and a0 are standard quotients. % Solves a3*x**3+a2*x**2+a1*x+a0=0 for x. % Returns a list of standard quotient solutions. % See Abramowitz and Stegun, Sect. 3.8.2, for details. begin scalar q,r,sm,sp,s1,s2,x; a2 := quotsq(a2,a3); a1 := quotsq(a1,a3); a0 := quotsq(a0,a3); q := subtrsq(quotsqf(a1,3),quotsqf(exptsq(a2,2),9)); r := subtrsq(quotsqf(subtrsq(multsq(a1,a2),multfq(3,a0)),6), quotsqf(exptsq(a2,3),27)); if null numr q or not !*trigform or not all_real(a0,a1,a2) then go to cbr; % this section uses trig functions, but only when a0,a1,a2 are real. x := sqrtq negsq addsq(exptsq(q,3),exptsq(r,2)); if one_real simp list('times,'i,mk!*sq x) and not pos_num q then x := negsq x; s1 := quotsqf(atan2q(x,r),3); s2 := negsq sqrtq negsq q; sp := negsq multfq(2,multsq(s2,cossq s1)); sm := multsq(simp '(sqrt 3),multsq(s2,sinsq s1)); % sp := -2*sqrt(-q)*cos(atan2(sqrt( - q**3 - r**2),r)/3)$ % sm := - sqrt(-q)*sqrt(3)*sin(atan2(sqrt( - q**3 - r**2),r)/3)$ go to com; cbr: x := sqrtq addsq(exptsq(q,3),exptsq(r,2)); s1 := cubertq addsq(r,x); s2 := if numr s1 then negsq quotsq(q,s1) else cubertq subtrsq(r,x); % This optimization only works if s1 is non zero. sp := addsq(s1,s2); sm := quotsqf(multsq(simp '(times i (sqrt 3)),subtrsq(s1,s2)),2); com: x := subtrsq(sp,quotsqf(a2,3)); sp := negsq addsq(quotsqf(sp,2),quotsqf(a2,3)); return list(subs2!* x,subs2!* addsq(sp,sm), subs2!* subtrsq(sp,sm)) end; symbolic procedure pos_num a; begin scalar r,dmode,!*msg,!*numval; dmode := dmode!*; !*numval := t; on rounded,complex; a := resimp a; a := real_1 a and (numr simp list('sgn,mk!*sq a)=1); off rounded,complex; if dmode then onoff(get(dmode,'dname),t); return a end; symbolic procedure sinsq a; simpiden list('sin,mk!*sq subs2!* a); symbolic procedure cossq a; simpiden list('cos,mk!*sq subs2!* a); symbolic procedure all_real(a,b,c); begin scalar r,dmode,!*msg,!*numval; dmode := dmode!*; !*numval := t; on complex,rounded; a := resimp a; b := resimp b; c := resimp c; a := real_1 a and real_1 b and real_1 c; off rounded,complex; if dmode then onoff(get(dmode,'dname),t); return a end; symbolic procedure real_1 x; numberp denr x and domainp numr x and null numr impartsq x; symbolic procedure one_real a; begin scalar r,dmode,!*msg,!*numval; dmode := dmode!*; !*numval := t; on complex,rounded; a := real_1 resimp a; off rounded,complex; if dmode then onoff(get(dmode,'dname),t); return a end; symbolic procedure atan2q(b,a); % Used by solvecubic to set up trig form expressions for atan2 in % terms of atan and, where necessary, a bias of +/- pi; or to call % atan2 directly if numerical solution is called for. ((begin scalar !*msg,x,y,r,dmode,q,fg,s1,s2,s3,s4,s5; y := b := simp!*(b := mk!*sq subs2!* b); x := a := simp!*(a := mk!*sq subs2!* a); if domainp numr y and domainp numr x and numberp denr y and numberp denr x then go to aret; dmode := dmode!*; on complex,rounded; y := resimp b; x := resimp a; if not(domainp numr y and domainp numr x and numberp denr y and numberp denr x) then go to ret; q := sqrtq addsq(exptsq(x,2),exptsq(y,2)); x := quotsq(x,q); y := quotsq(y,q); if null numr x then <>; s3 := t; x := numr simp list('sgn,list('repart,mk!*sq x)); if x=-1 then <>; ret: off rounded,complex; if dmode then onoff(get(dmode,'dname),t); if s1 then fg := quotsqf(simp 'pi,2); if s2 then fg := negsq fg; if s3 then fg := simpiden list('atan,mk!*sq quotsq(b,a)); if s4 then fg := subtrsq(fg,simp 'pi); if s5 then fg := addsq(fg,simp 'pi); aret: return if fg then fg else simpiden list('atan2,mk!*sq b,mk!*sq a) end) where !*numval=t); symbolic procedure solvequartic(a4,a3,a2,a1,a0); % Solve the quartic equation a4*x**4+a3*x**3+a2*x**2+a1*x+a0 = 0, % where the ai are standard quotients, using technique described in % Section 3.8.3 of Abramowitz and Stegun; begin scalar x,y,yy,cx,z,s,l,zz1,zz2,r,dmode,neg,!*msg,!*numval; % Convert equation to monomial form. dmode := dmode!*; a3 := quotsq(a3,a4); a2 := quotsq(a2,a4); a1 := quotsq(a1,a4); a0 := quotsq(a0,a4); % Build and solve the resultant cubic equation. We select the % real root if there is only one; or if there are three, we choose % one that yields real coefficients for the quadratics. If no % roots are known to be real, we use an arbitrary one. yy := subtrsq(exptsq(a3,2),multfq(4,a2)); x := solvecubic(!*f2q 1, negsq a2, subs2!* subtrsq(multsq(a1,a3),multfq(4,a0)), subs2!* negsq addsq(exptsq(a1,2), multsq(a0,yy))); cx := car x; % Now check for real roots of the cubic. for each rr in x do if one_real rr then s := append(s,list rr); x := if (l := length s)=1 then car s else cx; % Now solve the two equivalent quadratic equations. a3 := quotsqf(a3,2); yy := quotsqf(yy,4); % select real coefficient for quadratic if possible. y := addsq(yy,x); if l<2 then go to zz; loop: if not pos_num negsq y then go to zz else if l=1 then <>; l := l-1; s := cdr s; x := car s; y := addsq(yy,x); go to loop; zz: y := sqrtq y; x := quotsqf(x,2); z := sqrtq subtrsq(exptsq(x,2),a0); % the following test is needed, according to some editions of % Abramowitz and Stegun, to select the correct signs % (for the terms z) in the quadratics to produce correct roots. % Unfortunately, this test may fail for coefficients which are not % numeric because of the inability to recognize zero. !*numval := t; on rounded,complex; if null numr (zz1 := resimp subtrsq(a1,addsq(multsq(subtrsq(a3,y),addsq(x,z)), multsq(addsq(a3,y),subtrsq(x,z))))) then go to rst; if null numr (zz2 := resimp subtrsq(a1,addsq(multsq(subtrsq(a3,y),subtrsq(x,z)), multsq(addsq(a3,y),addsq(x,z))))) then <>; if domainp numr zz1 and domainp numr zz2 and numberp denr zz1 and numberp denr zz2 and numr simp list('sgn,list('difference,list('norm,mk!*sq zz1), list('norm,mk!*sq zz2)))=1 then neg := t; rst: off rounded,complex; if dmode then onoff(get(dmode,'dname),t); if neg then z := negsq z; return append(solvequadratic(!*f2q 1,subtrsq(a3,y),subtrsq(x,z)), solvequadratic(!*f2q 1,addsq(a3,y),addsq(x,z))) end; endmodule; end;