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prellesingerODE.mac
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prellesingerODE.mac
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/*****************************************************************************/
/* */
/* O D E F I */
/* */
/* Differential Equation solver using First Integrals */
/* Based on the Prelle-Singer Algorithm and the ODEFI code */
/* Original author: R. Shtokhamer, 1988 */
/* Re-implementation: N. Beishuizen, 2014-2015 */
/* To do: 1. run kamke database, check nr of equations solved and time */
/* 2. improve cases that can be solved */
/* May 14, 2015: all cases mentioned in Man,1993 can be solved except the
cases with the highest N and longest running times
additionally, we can solve cases with arbitrary functions, like kamke 1.35
*/
/* 3. extended prelle-singer method
4. system of ode's
5. second order ode's
6. nth order ode's
*/
/*****************************************************************************/
/*****************************************************************************/
/* */
/* Extending the ODE package. */
/* */
/* Includes: The Prelle-Singer method, and a subroutine for finding special */
/* forms of an integrating factor. */
/*****************************************************************************/
/* GLOBALS: */
/* ----- THE DEFAULT SETTINGS ----- */
/* */
define_variable(type_of_intfactor ,3 ,fixnum)$ /* 1=heuristics only, 2=prelle-singer only, 3=1+2*/
define_variable(elementary ,4 ,fixnum)$
define_variable(primitiveflag ,false ,boolean)$
define_variable(LISTOFTERMS ,false ,any)$
define_variable(MAXTERMS ,6 ,fixnum)$ /* the maximum number in the monomials 1+x + y +x^2 + xy + y^2 +..*/
define_variable(INTFACTOR_FI ,false ,any)$ /* global variable containing the integrating factor */
define_variable(METHOD_FI ,false ,any)$ /* global variable containing the method used, prelle-singer or one of the heuristics */
define_variable(ODEFI_GRAD_LIST ,[] ,any)$ /* our own definitions of gradients of functions */
define_variable(Y_X_DEPEND ,false ,boolean)$ /* we have y(x) and diff(y(x),x) instead of y and diff(y,x) */
/* */
define_variable(nil,[],any_check,"nil has to bound to []")$
put('nil,lambda([x],if x # [] then error("Don't reset nil")),'value_check)$
/* *************************************************************************** */
/* *************************************************************************** */
prellesingerODE(expr,_y ,_x):= block([_a:0,_b,_p:1,_q:1,res:false,%x%,%y%,mon2nspace:10,newres,oldres],local(%y%,%x%),declare(mon2nspace,integer),
/* check the input and return the ODE, the dependent variable, the independent variable and the quotient M,N of dy/dx=M/N */
[res,%y%,%x%,[_q,_p]]: checkPSInput(expr,_y,_x,'firstorder),
if (res=false) then return(false),
print("ode = ",expr),
print("y = ",%y%),
print("x = ",%x%),
print("_M = ",_q),
print("_N = ",_p),
/* ----- Get the integrating factor ------- */
/* ---------------------------------------- */
res:intfactor_control(_p,_q,%x%,%y%),
/* ---------------------------------------- */
print("res = ",res),
if res=false or (res[1]=false and res[2]=false)then (remarray(vfunv,vifunv),return(false)),
print("some solution found...",res),
if first(res) #false then ( /* " Rational Solution obtained " */
print("rational solution"),
res: rootscontract(first(res)) - %c
)
else if second(res) # false then ( /* "integrating factor obtained " */
print("integrating factor = ",grind(INTFACTOR_FI)),
/* do some simplification here (should be moved to end of intfactor_control)*/
INTFACTOR_FI: ratsimp(trigsimp(INTFACTOR_FI)),
/* rootscontract converts products of roots into roots of products, simplifies an expression like 1/(sqrt(x-%i)sqrt(x+%i)) = 1/(sqrt(x^2+1)) */
/* note that abs(x) can also be put into the root when x is real*/
INTFACTOR_FI: rootscontract(INTFACTOR_FI),
INTFACTOR_FI: radcan(INTFACTOR_FI), /* will simplify e.g. exp(log(a*x+b)) */
print("MU = ",INTFACTOR_FI),
res: integrate(mysimp(-INTFACTOR_FI*_q),%x%),
print("res = ",res),
res: res + integrate(INTFACTOR_FI*_p,%y%) - integrate(trigsimp(ratsimp(diff(res,%y%))),%y%) +%c,
/* FIXME: merge with cleanupsolution */
if not freeof(log,res) then res:logcontract(mysimp(res))
),
print("treating grad_list"),
if ODEFI_GRAD_LIST # nil then (
res:subst(ODEFI_GRAD_LIST,res),
/*_p:subst(ODEFI_GRAD_LIST,_p),*/ /* we never use this again, why substitute back? */
/*_q:subst(ODEFI_GRAD_LIST,_q),*/
INTFACTOR_FI:subst(ODEFI_GRAD_LIST,INTFACTOR_FI),
remlet(all,defi_let_package)
),
print("before res = ",grind(res)),
res: cleanupODESolution(res,%y%,%x%),
print("after res = ",grind(res)),
/* substitute the dependency back */
if Y_X_DEPEND then (
res:subst(_y,%y%,subst(_x,%x%,res)),
LISTOFTERMS:subst(_y,%y%,subst(_x,%x%,LISTOFTERMS)),
INTFACTOR_FI:subst(_y,%y%,subst(_x,%x%,INTFACTOR_FI))
),
print("returning res = ",grind(res)),
return(res)
)$
/* */
/*****************************************************************************/
intfactor_control(_N,_M,vx,vy):=
/* */
/* This is the controlling routine for finding integrating factors */
/* It chooses type of algorithm applied depending on the flags */
/* Applies heuristics as well as the P-S algorithm */
/* Returns a rational First-Integral if such found using the P-S method */
/* The "Prelle-Singer" algorithm is adopted to treat transcendentals */
/* LOCAL VARIABLES: */
/* degree = bound on the order of monomial in solution of df/f */
/* nterms = number of terms allowed or list (of list) of monomials */
/* trlst = list of the transcendentals appearing in grad */
/* prlst = some other symbolic parameters appearing in p,q, in the */
/* case that prlst # nil consistency conditions can be */
/* obtained to enable solutions of the problem */
/* THIS OPTION IS NOT IMPLEMENTED IN THIS VERSION SO */
/* prlst is set to nil */
/* n2mon and mon2n are two function to be supplied by the user */
/* (a crude version is supplied ) */
/* n2mon is bijection from integers to monomials */
/* mon2n is inverse of n2mon */
/* OUTPUT: */
/* false, if no integrating factor or rational solution are found. */
/* otherwise the result is set to [ratres,integfactor] */
/* where a) in case that rational first-integral is found */
/* then ratres is set to the solution and integfactor is false */
/* b) else ratres is false and integfactor is the INTFACTOR */
/* The global variable METHOD_FI describes the method used */
/* In the case that a "rational" first-integral is found */
/* using the Prelle-Singer method, firstintegral is set to the */
/* solution and METHOD_FI to:" Prelle-Singer rational first integral" */
/* */
block([_f,_g,_p,solutions,_sol,primitive,sol1,heursol:false,lg,defpol,gradnew,
algexact:true,ratalgdenom:true,algebraic:true,
exptsubst:true,a,b,trlst,prlst:nil,degree,lst,nterms,
rootscondmode: 'all,radexpand:'all, rhs,newtr], declare([lg],integer),
/* dy/dx = M/N ==> Ndy = Mdx Mdx - Ndy = 0 */
print("M = ",_M),
print("N = ",_N),
/* ----------------------------------------------------------------------------- */
/* ----- try heuristics, when type_of_intfactor=2, only P-S method is used ----- */
if (type_of_intfactor#2) then (
print("trying heuristics..."),
[METHOD_FI,INTFACTOR_FI]:specialintfactor(_M,_N,vx,vy),
print("method =", METHOD_FI),
print("mu = ", INTFACTOR_FI),
if METHOD_FI#false and INTFACTOR_FI #false then return([false,INTFACTOR_FI])
),
/* ----------------------------------------------------------------------------- */
/* ----- if an integrating factor was found then we can stop now ......... ----- */
/* ----------------------------------------------------------------------------- */
/* ----------------------------------------------------------------------------- */
/* ----- compute the basis of functions */
trlst : BasisFunctions(_M,_N,vy,vx),
print("----- BASIS OF FUNCTIONS = ",trlst),
/* ----------------------------------------------------------------------------- */
/* nijso moved out of the loop */
/* Only the trivial extension F(i) is checked, Future releases will include */
/* general treatment of algebraic extensions */
/* */
if freeof(%i,_N) and freeof(%i,_M) then (
primitive:nil,
defpol:nil
) else (
primitive:%i,
block([simp:false],defpol:%i^2+1)
),
/* nijso added loop over degrees */
exittheloop:false,
degree:5,
while degree <= MAXTERMS do (
print("---------------------------"),
print("degree = ",degree),
print("---------------------------"),
/* this finds -(dP/dx + dQ/dy) for the ode Qdx=Pdy or dy/dx=Q/P */
rhs:newgradient([_N,_M],[vx,vy],trlst,degree),
print("gradients: ",rhs),
newtr:first(rhs), /* List of transcendentals */
gradnew:second(rhs) ,
rhs:third(rhs) , /* this is the actual -(dP/dx+dQ/dy) term */
/* */
/* lst is the list of transcendentals */
/*degree:if MAXTERMS < 0 then - MAXTERMS else max(MAXTERMS,length(lst)+1), */
/*degree:if MAXTERMS < 0 then - MAXTERMS else max(MAXTERMS,length(trlst)+1), */
/*degree:if MAXTERMS < 0 then - MAXTERMS else MAXTERMS, */
nterms:if LISTOFTERMS=false then degree else LISTOFTERMS,
/* */
if listp(nterms) then (
/* Transform the list nterms into list of numbers identifying terms */
b:trlst,
for a in newtr do (
nterms:ratsubst(a,first(b),nterms),
b:rest(b)
),
nterms :ordl1(nterms,newtr)
),
print("nterms: ",nterms),
/* nterms is number of terms in polynomial, i.e.
f = f1 + f2*x + f3*y + f4*x^2 + f5*xy + f6*y^2 + f7*x^3 + f8*x^2y + f9*xy^2 + f10*y^3 */
/* when degree is 3 (MAXTERMS) then the above polynomial is the largest that will be tested */
/* */
/* Now solve the Dp/p problem (The Darboux polynomials) */
/* Construct all monic polynomials fi of degree <= MAXTERMS */
/*solutions:prelle_singer_dfovf(degree,nterms,gradnew,newtr,prlst, */
solutions:prelle_singer_dfovf(degree,nterms,gradnew,newtr,prlst,
primitive,defpol,n2mon,mon2n),
print("--------------------------------------------------------------"),
print("solutions: ",solutions),
print("solutions: ",grind(solutions)),
print("--------------------------------------------------------------"),
primitive:second(solutions), /* */
defpol:third(solutions),
/* note that we re-use the variable p == _N here*/
_sol:rest(solutions,3),
solutions:first(solutions),
/*print("heursol = ",heursol),*/ /* we only arrive here when heursol=false*/
if solutions=nil then (
METHOD_FI:concat("No Solution Found to the Dp/p Problem.
Try to increase MAXTERMS. ")
) else (
lg:first(_sol),
/* this is the right hand side -(P_x +Q_y) */
print("r1Df/f + r2Df/f = rhs: ",rhs),
/* this is the list of f's */
print("trlst: ",trlst),
print("newtr: ",newtr),
print("primitive: ",primitive),
print("defpol: ",defpol),
print("lg: ",lg),
/* */
/* Check the dependencies */
/* */
_sol:prelle_singer_dep(rhs,trlst,newtr,vx,vy,solutions,
primitive,defpol,lg,n2mon,mon2n),
print("returning from prelle_singer_dep into intfactor_control "),
if _sol#false then (
/* found a integrating factor */
print("found a integrating factor: N= ",_sol),
sol1:first(_sol),
METHOD_FI: if sol1 # false then " Prelle-Singer Rational Solution" else" Prelle-Singer Integrating Factor",
INTFACTOR_FI:if sol1 # false then -diff(sol1,vx)/_M else second(_sol),
exittheloop:true,
print("exittheloop 1"),
return(false)
),
if exittheloop then ( print("exittheloop 2"),return(false) )
),
if exittheloop then ( print("exittheloop 3"),return(false) ),
degree:degree+1,
remarray(vfunv,vifunv)
), /* end loop over N. If we exit here, then we did not find a solution */
print("exiting function"),
return(_sol)
)$
/* */
matchdeclare([%a%,%b%,%n%,%m%] ,freeof(x,y) ,%c%,freeof(x) ,%e%,freeof(y));
notfreeofxy(x,y,z):=not freeof(x,y,z);
matchdeclare(%d%,notfreeofxy(x,y));
defmatch(defi_type6_0, %a% + %d%, x,y);
defmatch(defi_type6_1, %e%*y^%n%, y);
defmatch(defi_type6_2, %b%*x^%m%, x);
defi_type6(exp,x,y):=block([a,b,c,d],
if (a:defi_type6_0(exp,x,y)) # false and (b:defi_type6_1(%d%,y))# false
and (c:defi_type6_2(%e%,x))# false then
subst(a,subst(b,subst(c,[[%a%,%b%],[%m%,%n%]]))) else false
)$
/*****************************************************************************/
/*****************************************************************************/
/* This function creates the basis of functions for which we take the multivariate polynomials */
/*****************************************************************************/
BasisFunctions(_M,_N,_y,_x):=block([_LBasisFunctions,_TrimmedList:nil,_a,sol1],
/*****************************************************************************/
print("computing basis functions"),
/* ----- We continue with the Prelle-Singer algorithm ... ----- */
/* The field is assumed to be generated by the kernels in p and q and by */
/* the derivatives of those kernels */
/* */
_LBasisFunctions:showratvars([_N,_M]), /* list of all unknowns */
/* now add to the list the derivatives wrt the dependent and independent variable */
/* nijso: actually, we have to do this again to make sure that derivatives of derivatives are now in the list... we dont do this!*/
_LBasisFunctions:append(map(lambda([_a],diff(_a,_x)),_LBasisFunctions), map(lambda([_a],diff(_a,_y)),_LBasisFunctions), _LBasisFunctions),
_LBasisFunctions:showratvars(_LBasisFunctions),
/* remove everything that is not depending on x,y */
for _a in _LBasisFunctions do
if not freeof(_x,_y,_a) or diff(_a,_x)#0 or diff(_a,_y)#0 then
_TrimmedList:cons(_a,_TrimmedList),
/* */
/* This section comes to insure proper substitution (no internal substitutions)*/
/* inside the transcendentals ! */
_LBasisFunctions:nil,
/* check for double entries (we need to keep the correct order) */
while _TrimmedList # nil do(
_a:first(_TrimmedList),
_TrimmedList:rest(_TrimmedList),
if freeof(_a,_TrimmedList) then _LBasisFunctions:endcons(_a,_LBasisFunctions) else _TrimmedList:endcons(_a,_TrimmedList)
),
/* trimmedlist is now empty, basisfunctions contains everything*/
/* */
/* It is assumed that all entries in trlst are algebraically independent! */
/* One may use the Rothstein-Caviness algorithm to determine algebraic */
/* dependence (see Siam.J.Comp. Vol 8 No.3. Aug 1979. ) */
/* */
_TrimmedList:_LBasisFunctions,
/* ------------------------------------------------------------------------------ */
/* we are finished with constructing the list of variables... */
/* */
/* Now the Prelle-Singer method */
/* */
/* Check if a closed differential field */
sol1:map(lambda([z],diff(z,_x)),_TrimmedList),
sol1:append(map(lambda([z],diff(z,_y)), _TrimmedList),sol1),
sol1:showratvars(sol1),
print("sol1 : ",sol1),
for _a in sol1 do if (not freeof(_x,_y,_a) or radcan(diff(_a,_x))#0
or radcan(diff(_a,_y))#0) and (not member(_a,_TrimmedList))
then (print (_a," is non constant in the field R", _TrimmedList),
print(" The result may be WRONG! "),
let([_a,0],defi_let_package)
),
/* we check if it is a closed differential field, but if it isn't then we don't do anything about it! */
/* when does this happen? e.g. with unknown functions f(x) and their derivatives df/dx */
return(_TrimmedList)
)$
/*****************************************************************************/
/*****************************************************************************/
prelle_singer_dfovf(degree,nterms,grad,vrlst,prlst,primitive,defpol,fun,ifun):=
/*****************************************************************************/
/* Solving the Dp/p problem of many variable case */
/* Input : */
/* degree = maximal order of monomial appearing if p */
/* nterms = maximal number of terms allowed */
/* or list of terms to be examined */
/* grad = list of polynomials (D = sum of pi*diff(.,xi)) */
/* vrlst = list of the variables (transcendentals) */
/* prlst = list of parameters appearing in grad */
/* primitive= primitive element of the algebraic extension */
/* defpol = The irreducible polynomial defining primitive */
/* fun = bijection from positive integers onto monomials */
/* ifun = inverse of fun */
/* output: */
/* list [ll,primitive,defpol,lg], where */
/* ll=list of [[f,g], ... ,], where Df=f*g, */
/* primitive = possible a new primitive element */
/* defpol = defining polynomial */
/* ng = maximal number of terms in the g's */
/* */
/* Arrays used : vgrad = vector of distributive representation of grad */
/* vf = .. symbols vf [i] */
/* vg = .. .. vg[i] */
/* veq = .. of equations */
/* vgmon = vector of monomials in g */
/*****************************************************************************/
block([ngr,i,ng,m,nmax,n,nf,
a,b,c,d,e,glist,flist,fpol1,gpol1,cof,var,
monom,fpol,gpol,lindex,nlindex,res,eql,nleql,
nterms1,solvedlist,savenleq,saveleq,list_of_terms,nfm,
zerolist,nozerolist,endloop:false,nofsol:0,lc,
oprimitive,odefpol,overpol,oldofnew:nil,irred] ,
/* */
declare([ngr,i,ng,m,nmax,n,nf,nfm,degree],integer),
/*declare([vg,vgmon,vf,vgrad,veq] ,special), */
print("entering prellesinger_dfovf"),
print("maximal order of monomial : ",degree),
print("maximum number of terms allowed:",nterms),
print("list of polynomials : ",grad),
print("list of transcendentals : ", vrlst),
print("list of parameters in poly list : ",prlst),
print("primitive element of the algebraic extension : ",primitive),
print("irreducible polynomial defining primitive : ", defpol),
/* comment of Rami on nterms:nterms-1: degree-1 (BUG?) */
/* is nterms an integer or a rational number? */
if numberp(nterms) then nterms:nterms-1,
print("nterms = ",nterms),
/* Because the coefficient of the highest term if f (in the df/f problem) */
/* can be set to 1 */
res :nil,
ngr:length(grad),
/* declare the array vgrad */
array(vgrad,ngr),
/* */
print("calling pol2dist"),
for i:1 thru ngr do (
a:first(grad),
vgrad[i]:pol2dist(a,vrlst,vrlst,ifun),
grad:rest(grad)
),
/* */
print("vgrad = ",listarray(vgrad)),
/* vgrad has the following form for the polynomials:
[index,term,coefficient], where index is the index to the vfunv array. term is the poly term in terms of transcendentals
and coefficient is the constant prefactor in Complex C
*/
print("determining highest monomial in g"),
/* note, g is the remainder term of the division: Df/f = g */
/* Nijso: if N=2, then we only have to look in monomials 1..6 */
/* if N=3, then we only have to look in monomials 1..10*/
/* if N=4, then we only have to look in monomials 1..15*/
/* Determine the highest monomial appearing in g */
/* but... we know this, right? */
b:vrlst,
ng:0,
for i:1 thru ngr do (
print("---- i=",i,"/",ngr),
a:first(vgrad[i]),
cof:third(a),
var:first(b),
print("a=",a),
print("cof=",cof),
print("var=",var),
/* nijso: new degree bound for g from Y.K. Man 1993: deg(Df) - deg(f) */
print("degree of f = ",degree),
if cof # 0 then (
monom:second(a), /* this is the maximal monomial in A, M(A)*/
m:first(a), /* the index, or the maximum order in terms of the monomials */
print(" monom=",monom," ,m=",m),
if hipow(monom,var)>0 then m:apply(ifun,[ratsimp(monom/var),vrlst]),
print("m = ",m), /* this is the degree of A, deg(A)=Phi^-1(M(A)) */
ng:max(ng,m)
),
b:rest(b)
),
/* g gives the index in the list of polynomials with the highest monomial occuring in the ode*/
print("highest monomial in g = ",ng),
print("monomial = ",monom),
/* Initialize */
array(vg,ng),array(vgmon,ng),array(vf,degree),
/* */
flist:[vf[1]], /* flist will hold the list of all unknowns vf(1] .. vf[.] */
fpol : vf[1],
glist:nil,
gpol: 0,
/* */
print("glist"),
for i:1 thru ng do (
glist:cons(vg[i],glist),
b:apply(fun,[i,vrlst]),
vgmon[i]:b,
gpol:gpol+b*vg[i]
),
print("glist = ",glist),
print("vgmon = ",listarray(vgmon)),
print("gpol = ",gpol), /* this is the first polynomial f */
/* find nmax the maximal monomial appearing in f*g */
print("find nmax for degree=",degree),
nmax:apply(ifun,[ratsimp(vgmon[ng]*apply(fun,[degree,vrlst])),vrlst]),
print("nmax = ",nmax),
/* */
array(veq,nmax),
/* */
/* initialize lindex = list of integers from 1 to nmax */
/* and set all the equations veq[i] to zero */
lindex: nil,
for i:1 thru nmax do (veq[i] :0,lindex:cons(i,lindex)),
/* */
for i:1 thru ng do veq[i]:vg[i]*vf[1],
/*print("veq = ",listarray(veq)), */
print("lindex = ",lindex),
/* */
/* Start the main loop : */
/* */
/* nijso: we move the loop outside of the darboux function*/
for nf : 2 thru degree do (
monom: apply(fun,[nf,vrlst]),
print("monom = ",monom),
/* */
/* Depending on "nterms" prepare the list "list_of_terms" */
/* */
print("--------------------"),
print("----- N=",nf," -----"),
print("--------------------"),
print("prepare listofterms"),
print("nterms = ",nterms),
if numberp(nterms) then (
print(" --- numberp(nterms) is true"),
solvedlist:nil,
nterms1: nterms ,
nfm:nf-1,
print("nterms1 = ",nterms),
print("nfm = ",nfm),
if nterms1 > nfm then nterms1:nfm,
list_of_terms:combinations(flist,nfm,nterms1),
print("-- list of terms = ",list_of_terms)
) /* rami put cross - wrong here */
else(
print(" --- numberp(nterms) is false"),
list_of_terms:nil,
c:nterms,
while c#nil do(
a:first(c),
/* *rami put a cross through the following code: */
nfm:first(a),
if nfm = nf /* only lists whose highest order term is nf examined */
then(
list_of_terms: cons(combinations1(flist,rest(a)),list_of_terms) ,
nterms: rest (c)
)
else if nfm>nf then c:[nil]
else error(print(" incompatible terms ",a," for nf= ",nf)),
c:rest(c)
),
if nterms=nil then endloop:true
),
print("nterms = ",nterms),
print("endloop = ",endloop),
print("list of terms = ",list_of_terms),
/* rami ends the cross through the code here */
/* update the equations except the one arising from vf[nf]*g */
/* NB: here we start constructing the matrix F */
c:vrlst, /* for example [x,y] */
print("update equations"),
/*print("veqn = ",listarray(veq)),*/
for i:1 thru ngr do (
var:first(c),
b:vgrad[i],
m:hipow(monom,var),
if m>0 then (
d:ratsimp(monom/var),
while b # nil do (
e:first(b),
a:second(e),
cof:third(e),
n:apply(ifun,[a*d,vrlst]),
veq[n] : veq[n]-cof*m*vf[nf] ,
/*print("veq=",listarray(veq)),*/
b:rest(b)
)
),
c:rest(c)
),
/*
print("vrlist of transcendentals = ",vrlst),
print("vgrad = ",listarray(vgrad)),
print("veq = ",veq, listarray(veq)),
*/
/* */
print("pick linear equations"),
/* Pick up the linear equations */
n:apply(ifun,[monom*vgmon[ng],vrlst]),
eql:nil,
nlindex : rest(lindex,nmax-n),
for i:1 thru ng do (
m:apply(ifun,[monom*vgmon[i],vrlst]),
eql:cons(-ratsubst(1,vf[nf],veq[m]),eql),/* we eliminate the highest vf[i], i.e. vf[ng]*/
/* print("eql = ",eql," ",nf," ",vf[nf]," ",veq[m]),*/
nlindex:delete(m,nlindex)
),
/*print("eql = ",eql),*/
saveleq : eql,
/* */
/* Collect the nonlinear equations */
print("pick nonlinear equations"),
nleql:nil ,
while nlindex # nil do (
m:first(nlindex),
a:ratsubst(1,vf[nf] ,veq[m]),
if a#0 then nleql:cons(a,nleql),
nlindex:rest(nlindex)
),
/* print("nleql = ",nleql),*/
/* */
savenleq:nleql,
/* */
/* */
/* Loop on members in the list "list_of_terms" */
/* rami comments: no while*/
/* rami comments: nozerolist:flist ???nolist:nil*/
print("loop over listofterms",list_of_terms),
while list_of_terms # nil do (
print("insert list_of_terms"),
print(first(list_of_terms)),
zerolist: first (list_of_terms),
nozerolist:first(zerolist),
zerolist: second(zerolist),
/* -------------------------------- */
/* -------------------------------- */
/* -------------------------------- */
/* */
/* solve the linear equations */
/* */
print("solve linear equations, ",zerolist," ",saveleq," ",glist),
eql: subst(zerolist,saveleq),
/*print("eql = ",eql), */
eql: lsolveforg(eql,glist),
/*print("eql = ",eql), */
/* -------------------------------- */
/* -------------------------------- */
/* -------------------------------- */
/* */
/* Solve the set of nonlinear equations in nleql */
/* save the old primitive and defining polynomial */
print("solve nonlinear equations"),
oprimitive:primitive,
odefpol:defpol,
/* rami crosses appendzerolist here */
nleql : subst (append(zerolist,eql),savenleq),
/*print("nleql = ",nleql), */
nleql : nlsolverpp(delete(0,nleql),nozerolist,primitive,defpol),
/*print("nleql = ",nleql), */
/* */
/*Resolve the output */
oldofnew:first(rest(nleql,3)),
defpol:third(nleql),
primitive:second(nleql),
nleql:first(nleql),
/*print("nleql,first = ",nleql), */
/* rami comments: not necessary */
if numberp(nterms) then if (terms < (degree-1)) then (
/* examine redundant solutions */
lc : solvedlist,
c : nleql,b:nil,
while c#nil do(
a:first(c),
m:0, i:1,
while a # nil do (
if rhs(first(a))#0 then m:m+2^first(lhs(first(a))) else i:0,
a:rest(a)
),
if i=1 then b:cons(first(c),b)
else if not member(m,lc) then (
solvedlist:cons(m,solvedlist),
b: cons(first(c),b)
),
c:rest(c)
),
nleql:reverse(b)
),
/*print("nleql,reverse = ",nleql), */
/* */
/* Redefine the solutions in terms of the new primitive element */
/* */
/* rami remarks: X bug check!*/
if defpol # nil and primitive # oprimitive and oldofnew #nil
then mysip(subst(oldofnew,oprimitive,nleql)),
/* */
/* Now construct the actual polynomials */
/*print("construct polynomials, nleq = ",nleql),*/
c:res,
print("c-res = ",res,",fpol= ",fpol,",monom= ",monom,",zerolist= ",zerolist),
/* rami crosses zerolist */
/* rami wants to move the while loop to after 'if domainp line */
print("---------- BEGIN eliminate nonlinear equations ----------"),
while nleql # nil do (
print("----- nleql = ",nleql),
a:first(nleql),
print("a = ",a),
fpol1:rat(mysimp(subst(append(a,zerolist),fpol)+monom)),
print("fpol1=",fpol1," ",fpol," ",monom),
b:subst(a,eql),
print("a = ",a),
print("eql= ",eql),
print("b = ",b),
print("gpol = ",gpol),
gpol1:rat(mysimp(subst(b,gpol))),
print("gpol1 = ",gpol1),
/* */
/* Now check for irreducability is performed */
/* Irreducability in Q or Q(i) only, is checked */
/* */
print("check irreducibility"),
b:c,irred:true,
while b # nil do (
a:first(first(b)),
a:mysimp(fpol1/a),
if domainp(denom(a),vrlst) then (b : [1] ,irred:false),
b:rest(b)
),
print("irred = ",irred),
if irred then (
print("irreduceable"),
nofsol :nofsol+1, /* Check Lemma 1 */
print("nofsol = ",nofsol," ",ng),
/* we have removed the loop inside the darboux function */
if nofsol > ng then (endloop:true,nleql:[1]),
print("res = ",res,", ",fpol1,", ",gpol1),
res:cons([fpol1,gpol1],res)
),
nleql:rest(nleql)
),
print("nleql = ",nleql),
print("res = ",res),
/* */
list_of_terms:rest(list_of_terms)
), /* end loop over list_of_terms... rami has crossed this list_of_terms */
print("---------- END eliminate nonlinear equations ----------"),
print("list_of_terms = ",list_of_terms),
/* */
/* Update veq , fpol and flist */
for i:1 thru ng do (
m:apply(ifun,[vgmon[i]*monom,vrlst]),
veq [m] : veq[m]+vf[nf]*vg[i]
),
/*print("veq = ",listarray(veq)),*/
print("fpol=",fpol,", monom=",monom,", vf=",vf[nf]),
fpol:rat(fpol+monom*vf[nf]),
flist:cons(vf[nf],flist),
print("flist= ",flist)
/* but how do we know we can exit the loop when we have found 2 independent solutions? */
/*if endloop then nf:degree */
/* the loop has been moved outside of the darboux funxtion*/
), /* End of the main loop over nf */
print("end of loop over listofterms"),
remarray(vf,vg,veq,vgmon,vgrad),
print("----- exiting prellesinger_dfovf ----- "),
print("result = ",res," ",primitive," ",defpol," ",ng),
return ([reverse(res),primitive,defpol,ng])
)$
/*****************************************************************************/
/* Checking linear dependence of the solutions */
/* returned by the subroutine prelle_singer_dfovf. */
/* Constructing rational first-integral or an integrating factor */
/*****************************************************************************/
prelle_singer_dep(rhs,trlst,newtr,vx,vy,
solutions,primitive,defpol,lg,n2mon,mon2n):=
/* INPUT: */
/* rhs = -(diff(p,vx)+diff(q,vy) (generalization of) the right hand */
/* side of the equation Dp/p = rhs, for transcendentals. */
/* trlst = list of transcendentals */
/* newtr = The new names of transcendentals */
/* solutions = solutions returned by prelle_singer_dfovf */
/*OUTPUT: */
/* false if solution is not found */
/* [first-integral, ... ] if "rational" solution found */
/* [false,integrating-factor] otherwise */
/*****************************************************************************/
block([_a,_b,s,k,i,j,l,fpol,nofsol,exptsubst:true,indlist:nil,
gpol,lm,m,n,solved:false,degreeprim:0],
declare([i,j,k,l,lm,nofsol,m,n,degreeprim] ,integer),
print("----- entering prelle singer dep ----- "),
print("rhs = ",rhs),
print("trlst = ",trlst),
print("newtr = ",newtr),
print("vx = ",vx),
print("vy = ",vx),
print("solutions = ",solutions),
print("primitive = ",primitive),
print("defpol = ",defpol),
print("lg = ",lg),
/* first we check if the g's are linearly dependent. If they are, then sum(c_i*g_i)=0 for some nonzero c_i */
/* so we have to collect all the different terms, i.e. tr2*tr3*tr4 etc and check if we can find coefficients */
/* such that we get a+b+c = 0, b-c+d=0, etc...*/
/* */
degreeprim: 1,
if primitive='%i then (
print("%i is a primitive!"),
degreeprim:2,
block([simp:false] , defpol:%i^2+1)
) else if defpol # nil then
degreeprim:hipow(defpol,primitive),
degreeprim:degreeprim-1,
print("primitiveflag: ",primitiveflag),
print("elementary:",elementary,evenp(elementary)),
if primitiveflag=false or evenp(elementary) then (
defpol=nil,
degreeprim=0,
primitive=nil
),
print("defpol = ",defpol),
print("degreeprim = ",degreeprim),
print("primitive = ",primitive),
nofsol:length(solutions),
array(gvec,nofsol,degreeprim,lg),
array(fun,nofsol),
array (per_vec,degreeprim,lg),
/* set to zero all vectors */
for l:0 thru nofsol do (
fun[l]:0,
for i:0 thru degreeprim do
for j:1 thru lg do gvec[l,i,j]:0
),
/* Prepare the substitution */
trlst:map("=",newtr,trlst),
/* --- construct the gvec --- */
l:0,
while solutions # nil do (
l:l+1,
gpol:first(solutions),
fpol:first(gpol),
gpol:second(gpol),
print("fpol:",fpol),
print("gpol:",gpol),
/* If gpol = 0 then solution found (this is step 2 in Shtokhamer) */
/* Because this means that we have 0*g1 + 0*g2 + ... + c*gpol + ... + 0*gn =0 */
/* At this stage return the result and dont search any further */
/* We find these trivial solutions while we are building the gvec array for further analysis*/
if gpol=0 then /* check if trivial solution or not */
if ((letsimp(diff(fpol:subst(trlst,fpol),vx),defi_let_package) #0) or
(letsimp(diff(fpol,vy),defi_let_package)#0) )
then (
remarray(gvec,fun,per_vec),
solved:true,
return(1) /* we exit the gvec construction routine */
)
else (
l:l-1,
nofsol:nofsol-1
)
else (
fun[l] :fpol,
print("gpol:",gpol),
print("newtr:",gpol),
gpol:pol2dist(gpol,newtr,newtr,mon2n),/* pol2dist rewrites the gpol as a list [[index_i,variable_i,coeff_i],[],[]] */
/* where variable is tr1,tr2,tr1*tr2,etc, coeff is the constant factor and index */
/* is the index to the polynomial array [tr1,tr2,tr3,tr1^2,tr1*tr2,tr2^2,...] */
print("fun:",fun[l]),
print("gpol:",gpol),
while gpol # nil do (
_a:first(gpol), /* takes the first gpol from the list */
i:mode_identity(fixnum,first(_a)), /* takes the index from gpol_i */
_a:third(_a), /* takes the coefficient */
if _a # 0 then
if primitive # nil then
for k:0 thru degreeprim do /* loop over all powers */
gvec[l,k,i]:ratcoeff(_a,primitive,k) /* returns coefficient _a of _a*primitive^k */
/* so gvec[l,k,i] contains for all "l" solutions the */
/* coefficients of the "kth" power */
else
gvec[l,0,i]:_a, /* if primitive is nil then naturally we place it at position 0, but why create an extra else for it? */
gpol:rest(gpol)
)
),
solutions:rest(solutions)
), /* --- end of construction of gvec --- */
print("solutions: ",solutions),
/*print("gvec = ",listarray(gvec)),*/
print("solved = ",solved),
/* at this point we have only found the simple linearly independents solutions where one of the g's is zero*/
if solved then return([fpol,false]),
/* */
/* at this point we are ready to test the linear dependence of all g's (if there is a g=0, then we have a solution and we exited before) */
/* examine now the rhs? -(diff(p,x)+diff(q,y)) */
/* */
_a:rhs,
_a:pol2dist(_a,newtr,newtr,mon2n),
s:first(_a), /* redundant with s:first a immediately down? (only 4 occurences of variable s)*/
/* /* we never enter here...*/
if(first(s) > lg) then (
remarray(gvec,fun,per_vec),
print(" BUG in prelle_singer_dep"), return(false)
),
*/
/* */
while _a # nil do (
s:first(_a),
i:mode_identity(fixnum,first(s)),
rhs:third(s),
if rhs # 0 then
if primitive # nil then
for k:0 thru degreeprim do
gvec[0,k,i]:ratcoef(rhs,primitive,k) /* right-hand-side is put at location 0 */
else gvec[0,0,i]:rhs,
_a:rest(_a)
),
indlist: nil,
for j:1 thru nofsol do(
print("------ j: ",j,"/",nofsol),