/*
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 * $OpenXM: OpenXM_contrib2/asir2018/lib/ratint,v 1.1 2018/09/19 05:45:08 noro Exp $
*/
/*
 * rational function integration (Trager's algorithm)
 * load("gr"); load("sp");
 * toplevel : ratint(F,V)
 * returns a complicated list s.t. 
 * [<rational part>, [[root*log(poly),defpoly],...].
 */

#define FIRST(a) car(a)
#define SECOND(a) car(cdr(a))
#define THIRD(a) car(cdr(cdr(a)))


def substv(P,Sl)
{
	for ( A = P; Sl != []; Sl = cdr(Sl) )
		A = subst(A,FIRST(car(Sl)),SECOND(car(Sl)));
	return A;
}
			
def co(X,V,D)
{
	for ( I = 0; I < D; I++ )
		X = diff(X,V);
	return sdiv(subst(X,V,0),fac(D));
}

def solve(El,Vl)
/*
 * El : list of linear forms
 * Vl : list of variable
 */
{
	N = length(El); M = length(Vl);
	Mat = newmat(N,M+1); 
	W = newvect(M+1); Index = newvect(N); Vs = newvect(M);
	for ( I = 0, Tl = Vl; I < M; Tl = cdr(Tl), I++ )
		Vs[I] = car(Tl);
	for ( I = 0, Tl = El; I < N; Tl = cdr(Tl), I++ ) {
		ltov(car(Tl),Vl,W);
		for ( J = 0; J <= M; J++ )
			Mat[I][J] = W[J];
	}
	Tl = solvemain(Mat,Index,N,M); L = car(Tl); D = car(cdr(Tl));
	if ( L < 0 )
		return [];
	for ( I = L - 1, S = []; I >= 0; I-- ) {
		for ( J = Index[I]+1, A = 0; J < M; J++ ) {
			A += Mat[I][J]*Vs[J];
		}
		S = cons([Vs[Index[I]],-red((A+Mat[I][M])/D)],S);
	}
	return S;
}

def solvemain(Mat,Index,N,M) 
/* 
 *	Mat : matrix of size Nx(M+1)
 *	Index : vector of length N
 */
{
	for ( J = 0, L = 0, D = 1; J < M; J++ ) {
		for ( I = L; I < N && !Mat[I][J]; I++ );
		if ( I == N )
			continue;
		Index[L] = J;
		for ( K = 0; K <= M; K++ ) {
			T = Mat[I][K]; Mat[I][K] = Mat[L][K]; Mat[L][K] = T;
		}
		for ( I = L + 1, V = Mat[L][J]; I < N; I++ )
			for ( K = J, U = Mat[I][J]; K <= M; K++ )
				Mat[I][K] = sdiv(Mat[I][K]*V-Mat[L][K]*U,D);
		D = V; L++;
	}
	for ( I = L; I < N; I++ )
		for ( J = 0; J <= M; J++ )
			if ( Mat[I][J] )
				return -1;
	for ( I = L - 2, W = newvect(M+1); I >= 0; I-- ) {
		for ( J = 0; J <= M; J++ )
			W[J] = 0;
		for ( G = I + 1; G < L; G++ )
			for ( H = Index[G], U = Mat[I][H]; H <= M; H++ )
				W[H] += Mat[G][H]*U;
		for ( J = Index[I], U = Mat[I][J]; J <= M; J++ )
			Mat[I][J] = sdiv(Mat[I][J]*D-W[J],U);
	}
	return [L,D];
}

def ltov(P,VL,W)
{
	for ( I = 0, L = VL; L != []; L = cdr(L), I++ ) {
		W[I] = co(P,car(L),1); P -= W[I]*car(L);
	}
	W[I] = P;
}

def makeucp(N,V) {
	for ( UCV = [], I = 0; I <= N; I++ )
		UCV = cons(uc(),UCV);
	for ( L = UCV, I = P = 0; I <= N; I++, L = cdr(L) )
		P += car(L)*V^I;
	return [P,UCV];
}

def ratint(F,V) {
	L = ratintsep(F,V);
	Rat = FIRST(L);
	if ( !SECOND(L) )
		return L;
	else {
		Pf = ratintpf(SECOND(L),V);
		for ( T = Pf, S = []; T != []; T = cdr(T) )
			S = cons(ratintlog(car(T),V),S);
		return [Rat,S];
	}
}

def ratintlog(F,V) {
	Nm = nm(F); Dn = dn(F);
	C = uc();
	R = res(V,ptozp(Nm-C*diff(Dn,V)),ptozp(Dn));
	Rc = FIRST(SECOND(fctr(R)));
	if ( deg(Rc,C) == 1 ) {
		VC = -co(Rc,C,0)/co(Rc,C,1);
		A = gcd(Nm-VC*diff(Dn,V),Dn);
		return [VC*log(A),0];
	} else {
		Root = newalg(Rc);
		A = gcda(subst(ptozp(Nm-C*diff(Dn,V)),C,Root),subst(ptozp(Dn),C,Root));
		return [Root*log(A),defpoly(Root)];
	}
}

def ratintsep(F,V) {
	B = dn(F); A = srem(nm(F),B); P = sdiv(nm(F)-R,B);
	IP = polyint(P,V);
	G = gcd(B,diff(B,x));
	if ( type(G) == 1 )
		return [IP,red(A/B)];
	H = sdiv(B,G);
	N = deg(B,V); M = deg(H,V);
	CL = makeucp(N-M-1,V); DL = makeucp(M-1,V);
	C = car(CL); CV = car(cdr(CL));
	D = car(DL); DV = car(cdr(DL));
	UCV = append(CV,DV);
	S = solveuc(A-(diff(C,V)*H-C*sdiv(H*diff(G,V),G)+D*G),V,UCV);
	C = substv(C,S); D = substv(D,S);
	return [IP+C/G,red(D/H)];
}

def polyint(P,V) {
	if ( !P )
		return 0;
	if ( type(P) == 1 )
		return P*V;
	for ( I = deg(P,V), T = 0; I >= 0; I-- )
		T += coef(P,I)/(I+1)*V^(I+1);
	return T;
}

def ratintpf(P,V) {
	NmP = nm(P); DnP = dn(P);
	DnPf = fctr(DnP); 
	L = length(DnPf) - 1;
	if ( L == 1 ) 
		return [P];

	Lc = FIRST(car(DnPf)); DnPf = cdr(DnPf);
	NmP = sdiv(NmP,Lc); DnP = sdiv(DnP,Lc); 
	Nm = newvect(L); Dn = newvect(L);
	for ( I = 0, F = DnPf; I < L; I++, F = cdr(F) )
		Dn[I] = FIRST(car(F));

	for ( I = 0, U = -NmP, Vl = []; I < L; I++ ) {
		CL = makeucp(deg(Dn[I],V)-1,V);
		Nm[I] = FIRST(CL); Vl = append(Vl,SECOND(CL));
		U += sdiv(DnP,Dn[I])*Nm[I];
	}

	S = solveuc(U,V,Vl);
	for ( I = 0, F = []; I < L; I++ )
		if ( T = substv(Nm[I],S) )
			F = cons(T/Dn[I],F);
	return F;
}

def solveuc(P,V,L) {
	for ( N = deg(P,V), E = [], I = N; I >= 0; I-- )
		if ( C = coef(P,I) )
			E = cons(C,E);
	EVG = eqsimp(E,L);
	if ( FIRST(EVG) == [] )
		return THIRD(EVG);
	else {
		S = solve(FIRST(EVG),SECOND(EVG));
		for ( T = S, G = THIRD(EVG); G != []; G = cdr(G) ) {
			VV = car(G); 
			T = cons([FIRST(VV),substv(SECOND(VV),S)],T);
		}
		return T;
	}
	
}

#if 0
def append(A,B) {
	return A == [] ? B : cons(car(A),append(cdr(A),B));
}
#endif

def eqsimp(E,Vs) {
	for ( Got = []; ; ) {
		if ( (VV = searchmonic(E,Vs)) == [] )
			return [E,Vs,Got];
		V = FIRST(VV); Val = SECOND(VV); 
		Vs = subtract(Vs,V);
		for ( T = []; E != []; E = cdr(E) )
			if ( S = subst(car(E),V,Val) )
				T = cons(S,T);
		E = T;
		for ( T = [VV]; Got != []; Got = cdr(Got) ) {
			VV1 = car(Got);
			T = cons([FIRST(VV1),subst(SECOND(VV1),V,Val)],T);
		}
		Got = T;
	}
}

def searchmonic(E,Vs) {
	for ( ; E != []; E = cdr(E) )
		for ( P = car(E), T = Vs; T != []; T = cdr(T) ) {
			V = car(T); C = diff(P,V);	
			if ( C == 1 )
				return [V,-(P-V)];
			else if ( C == -1 )
				return [V,P+V];
		}
	return [];
}

def subtract(S,E) {
	for ( T = []; S != []; S = cdr(S) )
		if ( car(S) == E )
			return append(T,cdr(S));
		else
			T = cons(car(S),T);
}
end$