/* 
 * A collection of functions designed for calculations involving
 * 	polynomials in one variable (by Ernest W. Bowen).
 *
 * On starting the program the independent variable has identifier x
 *	and name "x", i.e. the user can refer to it as x, the
 *	computer displays it as "x".  The name of the independent
 *	variable is stored as varname, so, for example, varname = "alpha"
 *	will change its name to "alpha".  At any time, the independent
 *	variable has only one name.  For some purposes, a name like
 *	"sin(t)" or "(a + b)" or "\lambda" might be useful;
 *	names like "*" or "-27" are legal but might give expressions
 *	that are difficult to intepret.
 *
 * Polynomial expressions may be constructed from numbers and the
 *	independent variable and other polynomials by the algebraic
 *	operations +, -, *, ^, and if the result is a polynomial /.
 *	The operations // and % are defined to have the quotient and
 *	remainder meanings as usually defined for polynomials.
 *
 * When polynomials are assigned to idenfifiers, it is convenient to
 *	think of the polynomials as values.  For example, p = (x - 1)^2
 *	assigns to p a polynomial value in the same way as q = (7 - 1)^2
 *	would assign to q a number value.  As with number expressions
 *	involving operations, the expression used to define the
 *	polynomial is usually lost; in the above example, the normal
 *	computer display for p will be  x^2 - 2x + 1.  Different
 *	identifiers may of course have the same polynomial value.
 * 
 * The polynomial we think of as a_0 + a_1 * x + ... + a_n * x^n,
 *	for number coefficients a_0, a_1, ... a_n may also be
 *	constructed as pol(a_0, a_1, ..., a_n).  Note that here the
 *	coefficients are to be in ascending power order.  The independent
 *	variable is pol(0,1), so to use t, say, as an identifier for
 *	this, one may assign  t = pol(0,1).  To simultaneously specify
 *	an identifier and a name for the independent variable, there is
 *	the instruction var, used as in identifier = var(name).  For
 *	example, to use "t" in the way "x" is initially, one may give
 *	the instruction  t = var("t").
 *
 * There are four parameters pmode, order, iod and ims for controlling
 *	the format in which polynomials are displayed.
 *	The parameter pmode may have values "alg" or "list": the
 *	former gives a display as an algebraic formula, while
 *	the latter only lists the coefficients.  Whether the terms or
 *	coefficients are in ascending or descending power order is
 *	controlled by order being "up" or "down".  If the
 *	parameter iod (for integer-only display), the polynomial
 *	is expressed in terms of a polynomial whose coefficients are
 *	integers with gcd = 1, the leading coefficient having positive
 *	real part, with where necessary a leading multiplying integer,
 *	a Gaussian integer multiplier if the coefficients are complex
 *	with a common complex factor, and a trailing divisor integer.
 *	If a non-zero value is assigned to the parameter ims,
 *	multiplication signs will be inserted where appropriate;
 *	this may be useful if the expression is to be copied to a
 *	program or a string to be used with eval. 
 *
 * For evaluation of polynomials the standard function is ev(p, t).
 *	If p is a polynomial and t anything for which the relevant
 *	operations can be performed, this returns the value of p
 *	at t.  The function ev(p, t) also accepts lists or matrices
 *	as possible values for p; each element of p is then evaluated
 *	at t.  For other p, t is ignored and the value of p is returned.
 *	If an identifier, a, say, is used for the polynomial, list or
 *	matrix p, the definition
 *			define a(t) = ev(a, t);
 *	permits a(t) to be used for the value of a at t as if the
 *	polynomial, list or matrix were a function.  For example,
 *	if a = 1 + x^2, a(2) will return the value 5, just as if
 *			define a(t) = 1 + t^2;
 *	had been used.   However, when the polynomial definition is
 *	used, changing the polynomial a will change a(t) to the value
 *	of the new polynomial at t.  For example,
 *	after 
 *		L = list(x, x^2, x^3, x^4);
		define a(t) = ev(a, t);
 *	the loop
 *		for (i = 0; i < 4; i++)
 *			print ev(L[[i]], 5);
 *	may be replaced by
 *		for (i = 0; i < 4; i++) {
 *			a = L[[i]];
 *			print a(5);
 *		} 
 * 	 
 * Matrices with polynomial elements may be added, subtracted and
 *	multiplied as long as the usual rules for compatibility are
 *	observed.  Also, matrices may be multiplied by polynomials,
 *	i.e. if p is a 	polynomial and A a matrix whose elements
 *	may be numbers or polynomials, p * A returns the matrix of
 *	the same shape as A with each element multiplied by p.
 *	Square matrices may also be 'substituted for the variable' in
 *	polynomials, e.g. if A is an m x m matrix, and
 *	p = x^2 + 3 * x + 2, ev(p, A) returns the same as
 *	A^2 + 3 * A + 2 * I, where I is the unit m x m matrix.  
 *	
 * On starting this program, three demonstration polynomials a, b, c
 *	have been defined.  The functions a(t), b(t), c(t) corresponding
 *	to a, b, c, and x(t) corresponding to x, have also been
 *	defined, so the usual function notation can be used for
 *	evaluations of a, b, c and x.  For x, as long as x identifies
 *	the independent variable, x(t) should return the value of t,
 *	i.e. it acts as an identity function.
 *	
 * Functions defined include:
 *
 *	monic(a) returns the monic multiple of a, i.e., if a != 0,
 *		the multiple of	a with leading coefficient 1	
 *	conj(a) returns the complex conjugate of a
 *	ispmult(a,b) returns 1 or 0 according as a is or is not
 *		a polynomial multiple of b
 *	pgcd(a,b) returns the monic gcd of a and b 
 *	pfgcd(a,b) returns a list of three polynomials (g, u, v)
 *		where g = pgcd(a,b) and g = u * a + v * b.
 *	plcm(a,b) returns the monic lcm of a and b
 *
 *	interp(X,Y,t) returns the value at t of the polynomial given
 *		by Newtonian divided difference interpolation, where
 *		X is a list of x-values, Y a list of corresponding
 *		y-values.  If t is omitted, the interpolating
 *		polynomial is returned.  A y-value may be replaced by
 *		list (y, y_1, y_2, ...), where y_1, y_2, ... are
 *		the reduced derivatives at the corresponding x;
 *		i.e. y_r is the r-th derivative divided by fact(r).
 *	mdet(A) returns the determinant of the square matrix A,
 *		computed by an algorithm that does not require
 *		inverses;  the built-in det function usually fails
 *		for matrices with polynomial elements.  
 *	D(a,n) returns the n-th derivative of a; if n is omitted,
 *		the first derivative is returned.
 *
 * A first-time user can see what the initially defined polynomials
 *	a, b and c are, and experiment with the algebraic operations
 *	and other functions that have been defined by giving
 *	instructions like:
 *			a
 *			b
 *			c
 *			(x^2 + 1) * a
 *			a^27
 *			a * b	
 *			a % b
 *			a // b
 *			a(1 + x)
 *			a(b)
 *			conj(c)
 *			g = pgcd(a, b)
 *			g
 *			a / g
 *			D(a)
 *			mat A[2,2] = {1 + x, x^2, 3, 4*x}
 *			mdet(A)
 *			D(A)
 *			A^2
 *			define A(t) = ev(A, t)
 *			A(2)
 *			A(1 + x)
 *			define L(t) = ev(L, t)
 *			L = list(x, x^2, x^3, x^4)
 *			L(5)
 *			a(L)
 *			interp(list(0,1,2,3), list(2,3,5,7))
 *			interp(list(0,1,2), list(0,list(1,0),2))
 *
 * One check on some of the functions is provided by the Cayley-Hamilton
 *	theorem:  if A is any m x m matrix and I the m x m unit matrix,
 *	and x is pol(0,1),
 *			ev(mdet(x * I - A), A)
 *	should return the zero m x m matrix.
 */

obj poly {p};

define pol() {
	local u,i,s;
	obj poly u;
	s = list();
	for (i=1; i<= param(0); i++) append (s,param(i));
	i=size(s) -1;
	while (i>=0 && s[[i]]==0) {i--; remove(s)}
	u.p = s;
	return u;
}

define ispoly(a) {
	local y;
	obj poly y;
	return istype(a,y);
}

define findlist(a) {
	if (ispoly(a)) return a.p;
	if (a) return list(a);
	return list();
}

pmode = "alg";	/* The other acceptable pmode is "list" */
ims = 0;	/* To be non-zero if multiplication signs to be inserted */
iod = 0;	/* To be non-zero for integer-only display */
order = "down"	/* Determines order in which coefficients displayed */

define poly_print(a) {
	local f, g, t;
	if (size(a.p) == 0) {
		print 0:;
		return;
	} 
	if (iod) {
		g = gcdcoeffs(a);
		t = a.p[[size(a.p) - 1]] / g;
		if (re(t) < 0) { t = -t; g = -g;}
		if (g != 1) {
			if (!isreal(t)) {
				if (im(t) > re(t)) g *= 1i;
				else if (im(t) <= -re(t)) g *= -1i;
			}
			if (isreal(g)) f = g;
			else f = gcd(re(g), im(g));
			if (num(f) != 1) {
				print num(f):;
				if (ims) print"*":;
			}
			if (!isreal(g)) {
				printf("(%d)", g/f);
				if (ims) print"*":;
			}
			if (pmode == "alg") print"(":;
			polyprint(1/g * a);
			if (pmode == "alg") print")":;
			if (den(f) > 1) print "/":den(f):;
			return;
		}
	}
	polyprint(a);
}

define polyprint(a) {
	local s,n,i,c;
	s = a.p;
	n=size(s) - 1;
	if (pmode=="alg") {
		if (order == "up") {
			i = 0;
			while (!s[[i]]) i++;
			pterm (s[[i]], i);
			for (i++ ; i <= n; i++) {
				c = s[[i]];
				if (c) {
					if (isreal(c)) {
						if (c > 0) print" + ":;
						else {
							print" - ":;
							c = -c;
						}
					} 
					else print " + ":;
					pterm(c,i);
				}
			}
			return;
		}
		if (order == "down") {
			pterm(s[[n]],n);
			for (i=n-1; i>=0; i--) {
				c = s[[i]];
				if (c) {
					if (isreal(c)) {
						if (c > 0) print" + ":;
						else {
							print" - ":;
							c = -c;
						}
					} 
					else print " + ":;
					pterm(c,i);
				}
			}
			return;
		}
		quit "order to be up or down";
	}
	if (pmode=="list") {
		plist(s);
		return;
	}
	print pmode,:"is unknown mode";
}
		

define poly_neg(a) {
	local s,i,y;
	obj poly y;
	s = a.p;
	for (i=0; i< size(s); i++) s[[i]] = -s[[i]];
	y.p = s;
	return y;
}

define poly_conj(a) {
	local s,i,y;
	obj poly y;
	s = a.p;
	for (i=0; i < size(s); i++) s[[i]] = conj(s[[i]]);
	y.p = s;
	return y;
}

define poly_inv(a) = pol(1)/a;	/* This exists only for a of zero degree */

define poly_add(a,b) {
	local sa, sb, i, y;
	obj poly y;
	sa=findlist(a); sb=findlist(b);
	if (size(sa) > size(sb)) swap(sa,sb);
	for (i=0; i< size(sa); i++) sa[[i]] += sb[[i]];	
	while (i < size(sb)) append (sa, sb[[i++]]);
	while (i > 0 && sa[[--i]]==0) remove (sa);
	y.p = sa;
	return y;
}

define poly_sub(a,b) {
	 return a + (-b);
}

define poly_cmp(a,b) {
	local sa, sb;
	sa = findlist(a);
	sb=findlist(b);
	return  (sa != sb);
}

define poly_mul(a,b) {
	local sa,sb,i, j, y;
	if (ismat(a)) swap(a,b);
	if (ismat(b)) {
		y = b;
		for (i=matmin(b,1); i <= matmax(b,1); i++)
			for (j = matmin(b,2); j<= matmax(b,2); j++)
				y[i,j] = a * b[i,j];
		return y;	
	}
	obj poly y;
	sa=findlist(a); sb=findlist(b);
	y.p = listmul(sa,sb);
	return y;
} 

define listmul(a,b) {
	local da,db, s, i, j, u;
	da=size(a)-1; db=size(b)-1;
	s=list();
	if (da >= 0 && db >= 0) {
		for (i=0; i<= da+db; i++) { u=0;
			for (j = max(0,i-db); j <= min(i, da); j++)
			u += a[[j]]*b[[i-j]]; append (s,u);}}
	return s;
}

define ev(a,t) {
	local v, i, j;
	if (ismat(a)) {
		v = a;
		for (i = matmin(a,1); i <= matmax(a,1); i++)
			for (j = matmin(a,2); j <= matmax(a,2); j++)
				v[i,j] = ev(a[i,j], t);
		return v;
	}
	if (islist(a)) {
		v = list();
		for (i = 0; i < size(a); i++)
			append(v, ev(a[[i]], t));
		return v;
	}
	if (!ispoly(a)) return a;
	if (islist(t)) {
		v = list();
		for (i = 0; i < size(t); i++)
			append(v, ev(a, t[[i]]));
		return v;
	}	
	if (ismat(t)) return evpm(a.p, t);
	return evp(a.p, t); 
}

define evp(s,t) {
	local n,v,i;
	n = size(s);
	if (!n) return 0;
	v = s[[n-1]];
	for (i = n - 2; i >= 0; i--) v=t * v +s[[i]];
	return v;
}

define evpm(s,t) {
	local m, n, V, i, I;
	n = size(s);
	m = matmax(t,1) - matmin(t,1);
	if (matmax(t,2) - matmin(t,2) != m) quit "Non-square matrix";
	mat V[m+1, m+1];
	if (!n) return V;
	mat I[m+1, m+1];
	matfill(I, 0, 1);
	V = s[[n-1]] * I;
	for (i = n - 2; i >= 0; i--) V = t * V + s[[i]] * I;
	return V;
}
pzero = pol(0);
x = pol(0,1); 
varname = "x";
define x(t) = ev(x, t);

define iszero(a) {
	if (ispoly(a))
		return !size(a.p);
	return a == 0;
}

define isstring(a) = istype(a, " ");

define var(name) {
	if (!isstring(name)) quit "Argument of var is to be a string";
	varname = name;
	return pol(0,1);
}

define pcoeff(a) {
		if (isreal(a)) print a:;
		else print "(":a:")":;
}

define pterm(a,n) {
	if (n==0) {
		pcoeff(a);
		return;
	}
	if (n==1) {
		if (a!=1) {
			pcoeff(a);
			if (ims) print"*":;
		}
		print varname:;
		return;
	}
	if (a!=1) {
		pcoeff(a);
		if (ims) print"*":;
	}
	print varname:"^":n:;
} 

define plist(s) {
	local i, n;
	n = size(s);
	print "( ":;
	if (order == "up") {
		for (i=0; i< n-1 ; i++)
			print s[[i]]:",",:;
		if (n) print s[[i]],")":;
		else print "0 )":;
	}
	else {
		if (n) print s[[n-1]]:;
		for (i = n - 2; i >= 0; i--)
			print ", ":s[[i]]:;
		print " )":;
	}
}

define deg(a) = size(a.p) - 1;

define polydiv(a,b) {
	local q, r, d, u, i, m, n, sa, sb, sq;
	obj poly q, r;
	sa=findlist(a); sb = findlist(b); sq = list();
	m=size(sa)-1; n=size(sb)-1;
	if (n<0) quit "Zero divisor";
	if (m<n) return list(pzero, a);
	d = sb[[n]]; 
	while ( m >= n) { u = sa[[m]]/d;
		for (i = 0; i< n; i++) sa[[m-n+i]] -= u*sb[[i]];
		push(sq,u); remove(sa); m--;
		while (m>=n && sa[[m]]==0) { m--; remove(sa); push(sq,0)}}
	while (m>=0 && sa[[m]]==0) { m--; remove(sa);}
	q.p = sq;  r.p = sa;
	return list(q, r);}
		
define poly_mod(a,b)  {
	local u;
	u=polydiv(a,b);
	return u[[1]];
}

define poly_quo(a,b) {
	local p;
	p = polydiv(a,b);
	return p[[0]];
}

define ispmult(a,b) = iszero(a % b);

define poly_div(a,b) {
	if (!ispmult(a,b)) quit "Result not a polynomial";
	return poly_quo(a,b);
}

define pgcd(a,b) {
	local r;
	if (iszero(a) && iszero(b)) return pzero;
	while (!iszero(b)) {
		r = a % b;
		a = b;
		b = r;
	}
	return monic(a);
}

define plcm(a,b) = monic( a * b // pgcd(a,b));
  
define pfgcd(a,b) {
	local u, v, u1, v1, s, q, r, d, w;
	u = v1 = pol(1); v = u1 = pol(0);
	while (size(b.p) > 0) {s = polydiv(a,b);
		q = s[[0]];
		a = b; b = s[[1]]; u -= q*u1; v -= -q*v1;
		swap(u,u1); swap(v,v1);}
	d=size(a.p)-1; if (d>=0 && (w= 1/a.p[[d]]) !=1)
		 { a *= w; u *= w; v *= w;}
	return list(a,u,v);
}
  
define monic(a) {
	local s, c, i, d, y;
	if (iszero(a)) return pzero;
	obj poly y;
	s = findlist(a);
	d = size(s)-1;
	for (i=0; i<=d; i++) s[[i]] /= s[[d]];
	y.p = s;
	return y;
}

define coefficient(a,n) = (n < size(a.p)) ? a.p[[n]] : 0;

define D(a, n) {
	local i,j,v;
	if (isnull(n)) n = 1;
	if (!isint(n) || n < 1) quit "Bad order for derivative";
 	if (ismat(a)) {
		v = a;
		for (i = matmin(a,1); i <= matmax(a,1); i++)
			for (j = matmin(a,2); j <= matmax(a,2); j++)
				v[i,j] = D(a[i,j], n);
		return v;
	}
	if (!ispoly(a)) return 0;
	return Dp(a,n);
}

define Dp(a,n) {
	local i, v;
	if (n > 1) return Dp(Dp(a, n-1), 1);
 	obj poly v;
	v.p=list();	
	for (i=1; i<size(a.p); i++) append (v.p, i*a.p[[i]]);
	return v;
}


define cgcd(a,b) {
	if (isreal(a) && isreal(b)) return gcd(a,b);
	while (a) {
		b -= bround(b/a) * a;
		swap(a,b);
	}
	if (re(b) < 0) b = -b; 
	if (im(b) > re(b)) b *= -1i;
	else if (im(b) <= -re(b)) b *= 1i;
	return b;
}

define gcdcoeffs(a) {
	local s,i,g, c;
	s = a.p;
	g=0;
	for (i=0; i < size(s) && g != 1; i++)
		if (c = s[[i]]) g = cgcd(g, c);
	return g;
}

define interp(X, Y, t) = evalfd(makediffs(X,Y), t);

define makediffs(X,Y) {
	local U, D, d, x, y, i, j, k, m, n, s;
	U = D = list();
	n = size(X);
	if (size(Y) != n) quit"Arguments to be lists of same size";
	for (i = n-1; i >= 0; i--) {
		x = X[[i]];
		y = Y[[i]];
		m = size(U);
		if (isnum(y)) {
			d = y;
			for (j = 0; j < m; j++) {
				d = D[[j]] = (D[[j]]-d)/(U[[j]] - x);
			}
			push(U, x);
			push(D, y);
		}
		else {
			s = size(y);
			for (k = 0; k < s ; k++) {
				d = y[[k]];
				for (j = 0; j < m; j++) {
					d = D[[j]] = (D[[j]] - d)/(U[[j]] - x);
				}
			}
			for (j=s-1; j >=0; j--) {
				push(U,x);
				push(D, y[[j]]);
			}
		}
	}
	return list(U, D);
}
	
define evalfd(T, t) {
	local U, D, n, i, v;
	if (isnull(t)) t = pol(0,1);
	U = T[[0]];
	D = T[[1]];
	n = size(U);
	v = D[[n-1]];
	for (i = n-2; i >= 0; i--) 
		v = v * (t - U[[i]]) + D[[i]];
	return v;
}


define mdet(A) {
	local n, i, j, k, I, J;
	n = matmax(A,1) - (i = matmin(A,1));
	if (matmax(A,2) - (j = matmin(A,2)) != n)
		quit "Non-square matrix for mdet";
	I = J = list();
	k = n + 1;
	while (k--) {
		append(I,i++);
		append(J,j++);
	}
	return M(A, n+1, I, J);
}

define M(A, n, I, J) {
	local v, J0, i, j, j1;
	if (n == 1) return A[ I[[0]], J[[0]] ];
	v = 0;
	i = remove(I);
	for (j = 0; j < n; j++) {
		J0 = J;
		j1 = delete(J0, j);
		v += (-1)^(n-1+j) * A[i, j1] * M(A, n-1, I, J0);
	}
	return v;
}

define mprint(A) {
	local i,j;
	if (!ismat(A)) quit "Argument to be a matrix";
	for (i = matmin(A,1); i <= matmax(A,1); i++) {
		for (j = matmin(A,2); j <= matmax(A,2); j++)
			printf("%8.4d ", A[i,j]);
		printf("\n");
	}
}
	
obj poly a;
obj poly b;
obj poly c;

define a(t) = ev(a,t);
define b(t) = ev(b,t);
define c(t) = ev(c,t);

a=pol(1,4,4,2,3,1);
b=pol(5,16,8,1);
c=pol(1+2i,3+4i,5+6i);

global lib_debug;
if (lib_debug >= 0) {
	print "obj poly {p} defined";
	print "pol() defined";
	print "poly_print(a) defined";
	print "poly_add(a, b) defined";
	print "poly_sub(a, b) defined";
	print "poly_mul(a, b) defined";
	print "poly_div(a, b) defined";
	print "poly_quo(a,b) defined";
	print "poly_mod(a,b) defined";
	print "poly_neg(a) defined";
	print "poly_conj(a) defined";
	print "poly_cmp(a,b) defined";
	print "iszero(a) defined";
	print "plist(a) defined";
	print "listmul(a,b) defined";
	print "ev(a,t) defined";
	print "evp(s,t) defined";
	print "ispoly(a) defined";
	print "isstring(a) defined";
	print "var(name) defined";
	print "pcoeff(a) defined";
	print "pterm(a,n) defined";
	print "deg(a) defined";
	print "polydiv(a,b) defined";
	print "D(a,n) defined";
	print "Dp(a,n) defined";
	print "pgcd(a,b) defined";
	print "plcm(a,b) defined";
	print "monic(a) defined";
	print "pfgcd(a,b) defined";
	print "interp(X,Y,x) defined";
	print "makediffs(X,Y) defined";
	print "evalfd(T,x) defined";
	print "mdet(A) defined";
	print "M(A,n,I,J) defined";
	print "mprint(A) defined";
}