We group here all functions which are specific to polynomials or power series. Many other functions which can be applied on these objects are described in the other sections. Also, some of the functions described here can be applied to other types.
If p is an integer greater than 2, returns a p-adic 0 of precision e. In all other cases, returns a power series zero with precision given by e v, where v is the X-adic valuation of p with respect to its main variable.
The library syntax is GEN ggrando().
GEN zeropadic(GEN p, long e) for a p-adic and
GEN zeroser(long v, long e) for a power series zero in variable v.
Deprecated alias for polresultantext
The library syntax is GEN polresultantext0(GEN A, GEN B, long v = -1) where v is a variable number.
Derivative of x with respect to the main
variable if v is omitted, and with respect to v otherwise. The derivative
of a scalar type is zero, and the derivative of a vector or matrix is done
componentwise. One can use x' as a shortcut if the derivative is with
respect to the main variable of x; and also use x", etc., for multiple
derivatives altough derivn is often preferrable.
By definition, the main variable of a t_POLMOD is the main variable among
the coefficients from its two polynomial components (representative and
modulus); in other words, assuming a polmod represents an element of
R[X]/(T(X)), the variable X is a mute variable and the derivative is
taken with respect to the main variable used in the base ring R.
? f = (x/y)^5; ? deriv(f) %2 = 5/y^5*x^4 ? f' %3 = 5/y^5*x^4 ? deriv(f, 'x) \\ same since 'x is the main variable %4 = 5/y^5*x^4 ? deriv(f, 'y) %5 = -5/y^6*x^5
This function also operates on closures, in which case the variable
must be omitted. It returns a closure performing a numerical
differentiation as per derivnum:
? f(x) = x^2; ? g = deriv(f) ? g(1) %3 = 2.0000000000000000000000000000000000000 ? f(x) = sin(exp(x)); ? deriv(f)(0) %5 = 0.54030230586813971740093660744297660373 ? cos(1) %6 = 0.54030230586813971740093660744297660373
The library syntax is GEN deriv(GEN x, long v = -1) where v is a variable number.
n-th derivative of x with respect to the main variable if v is omitted, and with respect to v otherwise; the integer n must be nonnegative. The derivative of a scalar type is zero, and the derivative of a vector or matrix is done componentwise. One can use x', x", etc., as a shortcut if the derivative is with respect to the main variable of x.
By definition, the main variable of a t_POLMOD is the main variable among
the coefficients from its two polynomial components (representative and
modulus); in other words, assuming a polmod represents an element of
R[X]/(T(X)), the variable X is a mute variable and the derivative is
taken with respect to the main variable used in the base ring R.
? f = (x/y)^5; ? derivn(f, 2) %2 = 20/y^5*x^3 ? f'' %3 = 20/y^5*x^3 ? derivn(f, 2, 'x) \\ same since 'x is the main variable %4 = 20/y^5*x^3 ? derivn(f, 2, 'y) %5 = 30/y^7*x^5
This function also operates on closures, in which case the variable
must be omitted. It returns a closure performing a numerical
differentiation as per derivnum:
? f(x) = x^10; ? g = derivn(f, 5) ? g(1) %3 = 30240.000000000000000000000000000000000 ? derivn(zeta, 2)(0) %4 = -2.0063564559085848512101000267299604382 ? zeta''(0) %5 = -2.0063564559085848512101000267299604382
The library syntax is GEN derivn(GEN x, long n, long v = -1) where v is a variable number.
Let v be a vector of variables, and d a vector of the same length,
return the image of x by the n-power (1 if n is not given) of the
differential operator D that assumes the value d[i] on the variable
v[i]. The value of D on a scalar type is zero, and D applies
componentwise to a vector or matrix. When applied to a t_POLMOD, if no
value is provided for the variable of the modulus, such value is derived
using the implicit function theorem.
Examples. This function can be used to differentiate formal expressions: if E = exp(X2) then we have E' = 2*X*E. We derivate X*exp(X2) as follows:
? diffop(E*X,[X,E],[1,2*X*E]) %1 = (2*X^2 + 1)*E
Let Sin and Cos be two function such that
Sin2+Cos2 = 1 and Cos' = -Sin.
We can differentiate Sin/Cos as follows,
PARI inferring the value of Sin' from the equation:
? diffop(Mod('Sin/'Cos,'Sin^2+'Cos^2-1),['Cos],[-'Sin])
%1 = Mod(1/Cos^2, Sin^2 + (Cos^2 - 1))
Compute the Bell polynomials (both complete and partial) via the Faa di Bruno formula:
Bell(k,n=-1)=
{ my(x, v, dv, var = i->eval(Str("X",i)));
v = vector(k, i, if (i==1, 'E, var(i-1)));
dv = vector(k, i, if (i==1, 'X*var(1)*'E, var(i)));
x = diffop('E,v,dv,k) / 'E;
if (n < 0, subst(x,'X,1), polcoef(x,n,'X));
}
The library syntax is GEN diffop0(GEN x, GEN v, GEN d, long n).
For n = 1, the function GEN diffop(GEN x, GEN v, GEN d) is also
available.
Replaces in x the formal variables by the values that
have been assigned to them after the creation of x. This is mainly useful
in GP, and not in library mode. Do not confuse this with substitution (see
subst).
If x is a character string, eval(x) executes x as a GP
command, as if directly input from the keyboard, and returns its
output.
? x1 = "one"; x2 = "two";
? n = 1; eval(Str("x", n))
%2 = "one"
? f = "exp"; v = 1;
? eval(Str(f, "(", v, ")"))
%4 = 2.7182818284590452353602874713526624978
Note that the first construct could be implemented in a
simpler way by using a vector x = ["one","two"]; x[n], and the second
by using a closure f = exp; f(v). The final example is more interesting:
? genmat(u,v) = matrix(u,v,i,j, eval( Str("x",i,j) ));
? genmat(2,3) \\ generic 2 x 3 matrix
%2 =
[x11 x12 x13]
[x21 x22 x23]
A syntax error in the evaluation expression raises an e_SYNTAX
exception, which can be trapped as usual:
? 1a
*** syntax error, unexpected variable name, expecting $end or ';': 1a
*** ^-
? E(expr) =
{
iferr(eval(expr),
e, print("syntax error"),
errname(e) == "e_SYNTAX");
}
? E("1+1")
%1 = 2
? E("1a")
syntax error
The library syntax is geval(GEN x).
p-adic factorization
of the polynomial pol to precision r, the result being a
two-column matrix as in factor. Note that this is not the same
as a factorization over ℤ/prℤ (polynomials over that ring do not form a
unique factorization domain, anyway), but approximations in ℚ/prℤ of
the true factorization in ℚp[X].
? factorpadic(x^2 + 9, 3,5) %1 = [(1 + O(3^5))*x^2 + O(3^5)*x + (3^2 + O(3^5)) 1] ? factorpadic(x^2 + 1, 5,3) %2 = [ (1 + O(5^3))*x + (2 + 5 + 2*5^2 + O(5^3)) 1] [(1 + O(5^3))*x + (3 + 3*5 + 2*5^2 + O(5^3)) 1]
The factors are normalized so that their leading coefficient is a power of p. The method used is a modified version of the round 4 algorithm of Zassenhaus.
If pol has inexact t_PADIC coefficients, this is not always
well-defined; in this case, the polynomial is first made integral by dividing
out the p-adic content, then lifted to ℤ using truncate
coefficientwise.
Hence we actually factor exactly a polynomial which is only p-adically
close to the input. To avoid pitfalls, we advise to only factor polynomials
with exact rational coefficients.
The library syntax is factorpadic(GEN f,GEN p, long r) . The function factorpadic0 is
deprecated, provided for backward compatibility.
Let w = [1,z,...,zN-1] from some primitive N-roots of unity z
where N is a power of 2, and P be a polynomial < N,
return the unnormalized discrete Fourier transform of P,
{ P(w[i]), 1 ≤ i ≤ N}. Also allow P to be a vector
[p0,...,pn] representing the polynomial ∑i pi Xi.
Composing fft and fftinv returns N times the original input
coefficients.
? w = rootsof1(4); fft(w, x^3+x+1) %1 = [3, 1, -1, 1] ? fftinv(w, %) %2 = [4, 4, 0, 4] ? Polrev(%) / 4 %3 = x^3 + x + 1 ? w = powers(znprimroot(5),3); fft(w, x^3+x+1) %4 = [Mod(3,5),Mod(1,5),Mod(4,5),Mod(1,5)] ? fftinv(w, %) %5 = [Mod(4,5),Mod(4,5),Mod(0,5),Mod(4,5)]
The library syntax is GEN FFT(GEN w, GEN P).
Let w = [1,z,...,zN-1] from some primitive N-roots of unity z
where N is a power of 2, and P be a polynomial < N,
return the unnormalized discrete Fourier transform of P,
{ P(1 / w[i]), 1 ≤ i ≤ N}. Also allow P to be a vector
[p0,...,pn] representing the polynomial ∑i pi Xi.
Composing
fft and fftinv returns N times the original input coefficients.
? w = rootsof1(4); fft(w, x^3+x+1) %1 = [3, 1, -1, 1] ? fftinv(w, %) %2 = [4, 4, 0, 4] ? Polrev(%) / 4 %3 = x^3 + x + 1 ? N = 512; w = rootsof1(N); T = random(1000 * x^(N-1)); ? U = fft(w, T); time = 3 ms. ? V = vector(N, i, subst(T, 'x, w[i])); time = 65 ms. ? exponent(V - U) %7 = -97 ? round(Polrev(fftinv(w,U) / N)) == T %8 = 1
The library syntax is GEN FFTinv(GEN w, GEN P).
formal integration of x with respect to the variable v (wrt. the main variable if v is omitted). Since PARI cannot represent logarithmic or arctangent terms, any such term in the result will yield an error:
? intformal(x^2)
%1 = 1/3*x^3
? intformal(x^2, y)
%2 = y*x^2
? intformal(1/x)
*** at top-level: intformal(1/x)
*** ^ — — — — --
*** intformal: domain error in intformal: residue(series, pole) != 0
The argument x can be of any type. When x is a rational function, we assume that the base ring is an integral domain of characteristic zero.
By definition, the main variable of a t_POLMOD is the main variable
among the coefficients from its two polynomial components
(representative and modulus); in other words, assuming a polmod represents an
element of R[X]/(T(X)), the variable X is a mute variable and the
integral is taken with respect to the main variable used in the base ring R.
In particular, it is meaningless to integrate with respect to the main
variable of x.mod:
? intformal(Mod(1,x^2+1), 'x) *** intformal: incorrect priority in intformal: variable x = x
The library syntax is GEN integ(GEN x, long v = -1) where v is a variable number.
Vector of p-adic roots of the polynomial pol congruent to the
p-adic number a modulo p, and with the same p-adic precision as a.
The number a can be an ordinary p-adic number (type t_PADIC, i.e. an
element of ℤp) or can be an integral element of a finite
unramified extension ℚp[X]/(T) of ℚp, given as a
t_POLMOD
Mod(A,T) at least one of whose coefficients is a t_PADIC and T
irreducible modulo p. In this case, the result is the vector of roots
belonging to the same extension of ℚp as a. The polynomial pol
should have exact coefficients; if not, its coefficients are first rounded
to ℚ or ℚ[X]/(T) and this is the polynomial whose roots we consider.
The library syntax is GEN padicappr(GEN pol, GEN a).
Also available is GEN Zp_appr(GEN f, GEN a) when a is a
t_PADIC.
Returns a vector of polynomials generating all the extensions of degree N of the field ℚp of p-adic rational numbers; N is allowed to be a 2-component vector [n,d], in which case we return the extensions of degree n and discriminant pd.
The list is minimal in the sense that two different polynomials generate nonisomorphic extensions; in particular, the number of polynomials is the number of classes of nonisomorphic extensions. If P is a polynomial in this list, α is any root of P and K = ℚp(α), then α is the sum of a uniformizer and a (lift of a) generator of the residue field of K; in particular, the powers of α generate the ring of p-adic integers of K.
If flag = 1, replace each polynomial P by a vector [P, e, f, d, c] where e is the ramification index, f the residual degree, d the valuation of the discriminant, and c the number of conjugate fields. If flag = 2, only return the number of extensions in a fixed algebraic closure (Krasner's formula), which is much faster.
The library syntax is GEN padicfields0(GEN p, GEN N, long flag).
Also available is
GEN padicfields(GEN p, long n, long d, long flag), which computes
extensions of ℚp of degree n and discriminant pd.
Returns the n-th
Chebyshev polynomial of the first kind Tn (flag = 1) or the second
kind Un (flag = 2), evaluated at a ('x by default). Both series of
polynomials satisfy the 3-term relation
Pn+1 = 2xPn - Pn-1,
and are determined by the initial conditions U0 = T0 = 1, T1 = x,
U1 = 2x. In fact Tn' = n Un-1 and, for all complex numbers z, we
have Tn(cos z) = cos (nz) and Un-1(cos z) = sin(nz)/sin z.
If n ≥ 0, then these polynomials have degree n. For n < 0,
Tn is equal to T-n and Un is equal to -U-2-n.
In particular, U-1 = 0.
The library syntax is GEN polchebyshev_eval(long n, long flag, GEN a = NULL).
Also available are
GEN polchebyshev(long n, long flag, long v),
GEN polchebyshev1(long n, long v) and
GEN polchebyshev2(long n, long v) for Tn and Un respectively.
Return a polynomial in ℤ[x] generating the Hilbert class field for the
imaginary quadratic discriminant D. If inv is 0 (the default),
use the modular j-function and return the classical Hilbert polynomial,
otherwise use a class invariant. The following invariants correspond to
the different values of inv, where f denotes Weber's function
weber, and wp,q the double eta quotient given by
wp,q = (η(x/p) η(x/q) )/(η(x) η(x/{pq}) )
The invariants wp,q are not allowed unless they satisfy the following technical conditions ensuring they do generate the Hilbert class field and not a strict subfield:
* if p ! = q, we need them both noninert, prime to the conductor of ℤ[sqrt{D}]. Let P, Q be prime ideals above p and q; if both are unramified, we further require that P± 1 Q± 1 be all distinct in the class group of ℤ[sqrt{D}]; if both are ramified, we require that PQ ! = 1 in the class group.
* if p = q, we want it split and prime to the conductor and the prime ideal above it must have order ! = 1, 2, 4 in the class group.
Invariants are allowed under the additional conditions on D listed below.
* 0 : j
* 1 : f, D = 1 mod 8 and D = 1,2 mod 3;
* 2 : f2, D = 1 mod 8 and D = 1,2 mod 3;
* 3 : f3, D = 1 mod 8;
* 4 : f4, D = 1 mod 8 and D = 1,2 mod 3;
* 5 : γ2 = j1/3, D = 1,2 mod 3;
* 6 : w2,3, D = 1 mod 8 and D = 1,2 mod 3;
* 8 : f8, D = 1 mod 8 and D = 1,2 mod 3;
* 9 : w3,3, D = 1 mod 2 and D = 1,2 mod 3;
* 10: w2,5, D ! = 60 mod 80 and D = 1,2 mod 3;
* 14: w2,7, D = 1 mod 8;
* 15: w3,5, D = 1,2 mod 3;
* 21: w3,7, D = 1 mod 2 and 21 does not divide D
* 23: w2,32, D = 1,2 mod 3;
* 24: w2,52, D = 1,2 mod 3;
* 26: w2,13, D ! = 156 mod 208;
* 27: w2,72, D ! = 28 mod 112;
* 28: w3,32, D = 1,2 mod 3;
* 35: w5,7, D = 1,2 mod 3;
* 39: w3,13, D = 1 mod 2 and D = 1,2 mod 3;
The algorithm for computing the polynomial does not use the floating point approach, which would evaluate a precise modular function in a precise complex argument. Instead, it relies on a faster Chinese remainder based approach modulo small primes, in which the class invariant is only defined algebraically by the modular polynomial relating the modular function to j. So in fact, any of the several roots of the modular polynomial may actually be the class invariant, and more precise assertions cannot be made.
For instance, while polclass(D) returns the minimal polynomial of
j(τ) with τ (any) quadratic integer for the discriminant D,
the polynomial returned by polclass(D, 5) can be the minimal polynomial
of any of γ2 (τ), ζ3 γ2 (τ) or
ζ32 γ2 (τ), the three roots of the modular polynomial
j = γ23, in which j has been specialised to j (τ).
The modular polynomial is given by j = ((f24-16)3 )/(f24) for Weber's function f.
For the double eta quotients of level N = p q, all functions are covered such that the modular curve X0+ (N), the function field of which is generated by the functions invariant under Γ0 (N) and the Fricke-Atkin-Lehner involution, is of genus 0 with function field generated by (a power of) the double eta quotient w. This ensures that the full Hilbert class field (and not a proper subfield) is generated by class invariants from these double eta quotients. Then the modular polynomial is of degree 2 in j, and of degree ψ (N) = (p+1)(q+1) in w.
? polclass(-163) %1 = x + 262537412640768000 ? polclass(-51, , 'z) %2 = z^2 + 5541101568*z + 6262062317568 ? polclass(-151,1) x^7 - x^6 + x^5 + 3*x^3 - x^2 + 3*x + 1
The library syntax is GEN polclass(GEN D, long inv, long x = -1) where x is a variable number.
Coefficient of degree n of the polynomial x, with respect to the main variable if v is omitted, with respect to v otherwise. If n is greater than the degree, the result is zero.
Naturally applies to scalars (polynomial of degree 0), as well as to rational functions whose denominator is a monomial. It also applies to power series: if n is less than the valuation, the result is zero. If it is greater than the largest significant degree, then an error message is issued.
The library syntax is GEN polcoef(GEN x, long n, long v = -1) where v is a variable number.
Deprecated alias for polcoef.
The library syntax is GEN polcoef(GEN x, long n, long v = -1) where v is a variable number.
n-th cyclotomic polynomial, evaluated at a ('x by default). The
integer n must be positive.
Algorithm used: reduce to the case where n is squarefree; to compute the
cyclotomic polynomial, use Φnp(x) = Φn(xp)/Φ(x); to compute
it evaluated, use Φn(x) = ∏d | n (xd-1)μ(n/d). In the
evaluated case, the algorithm assumes that ad - 1 is either 0 or
invertible, for all d | n. If this is not the case (the base ring has
zero divisors), use subst(polcyclo(n),x,a).
The library syntax is GEN polcyclo_eval(long n, GEN a = NULL).
The variant GEN polcyclo(long n, long v) returns the n-th
cyclotomic polynomial in variable v.
Returns a vector of polynomials, whose product is the product of distinct cyclotomic polynomials dividing f.
? f = x^10+5*x^8-x^7+8*x^6-4*x^5+8*x^4-3*x^3+7*x^2+3; ? v = polcyclofactors(f) %2 = [x^2 + 1, x^2 + x + 1, x^4 - x^3 + x^2 - x + 1] ? apply(poliscycloprod, v) %3 = [1, 1, 1] ? apply(poliscyclo, v) %4 = [4, 3, 10]
In general, the polynomials are products of cyclotomic polynomials and not themselves irreducible:
? g = x^8+2*x^7+6*x^6+9*x^5+12*x^4+11*x^3+10*x^2+6*x+3; ? polcyclofactors(g) %2 = [x^6 + 2*x^5 + 3*x^4 + 3*x^3 + 3*x^2 + 2*x + 1] ? factor(%[1]) %3 = [ x^2 + x + 1 1] [x^4 + x^3 + x^2 + x + 1 1]
The library syntax is GEN polcyclofactors(GEN f).
Degree of the polynomial x in the main variable if v is omitted, in the variable v otherwise.
The degree of 0 is -oo. The degree of a nonzero scalar is 0.
Finally, when x is a nonzero polynomial or rational function, returns the
ordinary degree of x. Raise an error otherwise.
The library syntax is GEN gppoldegree(GEN x, long v = -1) where v is a variable number.
Also available is
long poldegree(GEN x, long v), which returns -LONG_MAX if x = 0
and the degree as a long integer.
Discriminant of the polynomial pol in the main variable if v is omitted, in v otherwise. Uses a modular algorithm over ℤ or ℚ, and the subresultant algorithm otherwise.
? T = x^4 + 2*x+1; ? poldisc(T) %2 = -176 ? poldisc(T^2) %3 = 0
For convenience, the function also applies to types t_QUAD and
t_QFB:
? z = 3*quadgen(8) + 4; ? poldisc(z) %2 = 8 ? q = Qfb(1,2,3); ? poldisc(q) %4 = -8
The library syntax is GEN poldisc0(GEN pol, long v = -1) where v is a variable number.
Given a polynomial T with integer coefficients, return
[D, faD] where D is the discriminant of T and
faD is a cheap partial factorization of |D|: entries in its first
column are coprime and not perfect powers but need not be primes.
The factors are obtained by a combination of trial division, testing for
perfect powers, factorizations in coprimes, and computing Euclidean
remainder sequences for (T,T') modulo composite factors d of D
(which is likely to produce 0-divisors in ℤ/dℤ).
If flag is 1, finish the factorization using factorint.
? T = x^3 - 6021021*x^2 + 12072210077769*x - 8092423140177664432; ? [D,faD] = poldiscfactors(T); print(faD); D [3, 3; 7, 2; 373, 2; 500009, 2; 24639061, 2] %2 = -27937108625866859018515540967767467 ? T = x^3 + 9*x^2 + 27*x - 125014250689643346789780229390526092263790263725; ? [D,faD] = poldiscfactors(T); print(faD) [2, 6; 3, 3; 125007125141751093502187, 4] ? [D,faD] = poldiscfactors(T, 1); print(faD) [2, 6; 3, 3; 500009, 12; 1000003, 4]
The library syntax is GEN poldiscfactors(GEN T, long flag).
Reduced discriminant vector of the (integral, monic) polynomial f. This is the vector of elementary divisors of ℤ[α]/f'(α)ℤ[α], where α is a root of the polynomial f. The components of the result are all positive, and their product is equal to the absolute value of the discriminant of f.
The library syntax is GEN reduceddiscsmith(GEN f).
Returns the monic polynomial in variable v whose roots are the
components of the vector a with multiplicities, that is
∏i (x - ai).
? polfromroots([1,2,3]) %1 = x^3 - 6*x^2 + 11*x - 6 ? polfromroots([z, -z], 'y) %2 = y^2 - z^2
The library syntax is GEN polfromroots(GEN a, long v = -1) where v is a variable number.
Returns the Graeffe transform g of f, such that g(x2) = f(x) f(-x).
The library syntax is GEN polgraeffe(GEN f).
Given a prime p, an integral polynomial A whose leading coefficient is a p-unit, a vector B of integral polynomials that are monic and pairwise relatively prime modulo p, and whose product is congruent to A/lc(A) modulo p, lift the elements of B to polynomials whose product is congruent to A modulo pe.
More generally, if T is an integral polynomial irreducible mod p, and B is a factorization of A over the finite field 𝔽p[t]/(T), you can lift it to ℤp[t]/(T, pe) by replacing the p argument with [p,T]:
? { T = t^3 - 2; p = 7; A = x^2 + t + 1;
B = [x + (3*t^2 + t + 1), x + (4*t^2 + 6*t + 6)];
r = polhensellift(A, B, [p, T], 6) }
%1 = [x + (20191*t^2 + 50604*t + 75783), x + (97458*t^2 + 67045*t + 41866)]
? liftall( r[1] * r[2] * Mod(Mod(1,p^6),T) )
%2 = x^2 + (t + 1)
The library syntax is GEN polhensellift(GEN A, GEN B, GEN p, long e).
n-th Hermite polynomial Hn evaluated at a
('x by default), i.e.
Hn(x) = (-1)n ex^{2} (dn)/(dxn)e-x^{2}.
If flag is nonzero and n > 0, return [Hn-1(a), Hn(a)].
? polhermite(5) %1 = 32*x^5 - 160*x^3 + 120*x ? polhermite(5, -2) \\ H5(-2) %2 = 16 ? polhermite(5,,1) %3 = [16*x^4 - 48*x^2 + 12, 32*x^5 - 160*x^3 + 120*x] ? polhermite(5,-2,1) %4 = [76, 16]
The library syntax is GEN polhermite_eval0(long n, GEN a = NULL, long flag).
The variant GEN polhermite(long n, long v) returns the n-th
Hermite polynomial in variable v. To obtain Hn(a),
use GEN polhermite_eval(long n, GEN a).
Given the data vectors X and Y of the same length n (X containing the x-coordinates, and Y the corresponding y-coordinates), this function finds the interpolating polynomial P of minimal degree passing through these points and evaluates it at t. If Y is omitted, the polynomial P interpolates the (i,X[i]).
? v = [1, 2, 4, 8, 11, 13]; ? P = polinterpolate(v) \\ formal interpolation %1 = 7/120*x^5 - 25/24*x^4 + 163/24*x^3 - 467/24*x^2 + 513/20*x - 11 ? [ subst(P,'x,a) | a <- [1..6] ] %2 = [1, 2, 4, 8, 11, 13] ? polinterpolate(v,, 10) \\ evaluate at 10 %3 = 508 ? subst(P, x, 10) %4 = 508 ? P = polinterpolate([1,2,4], [9,8,7]) %5 = 1/6*x^2 - 3/2*x + 31/3 ? [subst(P, 'x, a) | a <- [1,2,4]] %6 = [9, 8, 7] ? P = polinterpolate([1,2,4], [9,8,7], 0) %7 = 31/3
If the goal is to extrapolate a function at a unique point, it is more efficient to use the t argument rather than interpolate formally then evaluate:
? x0 = 1.5; ? v = vector(20, i,random([-10,10])); ? for(i=1,10^3, subst(polinterpolate(v),'x, x0)) time = 352 ms. ? for(i=1,10^3, polinterpolate(v,,x0)) time = 111 ms. ? v = vector(40, i,random([-10,10])); ? for(i=1,10^3, subst(polinterpolate(v), 'x, x0)) time = 3,035 ms. ? for(i=1,10^3, polinterpolate(v,, x0)) time = 436 ms.
The threshold depends on the base field. Over small prime finite fields, interpolating formally first is more efficient
? bench(p, N, T = 10^3) =
{ my (v = vector(N, i, random(Mod(0,p))));
my (x0 = Mod(3, p), t1, t2);
gettime();
for(i=1, T, subst(polinterpolate(v), 'x, x0));
t1 = gettime();
for(i=1, T, polinterpolate(v,, x0));
t2 = gettime(); [t1, t2];
}
? p = 101;
? bench(p, 4, 10^4) \\ both methods are equivalent
%3 = [39, 40]
? bench(p, 40) \\ with 40 points formal is much faster
%4 = [45, 355]
As the cardinality increases, formal interpolation requires more points to become interesting:
? p = nextprime(2^128); ? bench(p, 4) \\ formal is slower %3 = [16, 9] ? bench(p, 10) \\ formal has become faster %4 = [61, 70] ? bench(p, 100) \\ formal is much faster %5 = [1682, 9081] ? p = nextprime(10^500); ? bench(p, 4) \\ formal is slower %7 = [72, 354] ? bench(p, 20) \\ formal is still slower %8 = [1287, 962] ? bench(p, 40) \\ formal has become faster %9 = [3717, 4227] ? bench(p, 100) \\ faster but relatively less impressive %10 = [16237, 32335]
If t is a complex numeric value and e is present, e will contain an
error estimate on the returned value. More precisely, let P be the
interpolation polynomial on the given n points; there exist a subset
of n-1 points and Q the attached interpolation polynomial
such that e = exponent(P(t) - Q(t)) (Neville's algorithm).
? f(x) = 1 / (1 + 25*x^2);
? x0 = 975/1000;
? test(X) =
{ my (P, e);
P = polinterpolate(X, [f(x) | x <- X], x0, &e);
[ exponent(P - f(x0)), e ];
}
\\ equidistant nodes vs. Chebyshev nodes
? test( [-10..10] / 10 )
%4 = [6, 5]
? test( polrootsreal(polchebyshev(21)) )
%5 = [-15, -10]
? test( [-100..100] / 100 )
%7 = [93, 97] \\ P(x0) is way different from f(x0)
? test( polrootsreal(polchebyshev(201)) )
%8 = [-60, -55]
This is an example of Runge's phenomenon: increasing the number of equidistant nodes makes extrapolation much worse. Note that the error estimate is not a guaranteed upper bound (cf %4), but is reasonably tight in practice.
Numerical stability. The interpolation is performed in a numerically stable way using ∏j ! = i (X[i] - X[j]) instead of Q'(X[i]) with Q = ∏i (x - X[i]). Centering the interpolation points X[i] around 0, thereby reconstructing P(x - m), for a suitable m will further reduce the numerical error.
The library syntax is GEN polint(GEN X, GEN Y = NULL, GEN t = NULL, GEN *e = NULL).
P being a monic irreducible polynomial with integer coefficients,
return 0 if P is not a class polynomial for the j-invariant,
otherwise return the discriminant D < 0 such that P = polclass(D).
? polisclass(polclass(-47)) %1 = -47 ? polisclass(x^5+x+1) %2 = 0 ? apply(polisclass,factor(poldisc(polmodular(5)))[,1]) %3 = [-16,-4,-3,-11,-19,-64,-36,-24,-51,-91,-99,-96,-84]~
The library syntax is long polisclass(GEN P).
Returns 0 if f is not a cyclotomic polynomial, and n > 0 if f = Φn, the n-th cyclotomic polynomial.
? poliscyclo(x^4-x^2+1) %1 = 12 ? polcyclo(12) %2 = x^4 - x^2 + 1 ? poliscyclo(x^4-x^2-1) %3 = 0
The library syntax is long poliscyclo(GEN f).
Returns 1 if f is a product of cyclotomic polynomial, and 0 otherwise.
? f = x^6+x^5-x^3+x+1; ? poliscycloprod(f) %2 = 1 ? factor(f) %3 = [ x^2 + x + 1 1] [x^4 - x^2 + 1 1] ? [ poliscyclo(T) | T <- %[,1] ] %4 = [3, 12] ? polcyclo(3) * polcyclo(12) %5 = x^6 + x^5 - x^3 + x + 1
The library syntax is long poliscycloprod(GEN f).
pol being a polynomial (univariate in the present version 2.17.0), returns 1 if pol is nonconstant and irreducible, 0 otherwise. Irreducibility is checked over the smallest base field over which pol seems to be defined.
The library syntax is long polisirreducible(GEN pol).
n-th Laguerre polynomial L(a)n of degree n and
parameter a evaluated at b ('x by default), i.e.
Ln(a)(x) =
(x-ae×)/(n!) (dn)/(dxn)(e-xxn+a).
If flag is 1, return [L(a)n-1(b), Ln(a)(b)].
The library syntax is GEN pollaguerre_eval0(long n, GEN a = NULL, GEN b = NULL, long flag).
To obtain the n-th Laguerre polynomial in variable v,
use GEN pollaguerre(long n, GEN a, GEN b, long v). To obtain
L(a)n(b), use GEN pollaguerre_eval(long n, GEN a, GEN b).
Leading coefficient of the polynomial or power series x. This is computed with respect to the main variable of x if v is omitted, with respect to the variable v otherwise.
The library syntax is GEN pollead(GEN x, long v = -1) where v is a variable number.
n-th Legendre polynomial Pn evaluated at a
('x by default), where
Pn(x) = (1)/(2n n!) (dn)/(dxn)(x2-1)n .
If flag is 1, return [Pn-1(a), Pn(a)].
The library syntax is GEN pollegendre_eval0(long n, GEN a = NULL, long flag).
To obtain the n-th Legendre polynomial Pn in variable v,
use GEN pollegendre(long n, long v). To obtain Pn(a),
use GEN pollegendre_eval(long n, GEN a).
Return the modular polynomial of prime level L in variables x and y
for the modular function specified by inv. If inv is 0 (the
default), use the modular j function, if inv is 1 use the
Weber-f function, and if inv is 5 use γ2 =
sqrt[3]{j}.
See polclass for the full list of invariants.
If x is given as Mod(j, p) or an element j of
a finite field (as a t_FFELT), then return the modular polynomial of
level L evaluated at j. If j is from a finite field and
derivs is nonzero, then return a triple where the
last two elements are the first and second derivatives of the modular
polynomial evaluated at j.
? polmodular(3) %1 = x^4 + (-y^3 + 2232*y^2 - 1069956*y + 36864000)*x^3 + ... ? polmodular(7, 1, , 'J) %2 = x^8 - J^7*x^7 + 7*J^4*x^4 - 8*J*x + J^8 ? polmodular(7, 5, 7*ffgen(19)^0, 'j) %3 = j^8 + 4*j^7 + 4*j^6 + 8*j^5 + j^4 + 12*j^2 + 18*j + 18 ? polmodular(7, 5, Mod(7,19), 'j) %4 = Mod(1, 19)*j^8 + Mod(4, 19)*j^7 + Mod(4, 19)*j^6 + ... ? u = ffgen(5)^0; T = polmodular(3,0,,'j)*u; ? polmodular(3, 0, u,'j,1) %6 = [j^4 + 3*j^2 + 4*j + 1, 3*j^2 + 2*j + 4, 3*j^3 + 4*j^2 + 4*j + 2] ? subst(T,x,u) %7 = j^4 + 3*j^2 + 4*j + 1 ? subst(T',x,u) %8 = 3*j^2 + 2*j + 4 ? subst(T'',x,u) %9 = 3*j^3 + 4*j^2 + 4*j + 2
The library syntax is GEN polmodular(long L, long inv, GEN x = NULL, long y = -1, long derivs) where y is a variable number.
Reciprocal polynomial of pol with respect to its main variable, i.e. the coefficients of the result are in reverse order; pol must be a polynomial.
? polrecip(x^2 + 2*x + 3) %1 = 3*x^2 + 2*x + 1 ? polrecip(2*x + y) %2 = y*x + 2
The library syntax is GEN polrecip(GEN pol).
Resultant of the two
polynomials x and y with exact entries, with respect to the main
variables of x and y if v is omitted, with respect to the variable v
otherwise. The algorithm assumes the base ring is a domain. If you also need
the u and v such that x*u + y*v = Res(x,y), use the
polresultantext function.
If flag = 0 (default), uses the algorithm best suited to the inputs, either the subresultant algorithm (Lazard/Ducos variant, generic case), a modular algorithm (inputs in ℚ[X]) or Sylvester's matrix (inexact inputs).
If flag = 1, uses the determinant of Sylvester's matrix instead; this should always be slower than the default.
If x or y are multivariate with a huge polynomial content, it is advisable to remove it before calling this function. Compare:
? a = polcyclo(7) * ((t+1)/(t+2))^100;
? b = polcyclo(11)* ((t+2)/(t+3))^100);
? polresultant(a,b);
time = 3,833 ms.
? ca = content(a); cb = content(b); \
polresultant(a/ca,b/cb)*ca^poldegree(b)*cb*poldegree(a); \\ instantaneous
The function only removes rational denominators and does not compute automatically the content because it is generically small and potentially very expensive (e.g. in multivariate contexts). The choice is yours, depending on your application.
The library syntax is GEN polresultant0(GEN x, GEN y, long v = -1, long flag) where v is a variable number.
Finds polynomials U and V such that A*U + B*V = R, where R is the resultant of U and V with respect to the main variables of A and B if v is omitted, and with respect to v otherwise. Returns the row vector [U,V,R]. The algorithm used (subresultant) assumes that the base ring is a domain.
? A = x*y; B = (x+y)^2; ? [U,V,R] = polresultantext(A, B) %2 = [-y*x - 2*y^2, y^2, y^4] ? A*U + B*V %3 = y^4 ? [U,V,R] = polresultantext(A, B, y) %4 = [-2*x^2 - y*x, x^2, x^4] ? A*U+B*V %5 = x^4
The library syntax is GEN polresultantext0(GEN A, GEN B, long v = -1) where v is a variable number.
Also available is
GEN polresultantext(GEN x, GEN y).
Complex roots of the polynomial T, given as a column vector where each
root is repeated according to its multiplicity and given as floating point
complex numbers at the current realprecision:
? polroots(x^2) %1 = [0.E-38 + 0.E-38*I, 0.E-38 + 0.E-38*I]~ ? polroots(x^3+1) %2 = [-1.00... + 0.E-38*I, 0.50... - 0.866...*I, 0.50... + 0.866...*I]~
The algorithm used is a modification of Schönhage's root-finding algorithm, due to and originally implemented by Gourdon. It runs in polynomial time in deg(T) and the precision. If furthermore T has rational coefficients, roots are guaranteed to the required relative accuracy. If the input polynomial T is exact, then the ordering of the roots does not depend on the precision: they are ordered by increasing |Im z|, then by increasing Re z; in case of tie (conjugates), the root with negative imaginary part comes first.
The library syntax is GEN roots(GEN T, long prec).
Return a sharp upper bound B for the modulus of the largest complex root of the polynomial T with complex coefficients with relative error τ. More precisely, we have |z| ≤ B for all roots and there exist one root such that |z0| ≥ B exp(-2τ). Much faster than either polroots or polrootsreal.
? T=poltchebi(500); ? vecmax(abs(polroots(T))) time = 5,706 ms. %2 = 0.99999506520185816611184481744870013191 ? vecmax(abs(polrootsreal(T))) time = 1,972 ms. %3 = 0.99999506520185816611184481744870013191 ? polrootsbound(T) time = 217 ms. %4 = 1.0098792554165905155 ? polrootsbound(T, log(2)/2) \\ allow a factor 2, much faster time = 51 ms. %5 = 1.4065759938190154354 ? polrootsbound(T, 1e-4) time = 504 ms. %6 = 1.0000920717983847741 ? polrootsbound(T, 1e-6) time = 810 ms. %7 = 0.9999960628901692905 ? polrootsbound(T, 1e-10) time = 1,351 ms. %8 = 0.9999950652993869760
The library syntax is GEN polrootsbound(GEN T, GEN tau = NULL).
Obsolete, kept for backward compatibility: use factormod.
The library syntax is GEN polrootsff(GEN x, GEN p = NULL, GEN a = NULL).
Vector of roots of the polynomial f over the finite field defined by the domain D as follows:
* D = p a prime: factor over 𝔽p;
* D = [T,p] for a prime p and T(y) an irreducible polynomial over 𝔽p: factor over 𝔽p[y]/(T) (as usual the main variable of T must have lower priority than the main variable of f);
* D a t_FFELT: factor over the attached field;
* D omitted: factor over the field of definition of f, which must be a finite field.
Multiple roots are not repeated.
? polrootsmod(x^2-1,2)
%1 = [Mod(1, 2)]~
? polrootsmod(x^2+1,3)
%2 = []~
? polrootsmod(x^2+1, [y^2+1,3])
%3 = [Mod(Mod(1, 3)*y, Mod(1, 3)*y^2 + Mod(1, 3)),
Mod(Mod(2, 3)*y, Mod(1, 3)*y^2 + Mod(1, 3))]~
? polrootsmod(x^2 + Mod(1,3))
%4 = []~
? liftall( polrootsmod(x^2 + Mod(Mod(1,3),y^2+1)) )
%5 = [y, 2*y]~
? t = ffgen(y^2+Mod(1,3)); polrootsmod(x^2 + t^0)
%6 = [y, 2*y]~
The library syntax is GEN polrootsmod(GEN f, GEN D = NULL).
Vector of p-adic roots of the polynomial pol, given to p-adic precision r; the integer p is assumed to be a prime. Multiple roots are not repeated. Note that this is not the same as the roots in ℤ/prℤ, rather it gives approximations in ℤ/prℤ of the true roots living in ℚp:
? polrootspadic(x^3 - x^2 + 64, 2, 4) %1 = [2^3 + O(2^4), 2^3 + O(2^4), 1 + O(2^4)]~ ? polrootspadic(x^3 - x^2 + 64, 2, 5) %2 = [2^3 + O(2^5), 2^3 + 2^4 + O(2^5), 1 + O(2^5)]~
As the second commands show, the first two roots are distinct in ℚp, even though they are equal modulo 24.
More generally, if T is an integral polynomial irreducible mod p and f has coefficients in ℚ[t]/(T), the argument p may be replaced by the vector [T,p]; we then return the roots of f in the unramified extension ℚp[t]/(T).
? polrootspadic(x^3 - x^2 + 64*y, [y^2+y+1,2], 5)
%3 = [Mod((2^3 + O(2^5))*y + (2^3 + O(2^5)), y^2 + y + 1),
Mod((2^3 + 2^4 + O(2^5))*y + (2^3 + 2^4 + O(2^5)), y^2 + y + 1),
Mod(1 + O(2^5), y^2 + y + 1)]~
If pol has inexact t_PADIC coefficients, this need not
well-defined; in this case, the polynomial is first made integral by
dividing out the p-adic content, then lifted to ℤ using truncate
coefficientwise. Hence the roots given are approximations of the roots of an
exact polynomial which is p-adically close to the input. To avoid pitfalls,
we advise to only factor polynomials with exact rational coefficients.
The library syntax is GEN polrootspadic(GEN f, GEN p, long r).
Real roots of the polynomial T with real coefficients, multiple
roots being included according to their multiplicity. If the polynomial
does not have rational coefficients, it is first rescaled and rounded.
The roots are given to a relative accuracy of realprecision.
If argument ab is
present, it must be a vector [a,b] with two components (of type
t_INT, t_FRAC or t_INFINITY) and we restrict to roots belonging
to that closed interval.
? \p9 ? polrootsreal(x^2-2) %1 = [-1.41421356, 1.41421356]~ ? polrootsreal(x^2-2, [1,+oo]) %2 = [1.41421356]~ ? polrootsreal(x^2-2, [2,3]) %3 = []~ ? polrootsreal((x-1)*(x-2), [2,3]) %4 = [2.00000000]~
The algorithm used is a modification of Uspensky's method (relying on
Descartes's rule of sign), following Rouillier and Zimmerman's article
"Efficient isolation of a polynomial real roots"
(https://hal.inria.fr/inria-00072518/). Barring bugs, it is guaranteed
to converge and to give the roots to the required accuracy.
Remark. If the polynomial T is of the form Q(xh) for some h ≥ 2 and ab is omitted, the routine will apply the algorithm to Q (restricting to nonnegative roots when h is even), then take h-th roots. On the other hand, if you want to specify ab, you should apply the routine to Q yourself and a suitable interval [a',b'] using approximate h-th roots adapted to your problem: the function will not perform this change of variables if ab is present.
The library syntax is GEN realroots(GEN T, GEN ab = NULL, long prec).
Number of distinct real roots of the real polynomial T. If
the argument ab is present, it must be a vector [a,b] with
two real components (of type t_INT, t_REAL, t_FRAC
or t_INFINITY) and we count roots belonging to that closed interval.
If possible, you should stick to exact inputs, that is avoid t_REALs in
T and the bounds a,b: the result is then guaranteed and we use a fast
algorithm (Uspensky's method, relying on Descartes's rule of sign, see
polrootsreal). Otherwise, the polynomial is rescaled and rounded first
and the result may be wrong due to that initial error. If only a or b is
inexact, on the other hand, the interval is first thickened using rational
endpoints and the result remains guaranteed unless there exist a root
very close to a nonrational endpoint (which may be missed or unduly
included).
? T = (x-1)*(x-2)*(x-3); ? polsturm(T) %2 = 3 ? polsturm(T, [-oo,2]) %3 = 2 ? polsturm(T, [1/2,+oo]) %4 = 3 ? polsturm(T, [1, Pi]) \\ Pi inexact: not recommended ! %5 = 3 ? polsturm(T*1., [0, 4]) \\ T*1. inexact: not recommended ! %6 = 3 ? polsturm(T^2, [0, 4]) \\ not squarefree: roots are not repeated! %7 = 3
The library syntax is long RgX_sturmpart(GEN T, GEN ab) or
long sturm(GEN T) (for the case ab = NULL). The function
long sturmpart(GEN T, GEN a, GEN b) is obsolete and deprecated.
Gives polynomials (in variable v) defining the (Abelian) subextensions of degree d of the cyclotomic field ℚ(ζn), where d | φ(n).
If there is exactly one such extension the output is a polynomial, else it is
a vector of polynomials, possibly empty. To get a vector in all cases,
use concat([], polsubcyclo(n,d)).
Each such polynomial is the minimal polynomial for a Gaussian period Trℚ(ζ_{f)/L} (ζf), where L is the degree d subextension of ℚ(ζn) and f | n is its conductor. In Galois-theoretic terms, L = ℚ(ζn)H, where H runs through all index d subgroups of (ℤ/nℤ)*.
The function galoissubcyclo allows to specify exactly which
sub-Abelian extension should be computed by giving H.
Complexity. Ignoring logarithmic factors, polsubcyclo runs
in time O(n). The function polsubcyclofast returns different, less
canonical, polynomials but runs in time O(d4), again ignoring logarithmic
factors; thus it can handle much larger values of n.
The library syntax is GEN polsubcyclo(long n, long d, long v = -1) where v is a variable number.
If 1 ≤ d ≤ 6 or a prime, finds an equation for the subfields of
ℚ(ζn) with Galois group Cd; the special value d = -4 provides
the subfields with group V4 = C2 x C2. Contrary to
polsubcyclo, the
output is always a (possibly empty) vector of polynomials. If s = 0 (default)
all signatures, otherwise s = 1 (resp., -1) for totally real
(resp., totally complex). Set exact = 1 for subfields of conductor n.
The argument n can be given as in arithmetic functions: as an integer, as a
factorization matrix, or (preferred) as a pair [N, factor(N)].
Comparison with polsubcyclo. First polsubcyclofast
does not usually return Gaussian periods, but ad hoc polynomials which do
generate the same field. Roughly speaking (ignoring
logarithmic factors), the complexity of polsubcyclo is independent of
d and the complexity of polsubcyclofast is independent of n.
Ignoring logarithmic factors, polsubcylo runs in time O(n) and
polsubcyclofast in time O(d4).
So the latter is much faster than polsubcyclo if n is large,
but gets slower as d increases and becomes unusable for d ≥ 40 or so.
? polsubcyclo(10^7+19,7); time = 1,852 ms. ? polsubcyclofast(10^7+19,7); time = 15 ms. ? polsubcyclo(10^17+21,5); \\ won't finish *** polsubcyclo: user interrupt after 2h ? polsubcyclofast(10^17+21,5); time = 3 ms. ? polsubcyclofast(10^17+3,7); time = 26 ms. ? polsubcyclo(10^6+117,13); time = 193 ms. ? polsubcyclofast(10^6+117,13); time = 50 ms. ? polsubcyclofast(10^6+199,19); time = 202 ms. ? polsubcyclo(10^6+199,19); \\ about as fast time = 3191ms. ? polsubcyclo(10^7+271,19); time = 2,067 ms. ? polsubcyclofast(10^7+271,19); time = 201 ms.
The library syntax is GEN polsubcyclofast(GEN n, long d, long s, long exact).
Forms the Sylvester matrix corresponding to the two polynomials x and y, where the coefficients of the polynomials are put in the columns of the matrix (which is the natural direction for solving equations afterwards). The use of this matrix can be essential when dealing with polynomials with inexact entries, since polynomial Euclidean division doesn't make much sense in this case.
The library syntax is GEN sylvestermatrix(GEN x, GEN y).
Creates the column vector of the symmetric powers of the roots of the polynomial x up to power n, using Newton's formula.
The library syntax is GEN polsym(GEN x, long n).
Deprecated alias for polchebyshev
The library syntax is GEN polchebyshev1(long n, long v = -1) where v is a variable number.
Given T ∈ 𝔽p[X] return the polynomial P ∈ ℤp[X] whose roots (resp. leading coefficient) are the Teichmuller lifts of the roots (resp. leading coefficient) of T, to p-adic precision r. If T is monic, P is the reduction modulo pr of the unique monic polynomial congruent to T modulo p such that P(Xp) = 0 (mod P(X),pr).
? T = ffinit(3, 3, 't) %1 = Mod(1,3)*t^3 + Mod(1,3)*t^2 + Mod(1,3)*t + Mod(2,3) ? P = polteichmuller(T,3,5) %2 = t^3 + 166*t^2 + 52*t + 242 ? subst(P, t, t^3) % (P*Mod(1,3^5)) %3 = Mod(0, 243) ? [algdep(a+O(3^5),2) | a <- Vec(P)] %4 = [x - 1, 5*x^2 + 1, x^2 + 4*x + 4, x + 1]
When T is monic and irreducible mod p, this provides
a model ℚp[X]/(P) of the unramified extension ℚp[X] / (T) where
the Frobenius has the simple form X mod P ⟼ Xp mod P.
The library syntax is GEN polteichmuller(GEN T, ulong p, long r).
Let T ∈ ℚ[x] be a nonzero polynomial; returns U monic in ℤ[x]
such that U(x) = C T(x/L) for some C,L ∈ ℚ. If the pointer argument
&L is present, set L to L.
? poltomonic(9*x^2 - 1/2) %1 = x^2 - 2 ? U = poltomonic(9*x^2 - 1/2, &L) %2 = x^2 - 2 ? L %3 = 6 ? U / subst(9*x^2 - 1/2, x, x/L) %4 = 4
This function does not compute discriminants or maximal orders and runs
with complexity almost linear in the input size. If T is already monic with
integer coefficient, poltomonic may still transform it if ℤ[x]/(T)
is contained in a trivial subring of the maximal order, generated by L x:
? poltomonic(x^2 + 4, &L) %5 = x^2 + 1 ? L %6 = 1/2
If T is irreducible, the functions polredabs
(exponential time) and polredbest (polynomial time) also find a monic
integral generating polynomial for the number field ℚ[x]/(T), with
explicit guarantees on its size, but are orders of magnitude slower.
The library syntax is GEN poltomonic(GEN T, GEN *L = NULL).
Creates Zagier's polynomial Pn(m) used in
the functions sumalt and sumpos (with flag = 1), see
"Convergence acceleration of alternating series", Cohen et al.,
Experiment. Math., vol. 9, 2000, pp. 3–12.
If m < 0 or m ≥ n, Pn(m) = 0. We have Pn := Pn(0) is Tn(2x-1), where Tn is the Legendre polynomial of the second kind. For n > m > 0, Pn(m) is the m-th difference with step 2 of the sequence nm+1Pn; in this case, it satisfies 2 Pn(m)(sin2 t) = (dm+1)/(dtm+1) (sin(2t)m sin(2(n-m)t)).
The library syntax is GEN polzag(long n, long m).
finds a linear relation between powers (1,s, ..., sp) of the series s, with polynomial coefficients of degree ≤ r. In case no relation is found, return 0.
? s = 1 + 10*y - 46*y^2 + 460*y^3 - 5658*y^4 + 77740*y^5 + O(y^6); ? seralgdep(s, 2, 2) %2 = -x^2 + (8*y^2 + 20*y + 1) ? subst(%, x, s) %3 = O(y^6) ? seralgdep(s, 1, 3) %4 = (-77*y^2 - 20*y - 1)*x + (310*y^3 + 231*y^2 + 30*y + 1) ? seralgdep(s, 1, 2) %5 = 0
The series main variable must not be x, so as to be able to express the result as a polynomial in x.
The library syntax is GEN seralgdep(GEN s, long p, long r).
Convolution (or Hadamard product) of the
two power series x and y; in other words if x = ∑ ak*Xk
and y = ∑ bk*Xk then serconvol(x,y) = ∑ ak*bk*Xk.
The library syntax is GEN convol(GEN x, GEN y).
Find a linear relation between the derivatives (s, s',..., sp) of the series s and 1, with polynomial coefficients of degree ≤ r. In case no relation is found, return 0, otherwise return [E,P] such that E(d)(S) = P where d is the standard derivation.
? S = sum(i=0, 50, binomial(3*i,i)*T^i) + O(T^51); ? serdiffdep(S, 3, 3) %2 = [(27*T^2 - 4*T)*x^2 + (54*T - 2)*x + 6, 0] ? (27*T^2 - 4*T)*S'' + (54*T - 2)*S' + 6*S %3 = O(T^50) ? S = exp(T^2) + T^2; ? serdiffdep(S, 3, 3) %5 = [x-2*T, -2*T^3+2*T] ? S'-2*T*S %6 = 2*T-2*T^3+O(T^17)
The series main variable must not be x, so as to be able to express the result as a polynomial in x.
The library syntax is GEN serdiffdep(GEN s, long p, long r).
x must be a power series with nonnegative exponents or a polynomial. If x = ∑ (ak/k!)*Xk then the result isi ∑ ak*Xk.
The library syntax is GEN laplace(GEN x).
Reverse power series of s, i.e. the series t such that t(s) = x; s must be a power series whose valuation is exactly equal to one.
? \ps 8 ? t = serreverse(tan(x)) %2 = x - 1/3*x^3 + 1/5*x^5 - 1/7*x^7 + O(x^8) ? tan(t) %3 = x + O(x^8)
The library syntax is GEN serreverse(GEN s).
Replace the simple variable y by the argument z in the "polynomial"
expression x. If z is a vector, return the vector of the evaluated
expressions subst(x, y, z[i]).
Every type is allowed for x, but if it is not a genuine polynomial (or power series, or rational function), the substitution will be done as if the scalar components were polynomials of degree zero. In particular, beware that:
? subst(1, x, [1,2; 3,4])
%1 =
[1 0]
[0 1]
? subst(1, x, Mat([0,1]))
*** at top-level: subst(1,x,Mat([0,1])
*** ^ — — — — — — --
*** subst: forbidden substitution by a non square matrix.
If x is a power series, z must be either a polynomial, a power series, or a rational function. If x is a vector, matrix or list, the substitution is applied to each individual entry.
Use the function substvec to replace several variables at once,
or the function substpol to replace a polynomial expression.
The library syntax is GEN gsubst(GEN x, long y, GEN z) where y is a variable number.
Replace the "variable" y by the argument z in the "polynomial"
expression x. Every type is allowed for x, but the same behavior
as subst above apply.
The difference with subst is that y is allowed to be any polynomial
here. The substitution is done moding out all components of x
(recursively) by y - t, where t is a new free variable of lowest
priority. Then substituting t by z in the resulting expression. For
instance
? substpol(x^4 + x^2 + 1, x^2, y) %1 = y^2 + y + 1 ? substpol(x^4 + x^2 + 1, x^3, y) %2 = x^2 + y*x + 1 ? substpol(x^4 + x^2 + 1, (x+1)^2, y) %3 = (-4*y - 6)*x + (y^2 + 3*y - 3)
The library syntax is GEN gsubstpol(GEN x, GEN y, GEN z).
Further, GEN gdeflate(GEN T, long v, long d) attempts to
write T(x) in the form t(xd), where x = polx(v), and returns
NULL if the substitution fails (for instance in the example %2
above).
v being a vector of monomials of degree 1 (variables), w a vector of expressions of the same length, replace in the expression x all occurrences of vi by wi. The substitutions are done simultaneously; more precisely, the vi are first replaced by new variables in x, then these are replaced by the wi:
? substvec([x,y], [x,y], [y,x]) %1 = [y, x] ? substvec([x,y], [x,y], [y,x+y]) %2 = [y, x + y] \\ not [y, 2*y]
As in subst, variables may be replaced
by a vector of values, in which case the cartesian product is returned:
? substvec([x,y], [x,y], [[1,2], 3]) %3 = [[1, 3], [2, 3]] ? substvec([x,y], [x,y], [[1,2], [3,4]]) %4 = [[1, 3], [2, 3], [1, 4], [2, 4]]
The library syntax is GEN gsubstvec(GEN x, GEN v, GEN w).
formal sum of the polynomial expression f with respect to the main variable if v is omitted, with respect to the variable v otherwise; it is assumed that the base ring has characteristic zero. In other words, considering f as a polynomial function in the variable v, returns F, a polynomial in v vanishing at 0, such that F(b) - F(a) = sumv = a+1b f(v):
? sumformal(n) \\ 1 + ... + n %1 = 1/2*n^2 + 1/2*n ? f(n) = n^3+n^2+1; ? F = sumformal(f(n)) \\ f(1) + ... + f(n) %3 = 1/4*n^4 + 5/6*n^3 + 3/4*n^2 + 7/6*n ? sum(n = 1, 2000, f(n)) == subst(F, n, 2000) %4 = 1 ? sum(n = 1001, 2000, f(n)) == subst(F, n, 2000) - subst(F, n, 1000) %5 = 1 ? sumformal(x^2 + x*y + y^2, y) %6 = y*x^2 + (1/2*y^2 + 1/2*y)*x + (1/3*y^3 + 1/2*y^2 + 1/6*y) ? x^2 * y + x * sumformal(y) + sumformal(y^2) == % %7 = 1
The library syntax is GEN sumformal(GEN f, long v = -1) where v is a variable number.
Taylor expansion around 0 of x with respect to
the simple variable t. x can be of any reasonable type, for example a
rational function. Contrary to Ser, which takes the valuation into
account, this function adds O(td) to all components of x.
? taylor(x/(1+y), y, 5) %1 = (y^4 - y^3 + y^2 - y + 1)*x + O(y^5) ? Ser(x/(1+y), y, 5) *** at top-level: Ser(x/(1+y),y,5) *** ^ — — — — — - *** Ser: main variable must have higher priority in gtoser.
The library syntax is GEN tayl(GEN x, long t, long precdl) where t is a variable number.
Returns all solutions of the equation
P(x,y) = a in integers x and y, where tnf was created with
thueinit(P). If present, sol must contain the solutions of
Norm(x) = a modulo units of positive norm in the number field
defined by P (as computed by bnfisintnorm). If there are infinitely
many solutions, an error is issued.
It is allowed to input directly the polynomial P instead of a tnf,
in which case, the function first performs thueinit(P,0). This is
very wasteful if more than one value of a is required.
If tnf was computed without assuming GRH (flag 1 in thueinit),
then the result is unconditional. Otherwise, it depends in principle of the
truth of the GRH, but may still be unconditionally correct in some
favorable cases. The result is conditional on the GRH if
a ! = ± 1 and P has a single irreducible rational factor, whose
attached tentative class number h and regulator R (as computed
assuming the GRH) satisfy
* h > 1,
* R/0.2 > 1.5.
Here's how to solve the Thue equation x13 - 5y13 = - 4:
? tnf = thueinit(x^13 - 5); ? thue(tnf, -4) %1 = [[1, 1]]
In this case, one checks that bnfinit(x^13 -5).no
is 1. Hence, the only solution is (x,y) = (1,1) and the result is
unconditional. On the other hand:
? P = x^3-2*x^2+3*x-17; tnf = thueinit(P); ? thue(tnf, -15) %2 = [[1, 1]] \\ a priori conditional on the GRH. ? K = bnfinit(P); K.no %3 = 3 ? K.reg %4 = 2.8682185139262873674706034475498755834
This time the result is conditional. All results computed using this particular tnf are likewise conditional, except for a right-hand side of ± 1. The above result is in fact correct, so we did not just disprove the GRH:
? tnf = thueinit(x^3-2*x^2+3*x-17, 1 /*unconditional*/); ? thue(tnf, -15) %4 = [[1, 1]]
Note that reducible or nonmonic polynomials are allowed:
? tnf = thueinit((2*x+1)^5 * (4*x^3-2*x^2+3*x-17), 1); ? thue(tnf, 128) %2 = [[-1, 0], [1, 0]]
Reducible polynomials are in fact much easier to handle.
Note. When P is irreducible without a real root, the default
strategy is to use brute force enumeration in time |a|1/deg P and
avoid computing a tough bnf attached to P, see thueinit.
Besides reusing a quantity you might need for other purposes, the
default argument sol can also be used to use a different strategy
and prove that there are no solutions; of course you need to compute a
bnf on you own to obtain sol. If there are solutions
this won't help unless P is quadratic, since the enumeration will be
performed in any case.
The library syntax is GEN thue(GEN tnf, GEN a, GEN sol = NULL).
Initializes the tnf corresponding to P, a nonconstant
univariate polynomial with integer coefficients.
The result is meant to be used in conjunction with thue to solve Thue
equations P(X / Y)Ydeg P = a, where a is an integer. Accordingly,
P must either have at least two distinct irreducible factors over ℚ,
or have one irreducible factor T with degree > 2 or two conjugate
complex roots: under these (necessary and sufficient) conditions, the
equation has finitely many integer solutions.
? S = thueinit(t^2+1); ? thue(S, 5) %2 = [[-2, -1], [-2, 1], [-1, -2], [-1, 2], [1, -2], [1, 2], [2, -1], [2, 1]] ? S = thueinit(t+1); *** at top-level: thueinit(t+1) *** ^ — — — — - *** thueinit: domain error in thueinit: P = t + 1
The hardest case is when deg P > 2 and P is irreducible with at least one real root. The routine then uses Bilu-Hanrot's algorithm.
If flag is nonzero, certify results unconditionally. Otherwise, assume
GRH, this being much faster of course. In the latter case, the result
may still be unconditionally correct, see thue. For instance in most
cases where P is reducible (not a pure power of an irreducible), or
conditional computed class groups are trivial or the right hand side
is ±1, then results are unconditional.
Note. The general philosophy is to disprove the existence of large
solutions then to enumerate bounded solutions naively. The implementation
will overflow when there exist huge solutions and the equation has degree
> 2 (the quadratic imaginary case is special, since we can stick to
bnfisintnorm, there are no fundamental units):
? thue(t^3+2, 10^30) *** at top-level: L=thue(t^3+2,10^30) *** ^ — — — — — -- *** thue: overflow in thue (SmallSols): y <= 80665203789619036028928. ? thue(x^2+2, 10^30) \\ quadratic case much easier %1 = [[-1000000000000000, 0], [1000000000000000, 0]]
Note. It is sometimes possible to circumvent the above, and in any
case obtain an important speed-up, if you can write P = Q(xd) for some
d > 1 and Q still satisfying the thueinit hypotheses. You can then
solve
the equation attached to Q then eliminate all solutions (x,y) such that
either x or y is not a d-th power.
? thue(x^4+1, 10^40); \\ stopped after 10 hours
? filter(L,d) =
my(x,y); [[x,y] | v<-L, ispower(v[1],d,&x)&&ispower(v[2],d,&y)];
? L = thue(x^2+1, 10^40);
? filter(L, 2)
%4 = [[0, 10000000000], [10000000000, 0]]
The last 2 commands use less than 20ms.
Note. When P is irreducible without a real root, the equation
can be solved unconditionnally in time |a|1/deg P. When this
latter quantity is huge and the equation has no solutions, this fact
may still be ascertained via arithmetic conditions but this now implies
solving norm equations, computing a bnf and possibly assuming the GRH.
When there is no real root, the code does not compute a bnf
(with certification if flag = 1) if it expects this to be an "easy"
computation (because the result would only be used for huge values of a).
See thue for a way to compute an expensive bnf on your own and
still get a result where this default cheap strategy fails.
The library syntax is GEN thueinit(GEN P, long flag, long prec).