Standard monadic or dyadic operators

+/-

The expressions +x and -x refer to monadic operators (the first does nothing, the second negates x).

The library syntax is GEN gneg(GEN x) for -x.


+

The expression x + y is the sum of x and y. Addition between a scalar type x and a t_COL or t_MAT y returns respectively [y[1] + x, y[2],...] and y + x {Id}. Other additions between a scalar type and a vector or a matrix, or between vector/matrices of incompatible sizes are forbidden.

The library syntax is GEN gadd(GEN x, GEN y).


+

The expression x + y is the sum of x and y. Addition between a scalar type x and a t_COL or t_MAT y returns respectively [y[1] + x, y[2],...] and y + x {Id}. Other additions between a scalar type and a vector or a matrix, or between vector/matrices of incompatible sizes are forbidden.

The library syntax is GEN gadd(GEN x, GEN y).


*

The expression x * y is the product of x and y. Among the prominent impossibilities are multiplication between vector/matrices of incompatible sizes, between a t_INTMOD or t_PADIC Restricted to scalars, * is commutative; because of vector and matrix operations, it is not commutative in general.

Multiplication between two t_VECs or two t_COLs is not allowed; to take the scalar product of two vectors of the same length, transpose one of the vectors (using the operator ~ or the function mattranspose, see Section [Label: se:linear_algebra]) and multiply a line vector by a column vector:

  ? a = [1,2,3];
  ? a * a
    ***   at top-level: a*a
    ***                  ^--
    *** _*_: forbidden multiplication t_VEC * t_VEC.
  ? a * a~
  %2 = 14

If x,y are binary quadratic forms, compose them; see also qfbnucomp and qfbnupow. If x,y are t_VECSMALL of the same length, understand them as permutations and compose them.

The library syntax is GEN gmul(GEN x, GEN y) for x * y. Also available is GEN gsqr(GEN x) for x * x.


/

The expression x / y is the quotient of x and y. In addition to the impossibilities for multiplication, note that if the divisor is a matrix, it must be an invertible square matrix, and in that case the result is x*y^{-1}. Furthermore note that the result is as exact as possible: in particular, division of two integers always gives a rational number (which may be an integer if the quotient is exact) and not the Euclidean quotient (see x \ y for that), and similarly the quotient of two polynomials is a rational function in general. To obtain the approximate real value of the quotient of two integers, add 0. to the result; to obtain the approximate p-adic value of the quotient of two integers, add O(p^k) to the result; finally, to obtain the Taylor series expansion of the quotient of two polynomials, add O(X^k) to the result or use the taylor function (see Section [Label: se:taylor]).

The library syntax is GEN gdiv(GEN x, GEN y) for x / y.


\

The expression x \y is the \idx{Euclidean quotient} of x and y. If y is a real scalar, this is defined as floor(x/y) if y > 0, and ceil(x/y) if y < 0 and the division is not exact. Hence the remainder x - (x\y)*y is in [0, |y|[.

Note that when y is an integer and x a polynomial, y is first promoted to a polynomial of degree 0. When x is a vector or matrix, the operator is applied componentwise.

The library syntax is GEN gdivent(GEN x, GEN y) for x \ y.


\/

The expression x \/ y evaluates to the rounded Euclidean quotient of x and y. This is the same as x \y except for scalar division: the quotient is such that the corresponding remainder is smallest in absolute value and in case of a tie the quotient closest to + oo is chosen (hence the remainder would belong to ]{-}|y|/2, |y|/2]).

When x is a vector or matrix, the operator is applied componentwise.

The library syntax is GEN gdivround(GEN x, GEN y) for x \/ y.


^

The expression x{^}n is powering. If the exponent is an integer, then exact operations are performed using binary (left-shift) powering techniques. In particular, in this case x cannot be a vector or matrix unless it is a square matrix (invertible if the exponent is negative). If x is a p-adic number, its precision will increase if v_p(n) > 0. Powering a binary quadratic form (types t_QFI and t_QFR) returns a reduced representative of the class, provided the input is reduced. In particular, x{^}1 is identical to x.

PARI is able to rewrite the multiplication x * x of two identical objects as x^2, or sqr(x). Here, identical means the operands are two different labels referencing the same chunk of memory; no equality test is performed. This is no longer true when more than two arguments are involved.

If the exponent is not of type integer, this is treated as a transcendental function (see Section [Label: se:trans]), and in particular has the effect of componentwise powering on vector or matrices.

As an exception, if the exponent is a rational number p/q and x an integer modulo a prime or a p-adic number, return a solution y of y^q = x^p if it exists. Currently, q must not have large prime factors. Beware that

  ? Mod(7,19)^(1/2)
  %1 = Mod(11, 19) /* is any square root */
  ? sqrt(Mod(7,19))
  %2 = Mod(8, 19)  /* is the smallest square root */
  ? Mod(7,19)^(3/5)
  %3 = Mod(1, 19)
  ? %3^(5/3)
  %4 = Mod(1, 19)  /* Mod(7,19) is just another cubic root */

If the exponent is a negative integer, an inverse must be computed. For non-invertible t_INTMOD, this will fail and implicitly exhibit a non trivial factor of the modulus:

  ? Mod(4,6)^(-1)
    ***   at top-level: Mod(4,6)^(-1)
    ***                         ^-----
    *** _^_: impossible inverse modulo: Mod(2, 6).

(Here, a factor 2 is obtained directly. In general, take the gcd of the representative and the modulus.) This is most useful when performing complicated operations modulo an integer N whose factorization is unknown. Either the computation succeeds and all is well, or a factor d is discovered and the computation may be restarted modulo d or N/d.

For non-invertible t_POLMOD, this will fail without exhibiting a factor.

  ? Mod(x^2, x^3-x)^(-1)
    ***   at top-level: Mod(x^2,x^3-x)^(-1)
    ***                               ^-----
    *** _^_: non-invertible polynomial in RgXQ_inv.
  ? a = Mod(3,4)*y^3 + Mod(1,4); b = y^6+y^5+y^4+y^3+y^2+y+1;
  ? Mod(a, b)^(-1);
    ***   at top-level: Mod(a,b)^(-1)
    ***                         ^-----
    *** _^_: impossible inverse modulo: Mod(0, 4).

In fact the latter polynomial is invertible, but the algorithm used (subresultant) assumes the base ring is a domain. If it is not the case, as here for Z/4Z, a result will be correct but chances are an error will occur first. In this specific case, one should work with 2-adics. In general, one can try the following approach

  ? inversemod(a, b) =
  { my(m);
    m = polsylvestermatrix(polrecip(a), polrecip(b));
    m = matinverseimage(m, matid(#m)[,1]);
    Polrev( vecextract(m, Str("..", poldegree(b))), variable(b) )
  }
  ? inversemod(a,b)
  %2 = Mod(2,4)*y^5 + Mod(3,4)*y^3 + Mod(1,4)*y^2 + Mod(3,4)*y + Mod(2,4)

This is not guaranteed to work either since it must invert pivots. See Section [Label: se:linear_algebra].

The library syntax is GEN gpow(GEN x, GEN n, long prec) for x{^}n.


cmp(x,y)

Gives the result of a comparison between arbitrary objects x and y (as -1, 0 or 1). The underlying order relation is transitive, the function returns 0 if and only if x  == =  y, and its restriction to integers coincides with the customary one. Besides that, it has no useful mathematical meaning.

In case all components are equal up to the smallest length of the operands, the more complex is considered to be larger. More precisely, the longest is the largest; when lengths are equal, we have matrix > vector > scalar. For example:

  ? cmp(1, 2)
  %1 = -1
  ? cmp(2, 1)
  %2 = 1
  ? cmp(1, 1.0)   \\ note that 1 == 1.0, but (1===1.0) is false.
  %3 = -1
  ? cmp(x + Pi, [])
  %4 = -1
This function is mostly useful to handle sorted lists or vectors of arbitrary objets. For instance, if v is a vector, the construction vecsort(v, cmp) is equivalent to Set(v).

The library syntax is GEN cmp_universal(GEN x, GEN y).


divrem(x,y,{v})

Creates a column vector with two components, the first being the Euclidean quotient (x \y), the second the Euclidean remainder (x - (x\y)*y), of the division of x by y. This avoids the need to do two divisions if one needs both the quotient and the remainder. If v is present, and x, y are multivariate polynomials, divide with respect to the variable v.

Beware that divrem(x,y)[2] is in general not the same as x % y; no GP operator corresponds to it:

  ? divrem(1/2, 3)[2]
  %1 = 1/2
  ? (1/2) % 3
  %2 = 2
  ? divrem(Mod(2,9), 3)[2]
   ***   at top-level: divrem(Mod(2,9),3)[2
   ***                 ^--------------------
   ***   forbidden division t_INTMOD \ t_INT.
  ? Mod(2,9) % 6
  %3 = Mod(2,3)

The library syntax is GEN divrem(GEN x, GEN y, long v = -1), where v is a variable number. Also available is GEN gdiventres(GEN x, GEN y) when v is not needed.


lex(x,y)

Gives the result of a lexicographic comparison between x and y (as -1, 0 or 1). This is to be interpreted in quite a wide sense: It is admissible to compare objects of different types (scalars, vectors, matrices), provided the scalars can be compared, as well as vectors/matrices of different lengths. The comparison is recursive.

In case all components are equal up to the smallest length of the operands, the more complex is considered to be larger. More precisely, the longest is the largest; when lengths are equal, we have matrix > vector > scalar. For example:

  ? lex([1,3], [1,2,5])
  %1 = 1
  ? lex([1,3], [1,3,-1])
  %2 = -1
  ? lex([1], [[1]])
  %3 = -1
  ? lex([1], [1]~)
  %4 = 0

The library syntax is GEN lexcmp(GEN x, GEN y).


max(x,y)

Creates the maximum of x and y when they can be compared.

The library syntax is GEN gmax(GEN x, GEN y).


max(x,y)

Creates the maximum of x and y when they can be compared.

The library syntax is GEN gmax(GEN x, GEN y).


shift(x,n)

Shifts x componentwise left by n bits if n >= 0 and right by |n| bits if n < 0. May be abbreviated as x << n or x >> (-n). A left shift by n corresponds to multiplication by 2^n. A right shift of an integer x by |n| corresponds to a Euclidean division of x by 2^{|n|} with a remainder of the same sign as x, hence is not the same (in general) as x \ 2^n.

The library syntax is GEN gshift(GEN x, long n).


shiftmul(x,n)

Multiplies x by 2^n. The difference with shift is that when n < 0, ordinary division takes place, hence for example if x is an integer the result may be a fraction, while for shifts Euclidean division takes place when n < 0 hence if x is an integer the result is still an integer.

The library syntax is GEN gmul2n(GEN x, long n).


sign(x)

sign (0, 1 or -1) of x, which must be of type integer, real or fraction.

The library syntax is GEN gsigne(GEN x).


vecmax(x,{&v})

If x is a vector or a matrix, returns the largest entry of x, otherwise returns a copy of x. Error if x is empty.

If v is given, set it to the index of a largest entry (indirect maximum), when x is a vector. If x is a matrix, set v to coordinates [i,j] such that x[i,j] is a largest entry. This flag is ignored if x is not a vector or matrix.

  ? vecmax([10, 20, -30, 40])
  %1 = 40
  ? vecmax([10, 20, -30, 40], &v); v
  %2 = 4
  ? vecmax([10, 20; -30, 40], &v); v
  %3 = [2, 2]

The library syntax is GEN vecmax0(GEN x, GEN *v = NULL). Also available is GEN vecmax(GEN x).


vecmin(x,{&v})

If x is a vector or a matrix, returns the smallest entry of x, otherwise returns a copy of x. Error if x is empty.

If v is given, set it to the index of a smallest entry (indirect minimum), when x is a vector. If x is a matrix, set v to coordinates [i,j] such that x[i,j] is a smallest entry. This is ignored if x is not a vector or matrix.

  ? vecmin([10, 20, -30, 40])
  %1 = -30
  ? vecmin([10, 20, -30, 40], &v); v
  %2 = 3
  ? vecmin([10, 20; -30, 40], &v); v
  %3 = [2, 1]

The library syntax is GEN vecmin0(GEN x, GEN *v = NULL). Also available is GEN vecmin(GEN x).


Comparison and Boolean operators

The six standard comparison operators <= , < , >= , > , == , != are available in GP. The result is 1 if the comparison is true, 0 if it is false. The operator == is quite liberal : for instance, the integer 0, a 0 polynomial, and a vector with 0 entries are all tested equal.

The extra operator == = tests whether two objects are identical and is much stricter than == : objects of different type or length are never identical.

For the purpose of comparison, t_STR objects are strictly larger than any other non-string type; two t_STR objects are compared using the standard lexicographic order.

GP accepts < > as a synonym for != . On the other hand, = is definitely not a synonym for == : it is the assignment statement.

The standard boolean operators || (inclusive or), && (and) and ! (not) are also available.