Transforms the object x into a column vector. The dimension of the resulting vector can be optionally specified via the extra parameter n.

If n is omitted or 0, the dimension depends on the type of x; the vector has a single component, except when x is

* a vector or a quadratic form (in which case the resulting vector is simply the initial object considered as a row vector),

* a polynomial or a power series. In the case of a polynomial, the coefficients of the vector start with the leading coefficient of the polynomial, while for power series only the significant coefficients are taken into account, but this time by increasing order of degree. In this last case, Vec is the reciprocal function of Pol and Ser respectively,

* a matrix (the column of row vector comprising the matrix is returned),

* a character string (a vector of individual characters is returned).

In the last two cases (matrix and character string), n is meaningless and must be omitted or an error is raised. Otherwise, if n is given, 0 entries are appended at the end of the vector if n > 0, and prepended at the beginning if n < 0. The dimension of the resulting vector is |n|.

Note that the function Colrev does not exist, use Vecrev.

The library syntax is GEN gtocol0(GEN x, long n). GEN gtocol(GEN x) is also available.

As Col(x, -n), then reverse the result. In particular, Colrev is the reciprocal function of Polrev: the coefficients of the vector start with the constant coefficient of the polynomial and the others follow by increasing degree.

The library syntax is GEN gtocolrev0(GEN x, long n). GEN gtocolrev(GEN x) is also available.

Transforms a (row or column) vector x into a list, whose components are the entries of x. Similarly for a list, but rather useless in this case. For other types, creates a list with the single element x. Note that, except when x is omitted, this function creates a small memory leak; so, either initialize all lists to the empty list, or use them sparingly.

The library syntax is GEN gtolist(GEN x = NULL). The variant GEN mklist(void) creates an empty list.

A "Map" is an associative array, or dictionary: a data type composed of a collection of (key, value) pairs, such that each key appears just once in the collection. This function converts the matrix [a_1,b_1;a_2,b_2;...;a_n,b_n] to the map a_i ⟼ b_i.

  ? M = Map(factor(13!));
  ? mapget(M,3)
  %2 = 5

If the argument x is omitted, creates an empty map, which may be filled later via mapput.

The library syntax is GEN gtomap(GEN x = NULL).

Transforms the object x into a matrix. If x is already a matrix, a copy of x is created. If x is a row (resp. column) vector, this creates a 1-row (resp. 1-column) matrix, unless all elements are column (resp. row) vectors of the same length, in which case the vectors are concatenated sideways and the attached big matrix is returned. If x is a binary quadratic form, creates the attached 2 x 2 matrix. Otherwise, this creates a 1 x 1 matrix containing x.

  ? Mat(x + 1)
  %1 =
  [x + 1]
  ? Vec( matid(3) )
  %2 = [[1, 0, 0]~, [0, 1, 0]~, [0, 0, 1]~]
  ? Mat(%)
  %3 =
  [1 0 0]
  
  [0 1 0]
  
  [0 0 1]
  ? Col( [1,2; 3,4] )
  %4 = [[1, 2], [3, 4]]~
  ? Mat(%)
  %5 =
  [1 2]
  
  [3 4]
  ? Mat(Qfb(1,2,3))
  %6 =
  [1 1]
  
  [1 3]

The library syntax is GEN gtomat(GEN x = NULL).

In its basic form, creates an intmod or a polmod (a mod b); b must be an integer or a polynomial. We then obtain a t_INTMOD and a t_POLMOD respectively:

  ? t = Mod(2,17); t^8
  %1 = Mod(1, 17)
  ? t = Mod(x,x^2+1); t^2
  %2 = Mod(-1, x^2+1)

If a % b makes sense and yields a result of the appropriate type (t_INT or scalar/t_POL), the operation succeeds as well:

  ? Mod(1/2, 5)
  %3 = Mod(3, 5)
  ? Mod(7 + O(3^6), 3)
  %4 = Mod(1, 3)
  ? Mod(Mod(1,12), 9)
  %5 = Mod(1, 3)
  ? Mod(1/x, x^2+1)
  %6 = Mod(-1, x^2+1)
  ? Mod(exp(x), x^4)
  %7 = Mod(1/6*x^3 + 1/2*x^2 + x + 1, x^4)

If a is a complex object, "base change" it to ℤ/bℤ or K[x]/(b), which is equivalent to, but faster than, multiplying it by Mod(1,b):

  ? Mod([1,2;3,4], 2)
  %8 =
  [Mod(1, 2) Mod(0, 2)]
  
  [Mod(1, 2) Mod(0, 2)]
  ? Mod(3*x+5, 2)
  %9 = Mod(1, 2)*x + Mod(1, 2)
  ? Mod(x^2 + y*x + y^3, y^2+1)
  %10 = Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1)*x + Mod(-y, y^2 + 1)

This function is not the same as x % y, the result of which has no knowledge of the intended modulus y. Compare

  ? x = 4 % 5; x + 1
  %1 = 5
  ? x = Mod(4,5); x + 1
  %2 = Mod(0,5)

Note that such "modular" objects can be lifted via lift or centerlift. The modulus of a t_INTMOD or t_POLMOD z can be recovered via z.mod.

The library syntax is GEN gmodulo(GEN a, GEN b).

Transforms the object t into a polynomial with main variable v. If t is a scalar, this gives a constant polynomial. If t is a power series with non-negative valuation or a rational function, the effect is similar to truncate, i.e. we chop off the O(X^k) or compute the Euclidean quotient of the numerator by the denominator, then change the main variable of the result to v.

The main use of this function is when t is a vector: it creates the polynomial whose coefficients are given by t, with t[1] being the leading coefficient (which can be zero). It is much faster to evaluate Pol on a vector of coefficients in this way, than the corresponding formal expression a_n X^n +...+ a_0, which is evaluated naively exactly as written (linear versus quadratic time in n). Polrev can be used if one wants x[1] to be the constant coefficient:

  ? Pol([1,2,3])
  %1 = x^2 + 2*x + 3
  ? Polrev([1,2,3])
  %2 = 3*x^2 + 2*x + 1

The reciprocal function of Pol (resp. Polrev) is Vec (resp.  Vecrev).

  ? Vec(Pol([1,2,3]))
  %1 = [1, 2, 3]
  ? Vecrev( Polrev([1,2,3]) )
  %2 = [1, 2, 3]

Warning. This is not a substitution function. It will not transform an object containing variables of higher priority than v.

  ? Pol(x + y, y)
    ***   at top-level: Pol(x+y,y)
    ***                 ^----------
    *** Pol: variable must have higher priority in gtopoly.

The library syntax is GEN gtopoly(GEN t, long v = -1) where v is a variable number.

Transform the object t into a polynomial with main variable v. If t is a scalar, this gives a constant polynomial. If t is a power series, the effect is identical to truncate, i.e. it chops off the O(X^k).

The main use of this function is when t is a vector: it creates the polynomial whose coefficients are given by t, with t[1] being the constant term. Pol can be used if one wants t[1] to be the leading coefficient:

  ? Polrev([1,2,3])
  %1 = 3*x^2 + 2*x + 1
  ? Pol([1,2,3])
  %2 = x^2 + 2*x + 3

The reciprocal function of Pol (resp. Polrev) is Vec (resp.  Vecrev).

The library syntax is GEN gtopolyrev(GEN t, long v = -1) where v is a variable number.

Creates the binary quadratic form ax^2+bxy+cy^2. If b^2-4ac > 0, initialize Shanks' distance function to D. Negative definite forms are not implemented, use their positive definite counterpart instead.

The library syntax is GEN Qfb0(GEN a, GEN b, GEN c, GEN D = NULL, long prec). Also available are GEN qfi(GEN a, GEN b, GEN c) (assumes b^2-4ac < 0) and GEN qfr(GEN a, GEN b, GEN c, GEN D) (assumes b^2-4ac > 0).

Transforms the object s into a power series with main variable v (x by default) and precision (number of significant terms) equal to d ≥ 0 (d = seriesprecision by default). If s is a scalar, this gives a constant power series in v with precision d. If s is a polynomial, the polynomial is truncated to d terms if needed

  ? Ser(1, 'y, 5)
  %1 = 1 + O(y^5)
  ? Ser(x^2,, 5)
  %2 = x^2 + O(x^7)
  ? T = polcyclo(100)
  %3 = x^40 - x^30 + x^20 - x^10 + 1
  ? Ser(T, 'x, 11)
  %4 = 1 - x^10 + O(x^11)

The function is more or less equivalent with multiplication by 1 + O(v^d) in theses cases, only faster.

If s is a vector, on the other hand, the coefficients of the vector are understood to be the coefficients of the power series starting from the constant term (as in Polrev(x)), and the precision d is ignored: in other words, in this case, we convert t_VEC / t_COL to the power series whose significant terms are exactly given by the vector entries. Finally, if s is already a power series in v, we return it verbatim, ignoring d again. If d significant terms are desired in the last two cases, convert/truncate to t_POL first.

  ? v = [1,2,3]; Ser(v, t, 7)
  %5 = 1 + 2*t + 3*t^2 + O(t^3)  \\ 3 terms: 7 is ignored!
  ? Ser(Polrev(v,t), t, 7)
  %6 = 1 + 2*t + 3*t^2 + O(t^7)
  ? s = 1+x+O(x^2); Ser(s, x, 7)
  %7 = 1 + x + O(x^2)  \\ 2 terms: 7 ignored
  ? Ser(truncate(s), x, 7)
  %8 = 1 + x + O(x^7)

The warning given for Pol also applies here: this is not a substitution function.

The library syntax is GEN gtoser(GEN s, long v = -1, long precdl) where v is a variable number.

Converts x into a set, i.e. into a row vector, with strictly increasing entries with respect to the (somewhat arbitrary) universal comparison function cmp. Standard container types t_VEC, t_COL, t_LIST and t_VECSMALL are converted to the set with corresponding elements. All others are converted to a set with one element.

  ? Set([1,2,4,2,1,3])
  %1 = [1, 2, 3, 4]
  ? Set(x)
  %2 = [x]
  ? Set(Vecsmall([1,3,2,1,3]))
  %3 = [1, 2, 3]

The library syntax is GEN gtoset(GEN x = NULL).

Converts its argument list into a single character string (type t_STR, the empty string if x is omitted). To recover an ordinary GEN from a string, apply eval to it. The arguments of Str are evaluated in string context, see Section se:strings.

  ? x2 = 0; i = 2; Str(x, i)
  %1 = "x2"
  ? eval(%)
  %2 = 0

This function is mostly useless in library mode. Use the pair strtoGEN/GENtostr to convert between GEN and char*. The latter returns a malloced string, which should be freed after usage.

Converts x to a string, translating each integer into a character.

  ? Strchr(97)
  %1 = "a"
  ? Vecsmall("hello world")
  %2 = Vecsmall([104, 101, 108, 108, 111, 32, 119, 111, 114, 108, 100])
  ? Strchr(%)
  %3 = "hello world"

The library syntax is GEN Strchr(GEN x).

Converts its argument list into a single character string (type t_STR, the empty string if x is omitted). Then perform environment expansion, see Section se:envir. This feature can be used to read environment variable values.

  ? Strexpand("$HOME/doc")
  %1 = "/home/pari/doc"

The individual arguments are read in string context, see Section se:strings.

Translates its arguments to TeX format, and concatenates the results into a single character string (type t_STR, the empty string if x is omitted).

The individual arguments are read in string context, see Section se:strings.

Transforms the object x into a row vector. The dimension of the resulting vector can be optionally specified via the extra parameter n.

If n is omitted or 0, the dimension depends on the type of x; the vector has a single component, except when x is

* a vector or a quadratic form: returns the initial object considered as a row vector,

* a polynomial or a power series: returns a vector consisting of the coefficients. In the case of a polynomial, the coefficients of the vector start with the leading coefficient of the polynomial, while for power series only the significant coefficients are taken into account, but this time by increasing order of degree. Vec is the reciprocal function of Pol for a polynomial and of Ser for a power series,

* a matrix: returns the vector of columns comprising the matrix,

* a character string: returns the vector of individual characters,

* a map: returns the vector of the domain of the map,

* an error context (t_ERROR): returns the error components, see iferr.

In the last four cases (matrix, character string, map, error), n is meaningless and must be omitted or an error is raised. Otherwise, if n is given, 0 entries are appended at the end of the vector if n > 0, and prepended at the beginning if n < 0. The dimension of the resulting vector is |n|. Variant: GEN gtovec(GEN x) is also available.

The library syntax is GEN gtovec0(GEN x, long n).

As Vec(x, -n), then reverse the result. In particular, Vecrev is the reciprocal function of Polrev: the coefficients of the vector start with the constant coefficient of the polynomial and the others follow by increasing degree.

The library syntax is GEN gtovecrev0(GEN x, long n). GEN gtovecrev(GEN x) is also available.

Transforms the object x into a row vector of type t_VECSMALL. The dimension of the resulting vector can be optionally specified via the extra parameter n.

This acts as Vec(x,n), but only on a limited set of objects: the result must be representable as a vector of small integers. If x is a character string, a vector of individual characters in ASCII encoding is returned (Strchr yields back the character string).

The library syntax is GEN gtovecsmall0(GEN x, long n). GEN gtovecsmall(GEN x) is also available.

Outputs the vector of the binary digits of |x|. Here x can be an integer, a real number (in which case the result has two components, one for the integer part, one for the fractional part) or a vector/matrix.

  ? binary(10)
  %1 = [1, 0, 1, 0]
  
  ? binary(3.14)
  %2 = [[1, 1], [0, 0, 1, 0, 0, 0, [...]]
  
  ? binary([1,2])
  %3 = [[1], [1, 0]]

By convention, 0 has no digits:

  ? binary(0)
  %4 = []

The library syntax is GEN binaire(GEN x).

Bitwise and of two integers x and y, that is the integer ∑_i (x_i and y_i) 2^i

Negative numbers behave 2-adically, i.e. the result is the 2-adic limit of bitand(x_n,y_n), where x_n and y_n are non-negative integers tending to x and y respectively. (The result is an ordinary integer, possibly negative.)

  ? bitand(5, 3)
  %1 = 1
  ? bitand(-5, 3)
  %2 = 3
  ? bitand(-5, -3)
  %3 = -7

The library syntax is GEN gbitand(GEN x, GEN y). Also available is GEN ibitand(GEN x, GEN y), which returns the bitwise and of |x| and |y|, two integers.

bitwise negation of an integer x, truncated to n bits, n ≥ 0, that is the integer ∑_{i = 0}^{n-1} not(x_i) 2^i. The special case n = -1 means no truncation: an infinite sequence of leading 1 is then represented as a negative number.

See Section se:bitand for the behavior for negative arguments.

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

Bitwise negated imply of two integers x and y (or not (x ==> y)), that is the integer ∑ (x_i and not(y_i)) 2^i

See Section se:bitand for the behavior for negative arguments.

The library syntax is GEN gbitnegimply(GEN x, GEN y). Also available is GEN ibitnegimply(GEN x, GEN y), which returns the bitwise negated imply of |x| and |y|, two integers.

bitwise (inclusive) or of two integers x and y, that is the integer ∑ (x_i or y_i) 2^i

See Section se:bitand for the behavior for negative arguments.

The library syntax is GEN gbitor(GEN x, GEN y). Also available is GEN ibitor(GEN x, GEN y), which returns the bitwise ir of |x| and |y|, two integers.

The function behaves differently according to whether n is present and positive or not. If n is missing, the function returns the (floating point) precision in bits of the PARI object x. If x is an exact object, the function returns +oo.

  ? bitprecision(exp(1e-100))
  %1 = 512                 \\ 512 bits
  ? bitprecision( [ exp(1e-100), 0.5 ] )
  %2 = 128                 \\ minimal accuracy among components
  ? bitprecision(2 + x)
  %3 = +oo                  \\ exact object

If n is present and positive, the function creates a new object equal to x with the new bit-precision roughly n. In fact, the smallest multiple of 64 (resp. 32 on a 32-bit machine) larger than or equal to n.

For x a vector or a matrix, the operation is done componentwise; for series and polynomials, the operation is done coefficientwise. For real x, n is the number of desired significant bits. If n is smaller than the precision of x, x is truncated, otherwise x is extended with zeros. For exact or non-floating point types, no change.

  ? bitprecision(Pi, 10)    \\ actually 64 bits ~ 19 decimal digits
  %1 = 3.141592653589793239
  ? bitprecision(1, 10)
  %2 = 1
  ? bitprecision(1 + O(x), 10)
  %3 = 1 + O(x)
  ? bitprecision(2 + O(3^5), 10)
  %4 = 2 + O(3^5)

The library syntax is GEN bitprecision0(GEN x, long n).

Outputs the n-th bit of x starting from the right (i.e. the coefficient of 2^n in the binary expansion of x). The result is 0 or 1.

  ? bittest(7, 0)
  %1 = 1 \\ the bit 0 is 1
  ? bittest(7, 2)
  %2 = 1 \\ the bit 2 is 1
  ? bittest(7, 3)
  %3 = 0 \\ the bit 3 is 0

See Section se:bitand for the behavior at negative arguments.

The library syntax is GEN gbittest(GEN x, long n). For a t_INT x, the variant long bittest(GEN x, long n) is generally easier to use, and if furthermore n ≥ 0 the low-level function ulong int_bit(GEN x, long n) returns bittest(abs(x),n).

Bitwise (exclusive) or of two integers x and y, that is the integer ∑ (x_i xor y_i) 2^i

See Section se:bitand for the behavior for negative arguments.

The library syntax is GEN gbitxor(GEN x, GEN y). Also available is GEN ibitxor(GEN x, GEN y), which returns the bitwise xor of |x| and |y|, two integers.

Ceiling of x. When x is in ℝ, the result is the smallest integer greater than or equal to x. Applied to a rational function, ceil(x) returns the Euclidean quotient of the numerator by the denominator.

The library syntax is GEN gceil(GEN x).

Same as lift, except that t_INTMOD and t_PADIC components are lifted using centered residues:

* for a t_INTMOD x ∈ ℤ/nℤ, the lift y is such that -n/2 < y ≤ n/2.

* a t_PADIC x is lifted in the same way as above (modulo p^padicprec(x)) if its valuation v is non-negative; if not, returns the fraction p^v centerlift(x p^{-v}); in particular, rational reconstruction is not attempted. Use bestappr for this.

For backward compatibility, centerlift(x,'v) is allowed as an alias for lift(x,'v).

The library syntax is centerlift(GEN x).

Returns the characteristic of the base ring over which x is defined (as defined by t_INTMOD and t_FFELT components). The function raises an exception if incompatible primes arise from t_FFELT and t_PADIC components.

  ? characteristic(Mod(1,24)*x + Mod(1,18)*y)
  %1 = 6

The library syntax is GEN characteristic(GEN x).

Extracts the n-th-component of x. This is to be understood as follows: every PARI type has one or two initial code words. The components are counted, starting at 1, after these code words. In particular if x is a vector, this is indeed the n-th-component of x, if x is a matrix, the n-th column, if x is a polynomial, the n-th coefficient (i.e. of degree n-1), and for power series, the n-th significant coefficient.

For polynomials and power series, one should rather use polcoeff, and for vectors and matrices, the [] operator. Namely, if x is a vector, then x[n] represents the n-th component of x. If x is a matrix, x[m,n] represents the coefficient of row m and column n of the matrix, x[m,] represents the m-th row of x, and x[,n] represents the n-th column of x.

Using of this function requires detailed knowledge of the structure of the different PARI types, and thus it should almost never be used directly. Some useful exceptions:

      ? x = 3 + O(3^5);
      ? component(x, 2)
      %2 = 81   \\ p^(p-adic accuracy)
      ? component(x, 1)
      %3 = 3    \\ p
      ? q = Qfb(1,2,3);
      ? component(q, 1)
      %5 = 1

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

Conjugate of x. The meaning of this is clear, except that for real quadratic numbers, it means conjugation in the real quadratic field. This function has no effect on integers, reals, intmods, fractions or p-adics. The only forbidden type is polmod (see conjvec for this).

The library syntax is GEN gconj(GEN x).

Conjugate vector representation of z. If z is a polmod, equal to Mod(a,T), this gives a vector of length degree(T) containing:

* the complex embeddings of z if T has rational coefficients, i.e. the a(r[i]) where r = polroots(T);

* the conjugates of z if T has some intmod coefficients;

if z is a finite field element, the result is the vector of conjugates [z,z^p,z^{p^2},...,z^{p^{n-1}}] where n = degree(T).

If z is an integer or a rational number, the result is z. If z is a (row or column) vector, the result is a matrix whose columns are the conjugate vectors of the individual elements of z.

The library syntax is GEN conjvec(GEN z, long prec).

Denominator of x. The meaning of this is clear when x is a rational number or function. If x is an integer or a polynomial, it is treated as a rational number or function, respectively, and the result is equal to 1. For polynomials, you probably want to use

  denominator( content(x) )

instead. As for modular objects, t_INTMOD and t_PADIC have denominator 1, and the denominator of a t_POLMOD is the denominator of its (minimal degree) polynomial representative.

If x is a recursive structure, for instance a vector or matrix, the lcm of the denominators of its components (a common denominator) is computed. This also applies for t_COMPLEXs and t_QUADs.

Warning. Multivariate objects are created according to variable priorities, with possibly surprising side effects (x/y is a polynomial, but y/x is a rational function). See Section se:priority.

The library syntax is GEN denom(GEN x).

Outputs the vector of the digits of |x| in base b, where x and b are integers (b = 10 by default). See fromdigits for the reverse operation.

  ? digits(123)
  %1 = [1, 2, 3, 0]
  
  ? digits(10, 2) \\ base 2
  %2 = [1, 0, 1, 0]

By convention, 0 has no digits:

  ? digits(0)
  %3 = []

The library syntax is GEN digits(GEN x, GEN b = NULL).

Floor of x. When x is in ℝ, the result is the largest integer smaller than or equal to x. Applied to a rational function, floor(x) returns the Euclidean quotient of the numerator by the denominator.

The library syntax is GEN gfloor(GEN x).

Fractional part of x. Identical to x-floor(x). If x is real, the result is in [0,1[.

The library syntax is GEN gfrac(GEN x).

Gives the integer formed by the elements of x seen as the digits of a number in base b (b = 10 by default). This is the reverse of digits:

  ? digits(1234,5)
  %1 = [1,4,4,1,4]
  ? fromdigits([1,4,4,1,4],5)
  %2 = 1234

By convention, 0 has no digits:

  ? fromdigits([])
  %3 = 0

The library syntax is GEN fromdigits(GEN x, GEN b = NULL).

If x is a t_INT, return the binary Hamming weight of |x|. Otherwise x must be of type t_POL, t_VEC, t_COL, t_VECSMALL, or t_MAT and the function returns the number of non-zero coefficients of x.

  ? hammingweight(15)
  %1 = 4
  ? hammingweight(x^100 + 2*x + 1)
  %2 = 3
  ? hammingweight([Mod(1,2), 2, Mod(0,3)])
  %3 = 2
  ? hammingweight(matid(100))
  %4 = 100

The library syntax is long hammingweight(GEN x).

Imaginary part of x. When x is a quadratic number, this is the coefficient of ω in the "canonical" integral basis (1,ω).

The library syntax is GEN gimag(GEN x).

Length of x; #x is a shortcut for length(x). This is mostly useful for

* vectors: dimension (0 for empty vectors),

* lists: number of entries (0 for empty lists),

* matrices: number of columns,

* character strings: number of actual characters (without trailing \0, should you expect it from C char*).

   ? #"a string"
   %1 = 8
   ? #[3,2,1]
   %2 = 3
   ? #[]
   %3 = 0
   ? #matrix(2,5)
   %4 = 5
   ? L = List([1,2,3,4]); #L
   %5 = 4

The routine is in fact defined for arbitrary GP types, but is awkward and useless in other cases: it returns the number of non-code words in x, e.g. the effective length minus 2 for integers since the t_INT type has two code words.

The library syntax is long glength(GEN x).

If v is omitted, lifts intmods from ℤ/nℤ in ℤ, p-adics from ℚ_p to ℚ (as truncate), and polmods to polynomials. Otherwise, lifts only polmods whose modulus has main variable v. t_FFELT are not lifted, nor are List elements: you may convert the latter to vectors first, or use apply(lift,L). More generally, components for which such lifts are meaningless (e.g. character strings) are copied verbatim.

  ? lift(Mod(5,3))
  %1 = 2
  ? lift(3 + O(3^9))
  %2 = 3
  ? lift(Mod(x,x^2+1))
  %3 = x
  ? lift(Mod(x,x^2+1))
  %4 = x

Lifts are performed recursively on an object components, but only by one level: once a t_POLMOD is lifted, the components of the result are not lifted further.

  ? lift(x * Mod(1,3) + Mod(2,3))
  %4 = x + 2
  ? lift(x * Mod(y,y^2+1) + Mod(2,3))
  %5 = y*x + Mod(2, 3)   \\  do you understand this one?
  ? lift(x * Mod(y,y^2+1) + Mod(2,3), 'x)
  %6 = Mod(y, y^2 + 1)*x + Mod(Mod(2, 3), y^2 + 1)
  ? lift(%, y)
  %7 = y*x + Mod(2, 3)

To recursively lift all components not only by one level, but as long as possible, use liftall. To lift only t_INTMODs and t_PADICs components, use liftint. To lift only t_POLMODs components, use liftpol. Finally, centerlift allows to lift t_INTMODs and t_PADICs using centered residues (lift of smallest absolute value).

The library syntax is GEN lift0(GEN x, long v = -1) where v is a variable number. Also available is GEN lift(GEN x) corresponding to lift0(x,-1).

Recursively lift all components of x from ℤ/nℤ to ℤ, from ℚ_p to ℚ (as truncate), and polmods to polynomials. t_FFELT are not lifted, nor are List elements: you may convert the latter to vectors first, or use apply(liftall,L). More generally, components for which such lifts are meaningless (e.g. character strings) are copied verbatim.

  ? liftall(x * (1 + O(3)) + Mod(2,3))
  %1 = x + 2
  ? liftall(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2))
  %2 = y*x + 2*z

The library syntax is GEN liftall(GEN x).

Recursively lift all components of x from ℤ/nℤ to ℤ and from ℚ_p to ℚ (as truncate). t_FFELT are not lifted, nor are List elements: you may convert the latter to vectors first, or use apply(liftint,L). More generally, components for which such lifts are meaningless (e.g. character strings) are copied verbatim.

  ? liftint(x * (1 + O(3)) + Mod(2,3))
  %1 = x + 2
  ? liftint(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2))
  %2 = Mod(y, y^2 + 1)*x + Mod(Mod(2*z, z^2), y^2 + 1)

The library syntax is GEN liftint(GEN x).

Recursively lift all components of x which are polmods to polynomials. t_FFELT are not lifted, nor are List elements: you may convert the latter to vectors first, or use apply(liftpol,L). More generally, components for which such lifts are meaningless (e.g. character strings) are copied verbatim.

  ? liftpol(x * (1 + O(3)) + Mod(2,3))
  %1 = (1 + O(3))*x + Mod(2, 3)
  ? liftpol(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2))
  %2 = y*x + Mod(2, 3)*z

The library syntax is GEN liftpol(GEN x).

Algebraic norm of x, i.e. the product of x with its conjugate (no square roots are taken), or conjugates for polmods. For vectors and matrices, the norm is taken componentwise and hence is not the L^2-norm (see norml2). Note that the norm of an element of ℝ is its square, so as to be compatible with the complex norm.

The library syntax is GEN gnorm(GEN x).

Numerator of x. The meaning of this is clear when x is a rational number or function. If x is an integer or a polynomial, it is treated as a rational number or function, respectively, and the result is x itself. For polynomials, you probably want to use

  numerator( content(x) )

instead.

In other cases, numerator(x) is defined to be denominator(x)*x. This is the case when x is a vector or a matrix, but also for t_COMPLEX or t_QUAD. In particular since a t_PADIC or t_INTMOD has denominator 1, its numerator is itself.

Warning. Multivariate objects are created according to variable priorities, with possibly surprising side effects (x/y is a polynomial, but y/x is a rational function). See Section se:priority.

The library syntax is GEN numer(GEN x).

Generates the k-th permutation (as a row vector of length n) of the numbers 1 to n. The number k is taken modulo n!, i.e. inverse function of permtonum. The numbering used is the standard lexicographic ordering, starting at 0.

The library syntax is GEN numtoperm(long n, GEN k).

Returns an object meaning + oo , for use in functions such as intnum. It can be negated (-oo represents - oo ), and compared to real numbers (t_INT, t_FRAC, t_REAL), with the expected meaning: + oo is greater than any real number and - oo is smaller.

The library syntax is GEN mkoo().

Returns the absolute p-adic precision of the object x; this is the minimum precision of the components of x. The result is +oo if x is an exact object (as a p-adic):

  ? padicprec((1 + O(2^5)) * x + (2 + O(2^4)), 2)
  %1 = 4
  ? padicprec(x + 2, 2)
  %2 = +oo
  ? padicprec(2 + x + O(x^2), 2)
  %3 = +oo

The function raises an exception if it encounters an object incompatible with p-adic computations:

  ? padicprec(O(3), 2)
   ***   at top-level: padicprec(O(3),2)
   ***                 ^-----------------
   *** padicprec: inconsistent moduli in padicprec: 3 != 2
  
  ? padicprec(1.0, 2)
   ***   at top-level: padicprec(1.0,2)
   ***                 ^----------------
   *** padicprec: incorrect type in padicprec (t_REAL).

The library syntax is GEN gppadicprec(GEN x, GEN p). Also available is the function long padicprec(GEN x, GEN p), which returns LONG_MAX if x = 0 and the p-adic precision as a long integer.

Given a permutation x on n elements, gives the number k such that x = numtoperm(n,k), i.e. inverse function of numtoperm. The numbering used is the standard lexicographic ordering, starting at 0.

The library syntax is GEN permtonum(GEN x).

The function behaves differently according to whether n is present and positive or not. If n is missing, the function returns the precision in decimal digits of the PARI object x. If x is an exact object, the function returns +oo.

  ? precision(exp(1e-100))
  %1 = 154                \\ 154 significant decimal digits
  ? precision(2 + x)
  %2 = +oo                \\ exact object
  ? precision(0.5 + O(x))
  %3 = 38                 \\ floating point accuracy, NOT series precision
  ? precision( [ exp(1e-100), 0.5 ] )
  %4 = 38                 \\ minimal accuracy among components

If n is present, the function creates a new object equal to x with a new floating point precision n: n is the number of desired significant decimal digits. If n is smaller than the precision of a t_REAL component of x, it is truncated, otherwise it is extended with zeros. For exact or non-floating point types, no change.

The library syntax is GEN precision0(GEN x, long n). Also available are GEN gprec(GEN x, long n) and long precision(GEN x). In both, the accuracy is expressed in words (32-bit or 64-bit depending on the architecture).

Returns a random element in various natural sets depending on the argument N.

* t_INT: returns an integer uniformly distributed between 0 and N-1. Omitting the argument is equivalent to random(2^31).

* t_REAL: returns a real number in [0,1[ with the same accuracy as N (whose mantissa has the same number of significant words).

* t_INTMOD: returns a random intmod for the same modulus.

* t_FFELT: returns a random element in the same finite field.

* t_VEC of length 2, N = [a,b]: returns an integer uniformly distributed between a and b.

* t_VEC generated by ellinit over a finite field k (coefficients are t_INTMODs modulo a prime or t_FFELTs): returns a "random" k-rational affine point on the curve. More precisely if the curve has a single point (at infinity!) we return it; otherwise we return an affine point by drawing an abscissa uniformly at random until ellordinate succeeds. Note that this is definitely not a uniform distribution over E(k), but it should be good enough for applications.

* t_POL return a random polynomial of degree at most the degree of N. The coefficients are drawn by applying random to the leading coefficient of N.

  ? random(10)
  %1 = 9
  ? random(Mod(0,7))
  %2 = Mod(1, 7)
  ? a = ffgen(ffinit(3,7), 'a); random(a)
  %3 = a^6 + 2*a^5 + a^4 + a^3 + a^2 + 2*a
  ? E = ellinit([3,7]*Mod(1,109)); random(E)
  %4 = [Mod(103, 109), Mod(10, 109)]
  ? E = ellinit([1,7]*a^0); random(E)
  %5 = [a^6 + a^5 + 2*a^4 + 2*a^2, 2*a^6 + 2*a^4 + 2*a^3 + a^2 + 2*a]
  ? random(Mod(1,7)*x^4)
  %6 = Mod(5, 7)*x^4 + Mod(6, 7)*x^3 + Mod(2, 7)*x^2 + Mod(2, 7)*x + Mod(5, 7)
  

These variants all depend on a single internal generator, and are independent from your operating system's random number generators. A random seed may be obtained via getrand, and reset using setrand: from a given seed, and given sequence of randoms, the exact same values will be generated. The same seed is used at each startup, reseed the generator yourself if this is a problem. Note that internal functions also call the random number generator; adding such a function call in the middle of your code will change the numbers produced.

Technical note. Up to version 2.4 included, the internal generator produced pseudo-random numbers by means of linear congruences, which were not well distributed in arithmetic progressions. We now use Brent's XORGEN algorithm, based on Feedback Shift Registers, see http://wwwmaths.anu.edu.au/~brent/random.html. The generator has period 2^{4096}-1, passes the Crush battery of statistical tests of L'Ecuyer and Simard, but is not suitable for cryptographic purposes: one can reconstruct the state vector from a small sample of consecutive values, thus predicting the entire sequence.

The library syntax is GEN genrand(GEN N = NULL).

Also available: GEN ellrandom(GEN E) and GEN ffrandom(GEN a).

Real part of x. In the case where x is a quadratic number, this is the coefficient of 1 in the "canonical" integral basis (1,ω).

The library syntax is GEN greal(GEN x).

If x is in ℝ, rounds x to the nearest integer (rounding to + oo in case of ties), then and sets e to the number of error bits, that is the binary exponent of the difference between the original and the rounded value (the "fractional part"). If the exponent of x is too large compared to its precision (i.e. e > 0), the result is undefined and an error occurs if e was not given.

Important remark. Contrary to the other truncation functions, this function operates on every coefficient at every level of a PARI object. For example truncate((2.4*X^2-1.7)/(X)) = 2.4*X, whereas round((2.4*X^2-1.7)/(X)) = (2*X^2-2)/(X). An important use of round is to get exact results after an approximate computation, when theory tells you that the coefficients must be integers.

The library syntax is GEN round0(GEN x, GEN *e = NULL). Also available are GEN grndtoi(GEN x, long *e) and GEN ground(GEN x).

Returns the absolute precision of x with respec to power series in the variable v; this is the minimum precision of the components of x. The result is +oo if x is an exact object (as a series in v):

  ? serprec(x + O(y^2), y)
  %1 = 2
  ? serprec(x + 2, x)
  %2 = +oo
  ? serprec(2 + x + O(x^2), y)
  %3 = +oo

The library syntax is GEN gpserprec(GEN x, long v) where v is a variable number. Also available is long serprec(GEN x, GEN p), which returns LONG_MAX if x = 0 and the series precision as a long integer.

This function simplifies x as much as it can. Specifically, a complex or quadratic number whose imaginary part is the integer 0 (i.e. not Mod(0,2) or 0.E-28) is converted to its real part, and a polynomial of degree 0 is converted to its constant term. Simplifications occur recursively.

This function is especially useful before using arithmetic functions, which expect integer arguments:

  ? x = 2 + y - y
  %1 = 2
  ? isprime(x)
    ***   at top-level: isprime(x)
    ***                 ^----------
    *** isprime: not an integer argument in an arithmetic function
  ? type(x)
  %2 = "t_POL"
  ? type(simplify(x))
  %3 = "t_INT"

Note that GP results are simplified as above before they are stored in the history. (Unless you disable automatic simplification with \y, that is.) In particular

  ? type(%1)
  %4 = "t_INT"

The library syntax is GEN simplify(GEN x).

Outputs the total number of bytes occupied by the tree representing the PARI object x.

The library syntax is long gsizebyte(GEN x). Also available is long gsizeword(GEN x) returning a number of words.

This function is DEPRECATED, essentially meaningless, and provided for backwards compatibility only. Don't use it!

outputs a quick upper bound for the number of decimal digits of (the components of) x, off by at most 1. More precisely, for a positive integer x, it computes (approximately) the ceiling of floor(1 + log_2 x) log_{10}2,

To count the number of decimal digits of a positive integer x, use #digits(x). To estimate (recursively) the size of x, use normlp(x).

The library syntax is long sizedigit(GEN x).

Truncates x and sets e to the number of error bits. When x is in ℝ, this means that the part after the decimal point is chopped away, e is the binary exponent of the difference between the original and the truncated value (the "fractional part"). If the exponent of x is too large compared to its precision (i.e. e > 0), the result is undefined and an error occurs if e was not given. The function applies componentwise on vector / matrices; e is then the maximal number of error bits. If x is a rational function, the result is the "integer part" (Euclidean quotient of numerator by denominator) and e is not set.

Note a very special use of truncate: when applied to a power series, it transforms it into a polynomial or a rational function with denominator a power of X, by chopping away the O(X^k). Similarly, when applied to a p-adic number, it transforms it into an integer or a rational number by chopping away the O(p^k).

The library syntax is GEN trunc0(GEN x, GEN *e = NULL). The following functions are also available: GEN gtrunc(GEN x) and GEN gcvtoi(GEN x, long *e).

Computes the highest exponent of p dividing x. If p is of type integer, x must be an integer, an intmod whose modulus is divisible by p, a fraction, a q-adic number with q = p, or a polynomial or power series in which case the valuation is the minimum of the valuation of the coefficients.

If p is of type polynomial, x must be of type polynomial or rational function, and also a power series if x is a monomial. Finally, the valuation of a vector, complex or quadratic number is the minimum of the component valuations.

If x = 0, the result is +oo if x is an exact object. If x is a p-adic numbers or power series, the result is the exponent of the zero. Any other type combinations gives an error.

The library syntax is GEN gpvaluation(GEN x, GEN p). Also available is long gvaluation(GEN x, GEN p), which returns LONG_MAX if x = 0 and the valuation as a long integer.

Return a variable name whose priority is higher than the priority of v (of all existing variables if v is omitted). This is a counterpart to varlower.

  ? Pol([x,x], t)
   ***   at top-level: Pol([x,x],t)
   ***                 ^------------
   *** Pol: incorrect priority in gtopoly: variable x <= t
  ? t = varhigher("t", x);
  ? Pol([x,x], t)
  %3 = x*t + x

This routine is useful since new GP variables directly created by the interpreter always have lower priority than existing GP variables. When some basic objects already exist in a variable that is incompatible with some function requirement, you can now create a new variable with a suitable priority instead of changing variables in existing objects:

  ? K = nfinit(x^2+1);
  ? rnfequation(K,y^2-2)
   ***   at top-level: rnfequation(K,y^2-2)
   ***                 ^--------------------
   *** rnfequation: incorrect priority in rnfequation: variable y >= x
  ? y = varhigher("y", x);
  ? rnfequation(K, y^2-2)
  %3 = y^4 - 2*y^2 + 9

Caution 1. The name is an arbitrary character string, only used for display purposes and need not be related to the GP variable holding the result, nor to be a valid variable name. In particular the name can not be used to retrieve the variable, it is not even present in the parser's hash tables.

  ? x = varhigher("#");
  ? x^2
  %2 = #^2

Caution 2. There are a limited number of variables and if no existing variable with the given display name has the requested priority, the call to varhigher uses up one such slot. Do not create new variables in this way unless it's absolutely necessary, reuse existing names instead and choose sensible priority requirements: if you only need a variable with higher priority than x, state so rather than creating a new variable with highest priority.

  \\ quickly use up all variables
  ? n = 0; while(1,varhigher("tmp"); n++)
   ***   at top-level: n=0;while(1,varhigher("tmp");n++)
   ***                             ^-------------------
   *** varhigher: no more variables available.
   ***   Break loop: type 'break' to go back to GP prompt
  break> n
  65510
  \\ infinite loop: here we reuse the same 'tmp'
  ? n = 0; while(1,varhigher("tmp", x); n++)

The library syntax is GEN varhigher(const char *name, long v = -1) where v is a variable number.

Gives the main variable of the object x (the variable with the highest priority used in x), and p if x is a p-adic number. Return 0 if x has no variable attached to it.

  ? variable(x^2 + y)
  %1 = x
  ? variable(1 + O(5^2))
  %2 = 5
  ? variable([x,y,z,t])
  %3 = x
  ? variable(1)
  %4 = 0

The construction

     if (!variable(x),...)

can be used to test whether a variable is attached to x.

If x is omitted, returns the list of user variables known to the interpreter, by order of decreasing priority. (Highest priority is initially x, which come first until varhigher is used.) If varhigher or varlower are used, it is quite possible to end up with different variables (with different priorities) printed in the same way: they will then appear multiple times in the output:

  ? varhigher("y");
  ? varlower("y");
  ? variable()
  %4 = [y, x, y]

Using v = variable() then v[1], v[2], etc. allows to recover and use existing variables.

The library syntax is GEN gpolvar(GEN x = NULL). However, in library mode, this function should not be used for x non-NULL, since gvar is more appropriate. Instead, for x a p-adic (type t_PADIC), p is gel(x,2); otherwise, use long gvar(GEN x) which returns the variable number of x if it exists, NO_VARIABLE otherwise, which satisfies the property varncmp(NO_VARIABLE, v) > 0 for all valid variable number v, i.e. it has lower priority than any variable.

Returns the list of all variables occuring in object x (all user variables known to the interpreter if x is omitted), sorted by decreasing priority.

  ? variables([x^2 + y*z + O(t), a+x])
  %1 = [x, y, z, t, a]

The construction

     if (!variables(x),...)

can be used to test whether a variable is attached to x.

If varhigher or varlower are used, it is quite possible to end up with different variables (with different priorities) printed in the same way: they will then appear multiple times in the output:

  ? y1 = varhigher("y");
  ? y2 = varlower("y");
  ? variables(y*y1*y2)
  %4 = [y, y, y]

The library syntax is GEN variables_vec(GEN x = NULL).

Also available is GEN variables_vecsmall(GEN x) which returns the (sorted) variable numbers instead of the attached monomials of degree 1.

Return a variable name whose priority is lower than the priority of v (of all existing variables if v is omitted). This is a counterpart to varhigher.

New GP variables directly created by the interpreter always have lower priority than existing GP variables, but it is not easy to check whether an identifier is currently unused, so that the corresponding variable has the expected priority when it's created! Thus, depending on the session history, the same command may fail or succeed:

  ? t; z;  \\ now t > z
  ? rnfequation(t^2+1,z^2-t)
   ***   at top-level: rnfequation(t^2+1,z^
   ***                 ^--------------------
   *** rnfequation: incorrect priority in rnfequation: variable t >= t

Restart and retry:

  ? z; t;  \\ now z > t
  ? rnfequation(t^2+1,z^2-t)
  %2 = z^4 + 1

It is quite annoying for package authors, when trying to define a base ring, to notice that the package may fail for some users depending on their session history. The safe way to do this is as follows:

  ? z; t;  \\ In new session: now z > t
  ...
  ? t = varlower("t", 'z);
  ? rnfequation(t^2+1,z^2-2)
  %2 = z^4 - 2*z^2 + 9
  ? variable()
  %3 = [x, y, z, t]

  ? t; z;  \\ In new session: now t > z
  ...
  ? t = varlower("t", 'z); \\ create a new variable, still printed "t"
  ? rnfequation(t^2+1,z^2-2)
  %2 = z^4 - 2*z^2 + 9
  ? variable()
  %3 = [x, y, t, z, t]

Now both constructions succeed. Note that in the first case, varlower is essentially a no-op, the existing variable t has correct priority. While in the second case, two different variables are displayed as t, one with higher priority than z (created in the first line) and another one with lower priority (created by varlower).

Caution 1. The name is an arbitrary character string, only used for display purposes and need not be related to the GP variable holding the result, nor to be a valid variable name. In particular the name can not be used to retrieve the variable, it is not even present in the parser's hash tables.

  ? x = varlower("#");
  ? x^2
  %2 = #^2

Caution 2. There are a limited number of variables and if no existing variable with the given display name has the requested priority, the call to varlower uses up one such slot. Do not create new variables in this way unless it's absolutely necessary, reuse existing names instead and choose sensible priority requirements: if you only need a variable with higher priority than x, state so rather than creating a new variable with highest priority.

  \\ quickly use up all variables
  ? n = 0; while(1,varlower("x"); n++)
   ***   at top-level: n=0;while(1,varlower("x");n++)
   ***                             ^-------------------
   *** varlower: no more variables available.
   ***   Break loop: type 'break' to go back to GP prompt
  break> n
  65510
  \\ infinite loop: here we reuse the same 'tmp'
  ? n = 0; while(1,varlower("tmp", x); n++)

The library syntax is GEN varlower(const char *name, long v = -1) where v is a variable number.