Interrupts execution of current *seq*, and
immediately exits from the n innermost enclosing loops, within the
current function call (or the top level loop); the integer n must be
positive. If n is greater than the number of enclosing loops, all
enclosing loops are exited.

Interrupt the program and enter the breakloop. The program continues when the breakloop is exited.

? f(N,x)=my(z=x^2+1);breakpoint();gcd(N,z^2+1-z); ? f(221,3) *** at top-level: f(221,3) *** ^-------- *** in function f: my(z=x^2+1);breakpoint();gcd(N,z *** ^-------------------- *** Break loop: type <Return> to continue; 'break' to go back to GP break> z 10 break> %2 = 13

(In the break loop) go down n frames. This allows to cancel a previous call to
`dbg_up`

.

In the break loop, return the error data of the current error, if any.
See `iferr`

for details about error data. Compare:

? iferr(1/(Mod(2,12019)^(6!)-1),E,Vec(E)) %1 = ["e_INV", "Fp_inv", Mod(119, 12019)] ? 1/(Mod(2,12019)^(6!)-1) *** at top-level: 1/(Mod(2,12019)^(6!)- *** ^-------------------- *** _/_: impossible inverse in Fp_inv: Mod(119, 12019). *** Break loop: type 'break' to go back to GP prompt break> Vec(dbg_err()) ["e_INV", "Fp_inv", Mod(119, 12019)]

(In the break loop) go up n frames. This allows to inspect data of the
parent function. To cancel a `dbg_up`

call, use `dbg_down`

Print the inner structure of `A`

, complete if `n`

is omitted, up
to level `n`

otherwise. This is useful for debugging. This is similar to
`\x`

but does not require `A`

to be an history entry. In particular,
it can be used in the break loop.

Evaluates *seq*, where
the formal variable X goes from a to b. Nothing is done if a > b.
a and b must be in **R**.

Evaluates *seq*,
where the formal variable n ranges over the composite numbers between the
non-negative real numbers a to b, including a and b if they are
composite. Nothing is done if a > b.

? forcomposite(n = 0, 10, print(n)) 4 6 8 9 10Omitting b means we will run through all composites

`>=`

a,
starting an infinite loop; it is expected that the user will break out of
the loop himself at some point, using `break`

or `return`

.
Note that the value of n cannot be modified within *seq*:

? forcomposite(n = 2, 10, n = []) *** at top-level: forcomposite(n=2,10,n=[]) *** ^--- *** index read-only: was changed to [].

Evaluates *seq*, where
the formal variable X ranges through the divisors of n
(see `divisors`

, which is used as a subroutine). It is assumed that
`factor`

can handle n, without negative exponents. Instead of n,
it is possible to input a factorization matrix, i.e. the output of
`factor(n)`

.

This routine uses `divisors`

as a subroutine, then loops over the
divisors. In particular, if n is an integer, divisors are sorted by
increasing size.

To avoid storing all divisors, possibly using a lot of memory, the following (much slower) routine loops over the divisors using essentially constant space:

FORDIV(N)= { my(P, E); P = factor(N); E = P[,2]; P = P[,1]; forvec( v = vector(#E, i, [0,E[i]]), X = factorback(P, v) \\ ... ); } ? for(i=1,10^5, FORDIV(i)) time = 3,445 ms. ? for(i=1,10^5, fordiv(i, d, )) time = 490 ms.

Evaluates *seq*, where the formal variable E = [*name*, M, G]
ranges through all elliptic curves of conductors from a to b. In this
notation *name* is the curve name in Cremona's elliptic curve database,
M is the minimal model, G is a **Z**-basis of the free part of the
Mordell-Weil group E(**Q**).

? forell(E, 1, 500, my([name,M,G] = E); \ if (#G > 1, print(name))) 389a1 433a1 446d1

The `elldata`

database must be installed and contain data for the
specified conductors.

The library syntax is

.**forell**(void *data, long (*call)(void*,GEN), long a, long b)

Evaluate *seq* over the partitions X = [x_1,...x_n] of the
integer k, i.e. increasing sequences x_1 `<=`

x_2... `<=`

x_n of sum
x_1+...+ x_n = k. By convention, 0 admits only the empty partition and
negative numbers have no partitions. A partition is given by a
`t_VECSMALL`

, where parts are sorted in nondecreasing order:

? forpart(X=3, print(X)) Vecsmall([3]) Vecsmall([1, 2]) Vecsmall([1, 1, 1])Optional parameters n and a are as follows:

***** n = *nmax* (resp. n = [*nmin*,*nmax*]) restricts
partitions to length less than *nmax* (resp. length between
*nmin* and nmax), where the *length* is the number of nonzero
entries.

***** a = *amax* (resp. a = [*amin*,*amax*]) restricts the parts
to integers less than *amax* (resp. between *amin* and
*amax*).

By default, parts are positive and we remove zero entries unless amin `<=`

0,
in which case X is of constant length *nmax*.

\\ at most 3 non-zero parts, all <= 4 ? forpart(v=5,print(Vec(v)),4,3) [1, 4] [2, 3] [1, 1, 3] [1, 2, 2] \\ between 2 and 4 parts less than 5, fill with zeros ? forpart(v=5,print(Vec(v)),[0,5],[2,4]) [0, 0, 1, 4] [0, 0, 2, 3] [0, 1, 1, 3] [0, 1, 2, 2] [1, 1, 1, 2]

The behaviour is unspecified if X is modified inside the loop.

The library syntax is

.**forpart**(void *data, long (*call)(void*,GEN), long k, GEN a, GEN n)

Evaluates *seq*,
where the formal variable p ranges over the prime numbers between the real
numbers a to b, including a and b if they are prime. More precisely,
the value of
p is incremented to `nextprime(p + 1)`

, the smallest prime strictly
larger than p, at the end of each iteration. Nothing is done if a > b.

? forprime(p = 4, 10, print(p)) 5 7Omitting b means we will run through all primes

`>=`

a,
starting an infinite loop; it is expected that the user will break out of
the loop himself at some point, using `break`

or `return`

.
Note that the value of p cannot be modified within *seq*:

? forprime(p = 2, 10, p = []) *** at top-level: forprime(p=2,10,p=[]) *** ^--- *** prime index read-only: was changed to [].

Evaluates *seq*,
where the formal variable X goes from a to b, in increments of s.
Nothing is done if s > 0 and a > b or if s < 0 and a < b. s must be in
**R**^* or a vector of steps [s_1,...,s_n]. In the latter case, the
successive steps are used in the order they appear in s.

? forstep(x=5, 20, [2,4], print(x)) 5 7 11 13 17 19

Evaluates *seq* for
each subgroup H of the *abelian* group G (given in
SNF form or as a vector of elementary divisors).

If *bound* is present, and is a positive integer, restrict the output to
subgroups of index less than *bound*. If *bound* is a vector
containing a single positive integer B, then only subgroups of index
exactly equal to B are computed

The subgroups are not ordered in any
obvious way, unless G is a p-group in which case Birkhoff's algorithm
produces them by decreasing index. A subgroup is given as a matrix
whose columns give its generators on the implicit generators of G. For
example, the following prints all subgroups of index less than 2 in G =
**Z**/2**Z** g_1 x **Z**/2**Z** g_2:

? G = [2,2]; forsubgroup(H=G, 2, print(H)) [1; 1] [1; 2] [2; 1] [1, 0; 1, 1]

The last one, for instance is generated by (g_1, g_1 + g_2). This
routine is intended to treat huge groups, when `subgrouplist`

is not an
option due to the sheer size of the output.

For maximal speed the subgroups have been left as produced by the algorithm. To print them in canonical form (as left divisors of G in HNF form), one can for instance use

? G = matdiagonal([2,2]); forsubgroup(H=G, 2, print(mathnf(concat(G,H)))) [2, 1; 0, 1] [1, 0; 0, 2] [2, 0; 0, 1] [1, 0; 0, 1]

Note that in this last representation, the index [G:H] is given by the
determinant. See `galoissubcyclo`

and `galoisfixedfield`

for
applications to Galois theory.

The library syntax is

.**forsubgroup**(void *data, long (*call)(void*,GEN), GEN G, GEN bound)

Let v be an n-component
vector (where n is arbitrary) of two-component vectors [a_i,b_i]
for 1 `<=`

i `<=`

n. This routine evaluates *seq*, where the formal
variables X[1],..., X[n] go from a_1 to b_1,..., from a_n to
b_n, i.e. X goes from [a_1,...,a_n] to [b_1,...,b_n] with respect
to the lexicographic ordering. (The formal variable with the highest index
moves the fastest.) If *flag* = 1, generate only nondecreasing vectors X, and
if *flag* = 2, generate only strictly increasing vectors X.

The type of X is the same as the type of v: `t_VEC`

or `t_COL`

.

Evaluates the expression sequence *seq1* if a is non-zero, otherwise
the expression *seq2*. Of course, *seq1* or *seq2* may be empty:

`if (a,`

evaluates *seq*)*seq* if a is not equal to zero
(you don't have to write the second comma), and does nothing otherwise,

`if (a,,`

evaluates *seq*)*seq* if a is equal to zero, and
does nothing otherwise. You could get the same result using the `!`

(`not`

) operator: `if (!a,`

.*seq*)

The value of an `if`

statement is the value of the branch that gets
evaluated: for instance

x = if(n % 4 == 1, y, z);sets x to y if n is 1 modulo 4, and to z otherwise.

Successive 'else' blocks can be abbreviated in a single compound `if`

as follows:

if (test1, seq1, test2, seq2, ... testn, seqn, seqdefault);is equivalent to

if (test1, seq1 , if (test2, seq2 , ... if (testn, seqn, seqdefault)...));For instance, this allows to write traditional switch / case constructions:

if (x == 0, do0(), x == 1, do1(), x == 2, do2(), dodefault());

**Remark.**
The boolean operators `&&`

and `||`

are evaluated
according to operator precedence as explained in Section [**Label: se:operators**], but,
contrary to other operators, the evaluation of the arguments is stopped
as soon as the final truth value has been determined. For instance

if (x != 0 && f(1/x), ...)

is a perfectly safe statement.

**Remark.** Functions such as `break`

and `next`

operate on
*loops*, such as `forxxx`

, `while`

, `until`

. The `if`

statement is *not* a loop. (Obviously!)

Evaluates the expression sequence *seq1*. If an error occurs,
set the formal parameter *E* set to the error data.
If *pred* is not present or evaluates to true, catch the error
and evaluate *seq2*. Both *pred* and *seq2* can reference *E*.
The error type is given by `errname(E)`

, and other data can be
accessed using the `component`

function. The code *seq2* should check
whether the error is the one expected. In the negative the error can be
rethrown using `error(E)`

(and possibly caught by an higher `iferr`

instance). The following uses `iferr`

to implement Lenstra's ECM factoring
method

? ecm(N, B = 1000!, nb = 100)= { for(a = 1, nb, iferr(ellmul(ellinit([a,1]*Mod(1,N)), [0,1]*Mod(1,N), B), E, return(gcd(lift(component(E,2)),N)), errname(E)=="e_INV" && type(component(E,2)) == "t_INTMOD")) } ? ecm(2^101-1) %2 = 7432339208719

The return value of `iferr`

itself is the value of *seq2* if an
error occurs, and the value of *seq1* otherwise. We now describe the
list of valid error types, and the associated error data *E*; in each
case, we list in order the components of *E*, accessed via
`component(E,1)`

, `component(E,2)`

, etc.

**Internal errors, "system" errors.**

***** `"e_ARCH"`

. A requested feature s is not available on this
architecture or operating system.
*E* has one component (`t_STR`

): the missing feature name s.

***** `"e_BUG"`

. A bug in the PARI library, in function s.
*E* has one component (`t_STR`

): the function name s.

***** `"e_FILE"`

. Error while trying to open a file.
*E* has two components, 1 (`t_STR`

): the file type (input, output,
etc.), 2 (`t_STR`

): the file name.

***** `"e_IMPL"`

. A requested feature s is not implemented.
*E* has one component, 1 (`t_STR`

): the feature name s.

***** `"e_PACKAGE"`

. Missing optional package s.
*E* has one component, 1 (`t_STR`

): the package name s.

**Syntax errors, type errors.**

***** `"e_DIM"`

. The dimensions of arguments x and y submitted
to function s does not match up.
E.g., multiplying matrices of inconsistent dimension, adding vectors of
different lengths,...
*E* has three component, 1 (`t_STR`

): the function name s, 2: the
argument x, 3: the argument y.

***** `"e_FLAG"`

. A flag argument is out of bounds in function s.
*E* has one component, 1 (`t_STR`

): the function name s.

***** `"e_NOTFUNC"`

. Generated by the PARI evaluator; tried to use a
`GEN`

x which is not a `t_CLOSURE`

in a function call syntax (as in
`f = 1; f(2);`

).
*E* has one component, 1: the offending `GEN`

x.

***** `"e_OP"`

. Impossible operation between two objects than cannot
be typecast to a sensible common domain for deeper reasons than a type
mismatch, usually for arithmetic reasons. As in `O(2) + O(3)`

: it is
valid to add two `t_PADIC`

s, provided the underlying prime is the same; so
the addition is not forbidden a priori for type reasons, it only becomes so
when inspecting the objects and trying to perform the operation.
*E* has three components, 1 (`t_STR`

): the operator name *op*,
2: first argument, 3: second argument.

***** `"e_TYPE"`

. An argument x of function s had an unexpected type.
(As in `factor("blah")`

.)
*E* has two components, 1 (`t_STR`

): the function name s,
2: the offending argument x.

***** `"e_TYPE2"`

. Forbidden operation between two objects than cannot be
typecast to a sensible common domain, because their types do not match up.
(As in `Mod(1,2) + Pi`

.)
*E* has three components, 1 (`t_STR`

): the operator name *op*,
2: first argument, 3: second argument.

***** `"e_PRIORITY"`

. Object o in function s contains
variables whose priority is incompatible with the expected operation.
E.g. `Pol([x,1], 'y)`

: this raises an error because it's not possible to
create a polynomial whose coefficients involve variables with higher priority
than the main variable. E has four components: 1 (`t_STR`

): the function
name s, 2: the offending argument o, 3 (`t_STR`

): an operator
*op* describing the priority error, 4 (`t_POL`

):
the variable v describing the priority error. The argument
satisfies `variable`

(x) *op* `variable`

(v).

***** `"e_VAR"`

. The variables of arguments x and y submitted
to function s does not match up. E.g., considering the algebraic number
`Mod(t,t^2+1)`

in `nfinit(x^2+1)`

.
*E* has three component, 1 (`t_STR`

): the function name s, 2
(`t_POL`

): the argument x, 3 (`t_POL`

): the argument y.

**Overflows.**

***** `"e_COMPONENT"`

. Trying to access an inexistent component in a
vector/matrix/list in a function: the index is less than 1 or greater
than the allowed length.
*E* has four components,
1 (`t_STR`

): the function name
2 (`t_STR`

): an operator *op* ( < or > ),
2 (`t_GEN`

): a numerical limit l bounding the allowed range,
3 (`GEN`

): the index x. It satisfies x *op* l.

***** `"e_DOMAIN"`

. An argument is not in the function's domain.
*E* has five components, 1 (`t_STR`

): the function name,
2 (`t_STR`

): the mathematical name of the out-of-domain argument
3 (`t_STR`

): an operator *op* describing the domain error,
4 (`t_GEN`

): the numerical limit l describing the domain error,
5 (`GEN`

): the out-of-domain argument x. The argument satisfies x
*op* l, which prevents it from belonging to the function's domain.

***** `"e_MAXPRIME"`

. A function using the precomputed list of prime
numbers ran out of primes.
*E* has one component, 1 (`t_INT`

): the requested prime bound, which
overflowed `primelimit`

or 0 (bound is unknown).

***** `"e_MEM"`

. A call to `pari_malloc`

or `pari_realloc`

failed. *E* has no component.

***** `"e_OVERFLOW"`

. An object in function s becomes too large to be
represented within PARI's hardcoded limits. (As in `2^2^2^10`

or
`exp(1e100)`

, which overflow in `lg`

and `expo`

.)
*E* has one component, 1 (`t_STR`

): the function name s.

***** `"e_PREC"`

. Function s fails because input accuracy is too low.
(As in `floor(1e100)`

at default accuracy.)
*E* has one component, 1 (`t_STR`

): the function name s.

***** `"e_STACK"`

. The PARI stack overflows.
*E* has no component.

**Errors triggered intentionally.**

***** `"e_ALARM"`

. A timeout, generated by the `alarm`

function.
*E* has one component (`t_STR`

): the error message to print.

***** `"e_USER"`

. A user error, as triggered by
`error`

(g_1,...,g_n).
*E* has one component, 1 (`t_VEC`

): the vector of n arguments given
to `error`

.

**Mathematical errors.**

***** `"e_CONSTPOL"`

. An argument of function s is a constant
polynomial, which does not make sense. (As in `galoisinit(Pol(1))`

.)
*E* has one component, 1 (`t_STR`

): the function name s.

***** `"e_COPRIME"`

. Function s expected coprime arguments,
and did receive x,y, which were not.
*E* has three component, 1 (`t_STR`

): the function name s,
2: the argument x, 3: the argument y.

***** `"e_INV"`

. Tried to invert a non-invertible object x in
function s.
*E* has two components, 1 (`t_STR`

): the function name s,
2: the non-invertible x. If x = `Mod`

(a,b)
is a `t_INTMOD`

and a is not 0 mod b, this allows to factor
the modulus, as `gcd`

(a,b) is a non-trivial divisor of b.

***** `"e_IRREDPOL"`

. Function s expected an irreducible polynomial,
and did receive T, which was not. (As in `nfinit(x^2-1)`

.)
*E* has two component, 1 (`t_STR`

): the function name s,
2 (`t_POL`

): the polynomial x.

***** `"e_MISC"`

. Generic uncategorized error.
*E* has one component (`t_STR`

): the error message to print.

***** `"e_MODULUS"`

. moduli x and y submitted to function s are
inconsistent. As in

nfalgtobasis(nfinit(t^3-2), Mod(t,t^2+1)

*E* has three component, 1 (`t_STR`

): the function s,
2: the argument x, 3: the argument x.

***** `"e_NEGVAL"`

. An argument of function s is a power series with
negative valuation, which does not make sense. (As in `cos(1/x)`

.)
*E* has one component, 1 (`t_STR`

): the function name s.

***** `"e_PRIME"`

. Function s expected a prime number,
and did receive p, which was not. (As in `idealprimedec(nf, 4)`

.)
*E* has two component, 1 (`t_STR`

): the function name s,
2: the argument p.

***** `"e_ROOTS0"`

. An argument of function s is a zero polynomial,
and we need to consider its roots. (As in `polroots(0)`

.) *E* has
one component, 1 (`t_STR`

): the function name s.

***** `"e_SQRTN"`

. Trying to compute an n-th root of x, which does
not exist, in function s. (As in `sqrt(Mod(-1,3))`

.)
*E* has two components, 1 (`t_STR`

): the function name s,
2: the argument x.

Interrupts execution of current seq, resume the next iteration of the innermost enclosing loop, within the current function call (or top level loop). If n is specified, resume at the n-th enclosing loop. If n is bigger than the number of enclosing loops, all enclosing loops are exited.

Returns from current subroutine, with
result x. If x is omitted, return the `(void)`

value (return no
result, like `print`

).

Evaluates *seq* until a is not
equal to 0 (i.e. until a is true). If a is initially not equal to 0,
*seq* is evaluated once (more generally, the condition on a is tested
*after* execution of the *seq*, not before as in `while`

).

While a is non-zero, evaluates the expression sequence *seq*. The
test is made *before* evaluating the seq, hence in particular if a
is initially equal to zero the *seq* will not be evaluated at all.