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Ceiling

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Ceiling[x]
    gives the smallest integer greater than or equal to x.
Ceiling[x, a]
    gives the smallest multiple of a greater than or equal to x.
Ceiling is Listable. Exact inputs return exact integers; Real / MPFR
inputs are rounded toward +Infinity at the input precision.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_ceiling (src/piecewise.c) is a thin wrapper over do_piecewise(res, OP_CEILING, ...), shared with Floor/Round/IntegerPart/FractionalPart. The 1-arg form dispatches to do_piecewise_1, which handles each exact kind directly: integers/bigints pass through; EXPR_REAL uses C ceil; EXPR_MPFR uses mpfr_ceil; exact Rational[p,q] (including bigint components) uses GMP mpz_cdiv_q for an exact integer result; Complex[re,im] recurses componentwise. For an exact-but-symbolic real numeric that the leaf branches cannot resolve, do_piecewise_numeric_exact numericalizes to MPFR at doubling precision (256 up to 2^16 bits) and accepts the ceiling only once two successive precisions agree — an interval-style certification that avoids mis-rounding values near an integer boundary, returning NULL rather than a wrong answer if it cannot converge. The 2-arg Ceiling[x, a] rewrites to a * Ceiling[x/a].

Data structures. Expr* leaves plus GMP mpz_t/mpq arithmetic and MPFR mpfr_t for the precision-escalation certification.

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Ceiling[7/2]
Out[1]= 4

In[2]:= Ceiling[-2.7]
Out[2]= -2

In[3]:= Ceiling[17, 5]
Out[3]= 20

In[1]:= Ceiling[Pi]
Out[1]= 4

In[1]:= Ceiling[E^2]
Out[1]= 8

In[1]:= Ceiling[100/7, 5]
Out[1]= 15

Notes

Ceiling[x] rounds toward +Infinity; the two-argument Ceiling[x, a] gives the smallest multiple of a at least x. Exact inputs return exact integers. Symbolic constants are resolved exactly: Ceiling[Pi] is 4 and Ceiling[E^2] is 8 (since e^2 ≈ 7.389). The two-argument form rounds up to a lattice point, so Ceiling[100/7, 5] (with 100/7 ≈ 14.29) gives the next multiple of 5, namely 15.