Subsections

• Recursion
• Reduction

It is often very useful to be able to refer to a sequence currently under construction, for example to define the sequence recursively. For this purpose the Self operator is available.

Self(n) : RngIntElt -> Elt

Self() : RngIntElt -> SeqEnum

This operator enables the user to refer to an already defined previous entry s[n] of the enumerated sequence s inside the sequence constructor, or the sequence s itself.

> s := [ i gt 2 select Self(i-2)+Self(i-1) else 1 : i in [1..100] ];
> &+s;
927372692193078999175

Instead of using a loop to apply the same binary associative operator to all elements of a complete enumerated sequence, it is possible to use the reduction operator &.

& o S : Op, SeqEnum -> Elt

Given a complete enumerated sequence S = [ a_1, a_2, ..., a_n ] of elements belonging to an algebraic structure U, and an (associative) operator o : U x U -> U, form the element a_1 o a_2 o a_3 o ... o a_n.

Currently, the following operators may be used to reduce sequences: +, *, and, or, join, meet, cat. An error will occur if the operator is not defined on U.

If S contains a single element a, then the value returned is a. If S is the null sequence (empty and no universe specified), then reduction over S leads to an error; if S is empty with universe U in which the operation is defined, then the result (or error) depends on the operation and upon U. The following table defines the return value:

          EMPTY     NULL
  &+       U!0      error
  &*       U!1      error
  &and    true      true
  &or     false     false
  &join   empty     null
  &meet   error     error
  &cat    empty     null