Given an m x n Matrix B of values in E, HermiteForm[E](B) returns the Hermite form H of B, an m x n Matrix of values in E.

The Hermite normal form Matrix H satisfies:

The uniqueness of H depends on the uniqueness of the remainder operation in E. For example, if E is the ring of integers, and a and b are integers, and the remainder of a / b is in the positive range 0..abs(b)-1, then H is unique. If Maple's irem function is used, as in the example below, because the remainder is in the range 1-abs(b)..abs(b)-1, H is not unique.

The (indexed) parameter E, which specifies the domain of computation, a Euclidean domain, must be a Maple table/module which has the following values/exports:

E[`0`]: a constant for the zero of the ring E

E[`1`]: a constant for the (multiplicative) identity of E

E[`+`]: a procedure for adding elements of E (nary)

E[`-`]: a procedure for negating and subtracting elements of E (unary and binary)

E[`*`]: a procedure for multiplying two elements of E (commutative)

E[`=`]: a boolean procedure for testing if two elements in F are equal

E[Quo]: a procedure which computes the quotient of a / b. E[Quo](a,b,'r') computes the quotient q of a / b and optionally assigns r the remainder satisfying a = b q + r.

E[Rem]: a procedure for finding the remainder of a / b. E[Rem(a,b,'q') computes the remainder r of a / b and optionally assigns q the quotient satisfying a = b q + r.

E[Gcdex]: a procedure for finding the gcd g of a and b, an element of E. E[Gcdex](a,b,'s','t') computes the gcd of a and b and optionally assigns s and t elements of E satisfying s a + t b = g.

E[UnitPart]: a procedure for returning the unit part of an element in E

E[EuclideanNorm]: a procedure for computing the Euclidean norm of an element in E, a non-negative integer.

For a,b in E, b non-zero, the remainder r and quotient q satisfy a = b q + r and r = 0 or EuclideanNorm(r) < EuclideanNorm(b).

For non-zero a,b in E, units u,v in E, the Euclidean norm satisfies:

EuclideanNorm(a b) >= EuclideanNorm(a)

EuclideanNorm(u) = EuclideanNorm(v)

EuclideanNorm(u a) = EuclideanNorm(a)

The Hermite form is computed by putting the principal block H[1..i,1..i] into Hermite form using elementary Row operations. This algorithm does at most O(n^4) operations in E.

Z[Quo] := proc(a,b,r) local q,rr; rr := Z[Rem](a,b,'q'); if nargs=3 then r := rr; end if; q; end proc:

Z[Rem] := proc(a,b,q) local r,qq; r := modp(a,b); if nargs=3 then q := iquo(a-r,b); end if; r; end proc: