This command applies a transformation to loop's index variables via the change of coordinates $\mathrm{newvars}=M·\mathrm{vars}$, where M is a unimodular matrix,  $\mathrm{vars}$ are the index variable(s) in loop, and $\mathrm{newvars}$ are the index variable(s) in the resulting transformed ForLoop.  The loop bounds and loop body in loop are updated according to this transformation.

The Matrix M is a square matrix with integer entries whose determinant is +1 or -1.  It must have the same number of rows and columns as there are elements in loop:-variables.

If the optional argument newvars is not supplied, the returned ForLoop will have the same names for its index variable(s) as the original loop, but with the transformation applied.  Otherwise, the returned loop's index variable(s) will have the names in newvars, ordered from the outermost to innermost loops.

$\mathrm{with}\left(\mathrm{CodeTools}\left[\mathrm{ProgramAnalysis}\right]\right)\:$

heateq := proc(A, m, n)
    local i1, j1;
    for i1 to m do
        for j1 from i1 + 1 to n do
            A[i1, j1] := A[i1 - 1, j1 - 1] + 2*A[i1 - 1, j1] + A[i1 - 1, j1 + 1] + 3:
        end do:
    end do:
end proc;

$\mathrm{heateq}≔\mathbf{proc}\left(A\,m\,n\right)\mathbf{local}\mathrm{i1}\,\mathrm{j1}\;\mathbf{for}\mathrm{i1}\mathbf{to}m\mathbf{do}\mathbf{for}\mathrm{j1}\mathbf{from}\mathrm{i1}\+1\mathbf{to}n\mathbf{do}A\[\mathrm{i1}\,\mathrm{j1}\]≔A\[\mathrm{i1}-1\,\mathrm{j1}-1\]\+2\*A\[\mathrm{i1}-1\,\mathrm{j1}\]\+A\[\mathrm{i1}-1\,\mathrm{j1}\+1\]\+3\mathbf{end\; do}\mathbf{end\; do}\mathbf{end\; proc}$

$\mathrm{loop}≔\mathrm{CreateLoop}\left(\mathrm{heateq}\right)\:$

$U≔\mathrm{Matrix}\left(\left[\left[1\,2\right]\,\left[2\,3\right]\right]\right)\:$$\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left(U\right)$

$\mathrm{-1}$

$\mathrm{transformed\_loop}≔\mathrm{UnimodularTransformation}\left(\mathrm{loop}\,U\,\left[\mathrm{i2}\,\mathrm{j2}\right]\right)\:$$\mathrm{GenerateProcedure}\left(\mathrm{transformed\_loop}\right)$

$\mathbf{proc}\left(A\,m\,n\right)\mathbf{local}\mathrm{i2}\,\mathrm{j2}\;\mathbf{for}\mathrm{i2}\mathbf{from}5\mathbf{to}\min \left(2\*n\+m\,3\*n-1\right)\mathbf{do}\mathbf{for}\mathrm{j2}\mathbf{from}\mathrm{ceil}\left(\max \left(1\/2\+3\/2\*\mathrm{i2}\,2\*\mathrm{i2}-n\right)\right)\mathbf{to}\mathrm{floor}\left(\min \left(3\/2\*\mathrm{i2}\+1\/2\*m\,5\/3\*\mathrm{i2}-1\/3\right)\right)\mathbf{do}A\[\−3\*\mathrm{i2}\+2\*\mathrm{j2}\,2\*\mathrm{i2}-\mathrm{j2}\]≔A\[\−1-3\*\mathrm{i2}\+2\*\mathrm{j2}\,2\*\mathrm{i2}-\mathrm{j2}-1\]\+2\*A\[\−1-3\*\mathrm{i2}\+2\*\mathrm{j2}\,2\*\mathrm{i2}-\mathrm{j2}\]\+A\[\−1-3\*\mathrm{i2}\+2\*\mathrm{j2}\,2\*\mathrm{i2}-\mathrm{j2}\+1\]\+3\mathbf{end\; do}\mathbf{end\; do}\;\mathbf{return}\mathbf{end\; proc}$

$\mathrm{transformed\_back}≔\mathrm{UnimodularTransformation}\left(\mathrm{transformed\_loop}\,U^{-1}\,\left[\mathrm{i1}\,\mathrm{j1}\right]\right)\:$

$\mathrm{GenerateProcedure}\left(\mathrm{transformed\_back}\right)$

$\mathbf{proc}\left(A\,m\,n\right)\mathbf{local}\mathrm{i1}\,\mathrm{j1}\;\mathbf{for}\mathrm{i1}\mathbf{to}\min \left(m\,\−1\+n\right)\mathbf{do}\mathbf{for}\mathrm{j1}\mathbf{from}\mathrm{i1}\+1\mathbf{to}n\mathbf{do}A\[\mathrm{i1}\,\mathrm{j1}\]≔A\[\mathrm{i1}-1\,\mathrm{j1}-1\]\+2\*A\[\mathrm{i1}-1\,\mathrm{j1}\]\+A\[\mathrm{i1}-1\,\mathrm{j1}\+1\]\+3\mathbf{end\; do}\mathbf{end\; do}\;\mathbf{return}\mathbf{end\; proc}$

The CodeTools[ProgramAnalysis][UnimodularTransformation] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.