Add support for negative indices for ring variables & submatrices

We count from the right when the index is negative, just like lists.

Author: d-torrance

M2/Macaulay2/m2/enginering.m2

@@ -448,6 +448,7 @@ Ring _ ZZ := RingElement => (R,i) -> (generators R)#i
 protect numallvars
 
 EngineRing _ ZZ := (R,i) -> (
+     if R.?numallvars and i < 0 then i += R.numallvars;
      if i < 0 or R.?numallvars and i >= R.numallvars then error("index ", toString i, " out of bounds 0 .. ", toString (R.numallvars-1));
      new R from R.RawRing_i
      )

M2/Macaulay2/m2/matrix.m2

@@ -442,10 +442,14 @@ Matrix _ ZZ := Vector => (m,i) -> (
      new target h from {h})
 
 -- given a map of free modules, find a submatrix of it
-submatrixFree = (m, rows, cols) -> map(ring m, if rows === null
-    then rawSubmatrix(raw cover m, listZZ cols)
-    else rawSubmatrix(raw cover m, listZZ rows,
-	if cols =!= null then listZZ cols else 0 .. numgens source m - 1))
+adjustindex := (I, n) -> apply(I, i -> if i < 0 then n + i else i)
+submatrixFree = (m, rows, cols) -> (
+    if rows =!= null then rows = adjustindex(listZZ rows, numRows m);
+    if cols =!= null then cols = adjustindex(listZZ cols, numColumns m);
+    map(ring m, if rows === null
+    then rawSubmatrix(raw cover m, cols)
+    else rawSubmatrix(raw cover m, rows,
+	if cols =!= null then cols else 0 .. numgens source m - 1)))
 -- given a module, find a part of the ambient module
 -- along with corresponding generators and relations
 sliceModule = (M, rows) -> (

M2/Macaulay2/tests/normal/rings.m2

@@ -7,6 +7,7 @@ assert( f == g )
 assert( class f === R )
 assert( class g === R )
 assert( class h === S )
+assert( S_-1 === c )
 
 --Date: Fri, 16 Apr 2004 11:43:07 +0000 (UTC)
 --From: Ben Richert <[email protected]>

M2/Macaulay2/tests/normal/submatrix.m2

@@ -19,8 +19,10 @@ N = coker matrix{{x}}
 m = map(N++R^1, N++R^1, sub(M, R))
 assert(m^{0} == map(N, N++R^1, {{a, b}}))
 assert(m^{1} == map(R^1, N++R^1, {{c, d}}))
+assert(m^{-1} == map(R^1, N++R^1, {{c, d}}))
 assert(m_{0} == map(N++R^1, N, {{a}, {c}}))
 assert(m_{1} == map(N++R^1, R^1, {{b}, {d}}))
+assert(m_{-1} == map(N++R^1, R^1, {{b}, {d}}))
 
 R = QQ[a..d];
 M = image vars R ++ coker vars R