exception Unnormalizable;

exception Misshaped;

(* normalize(a,L) divides each element of list L by a *)
fun normalize(a,nil) = nil
|   normalize(a,r::rs) = r/a::normalize(a,rs);

(* condense1(b,L1,L2) subtracts from the jth element of L2
	b times the jth element of L1 for all j and produces
	the resulting L2 *)
fun condense1(b:real,nil,nil) = nil
|   condense1(b,x::xs,y::ys) = (y-b*x)::condense1(b,xs,ys)
|   condense1(_) = raise Misshaped; (* one list is longer
			than the other *)

(* condense2(L,M) takes a list L and a list of lists M, and
	considers each list L' on M.  For L', we subtract from
	each element in the tail of L' the product of the head
	of L' and the corresponding element of list L *)
fun condense2(L,nil) = nil
|   condense2(L,(x::xs)::Rs) =
			condense1(x,L,xs)::condense2(L,Rs);

(* condense(M) applies pivotal condensation to matrix M *)
fun condense([[a]]) = a
|   condense((b::bs)::Rs) =
        if b <= 0.0 andalso b >= 0.0 then raise Unnormalizable
		(* above tests if b=0.0; recall that testing
		   equality with a real is illegal, as is using
		   a real in a pattern.  If b=0.0, then we cannot
		   normalize, because a division by 0.0 is called
		   for *)
        else
	    let
		val L = normalize(b,bs);
		val M' = condense2(L,Rs)
	    in
		b*condense(M')
	    end
|   condense(_) = raise Misshaped; (* the number of rows
			does not equal the length of each row *)