####BoundTensorNorm(tensor,order,dim) return an upper bound for the morm of  tensor, which is of dimension 'order' and order 'order', where tensor is a list
####NormofTensor(sys,var,k,mroot) return an upper bound for the norm of D`(sys)^k*mroot

SumofContents := proc (list, depth) 
	##a temp function
	local abstemp, i, listtemp, j; 
	if depth = 1 then 
		abstemp := 0; 
		for i in list do abstemp := abstemp+abs(i) end do; 
		return abstemp 
	else 
		listtemp := []; 
		for j in list do listtemp := [op(listtemp), SumofContents(j, depth-1)] end do; 
		return add(listtemp[i], i = 1 .. nops(listtemp)); 
	end if 
end proc;


BoundTensorNorm := proc (tensor, order, dim) 
	##a bound for the operation norm of tensor
	##tensor: a list of lists as a multi-dimensional matrix
	##order: order of tensor, that is the depth of the list.
	local i, maxtemp; 
	maxtemp := []; 
	for i in tensor do maxtemp := [op(maxtemp), SumofContents(i, order-1)] end do; 
	return dim^(order-1)*sqrt(dim)*max(maxtemp) 
end proc;


TensorofDerivative := proc (sys, var, k) 
	local listtemp, i, j, devtemp; 
	if k = 0 then return sys 
	else listtemp := []; 
		for i in sys do 
			devtemp := []; 
			for j in var do devtemp := [op(devtemp), diff(i, j)] end do; 
			listtemp := [op(listtemp), TensorofDerivative(devtemp, var, k-1)] 
		end do; 
		return listtemp 
	end if 
end proc;


NormofTensor := proc (sys, var, k, mroot) 
	##mroot is a list showing the root
	local tensortemp, i; 
	tensortemp := TensorofDerivative(sys, var, k); 
	for i to nops(var) do tensortemp := subs(var[i] = mroot[i], tensortemp) end do; 
	return BoundTensorNorm(tensortemp, k+1, nops(sys)) 
end proc;