determinant

determinant.sql

# Calculate determinant of X * X

select @determinant := sum(product * product_sign) 
from (
	select (xtx_1.value * xtx_2.value * xtx_3.value) 
	        product
# we need to weight the row_numbers according to the inverse of their natural order
# to estimate degree of permutation
	      , power(-1, dense_rank() over(
         	order by 3 * xtx_1.row_num + 2 * xtx_2.row_num + 1 * xtx_3.row_num) - 1)
         	product_sign
	from XtX xtx_1
	join XtX xtx_2
	  on xtx_1.row_num != xtx_2.row_num
	join XtX xtx_3
	  on xtx_1.row_num != xtx_3.row_num
	 and xtx_2.row_num != xtx_3.row_num
	where xtx_1.col_num = 1
	  and xtx_2.col_num = 2
	  and xtx_3.col_num = 3
	order by 3 * xtx_1.row_num + 2 * xtx_2.row_num + 1 * xtx_1.row_num
) determinant_components;

select @Determinant := sum(product * product_sign)

---

This line does not work in MS SSMS --- is there another way to phrase this?