If the FMM is then applied in a hierarchical manner, it can improve the complexity of matrix-vector products in an iterative solver from \ mathcal { O } ( N ^ 2 ) to \ mathcal { O } ( N ) in finite arithmetic, i . e ., given a tolerance \ epsilon, the matrix-vector product is guaranteed to be within a tolerance \ epsilon.
32.
If the FMM is then applied in a hierarchical manner, it can improve the complexity of matrix-vector products in an iterative solver from \ mathcal { O } ( N ^ 2 ) to \ mathcal { O } ( N ) in finite arithmetic, i . e ., given a tolerance \ epsilon, the matrix-vector product is guaranteed to be within a tolerance \ epsilon.
33.
The "'geometrical definition "'( sometimes referred to as static ) begins defining the line of nodes as the intersection of the planes " xy " and the " XY " ( it can also be defined as the common perpendicular to the axes " z " and " Z " and then written as the vector product N = Z x z ).
34.
Another way, in common with parallelograms, is to consider two adjacent sides as vectors, forming a bivector, so the area is the magnitude of the bivector ( the magnitude of the vector product of the two vectors ), which is the determinant of the two vectors'Cartesian coordinates : " K " = " x " 1 " y " 2 " x " 2 " y " 1.
