Newton's method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian

Probabilistic load margins of power systems embedded with wind farms

“…Such alternative characterizations have been studied [6], [4], [5]. We will use the bordered matrices technique to find a substitute function that avoids the above difficulties [2]. A bordered system ( , ) is constructed as follows:…”

“…In addition to power flow equations, an implicit test function [4], [5] is constructed in the extended system. In numerical aspect, we propose a new method with only one LU factorization is needed in each iteration [2]. The computational cost of our method compares very favourably with the costs of other proposed methods while preserving quadratic convergence.…”

An Efficient Algorithm for the Separable Nonlinear Least Squares Problem

“…For separable nonlinear systems with a rank-deficient A(y), the technique of bordered matrices [13][14][15][16] may be applied to produce a full rank matrix.…”

“…Theorem 2. Let P(y)A(y) = L(y)U(y) be the LU factorization of A(y) with a permutation matrix P(y) as in (13), and let the matrices M(y), S(y) and C(y) be defined as in (14)- (16), while M(y) is defined by (10).…”

“…Then, the thin QR factorization in (16) is cheap since the matrix Ψ(y) has only columns. More specifically, the LU factorization of the (N + ) × N dimensional A(y) in (13) costs (N + )N 2 − N 3 /3 ≈ 2N 3 /3 flops, with an additional O(N 2 ) comparisons to produce P(y) [12], while computing S(y) only costs O(N) flops and the thin QR factorization of the (N + ) × dimensional matrix S(y) takes only (N + ) 2 − 3 = O(N) flops [12].…”