Solving Separable Nonlinear Equations Using LU Factorization

Separable nonlinear equations have the form F(y,z) ≡ A (y)z + b(y) = 0, where the matrix A(y)∈ R m × N and the vector b(y) ∈ Rmare continuously differentiable functions of y ∈ Rn and z ∈ RN. We assume that m ≥ N + n, and F\u27(y,z) has full rank. We present a numerical method to compute the solution (y∗, z∗) for fully determined systems (m = N+ n) and compatible overdetermined systems (m \u3e N+ n). Our method reduces the original system to a smaller system f(y) = 0 of m − N ≥ n equations in y alone. The iterative process to solve the smaller system only requires the LU factorization of one m × m matrix per step, and the convergence is quadratic. Once y∗ has been obtained, z∗ is computed by direct solution of a linear system. Details of the numerical implementation are provided and several examples are presented

An Efficient Algorithm for the Separable Nonlinear Least Squares Problem

The nonlinear least squares problem m i n y , z ∥ A ( y ) z + b ( y ) ∥ , where A ( y ) is a full-rank ( N + ℓ ) × N matrix, y ∈ R n , z ∈ R N and b ( y ) ∈ R N + ℓ with ℓ ≥ n , can be solved by first solving a reduced problem m i n y ∥ f ( y ) ∥ to find the optimal value y * of y, and then solving the resulting linear least squares problem m i n z ∥ A ( y * ) z + b ( y * ) ∥ to find the optimal value z * of z. We have previously justified the use of the reduced function f ( y ) = C T ( y ) b ( y ) , where C ( y ) is a matrix whose columns form an orthonormal basis for the nullspace of A T ( y ) , and presented a quadratically convergent Gauss–Newton type method for solving m i n y ∥ C T ( y ) b ( y ) ∥ based on the use of QR factorization. In this note, we show how LU factorization can replace the QR factorization in those computations, halving the associated computational cost while also providing opportunities to exploit sparsity and thus further enhance computational efficiency

Bifurcation of solutions of separable parameterized equations into lines

Many applications give rise to separable parameterized equations of the form $A(y, mu)z+b(y, mu)=0$, where $y in mathbb{R}^n$, $z in mathbb{R}^N$ and the parameter $mu in mathbb{R}$; here $A(y, mu)$ is an $(N+n) imes N$ matrix and $b(y, mu) in mathbb{R}^{N+n}$. Under the assumption that $A(y,mu)$ has full rank we showed in [21] that bifurcation points can be located by solving a reduced equation of the form $f(y, mu)=0$. In this paper we extend that method to the case that $A(y,mu)$ has rank deficiency one at the bifurcation point. At such a point the solution curve $(y,mu,z)$ branches into infinitely many additional solutions, which form a straight line. A numerical method for reducing the problem to a smaller space and locating such a bifurcation point is given. Applications to equilibrium solutions of nonlinear ordinary equations and solutions of discretized partial differential equations are provided