Yun-Qiu Shen

Solving N + m nonlinear equations with only m nonlinear variables

We derive a method for solvingN+m nonlinear algebraic equations inN+m unknownsy≠Rm andz≠RN of the formA(y)z+b(y)=0, where the(N+m) × N matrixA(y) and vectorb(y) are continuously differentiable functions ofy alone. By exploiting properties of an orthonormal basis for null(AT(y)) the problem is reduced to solvingm nonlinear equations iny only. These equations are solved by Newtons method inm variables. Details of computational implementation and results are provided.ZusammenfassungEine Methode zur Lösung vonN+m nichtlinearen algebraischen Gleichungen mitN+m Variableny≠Rm undz≠RN non TypA(y)z+b(y)=0, in welchem die(N+m) × m MatrixA(y) und der Vektorb(y) stetig differenzierbare Funktionen der Variableny sind, wird hergeleitet. Durch Verwendung einer orthonormalen Basis im Nullraum vonAT(y) wird das Problem aufm nichtlineare Gleichungen allein mit der Variableny reduziert. Diese Gleichungen werden durch das Newton Verfahren (m Variable) gelöst. Einzelheiten der numerischen Rechnung werden beschrieben.

Solving nonlinear systems of equations with only one nonlinear variable

Abstract We describe a method for solving systems of N + 1 nonlinear equations in N + 1 unknowns y \te; R and z \te; R N of the form A(y)z + b(y) = 0, where the (N + 1) × N matrix A(y) and vector b(y) are functions of y alone. Such equations arise in minimax approximation. We reduce the problem to one equation in y only. An efficient quadratically convergent numerical technique based on Newtons method in one variable is used to solve this equation. Computational details and results are provided, and two generalizations are discussed.

Computation of a simple bifurcation point using one singular value decomposition nearby

Using an extended system to locate a simple bifurcation point via an iterative method usually requires a good choice of an initial point as well as several auxiliary vectors. The method we propose here requires a good choice of an initial point only. Our method is based upon an analysis of singular vectors of a singular value decomposition of a Jacobian matrix near the simple bifurcation point, which leads to the automatic determination of a certain type of auxiliary vectors in terms of the initial point. Numerical implementation via a Newton-like method is discussed and examples are provided.ZusammenfassungWird ein erweitertes System mit Hilfe eines Iterationsverfahrens zur Bestimmung eines einfachen Bifurkationspunktes benutzt, verlangt dies meistens ein gute Wahl eines Anfangspunktes, sowie mehrere zusätzliche Vektoren. Die Methode, die wir hier vorschlagen, verlangt nur eine gute Wahl eines Anfangspunktes. Unsere Methode basiert auf einer Analyse von singulären Vektoren einer Singulärwertzerlegung einer Jakobi-Matrix, die sich in der Nähe des einfachen Bifurkationspunktes befindet, was zur automatischen Bestimmung eines bestimmten Hilfsvektortypes führt bezüglich des Anfangspunktes. Numerische Implementierung über ein Newton-ähnliches Verfahren wird diskutiert und Beispiele werden angegeben.

Solving Separable Nonlinear Equations Using LU Factorization

Separable nonlinear equations have the form where the matrix and the vector are continuously differentiable functions of and . We assume that and has full rank. We present a numerical method to compute the solution for fully determined systems () and compatible overdetermined systems (). Our method reduces the original system to a smaller system of equations in alone. The iterative process to solve the smaller system only requires the LU factorization of one matrix per step, and the convergence is quadratic. Once has been obtained, is computed by direct solution of a linear system. Details of the numerical implementation are provided and several examples are presented.

Solving separable nonlinear equations with jacobians of rank deficiency one

Nonlinear systems of equations of the separable form A(y)z + b(y) = 0, with only one nonlinear variable y∈ℝ, can be reduced to a single nonlinear equation in y. We develop a technique for the case in which A(y) has rank deficiency one. The method requires only one LU factorization per iteration and is quadratically convergent. Numerical examples and applications are provided.

Solving separable nonlinear least squares problems using the QR factorization

Abstract We present a method for solving the separable nonlinear least squares problem min y , z ‖ F ( y , z ) ‖ , where F ( y , z ) ≡ A ( y ) z + b ( y ) with a full rank matrix A ( y ) ∈ R ( N + l ) × N , y ∈ R n , z ∈ R N and the vector b ( y ) ∈ R N + l , with small l ≥ n . We show how this problem can be reduced to a smaller equivalent problem min y ‖ f ( y ) ‖ where the function f has only l components. The reduction technique is based on the existence of a locally differentiable orthonormal basis for the nullspace of A T ( y ) . We use Newton’s method to solve the reduced problem. We show that successive iteration points are independent of the nullspace basis used at any particular iteration point; thus the QR factorization can be used to provide a local basis at each iteration. We show that the first and second derivative terms that arise are easily computed, so quadratic convergence is obtainable even for nonzero residual problems. For the class of problems with N much greater than n and l the main cost per iteration of the method is one QR factorization of A ( y ) . We provide a detailed algorithm and some numerical examples to illustrate the technique.

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.