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A new Kantorovich-type theorem for Newton's method

100%
Argyros I. K.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 2
151-157
EN
A new Kantorovich-type convergence theorem for Newton's method is established for approximating a locally unique solution of an equation F(x)=0 defined on a Banach space. It is assumed that the operator F is twice Fréchet differentiable, and that F', F'' satisfy Lipschitz conditions. Our convergence condition differs from earlier ones and therefore it has theoretical and practical value.
2
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Newton's iteration with a conjugate gradient based decomposition method for an elliptic PDE with a nonlinear boundary condition

100%
Koko J.
International Journal of Applied Mathematics and Computer Science
|
2004
|
tom 14
|
nr 1
13-18
EN
Newton's iteration is studied for the numerical solution of an elliptic PDE with nonlinear boundary conditions. At each iteration of Newton's method, a conjugate gradient based decomposition method is applied to the matrix of the linearized system. The decomposition is such that all the remaining linear systems have the same constant matrix. Numerical results confirm the savings with respect to the computational cost, compared with the classical Newton method with factorization at each step.
3
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Efficient numerical algorithms for balanced stochastic truncation

88%
Benner P., Quintana-Ortí E. S., Quintana-Ortí G.
International Journal of Applied Mathematics and Computer Science
|
2001
|
tom 11
|
nr 5
1123-1150
EN
We propose an efficient numerical algorithm for relative error model reduction based on balanced stochastic truncation. The method uses full-rank factors of the Gramians to be balanced versus each other and exploits the fact that for large-scale systems these Gramians are often of low numerical rank. We use the easy-to-parallelize sign function method as the major computational tool in determining these full-rank factors and demonstrate the numerical performance of the suggested implementation of balanced stochastic truncation model reduction.
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