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Preconditioning of truncated-Newton methods. (English) Zbl 0592.65038
The author discusses the problem of minimizing a real-valued function f(x) of n variables for large n. Truncated Newton methods are used to solve this problem. At each major iteration the Newton equation $$G^{(k)}p=g^{(k)}$$ is approximately solved for the direction p using a modified Lanczos method. Here $$g^{(k)}$$ and $$G^{(k)}$$ are, respectively, the gradient vector and the Hessian matrix of the second derivatives of f evaluated at the point $$x^{(k)}$$. A line search procedure is used to find the updated point $$x^{(k+1)}$$. Some preconditioning and scaling strategies are investigated. Numerical examples show that a carefully chosen truncated Newton method can perform well in comparison with conjugate gradient algorithms.
Reviewer: I.H.Mufti

##### MSC:
 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming 65H10 Numerical computation of solutions to systems of equations
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