Modeling of geochemical processes Numeric Algorithm Refreshment

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Modeling of geochemical processes Numeric Algorithm Refreshment J. Faimon

Modeling of geochemical processes Numeric mathematics Numerical Algorithms Taylor Series The terms of higher orders are often neglected! Newton method algoithm:

Modeling of geochemical processes Numeric mathematics Taylor Series of the Function of two Variables:

Modeling of geochemical processes Numeric mathematics A vector form: gradient Hessian

Modeling of geochemical processes Numeric mathematics Taylor Series of the System of the Functions of more Variables Two functions of two variables (only the first order terms; the first derivations) In vectors: Jakobian alternatively

Modeling of geochemical processes Numeric mathematics Newton-Raphson methods Equation system: fi (x1,x2...xn) = 0 f (x) = 0 Taylor Series (first order terms only): In matrix form (some matrix elements are vectors), it is:

Modeling of geochemical processes Numeric mathematics For f (xk+1) = 0, then it is: After rearranging: in vectors: where and Rearranging gives the algorithm: