MATLAB FINANCIAL DERIVATIVES TOOLBOX Manual de usuario descargar pdf (Pagina 107)

106

stable convergence but with extremely slow rate. Typical values of a

range between 0.01 and 0.15 depending on the faced problem.

Specifically, )x('f for a function with n variables is:













∂

)x(f

)x('f L

• Step #3: Increase: k

←

k+1 and if the current iteration index k is larger

than the maximum number of iterations or if )x('af

<e then stop and

return

x , otherwise go to Step #2 and perform one more iteration of

the algorithm (the e is the desire accuracy, usually set to a small

quantity such as 1e-6, whereas the maximum number of iterations

depends solely by the experience of the researcher).

The Newton Descent algorithm can be summarized as follows:

• Step #1: give an initial guess for the coordinates of the minimum point

(e.g. if the function under consideration has two unknowns x

and

, then x should be a two element row vector: x

x ,

x ])

• Step #2: perform an algorithm iteration, k, based on the following

formula:

11 −+

−= )]x(''f)[x('fxx

kkkk

for k=1,2,3,….

where )x(f

is the value of the function at k

iteration at point x,

)x('f

is a row vector that includes the first order partial derivatives of

f(.) w.r.t x’s at the k

iteration and

1−

)]x(''f[

is a square matrix that

represents the inverse of the second order partial derivatives of f(.) w.r.t

x’s (Hessian matrix) at the k

iteration.

1 2 ... 102 103 104 105 106 107 108 109 110 111 112 ... 118 119

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