MATLAB FINANCIAL DERIVATIVES TOOLBOX Uživatelský manuál stáhnout pdf (Strana 109)

108

To implement the previous algorithms, the user should be in a position to

find the gradient vector and the Hessian matrix. Although it is better to work

with the exact first and second order partial derivatives of the function via

their functional form, a more flexible, practical and quite accurate way is to

create two functions in Matlab that calculate these with two sided finite

differencing methods. That is, instead of evaluating a function at a point x

by using the functional form of the gradient vector and the Hessian matrix,

it is easier to find these information via numerical differentiation (two sided

finite differencing).

The centred and evenly spaced finite difference approximation of the first

order partial derivatives (gradient vector) of a function f at point x is (for

simplicity assume a two variable function with x

and x

to be the unknown

variable – the extension to the n dimensional case is trivial):

)x,hx(f)x,hx(f

)x(f

2121

−−+

≈

∂

)hx,x(f)hx,x(f

)x(f

2121

−−+

≈

∂

where h is given from a rule of thumb (see [4] page 103),

1 ∈= )|,xmax(|h

with

∈

to represent machine precision (this is the build-in function

s).

The centred and evenly spaced finite difference approximation of the second

order partial derivatives (Hessian matrix) of a function f at point x is (for

simplicity assume a two variable function with x

and x

to be the unknown

variable – the extension to the n dimensional case is trivial):

21212121

2222

)x,hx(f)x,hx(f)x,hx(f)x,hx(f

)x(f +−−−−++

≈

∂

1 2 ... 104 105 106 107 108 109 110 111 112 113 114 ... 118 119

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