Numerical Methods in Computational Finance. Daniel J. Duffy
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Название: Numerical Methods in Computational Finance

Автор: Daniel J. Duffy

Издательство: John Wiley & Sons Limited

Жанр: Ценные бумаги, инвестиции

Серия:

isbn: 9781119719724

isbn:

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      Definition 1.3 (O-Notation).

      An example is:

      We note that complexity analysis applies to both continuous and discrete functions.

      In general, we are interested in functions of two (or more) variables. We consider a function of the form:

      The variables x and y can take values in a given bounded or unbounded interval. First, we say that f (x, y) is continuous at (a, b) if the limit:

      exists and is equal to f (a, b). We now need definitions for the derivatives of f in the x and y directions.

      In general, we calculate the partial derivatives by keeping one variable fixed and differentiating with respect to the other variable; for example:

). We can think of these as ‘original’ and ‘transformed’ coordinate axes, respectively. Now define the function z(u,
) as follows:

      This can be seen as a function of a function. The result that we are interested in is the following: if z is a differentiable function of (u,

) and u,
are themselves continuous functions of x, y, with partial derivatives, then the following rule holds:

      We now wish to transform this equation into an equation in a circular region defined by the polar coordinates:

      The derivative in r is given by:

      and you can check that the derivative with respect to СКАЧАТЬ