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 The NLMIXED Procedure

## Finite Difference Approximations of Derivatives

The FD= and FDHESSIAN= options specify the use of finite difference approximations of the derivatives. The FD= option specifies that all derivatives are approximated using function evaluations, and the FDHESSIAN= option specifies that second-order derivatives are approximated using gradient evaluations.

Computing derivatives by finite difference approximations can be very time consuming, especially for second-order derivatives based only on values of the objective function (FD= option). If analytical derivatives are difficult to obtain (for example, if a function is computed by an iterative process), you might consider one of the optimization techniques that uses first-order derivatives only (QUANEW, DBLDOG, or CONGRA).

#### Forward Difference Approximations

The forward difference derivative approximations consume less computer time, but they are usually not as precise as approximations that use central difference formulas.
• For first-order derivatives, n additional function calls are required:
• For second-order derivatives based on function calls only (Dennis and Schnabel 1983, p. 80), n+n2/2 additional function calls are required for dense Hessian:
• For second-order derivatives based on gradient calls (Dennis and Schnabel 1983, p. 103), n additional gradient calls are required:

#### Central Difference Approximations

Central difference approximations are usually more precise, but they consume more computer time than approximations that use forward difference derivative formulas.
• For first-order derivatives, 2n additional function calls are required:

• For second-order derivatives based on function calls only (Abramowitz and Stegun 1972, p. 884), 2n+4n2/2 additional function calls are required.

• For second-order derivatives based on gradient calls, 2n additional gradient calls are required:

You can use the FDIGITS= = option to specify the number of accurate digits in the evaluation of the objective function. This specification is helpful in determining an appropriate interval size h to be used in the finite difference formulas.

The step sizes hj, j = 1, ... ,n are defined as follows.

• For the forward difference approximation of first-order derivatives using function calls and second-order derivatives using gradient calls, .
• For the forward difference approximation of second-order derivatives using only function calls and all central difference formulas, .
The value of is defined by the FDIGITS= option:
• If you specify the number of accurate digits using FDIGITS=r, is set to 10-r.
• If you do not specify the FDIGITS= option, is set to the machine precision .

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