Check an ADFun Sequence of Operations

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@Check an ADFun Sequence of Operations
Syntax
ok = FunCheck(f, g, x, r, a)

See Also CompareChange

Purpose
We use @(@ F : B^n \rightarrow B^m @)@ to denote the AD function corresponding to f . We use @(@ G : B^n \rightarrow B^m @)@ to denote the function corresponding to the C++ function object g . This routine check if @[@ F(x) = G(x) @]@ If @(@ F(x) \neq G(x) @)@, the operation sequence corresponding to f does not represents the algorithm used by g to calculate values for @(@ G @)@ (see Discussion below).

f
The FunCheck argument f has prototype



     ADFun<Base> f

Note that the ADFun object f is not const (see Forward below).

g
The FunCheck argument g has prototype



     Fun &g

( Fun is defined the properties of g ). The C++ function object g supports the syntax



     y = g(x)

which computes @(@ y = G(x) @)@.

x
The g argument x has prototype



     const Vector &x

(see Vector below) and its size must be equal to n , the dimension of the domain space for f .

y
The g result y has prototype



     Vector y

and its value is @(@ G(x) @)@. The size of y is equal to m , the dimension of the range space for f .

x
The FunCheck argument x has prototype



     const Vector &x

and its size must be equal to n , the dimension of the domain space for f . This specifies that point at which to compare the values calculated by f and G .

r
The FunCheck argument r has prototype



     const Base &r

It specifies the relative error the element by element comparison of the value of @(@ F(x) @)@ and @(@ G(x) @)@.

a
The FunCheck argument a has prototype



     const Base &a

It specifies the absolute error the element by element comparison of the value of @(@ F(x) @)@ and @(@ G(x) @)@.

ok
The FunCheck result ok has prototype



     bool ok

It is true, if for @(@ i = 0 , \ldots , m-1 @)@ either the relative error bound is satisfied @[@ | F_i (x) - G_i (x) | \leq r ( | F_i (x) | + | G_i (x) | ) @]@ or the absolute error bound is satisfied @[@ | F_i (x) - G_i (x) | \leq a @]@ It is false if for some @(@ (i, j) @)@ neither of these bounds is satisfied.

Vector
The type Vector must be a SimpleVector class with elements of type Base . The routine CheckSimpleVector will generate an error message if this is not the case.

FunCheck Uses Forward
After each call to Forward , the object f contains the corresponding Taylor coefficients . After FunCheck, the previous calls to Forward are undefined.

Discussion
Suppose that the algorithm corresponding to g contains



     if( x >= 0 )

          y = exp(x)

     else y = exp(-x)

where x and y are AD<double> objects. It follows that the AD of double operation sequence depends on the value of x . If the sequence of operations stored in f corresponds to g with @(@ x \geq 0 @)@, the function values computed using f when @(@ x < 0 @)@ will not agree with the function values computed by @(@ g @)@. This is because the operation sequence corresponding to g changed (and hence the object f does not represent the function @(@ G @)@ for this value of x ). In this case, you probably want to re-tape the calculations performed by g with the independent variables equal to the values in x (so AD operation sequence properly represents the algorithm for this value of independent variables).

Example
The file fun_check.cpp contains an example and test of this function. It returns true if it succeeds and false otherwise.

Input File: cppad/core/fun_check.hpp