automatic_differentiation(n) 1.0 "flibs"

NAME

automatic_differentiation - Implement the "automatic differentation" technique

    TABLE OF CONTENTS
    SYNOPSIS
    DESCRIPTION
    EXAMPLE
    DATA TYPES AND ROUTINES
    COPYRIGHT

SYNOPSIS

use automatic_differentiation

type(AUTODERIV)

x = derivvar( value )

DESCRIPTION

The automatic_differentiation module defines a derived type that enables you (via overloading common mathematical functions and operators) to:

Define a mathematical function in the usual way
Evaluate that function and its first derivative at the same time

without having to program that first derivative.

The module uses real numbers of the kind "wp" as defined in the auxiliary module select_precision.

(I was pointed to this technique by Simon Geard, a couple of years ago, in the context of one of the many language shootouts on the Internet.)

EXAMPLE

In numerical methods like Newton-Raphson for finding a root of the equation "f(x) = 0", you need the first derivative:

x(k+1) = x(k) - f(x(k)) / f'(x(k))

Rather than implementing the function and its first derivatives as separate functions, using this module you can dispense with manually determining the first derivative:

program root use automatic_differentation type(AUTODERIV) :: x type(AUTODERIV) :: res integer :: i ! ! First estimate ! x = derivvar( 1.0_wp ) do i = 1,10 res = f(x) x = x - res.v / res.dv enddo write(*,*) 'Root: ', x.v contains type(AUTODERIV) function f(x) type(AUTODERIV) :: x f = x + cos(x) end function f end program

DATA TYPES AND ROUTINES

The module defines a single data type, AUTODERIV, and one specific function, derivvar().

use automatic_differentiation

The name of the module

type(AUTODERIV)

The type has two fields:

real(wp) v: The value of the variable/function (or any intermediate result)
real(wp) dv: The first derivative

x = derivvar( value )

Use this function to properly initialise the program variable x as a "mathematical" variable. (It sets the derivative of x to 1, because otherwise it would be regarded as a constant).

real(wp) value: The value of the "mathematical" variable.