Class that computes partial derivatives by automatic differentiation. More...

#include <AutoDiff.h>

Inheritance diagram for casacore::AutoDiff< T >:

Public Types
typedef T	value_type

typedef value_type &	reference

typedef const value_type &	const_reference

typedef value_type *	iterator

typedef const value_type *	const_iterator

Public Member Functions
	AutoDiff ()
	Construct a constant with a value of zero. More...

	AutoDiff (const T &v)
	Construct a constant with a value of v. More...

	AutoDiff (const T &v, const uInt ndiffs, const uInt n)
	A function f(x0,x1,...,xn,...) with a value of v. More...

	AutoDiff (const T &v, const uInt ndiffs)
	A function f(x0,x1,...,xn,...) with a value of v. More...

	AutoDiff (const AutoDiff< T > &other)
	Construct one from another. More...

	AutoDiff (const T &v, const Vector< T > &derivs)
	Construct a function f(x0,x1,...,xn) of a value v and a vector of derivatives derivs(0) = df/dx0, derivs(1) = df/dx1,... More...

	~AutoDiff ()

AutoDiff< T > &	operator= (const T &v)
	Assignment operator. More...

AutoDiff< T > &	operator= (const AutoDiff< T > &other)
	Assign one to another. More...

void	operator*= (const AutoDiff< T > &other)
	In-place mathematical operators. More...

void	operator/= (const AutoDiff< T > &other)

void	operator+= (const AutoDiff< T > &other)

void	operator-= (const AutoDiff< T > &other)

void	operator*= (const T other)

void	operator/= (const T other)

void	operator+= (const T other)

void	operator-= (const T other)

T &	value ()
	Returns the value of the function. More...

const T &	value () const

const Vector< T > &	derivatives () const
	Returns a vector of the derivatives of an AutoDiff. More...

Vector< T > &	derivatives ()

void	derivatives (Vector< T > &res) const

T &	derivative (uInt which)
	Returns a specific derivative. More...

const T &	derivative (uInt which) const

T &	deriv (uInt which)

const T &	deriv (uInt which) const

uInt	nDerivatives () const
	Return total number of derivatives. More...

Bool	isConstant () const
	Is it a constant, i.e., with zero derivatives? More...

Private Attributes
T	val_p
	The function value. More...

uInt	nd_p
	The number of derivatives. More...

Vector< T >	grad_p
	The derivatives. More...

Detailed Description

template<class T>
class casacore::AutoDiff< T >

Class that computes partial derivatives by automatic differentiation.

Intended use:

Public interface

Review Status

Reviewed By:: UNKNOWN
Date Reviewed:: before2004/08/25
Test programs:: tAutoDiff
Demo programs:: dAutoDiff

Prerequisite

Etymology

Class that computes partial derivatives by automatic differentiation, thus AutoDiff.

Synopsis

Class that computes partial derivatives by automatic differentiation. It does this by storing the value of a function and the values of its first derivatives with respect to its independent parameters. When a mathematical operation is applied to an AutoDiff object, the derivative values of the resulting new object are computed according to chain rules of differentiation.

Suppose we have a function f(x0,x1,...,xn) and its differential is

df = (df/dx0)*dx0 + (df/dx1)*dx1 +... + (df/dxn)*dxn

We can build a class that has the value of the function, f(x0,x1,...,xn), and the values of the derivatives, (df/dx0), (df/dx1), ..., (df/dxn) at (x0,x1,...,xn), as class members.

Now if we have another function, g(x0,x1,...,xn) and its differential is dg = (dg/dx0)*dx0 + (dg/dx1)*dx1 +... + (dg/dxn)*dxn, since

 d(f+g) = df + dg, 
 d(f*g) = g*df + f*dg, 
 d(f/g) = df/g - fdg/g^2,
 dsin(f) = cos(f)df, 
...,

we can calculate

d(f+g), d(f*g),...,

based on our information on

df/dx0, df/dx1,..., dg/dx0, dg/dx1,..., dg/dxn.

All we need to do is to define the operators and derivatives of common mathematical functions.

To be able to use the class as an automatic differentiator of a function object, we need a templated function object, i.e. an object with:

a template <class T> T operator()(const T)
or multiple variable input like: template <class T> T operator()(const Vector<T> &)
all variables and constants used in the calculation of the function value should have been typed with T

A simple example of such a function object could be:

template <class T> f {
public:
T operator()(const T &x, const T &a, const T &b) {
return a*b*x; }
};
// Instantiate the following versions:
template class f<Double>;
template class f<AutoDiff<Double> >;

A call with values will produce the function value:

cout << f(7.0, 2.0, 3.0) << endl;
// will produce the value at x=7 for a=2; b=3:
42
// But a call indicating that we want derivatives to a and b:
cout << f(AutoDiff<Double>(7.0), AutoDiff<Double>(2.0, 2, 0),
AutoDiff<Double>(3.0, 2, 1)) << endl;
// will produce the value at x=7 for a=2; b=3:
// and the partial derivatives wrt a and b at x=7:
(42, [21, 14])
// The following will calculate the derivate wrt x:
cout << f(AutoDiff<Double>(7.0, 1, 0), AutoDiff<Double>(2.0),
AutoDiff<Double>(3.0)) << endl;
(42,[6])

In actual practice, there are a few rules to obey for the structure of the function object if you want to use the function object and its derivatives in least squares fitting procedures in the Fitting module. The major one is to view the function object having 'fixed' and 'variable' parameters. I.e., rather than viewing the function as depending on parameters a, b, x (f(a,b,x)), the function is considered to be f(x; a,b), where a, b are 'fixed' parameters, and x a variable parameter. Fixed parameters should be contained in a FunctionParam container object; while the variable parameter(s) are given in the function operator(). See Function class for details.

A Gaussian spectral profile would in general have the center frequency, the width and the amplitude as fixed parameters, and the frequency as a variable. Given a spectrum, you would solve for the fixed parameters, given spectrum values. However, in other cases the role of the parameters could be reversed. An example could be a whole stack of observed (in the laboratory) spectra at different temperatures at one frequency. In that case the width would be the variable parameter, and the frequency one of the fixed (and to be solved for)parameters.

Since the calculation of the derivatives is done with simple overloading, the calculation of second (and higher) derivatives is easy. It should be noted that higher deivatives are inefficient in the current incarnation (there is no knowledge e.g. about symmetry in the Jacobian). However, it is a very good way to get the correct answers of the derivatives. In practice actual production code will be better off with specialization of the f<AutoDiff<> > implementation.

The AutoDiff class is the class the user communicates with. Alias classes (AutoDiffA and AutoDiffX) exists to make it possible to have different incarnations of a templated method (e.g. a generic one and a specialized one). See the dAutoDiff demo for an example of its use.

All operators and functions are declared in (see (file=AutoDiffMath.h)) AutoDiffMath. The output operator in (see (file=AutoDiffIO.h))AutoDiffIO.

Example

// First a simple example.
// We have a function of the form f(x,y,z); and want to know the
// value of the function for x=10; y=20; z=30; and for
// the derivatives at those point.
// Specify the values; and indicate 3 derivatives:
AutoDiff<Double> x(10.0, 3, 0); 
AutoDiff<Double> y(20.0, 3, 1); 
AutoDiff<Double> z(30.0, 3, 2);
// The result will be:
AutoDiff<Double> result = x*y + sin(z);
cout << result.value() << endl;
// 199.012
cout << result.derivatives() << endl;
// [20, 10, 0.154251]
// Note: sin(30) = -0.988; cos(30) =  0.154251;

See for an extensive example the demo program dAutoDiff. It is based on the example given above, and shows also the use of second derivatives (which is just using AutoDiff<AutoDiff<Double> > as template argument).

// The function, with fixed parameters a,b:
template <class T> class f {
public:
T operator()(const T& x) { return a_p*a_p*a_p*b_p*b_p*x; }
void set(const T& a, const T& b) { a_p = a; b_p = b; }
private:
T a_p;
T b_p;
};
// Call it with different template arguments:
Double a0(2), b0(3), x0(7);
f<Double> f0; f0.set(a0, b0);
cout << "Value:     " << f0(x0) << endl;
AutoDiff<Double> a1(2,2,0), b1(3,2,1), x1(7);
f<AutoDiff<Double> > f1; f1.set(a1, b1);
cout << "Diff a,b:   " << f1(x1) << endl;
AutoDiff<Double> a2(2), b2(3), x2(7,1,0);
f<AutoDiff<Double> > f2; f2.set(a2, b2);
cout << "Diff x:     " << f2(x2) << endl;
AutoDiff<AutoDiff<Double> > a3(AutoDiff<Double>(2,2,0),2,0),
b3(AutoDiff<Double>(3,2,1),2,1), x3(AutoDiff<Double>(7),2);
f<AutoDiff<AutoDiff<Double> > > f3; f3.set(a3, b3);
cout << "Diff2 a,b:  " << f3(x3) << endl;
AutoDiff<AutoDiff<Double> > a4(AutoDiff<Double>(2),1),
b4(AutoDiff<Double>(3),1),
x4(AutoDiff<Double>(7,1,0),1,0);
f<AutoDiff<AutoDiff<Double> > > f4; f4.set(a4, b4);
cout << "Diff2 x:    " << f4(x4) << endl;
// Result will be:
// Value:     504
// Diff a,b:  (504, [756, 336])
// Diff x:    (504, [72])
// Diff2 a,b: ((504, [756, 336]), [(756, [756, 504]), (336, [504, 112])])
// Diff2 x:   ((504, [72]), [(72, [0])])
// It needed the template instantiations definitions:
template class f<Double>;
template class f<AutoDiff<Double> >;
template class f<AutoDiff<AutoDiff<Double> > >;

Motivation

The creation of the class was motivated by least-squares non-linear fit where partial derivatives of a fitted function are needed. It would be tedious to create functionals for all partial derivatives of a function.

Template Type Argument Requirements (T)

any class that has the standard mathematical and comparisons defined

To Do

Nothing I know

Definition at line 257 of file AutoDiff.h.

Member Typedef Documentation

template<class T>

typedef const value_type* casacore::AutoDiff< T >::const_iterator

Definition at line 264 of file AutoDiff.h.

template<class T>

typedef const value_type& casacore::AutoDiff< T >::const_reference

Definition at line 262 of file AutoDiff.h.

template<class T>

typedef value_type* casacore::AutoDiff< T >::iterator

Definition at line 263 of file AutoDiff.h.

template<class T>

typedef value_type& casacore::AutoDiff< T >::reference

Definition at line 261 of file AutoDiff.h.

template<class T>

typedef T casacore::AutoDiff< T >::value_type

Definition at line 260 of file AutoDiff.h.

Constructor & Destructor Documentation

template<class T>

casacore::AutoDiff< T >::AutoDiff ( )

Construct a constant with a value of zero.

Zero derivatives.

template<class T>

casacore::AutoDiff< T >::AutoDiff ( const T & v )

Construct a constant with a value of v.

Zero derivatives.

template<class T>

casacore::AutoDiff< T >::AutoDiff	(	const T &	v,
		const uInt	ndiffs,
		const uInt	n
	)

A function f(x0,x1,...,xn,...) with a value of v.

The total number of derivatives is ndiffs, the nth derivative is one, and all others are zero.

template<class T>

casacore::AutoDiff< T >::AutoDiff	(	const T &	v,
		const uInt	ndiffs
	)

A function f(x0,x1,...,xn,...) with a value of v.

The total number of derivatives is ndiffs. All derivatives are zero.

template<class T>

casacore::AutoDiff< T >::AutoDiff ( const AutoDiff< T > & other )

Construct one from another.

template<class T>

casacore::AutoDiff< T >::AutoDiff	(	const T &	v,
		const Vector< T > &	derivs
	)

Construct a function f(x0,x1,...,xn) of a value v and a vector of derivatives derivs(0) = df/dx0, derivs(1) = df/dx1,...

template<class T>

casacore::AutoDiff< T >::~AutoDiff ( )

Member Function Documentation

template<class T>

T& casacore::AutoDiff< T >::deriv ( uInt which )

inline

Definition at line 329 of file AutoDiff.h.

References casacore::AutoDiff< T >::grad_p.

template<class T>

const T& casacore::AutoDiff< T >::deriv ( uInt which ) const

inline

Definition at line 330 of file AutoDiff.h.

References casacore::AutoDiff< T >::grad_p.

template<class T>

T& casacore::AutoDiff< T >::derivative ( uInt which )

inline

Returns a specific derivative.

The second set does not check for a valid which; the first set does through Vector addressing.

Definition at line 327 of file AutoDiff.h.

References casacore::AutoDiff< T >::grad_p.

template<class T>

const T& casacore::AutoDiff< T >::derivative ( uInt which ) const

inline

Definition at line 328 of file AutoDiff.h.

References casacore::AutoDiff< T >::grad_p.

template<class T>

const Vector<T>& casacore::AutoDiff< T >::derivatives ( ) const

inline

Returns a vector of the derivatives of an AutoDiff.

Definition at line 319 of file AutoDiff.h.

References casacore::AutoDiff< T >::grad_p.

template<class T>

Vector<T>& casacore::AutoDiff< T >::derivatives ( )

inline

Definition at line 320 of file AutoDiff.h.

References casacore::AutoDiff< T >::grad_p.

template<class T>

void casacore::AutoDiff< T >::derivatives ( Vector< T > & res ) const

template<class T>

Bool casacore::AutoDiff< T >::isConstant ( ) const

inline

Is it a constant, i.e., with zero derivatives?

Definition at line 337 of file AutoDiff.h.

References casacore::AutoDiff< T >::nd_p.

template<class T>

uInt casacore::AutoDiff< T >::nDerivatives ( ) const

inline

Return total number of derivatives.

Definition at line 334 of file AutoDiff.h.

References casacore::AutoDiff< T >::nd_p.

template<class T>

void casacore::AutoDiff< T >::operator*= ( const AutoDiff< T > & other )

In-place mathematical operators.

template<class T>

void casacore::AutoDiff< T >::operator*= ( const T other )

template<class T>

void casacore::AutoDiff< T >::operator+= ( const AutoDiff< T > & other )

template<class T>

void casacore::AutoDiff< T >::operator+= ( const T other )

template<class T>

void casacore::AutoDiff< T >::operator-= ( const AutoDiff< T > & other )

template<class T>

void casacore::AutoDiff< T >::operator-= ( const T other )

template<class T>

void casacore::AutoDiff< T >::operator/= ( const AutoDiff< T > & other )

template<class T>

void casacore::AutoDiff< T >::operator/= ( const T other )

template<class T>

AutoDiff<T>& casacore::AutoDiff< T >::operator= ( const T & v )

Assignment operator.

Assign a constant to variable. All derivatives are zero.

Referenced by casacore::AutoDiffA< T >::operator=(), and casacore::AutoDiffX< T >::operator=().

template<class T>

AutoDiff<T>& casacore::AutoDiff< T >::operator= ( const AutoDiff< T > & other )

Assign one to another.

template<class T>

T& casacore::AutoDiff< T >::value ( void )

inline

Returns the value of the function.

Definition at line 313 of file AutoDiff.h.

References casacore::AutoDiff< T >::val_p.

Referenced by casacore::FunctionTraits_P< AutoDiff< T > >::getValue(), casacore::FunctionTraits_PA< AutoDiffA< T > >::getValue(), and casacore::FunctionTraits_PX< AutoDiffX< T > >::getValue().

template<class T>

const T& casacore::AutoDiff< T >::value ( void ) const

inline

Definition at line 314 of file AutoDiff.h.

References casacore::AutoDiff< T >::val_p.

Member Data Documentation

template<class T>

Vector<T> casacore::AutoDiff< T >::grad_p

private

The derivatives.

Definition at line 346 of file AutoDiff.h.

Referenced by casacore::AutoDiff< T >::deriv(), casacore::AutoDiff< T >::derivative(), and casacore::AutoDiff< T >::derivatives().

template<class T>

uInt casacore::AutoDiff< T >::nd_p

private

The number of derivatives.

Definition at line 344 of file AutoDiff.h.

Referenced by casacore::AutoDiff< T >::isConstant(), and casacore::AutoDiff< T >::nDerivatives().

template<class T>

T casacore::AutoDiff< T >::val_p

private

The function value.

Definition at line 342 of file AutoDiff.h.

Referenced by casacore::AutoDiff< T >::value().

The documentation for this class was generated from the following file:

casa/casacore/scimath/Mathematics/AutoDiff.h

Public Types

Public Member Functions

Private Attributes

Detailed Description

template<class T> class casacore::AutoDiff< T >

Intended use:

Review Status

Prerequisite

Etymology

Synopsis

Example

Motivation

Template Type Argument Requirements (T)

To Do

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<class T>
class casacore::AutoDiff< T >