casa  \$Rev:20696\$
casa::AutoDiff< T > Class Template Reference

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

`#include <AutoDiff.h>`

Inheritance diagram for casa::AutoDiff< T >: List of all members.

## Public Types

typedef T value_type
typedef value_typereference
typedef const value_typeconst_reference
typedef value_typeiterator
typedef const value_typeconst_iterator

## Public Member Functions

AutoDiff ()
Construct a constant with a value of zero.
AutoDiff (const T &v)
Construct a constant with a value of v.
AutoDiff (const T &v, const uInt ndiffs, const uInt n)
A function f(x0,x1,...,xn,...) with a value of v.
AutoDiff (const T &v, const uInt ndiffs)
A function f(x0,x1,...,xn,...) with a value of v.
AutoDiff (const AutoDiff< T > &other)
Construct one from another.
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, ...
~AutoDiff ()
AutoDiff< T > & operator= (const T &v)
Assignment operator.
AutoDiff< T > & operator= (const AutoDiff< T > &other)
Assign one to another.
void operator*= (const AutoDiff< T > &other)
Assignment operators.
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)
AutoDiffRep< T > * theRep ()
Returns the pointer to the structure of value and derivatives.
const AutoDiffRep< T > * theRep () const
T & value ()
Returns the value of the function.
const T & value () const
Vector< T > derivatives () const
Returns a vector of the derivatives of an AutoDiff.
void derivatives (Vector< T > &res) const
T & derivative (uInt which)
Returns a specific derivative.
const T & derivative (uInt which) const
T & deriv (uInt which)
const T & deriv (uInt which) const
uInt nDerivatives () const
Bool isConstant () const
Is it a constant, i.e., with zero derivatives?
const AutoDiff< T > & ref ()
Indicate that we are going to use a temporary value for the last time (e.g.

## Private Member Functions

void release ()
Release a struct of value and derivative data.

## Private Attributes

AutoDiffRep< T > * rep_p
Value representation.

## Static Private Attributes

static ObjectPool< AutoDiffRep
< T >, uInt
theirPool
Pool of data blocks.
static Mutex theirMutex

## Detailed Description

### template<class T> class casa::AutoDiff< T >

Class that computes partial derivatives by automatic differentiation.

Public interface

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

### 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,)
```

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 . The actual structure of the data block used by `AutoDiff` is described in AutoDiffRep .

### 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, )
// Diff2 a,b: ((504, [756, 336]), [(756, [756, 504]), (336, [504, 112])])
// Diff2 x:   ((504, ), [(72, )])

// 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 265 of file AutoDiff.h.

## Member Typedef Documentation

template<class T>
 typedef const value_type* casa::AutoDiff< T >::const_iterator

Definition at line 272 of file AutoDiff.h.

template<class T>
 typedef const value_type& casa::AutoDiff< T >::const_reference

Definition at line 270 of file AutoDiff.h.

template<class T>
 typedef value_type* casa::AutoDiff< T >::iterator

Definition at line 271 of file AutoDiff.h.

template<class T>
 typedef value_type& casa::AutoDiff< T >::reference

Definition at line 269 of file AutoDiff.h.

template<class T>
 typedef T casa::AutoDiff< T >::value_type

Definition at line 268 of file AutoDiff.h.

## Constructor & Destructor Documentation

template<class T>
 casa::AutoDiff< T >::AutoDiff ( )

Construct a constant with a value of zero.

Zero derivatives.

template<class T>
 casa::AutoDiff< T >::AutoDiff ( const T & v )

Construct a constant with a value of v.

Zero derivatives.

template<class T>
 casa::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>
 casa::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>
 casa::AutoDiff< T >::AutoDiff ( const AutoDiff< T > & other )

Construct one from another.

template<class T>
 casa::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>
 casa::AutoDiff< T >::~AutoDiff ( )

## Member Function Documentation

template<class T>
 T& casa::AutoDiff< T >::deriv ( uInt which ) ` [inline]`

Definition at line 342 of file AutoDiff.h.

template<class T>
 const T& casa::AutoDiff< T >::deriv ( uInt which ) const` [inline]`

Definition at line 343 of file AutoDiff.h.

template<class T>
 T& casa::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 340 of file AutoDiff.h.

template<class T>
 const T& casa::AutoDiff< T >::derivative ( uInt which ) const` [inline]`

Definition at line 341 of file AutoDiff.h.

template<class T>
 Vector casa::AutoDiff< T >::derivatives ( ) const

Returns a vector of the derivatives of an AutoDiff.

template<class T>
 void casa::AutoDiff< T >::derivatives ( Vector< T > & res ) const
template<class T>
 Bool casa::AutoDiff< T >::isConstant ( ) const` [inline]`

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

Definition at line 350 of file AutoDiff.h.

template<class T>
 uInt casa::AutoDiff< T >::nDerivatives ( ) const` [inline]`

Definition at line 347 of file AutoDiff.h.

template<class T>
 void casa::AutoDiff< T >::operator*= ( const AutoDiff< T > & other )

Assignment operators.

template<class T>
 void casa::AutoDiff< T >::operator*= ( const T other )
template<class T>
 void casa::AutoDiff< T >::operator+= ( const AutoDiff< T > & other )
template<class T>
 void casa::AutoDiff< T >::operator+= ( const T other )
template<class T>
 void casa::AutoDiff< T >::operator-= ( const AutoDiff< T > & other )
template<class T>
 void casa::AutoDiff< T >::operator-= ( const T other )
template<class T>
 void casa::AutoDiff< T >::operator/= ( const AutoDiff< T > & other )
template<class T>
 void casa::AutoDiff< T >::operator/= ( const T other )
template<class T>
 AutoDiff& casa::AutoDiff< T >::operator= ( const T & v )

Assignment operator.

Assign a constant to variable. All derivatives are zero.

Reimplemented in casa::AutoDiffA< T >, and casa::AutoDiffX< T >.

template<class T>
 AutoDiff& casa::AutoDiff< T >::operator= ( const AutoDiff< T > & other )

Assign one to another.

Reimplemented in casa::AutoDiffA< T >, and casa::AutoDiffX< T >.

template<class T>
 const AutoDiff& casa::AutoDiff< T >::ref ( ) ` [inline]`

Indicate that we are going to use a temporary value for the last time (e.g.

at the of a function returning by value). This way superfluous copying can be circumvented.

Definition at line 355 of file AutoDiff.h.

template<class T>
 void casa::AutoDiff< T >::release ( ) ` [inline, private]`

Release a struct of value and derivative data.

Definition at line 368 of file AutoDiff.h.

template<class T>
 AutoDiffRep* casa::AutoDiff< T >::theRep ( ) ` [inline]`

Returns the pointer to the structure of value and derivatives.

Definition at line 321 of file AutoDiff.h.

template<class T>
 const AutoDiffRep* casa::AutoDiff< T >::theRep ( ) const` [inline]`

Definition at line 322 of file AutoDiff.h.

template<class T>
 T& casa::AutoDiff< T >::value ( ) ` [inline]`

Returns the value of the function.

Definition at line 327 of file AutoDiff.h.

template<class T>
 const T& casa::AutoDiff< T >::value ( ) const` [inline]`

Definition at line 328 of file AutoDiff.h.

## Member Data Documentation

template<class T>
 AutoDiffRep* casa::AutoDiff< T >::rep_p` [private]`

Value representation.

Definition at line 364 of file AutoDiff.h.

template<class T>
 Mutex casa::AutoDiff< T >::theirMutex` [static, private]`

Definition at line 362 of file AutoDiff.h.

Referenced by casa::AutoDiff< Double >::release().

template<class T>
 ObjectPool, uInt> casa::AutoDiff< T >::theirPool` [static, private]`

Pool of data blocks.

Definition at line 360 of file AutoDiff.h.

Referenced by casa::AutoDiff< Double >::release().

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