Simbody  3.5
 All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Friends Macros Modules Pages
SimTK::Differentiator Class Reference

Given a function f(y), where f, y or both can be vectors, calculate the derivative (gradient, Jacobian) df/dy. More...

Classes

class  Function
 This abstract class defines a function to be differentiated (repeatedly) by a Differentiator object. More...
 
class  GradientFunction
 Derive a concrete class from this one if you have a scalar function of multiple variables that you want to differentiate. More...
 
class  JacobianFunction
 Derive a concrete class from this one if you have a set of functions (i.e., a vector-valued function) of multiple variables that you want to differentiate. More...
 
class  ScalarFunction
 Derive a concrete class from this one if you have a scalar function of a single scalar variable that you want to differentiate. More...
 

Public Types

enum  Method {
  UnspecifiedMethod =0,
  ForwardDifference =1,
  CentralDifference =2
}
 

Public Member Functions

virtual ~Differentiator ()
 
 Differentiator (const Function &f, Method defaultMethod=UnspecifiedMethod)
 
DifferentiatorsetDefaultMethod (Method)
 
Method getDefaultMethod () const
 
void calcDerivative (Real y0, Real fy0, Real &dfdy, Method=UnspecifiedMethod) const
 
void calcGradient (const Vector &y0, Real fy0, Vector &gf, Method=UnspecifiedMethod) const
 
void calcJacobian (const Vector &y0, const Vector &fy0, Matrix &dfdy, Method=UnspecifiedMethod) const
 
Real calcDerivative (Real y0, Method=UnspecifiedMethod) const
 
Vector calcGradient (const Vector &y0, Method=UnspecifiedMethod) const
 
Matrix calcJacobian (const Vector &y0, Method=UnspecifiedMethod) const
 
void resetAllStatistics ()
 
int getNumDifferentiations () const
 
int getNumDifferentiationFailures () const
 
int getNumCallsToUserFunction () const
 

Static Public Member Functions

static bool isValidMethod (Method)
 
static const char * getMethodName (Method)
 
static int getMethodOrder (Method)
 

Detailed Description

Given a function f(y), where f, y or both can be vectors, calculate the derivative (gradient, Jacobian) df/dy.

Calculation is done using numerical differencing, which should be considered a last resort for cases in which the analytic derivative is unavailable. (Note that you can obtain an analytic gradient automatically from the source code for f using automatic differentiation methods like complex step derivatives, ADIFOR, etc.).

Theory and Implementation

The SimTK::Differentiator class uses methods adapted from the book Practical Optimization by Gill, Murray, and Wright (1981), section 8.6 (339ff) and Numerical Recipies in C++ 2nd ed. (2002) section 5.7 (192ff). Here is a summary:

  • We want to differentiate a function f(y) whose estimated relative accuracy eps is known (e.g. eps=1e-6). (We'll treat y as a scalar here but for vector y this is done for one element yi at a time.)
  • We need to know what perturbation h to use for calculating an estimate of df/dy that optimally balances roundoff error (h too small) with truncation error (h too big).
  • First guess at h depends on the order of the numerical method: either forward difference (1st order) or central difference (2nd order). For 1st order, h0=eps^(1/2); for 2nd order h0=eps^(1/3).
  • Now we have to make sure that we can compute y+h reliably. If y is very large, we can not allow h to be too small. We calculate a scaled perturbation h1=h0*max(y, 0.1). The 0.1 allows a small y to pull down the step size a little; but it is dangerous to go much lower because a very small y might just be zero plus noise.
  • Finally, the step size should be exactly representable as a power of 2. Conceptually, this is just h=(y+h1)-y although one must be careful to stop the compiler from cleverly "simplifying" this expression. Differentiator uses a C++ volatile variable for that purpose.

Then the derivative, gradient element, or Jacobian column is computed as df/dy=[f(x+h)-f(x)]/h (1st order) or df/dy=[f(x+h)-f(x-h)]/(2h) (2nd order).

Member Enumeration Documentation

Enumerator
UnspecifiedMethod 
ForwardDifference 
CentralDifference 

Constructor & Destructor Documentation

virtual SimTK::Differentiator::~Differentiator ( )
virtual
SimTK::Differentiator::Differentiator ( const Function f,
Method  defaultMethod = UnspecifiedMethod 
)
explicit

Member Function Documentation

static bool SimTK::Differentiator::isValidMethod ( Method  )
static
static const char* SimTK::Differentiator::getMethodName ( Method  )
static
static int SimTK::Differentiator::getMethodOrder ( Method  )
static
Differentiator& SimTK::Differentiator::setDefaultMethod ( Method  )
Method SimTK::Differentiator::getDefaultMethod ( ) const
void SimTK::Differentiator::calcDerivative ( Real  y0,
Real  fy0,
Real dfdy,
Method  = UnspecifiedMethod 
) const
void SimTK::Differentiator::calcGradient ( const Vector y0,
Real  fy0,
Vector gf,
Method  = UnspecifiedMethod 
) const
void SimTK::Differentiator::calcJacobian ( const Vector y0,
const Vector fy0,
Matrix dfdy,
Method  = UnspecifiedMethod 
) const
Real SimTK::Differentiator::calcDerivative ( Real  y0,
Method  = UnspecifiedMethod 
) const
Vector SimTK::Differentiator::calcGradient ( const Vector y0,
Method  = UnspecifiedMethod 
) const
Matrix SimTK::Differentiator::calcJacobian ( const Vector y0,
Method  = UnspecifiedMethod 
) const
void SimTK::Differentiator::resetAllStatistics ( )
int SimTK::Differentiator::getNumDifferentiations ( ) const
int SimTK::Differentiator::getNumDifferentiationFailures ( ) const
int SimTK::Differentiator::getNumCallsToUserFunction ( ) const

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