Simbody  3.7
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...


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

◆ Method


Constructor & Destructor Documentation

◆ ~Differentiator()

virtual SimTK::Differentiator::~Differentiator ( )

◆ Differentiator()

SimTK::Differentiator::Differentiator ( const Function f,
Method  defaultMethod = UnspecifiedMethod 

Member Function Documentation

◆ isValidMethod()

static bool SimTK::Differentiator::isValidMethod ( Method  )

◆ getMethodName()

static const char* SimTK::Differentiator::getMethodName ( Method  )

◆ getMethodOrder()

static int SimTK::Differentiator::getMethodOrder ( Method  )

◆ setDefaultMethod()

Differentiator& SimTK::Differentiator::setDefaultMethod ( Method  )

◆ getDefaultMethod()

Method SimTK::Differentiator::getDefaultMethod ( ) const

◆ calcDerivative() [1/2]

void SimTK::Differentiator::calcDerivative ( Real  y0,
Real  fy0,
Real dfdy,
Method  = UnspecifiedMethod 
) const

◆ calcGradient() [1/2]

void SimTK::Differentiator::calcGradient ( const Vector y0,
Real  fy0,
Vector gf,
Method  = UnspecifiedMethod 
) const

◆ calcJacobian() [1/2]

void SimTK::Differentiator::calcJacobian ( const Vector y0,
const Vector fy0,
Matrix dfdy,
Method  = UnspecifiedMethod 
) const

◆ calcDerivative() [2/2]

Real SimTK::Differentiator::calcDerivative ( Real  y0,
Method  = UnspecifiedMethod 
) const

◆ calcGradient() [2/2]

Vector SimTK::Differentiator::calcGradient ( const Vector y0,
Method  = UnspecifiedMethod 
) const

◆ calcJacobian() [2/2]

Matrix SimTK::Differentiator::calcJacobian ( const Vector y0,
Method  = UnspecifiedMethod 
) const

◆ resetAllStatistics()

void SimTK::Differentiator::resetAllStatistics ( )

◆ getNumDifferentiations()

int SimTK::Differentiator::getNumDifferentiations ( ) const

◆ getNumDifferentiationFailures()

int SimTK::Differentiator::getNumDifferentiationFailures ( ) const

◆ getNumCallsToUserFunction()

int SimTK::Differentiator::getNumCallsToUserFunction ( ) const

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