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Interpolant Class Reference

The main class for interpolation of functors. More...

#include <interpolant.hh>

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Classes

class  FuncError_lp
 
class  FuncError_wp1
 

Public Member Functions

 Interpolant ()
 Default constructor. More...
 
template<template< class > class Func, class Type >
 Interpolant (Func< Type > *func, bool interpolate_derivative=false)
 Constructor with functor setting. More...
 
virtual ~Interpolant (void)
 Destructor. More...
 
Evaluation.
double val (double x)
 Returns interpolated value. More...
 
double val_test (double x)
 Do NOT use, only for testing purpose. More...
 
double val_p1 (double x)
 Do NOT use, unless you are 100% sure. More...
 
DiffValue diff (double x)
 Returns interpolated value of the derivation. More...
 
double f_val (double x)
 Returns value of the original functor. More...
 
DiffValue f_diff (double x)
 Returns 1st derivative of original functor using FADBAD. More...
 
double f_diffn (double x, unsigned int n)
 
- Public Member Functions inherited from InterpolantBase
 InterpolantBase ()
 Default constructor. More...
 
virtual ~InterpolantBase ()
 Destructor. More...
 
double error ()
 Returns error of the interpolation. More...
 
double bound_a () const
 Returns left boundary of the interval. More...
 
double bound_b () const
 Returns right boundary of the interval. More...
 
unsigned int size () const
 Returns the size of the interpolation table. More...
 
void set_interval (double bound_a, double bound_b)
 Sets the interpolation interval boundaries. More...
 
void set_size (unsigned int size)
 Sets size of the interpolation table. It is also equal to the number of intervals. More...
 
void set_norm (ErrorNorm::Type norm_type=ErrorNorm::max, double p=2)
 Sets the type of norm used for computing estimate of the error of the interpolation. More...
 
void set_size_automatic (double user_tol, unsigned int init_size, unsigned int max_size=default_max_size)
 Sets automatic choice of the size of the table. More...
 
void set_extrapolation (Extrapolation::Type extrapolation)
 Sets the type of extrapolation. Functor type is default. More...
 
EvalStatistics statistics () const
 Returns structure with evaluation statistics. More...
 
void reset_stat ()
 Resets all measured statistics. More...
 
void check_stats_and_reinterpolate (double percentage=0.3)
 Can be called to check automatically the evaluation statistics and possibly reinterpolate. More...
 

Interpolation.

FunctorBase< double > * func
 Pointer to original functor with double type. More...
 
FunctorBase< B< double > > * func_diff
 Pointer to original functor with FADBAD B type. More...
 
FunctorBase< T< double > > * func_diffn
 Pointer to original functor with FADBAD T type. More...
 
bool interpolate_derivative
 Is true if we want to interpolate the derivative too. More...
 
std::vector< double > f_vec
 Vector of function values at nodes. More...
 
std::vector< double > df_vec
 Vector of function derivatives values at nodes. More...
 
virtual int interpolate ()
 Creates piecewise interpolation with polynomials of selected degree. More...
 
template<template< class > class Func, class Type >
void set_functor (Func< Type > *func, bool interpolate_derivative=false)
 Sets the functor. More...
 
void compute_error (double tol, std::vector< double > &f, std::vector< double > &df)
 Creates piecewise linear interpolation. More...
 
void compute_error (double tol, double p, ErrorNorm::Type norm_type)
 Computes estimate of interpolation error with given norm. More...
 
void swap_middle_values (std::vector< double > &f, std::vector< double > &df)
 
void compute_values ()
 Creates table of nodes and function values. More...
 
DiffValue diff_p1 (double x)
 Finds interval on which x lies. More...
 

Additional Inherited Members

- Protected Member Functions inherited from InterpolantBase
long fact (long x)
 Recursive factorial function (used in Taylor row expansion in n-th derivative computation). More...
 
void check_all ()
 Checks that the parameters are set before interpolation. More...
 
- Protected Attributes inherited from InterpolantBase
double bound_a_
 Left interval boundary. More...
 
double bound_b_
 Right interval boundary. More...
 
double step
 Chosen interpolation step. More...
 
double a_div_step
 bound_ divided by step - precomputed value for evaluation More...
 
unsigned int size_
 Number of dividing intervals. More...
 
unsigned int n_nodes
 Number of nodes in the interval $(a,b)$. More...
 
double user_tol
 User set tolerance which is used during automatic step choice. More...
 
unsigned int max_size
 Maximal size of the interpolation table. More...
 
bool automatic_size
 Is true if step/size should be chosen automatically. More...
 
ErrorNorm::Type norm_type
 Type of norm used to compute error of the interpolation. More...
 
double p
 Exponent used in norms $ L_p $ and $ W_p^1 $ when computing error. More...
 
double error_
 Error of the interpolation. More...
 
Extrapolation::Type extrapolation
 
std::vector< bool > checks
 Vector of boolean values telling us which parameters are set or not. More...
 
EvalStatistics stats
 Structure which keeps evaluation statistics. See InterpolantBase::eval_statistics. More...
 
- Static Protected Attributes inherited from InterpolantBase
static const unsigned int n_derivatives = 10
 Defines how many derivatives we allow to be returned from Taylor's coeficients. More...
 
static const unsigned int default_max_size = 10*1000
 Default maximal size of the interpolation table. More...
 
static const double simpson_tolerance = 1e-10
 Tolerance in Adaptive Simpson intergration. More...
 

Detailed Description

The main class for interpolation of functors.

This class is a wrapper of a functor (defined by class FunctorBase) which provides computation of derivates and interpolation of the functor. We will describe the functionality in the following paragrahphs.

Evaluation of a functor and its derivatives

Beside the interpolation, the class provides calling of the functor itself. All functions evaluating directly the functor starts with 'f_', e.g. function f_val computes the functor value.

We use the FADBAD++ library (website) to compute derivatives of the functor. First derivatives can be obtained by function f_diff which uses backward automatic differentiation. Higher order derivatives can be obtained via f_diffn which uses Taylor expansion method.

Interpolation

Sofar we provide only a piecewise linear interpolation. Both the function and its derivative can be interpolated. User has to provide the functor object to the constructor or later to the function set_functor. Then one must set at least the interval $ [a,b]$ by set_interval and the size of the interpolation table by set_size before calling interpolate. Otherwise assert will crash the program in debug build.

When user wants the size to be chosen automatically then set_size_automatic must be called at first. An initial guess of size of the interpolation table and a tolerance which is assumed to be the maximum relative difference between the function and the interpolant (sum of value and derivative difference) must be provided. Futher user can set the maximum size of the interpolation (default 1000) table which will not be exceeded and the type of norm in which the estimate of the error of interpolation will be computed (enumerated in ErrorNorm, default maximal norm). See one of the following paragraphs to learn how we deal with the error of the interpolation.

Outside the given interval the interpolation is extrapolated. User can choose among several types of extrapolation which are enumerated in Extrapolation. Constant and linear extrapolation use the first and last interpolating polynomials to compute the value. Functor type means that the functor itself will be evaluated outside the interval. Type of extrapolation can be changed by set_extrapolation function. Calling functor is the default type of extrapolation.

Error of the interpolation

We are interested in the maximum difference between the function value and its interpolation. When interpolating also the derivative, we need to compute the difference between derivative and its interpolation too. Therefore we would like to compute something like the L-infinity norm, relative to function value and its derivative. Let $ f(x), f'(x) $ be the function and its derivative and $ g(x), g'(x)$ interpolation of the function and interpolation of the derivative on the interval $ I $. We suggest an approximation of the infinity norm:

\[ \|f-g\|_\infty = \sup_{x\in I} F(x) = \sup_{x\in I} \left( \frac{|f(x)-g(x)|}{|f(x)| + tol} + \frac{|f'(x)-g'(x)|}{|f'(x)| + tol} \right) \approx \max_{x\in J} \left( \frac{|f(x)-g(x)|}{|f(x)| + tol} + \frac{|f'(x)-g'(x)|}{|f'(x)| + tol} \right) \]

where $ J $ is set of points that lies in the middle of two neighboring nodes. $ tol $ is a given tolerance for zero (avoids zero division). When computing the size of the interpolation table automatically, we compare the user tolerance $ TOL $ given by set_size_automatic funtion directly with the estimated value of the supremum norm.

This approach is fast and when computing the size automatically, we use the computed values (middle points, function and derivative values) in next interpolation. On the other hand this does not work good around zero points of the function where the value is small and the fraction becomes large or does not converge when dividing the intervals ( $ x^2 $ is a good example).

Therefore we provide also error estimation with $ L_p $ and $ W_p^1 $ norm. Then the user given tolerance is compared as it is stated in the following inequation:

\[ \left( \frac{\int \limits_I F^p(x) \,\rm{d} x}{|I|} \right)^{\frac{1}{p}} \leq TOL. \]

This approach is much more expensive but can give better results.

TODO: Improve error computation. Suggest more robust method.

Evaluation statistics

We collect some statistics during the evaluation of the interpolant in the struct InterpolantBase::EvalStatistics – total number of calls, number of evaluations inside the interpolation interval and number of evaluation outside the interval. One can use these statistics to decide whether it is necessary to widen the interval and recompute the interpolation. Function check_stats_and_reinterpolate is provided to do this automatically according to the given percentage of evaluation outside the interval.

Definition at line 291 of file interpolant.hh.

Constructor & Destructor Documentation

◆ Interpolant() [1/2]

Interpolant::Interpolant ( )

Default constructor.

Definition at line 155 of file interpolant.cc.

◆ Interpolant() [2/2]

template<template< class > class Func, class Type >
Interpolant::Interpolant ( Func< Type > *  func,
bool  interpolate_derivative = false 
)

Constructor with functor setting.

Parameters
funcis the pointer to functor
interpolate_derivativeis true when derivate is also interpolated
Template Parameters
Funcis the functor type
Typeis the template type of the functor (e.g. double)

Definition at line 56 of file interpolant_impl.hh.

◆ ~Interpolant()

Interpolant::~Interpolant ( void  )
virtual

Destructor.

Definition at line 164 of file interpolant.cc.

Member Function Documentation

◆ compute_error() [1/2]

void Interpolant::compute_error ( double  tol,
double  p,
ErrorNorm::Type  norm_type 
)
protected

Computes estimate of interpolation error with given norm.

Definition at line 490 of file interpolant.cc.

◆ compute_error() [2/2]

void Interpolant::compute_error ( double  tol,
std::vector< double > &  f,
std::vector< double > &  df 
)
protected

Creates piecewise linear interpolation.

Computes estimate of interpolation error in maximum norm.

Definition at line 410 of file interpolant.cc.

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◆ compute_values()

void Interpolant::compute_values ( )
protected

Creates table of nodes and function values.

Creates vector of nodes according to the table size and computes function and derivative values at the nodes.

Definition at line 335 of file interpolant.cc.

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◆ diff()

DiffValue Interpolant::diff ( double  x)
inline

Returns interpolated value of the derivation.

Parameters
xis the point at which we evaluate the interpolation

Definition at line 157 of file interpolant_impl.hh.

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◆ diff_p1()

DiffValue Interpolant::diff_p1 ( double  x)
inlineprotected

Finds interval on which x lies.

Function that evaluates the derivative of P1 interpolant at x.

Definition at line 281 of file interpolant_impl.hh.

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◆ f_diff()

DiffValue Interpolant::f_diff ( double  x)
inline

Returns 1st derivative of original functor using FADBAD.

Parameters
xis the point at which we evaluate the original functor

Definition at line 213 of file interpolant_impl.hh.

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◆ f_diffn()

double Interpolant::f_diffn ( double  x,
unsigned int  n 
)

Returns n-th derivative of original functor using FADBAD. Uses coeficients in Taylor's row.

Parameters
xis the point at which we evaluate the original functor
nis the order of the derivative we want

Definition at line 171 of file interpolant.cc.

◆ f_val()

double Interpolant::f_val ( double  x)
inline

Returns value of the original functor.

Parameters
xis the point at which we evaluate the original functor

Definition at line 208 of file interpolant_impl.hh.

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◆ interpolate()

int Interpolant::interpolate ( )
virtual

Creates piecewise interpolation with polynomials of selected degree.

Implements InterpolantBase.

Definition at line 189 of file interpolant.cc.

◆ set_functor()

template<template< class > class Func, class Type >
void Interpolant::set_functor ( Func< Type > *  func,
bool  interpolate_derivative = false 
)

Sets the functor.

Can be used when the functor is not set in the constructor.

Parameters
funcis the pointer to functor
interpolate_derivativeis true when derivate is also interpolated
Template Parameters
Funcis the functor type
Typeis the template type of the functor (e.g. double)

Definition at line 71 of file interpolant_impl.hh.

◆ swap_middle_values()

void Interpolant::swap_middle_values ( std::vector< double > &  f,
std::vector< double > &  df 
)
protected

Definition at line 375 of file interpolant.cc.

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◆ val()

double Interpolant::val ( double  x)
inline

Returns interpolated value.

Parameters
xis the point at which we evaluate the interpolation

Definition at line 84 of file interpolant_impl.hh.

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◆ val_p1()

double Interpolant::val_p1 ( double  x)
inline

Do NOT use, unless you are 100% sure.

This function evaluates the P1 interpolant at x. It does not check the interval, does not provide extrapolation and does not collect statistics.

Can be used to POSSIBLY speed the evaluation just a little bit, if you are absolutely sure that you evaluate the interpolant only on the given interval and do not want to collect statistics.

Same can be done with diff_p1 if it is made public.

Used in unit_test benchmark to compare with val function.

Definition at line 250 of file interpolant_impl.hh.

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◆ val_test()

double Interpolant::val_test ( double  x)
inline

Do NOT use, only for testing purpose.

TODO: After testing it can be removed and val_p1 can be made private again.

Definition at line 137 of file interpolant_impl.hh.

Member Data Documentation

◆ df_vec

std::vector<double> Interpolant::df_vec
protected

Vector of function derivatives values at nodes.

Definition at line 390 of file interpolant.hh.

◆ f_vec

std::vector<double> Interpolant::f_vec
protected

Vector of function values at nodes.

Definition at line 389 of file interpolant.hh.

◆ func

FunctorBase<double>* Interpolant::func
protected

Pointer to original functor with double type.

Definition at line 382 of file interpolant.hh.

◆ func_diff

FunctorBase<B<double> >* Interpolant::func_diff
protected

Pointer to original functor with FADBAD B type.

Definition at line 383 of file interpolant.hh.

◆ func_diffn

FunctorBase<T<double> >* Interpolant::func_diffn
protected

Pointer to original functor with FADBAD T type.

Definition at line 384 of file interpolant.hh.

◆ interpolate_derivative

bool Interpolant::interpolate_derivative
protected

Is true if we want to interpolate the derivative too.

Definition at line 386 of file interpolant.hh.

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