NumericalApproximation provides various numerical analysis methods of interpolation, extrapolation and approximation of a tabulated set of x, y data. More...

#include <NumericalApproximation.h>

Inheritance diagram for Isis::NumericalApproximation:

Collaboration diagram for Isis::NumericalApproximation:

Public Types
enum	InterpType { Linear, Polynomial, PolynomialNeville, CubicNatural, CubicClamped, CubicNatPeriodic, CubicNeighborhood, CubicHermite, Akima, AkimaPeriodic }
	This enum defines the types of interpolation supported in this class. More...

enum	ExtrapType { ThrowError, Extrapolate, NearestEndpoint }
	This enum defines the manner in which a value outside of the domain should be handled if passed to the Evaluate() method. More...

Public Member Functions
	NumericalApproximation (const NumericalApproximation::InterpType &itype=CubicNatural)
	Default constructor creates NumericalApproximation object. More...

	NumericalApproximation (unsigned int n, double x, double y, const NumericalApproximation::InterpType &itype=CubicNatural)
	Constructs NumericalApproximation object, sets interpolation type, populates the data set with pointer values. More...

	NumericalApproximation (const vector< double > &x, const vector< double > &y, const NumericalApproximation::InterpType &itype=CubicNatural)
	Constructs NumericalApproximation object, sets interpolation type, populates the data set with vector values. More...

	NumericalApproximation (const NumericalApproximation &dint)
	Copy constructor creates NumericalApproximation object and sets the copies the class variable values and state of input NumericalApproximation object to the new object. More...

NumericalApproximation &	operator= (const NumericalApproximation &numApMeth)
	NumericalApproximation assigmment operator sets this object "equal to" another. More...

virtual	~NumericalApproximation ()
	Destructor deallocates memory being used. More...

string	Name () const
	Get name of interpolating function assigned to object. More...

InterpType	InterpolationType ()
	Returns the enumerated type of interpolation chosen. More...

int	MinPoints ()
	Minimum number of points required by interpolating function. More...

int	MinPoints (NumericalApproximation::InterpType itype)
	Minimum number of points required by interpolating function. More...

void	AddData (const double x, const double y)
	Add a datapoint to the set. More...

void	AddData (unsigned int n, double x, double y)
	Add set of points to the data set using pointer arrays. More...

void	AddData (const vector< double > &x, const vector< double > &y)
	Add set of points to the data set using vectors. More...

void	SetCubicClampedEndptDeriv (const double yp1, const double ypn)
	Sets the values for the first derivatives of the endpoints of the data set. More...

void	AddCubicHermiteDeriv (unsigned int n, double *fprimeOfx)
	Adds values for the first derivatives of the data points. More...

void	AddCubicHermiteDeriv (const vector< double > &fprimeOfx)
	Adds values for the first derivatives of the data points. More...

void	AddCubicHermiteDeriv (const double fprimeOfx)
	Adds value of a first derivative of a data point. More...

double	DomainMinimum ()
	Input data domain minimum value. More...

double	DomainMaximum ()
	Input data domain maximum value. More...

bool	Contains (double x)
	Returns whether the passed value is an element of the set of x-values in the data set. More...

unsigned int	Size ()
	Returns the number of the coordinates added to the data set. More...

double	Evaluate (const double a, const ExtrapType &etype=ThrowError)
	Calculates interpolated or extrapolated value of tabulated data set for given domain value. More...

vector< double >	Evaluate (const vector< double > &a, const ExtrapType &etype=ThrowError)
	Calculates interpolated value for given set of domain values. More...

vector< double >	PolynomialNevilleErrorEstimate ()
	Retrieves the error estimate for the Neville's polynomial interpolation type. More...

vector< double >	CubicClampedSecondDerivatives ()
	Retrieves the second derivatives of the data set. More...

double	EvaluateCubicHermiteFirstDeriv (const double a)
	Approximates the first derivative of the data set function evaluated at the given domain value for CubicHermite interpolation type. More...

double	EvaluateCubicHermiteSecDeriv (const double a)
	Approximates the second derivative of the data set function evaluated at the given domain value for CubicHermite interpolation type. More...

double	GslFirstDerivative (const double a)
	Approximates the first derivative of the data set function evaluated at the given domain value for GSL supported interpolation types. More...

double	BackwardFirstDifference (const double a, const unsigned int n=3, const double h=0.1)
	Uses an n point backward first difference formula to approximate the first derivative evaluated at a given domain value. More...

double	ForwardFirstDifference (const double a, const unsigned int n=3, const double h=0.1)
	Uses an n point forward first difference formula to approximate the first derivative evaluated at a given domain value. More...

double	CenterFirstDifference (const double a, const unsigned int n=5, const double h=0.1)
	Uses an n point center first difference formula to approximate the first derivative evaluated at a given domain value. More...

double	GslSecondDerivative (const double a)
	Approximates the second derivative of the interpolated data set function evaluated at the given domain value for GSL supported interpolation types. More...

double	BackwardSecondDifference (const double a, const unsigned int n=3, const double h=0.1)
	Uses an n point backward second difference formula to approximate the second derivative evaluated at a given domain value. More...

double	ForwardSecondDifference (const double a, const unsigned int n=3, const double h=0.1)
	Uses an n point forward second difference formula to approximate the second derivative evaluated at a given domain value. More...

double	CenterSecondDifference (const double a, const unsigned int n=5, const double h=0.1)
	Uses an n point center second difference formula to approximate the second derivative evaluated at a given domain value. More...

double	GslIntegral (const double a, const double b)
	Approximates the integral of the data set function evaluated on the given interval for GSL supported interpolation types. More...

double	TrapezoidalRule (const double a, const double b)
	Uses the trapezoidal rule to approximate the integral of the interpolated data set function on the interval (a, b). More...

double	Simpsons3PointRule (const double a, const double b)
	Uses Simpson's 3-point rule to approximate the integral of the interpolated data set function on the interval (a,b). More...

double	Simpsons4PointRule (const double a, const double b)
	Uses Simpson's 4-point rule to approximate the integral of the interpolated data set function on the interval (a,b). More...

double	BoolesRule (const double a, const double b)
	Uses Boole's Rule to approximate the integral of the interpolated data set function on the interval (a,b). More...

double	RefineExtendedTrap (double a, double b, double s, unsigned int n)
	Calculates refinements extended trapezoidal rule to approximate the integral of the interpolated data set function on the interval (a,b). More...

double	RombergsMethod (double a, double b)
	Uses Romberg's method to approximate the integral of the interpolated data set function on the interval (a,b). More...

void	Reset ()
	Resets the state of the object. More...

void	Reset (NumericalApproximation::InterpType itype)
	Resets the state of the object and resets interpolation type. More...

void	SetInterpType (NumericalApproximation::InterpType itype)
	Sets interpolation type. More...

Protected Types
typedef const gsl_interp_type *	InterpFunctor
	GSL Interpolation specs. More...

typedef map< InterpType, InterpFunctor >	FunctorList
	Set up a std::map of GSL interpolator functors. List of function types. More...

typedef FunctorList::const_iterator	FunctorConstIter
	GSL Iterator. More...

Protected Member Functions
void	Init (NumericalApproximation::InterpType itype)
	Initializes the object upon instantiation. More...

bool	GslInterpType (NumericalApproximation::InterpType itype) const
	Returns whether an interpolation type is adapted from the GSL library. More...

void	GslAllocation (unsigned int npoints)
	Allocates GSL interpolation functions. More...

void	GslDeallocation ()
	Deallocate GSL interpolator resources, if used. More...

InterpFunctor	GslFunctor (NumericalApproximation::InterpType itype) const
	Search for a GSL interpolation function. More...

void	GslIntegrityCheck (int gsl_status, const char *src, int line)
	Checks the status of the GSL interpolation operations. More...

void	ValidateDataSet ()
	Validates the data set before computing interpolation. More...

bool	InsideDomain (const double a)
	Returns whether the passed value is greater than or equal to the domain minimum and less than or equal to the domain maximum. More...

bool	GslComputed () const
	Returns whether a GSL interpolation computation of the data set has been performed. More...

void	ComputeGsl ()
	Computes the GSL interpolation for a set of (x,y) data points. More...

void	ComputeCubicClamped ()
	Computes the cubic clamped interpolation for a set of (x,y) data points, given the first derivatives of the endpoints of the data set. More...

double	ValueToExtrapolate (const double a, const ExtrapType &etype)
	Returns the domain value at which to evaluate. More...

double	EvaluateCubicNeighborhood (const double a)
	Performs cubic spline interpolation for a neighborhood about a. More...

vector< double >	EvaluateCubicNeighborhood (const vector< double > &a, const ExtrapType &etype)
	Performs cubic spline interpolations for neighborhoods about each value of a. More...

double	EvaluateCubicClamped (const double a)
	Performs cubic spline interpolation with clamped boundary conditions, if possible. More...

double	EvaluateCubicHermite (const double a)
	Performs interpolation using the Hermite cubic polynomial. More...

double	EvaluatePolynomialNeville (const double a)
	Performs polynomial interpolation using Neville's algorithm. More...

vector< double >	EvaluateForIntegration (const double a, const double b, const unsigned int n)
	Evaluates data set in order to have enough data points to approximate the function to be integrated. More...

int	FindIntervalLowerIndex (const double a)
	Find the index of the x-value in the data set that is just below the input value, a. More...

void	ReportException (IException::ErrorType type, const string &method, const string &message, const char *filesrc, int lineno) const
	Generalized error report generator. More...

Protected Attributes
InterpType	p_itype
	Interpolation type. More...

vector< double >	p_x
	List of X values. More...

vector< double >	p_y
	List of Y values. More...

bool	p_dataValidated
	Flag variable to determine whether ValidateDataSet() has been called. More...

gsl_interp_accel *	p_acc
	Lookup accelorator. More...

gsl_spline *	p_interp
	Currently active interpolator. More...

bool	p_clampedEndptsSet
	Flag variable to determine whether SetCubicClampedEndptDeriv() has been called after all data was added for CubicClamped interpolation. More...

bool	p_clampedComputed
	Flag variable to determine whether ComputeCubicClamped() has been called. More...

double	p_clampedDerivFirstPt
	First derivative of first x-value, p_x[0]. This is only used for the CubicClamped interpolation type. More...

double	p_clampedDerivLastPt
	First derivative of last x-value, p_x[n-1]. This is only used for the CubicClamped interpolation type. More...

vector< double >	p_clampedSecondDerivs
	List of second derivatives evaluated at p_x values. This is only used for the CubicClamped interpolation type. More...

vector< double >	p_polyNevError
	Estimate of error for interpolation evaluated at x. This is only used for the PolynomialNeville interpolation type. 91 taken from AtmosModel. More...

vector< double >	p_fprimeOfx
	List of first derivatives corresponding to each x value in the data set (i.e. each value in p_x) More...

Static Protected Attributes
static FunctorList	p_interpFunctors
	Maintains list of interpolator options. More...

Detailed Description

NumericalApproximation provides various numerical analysis methods of interpolation, extrapolation and approximation of a tabulated set of x, y data.

This class contains a merged version of the Isis3 classes DataInterp and NumericalMethods. In addition, some methods from AtmosModel were moved to this class, the CubicNeighborhood interpolation type was adapted from the IDL routine called interpol (using the "/spline" keyword), and several differentiation and integration approximation methods were created.

NumericalApproximation contains 9 types of data interpolation routines. These routines act on x, y pairs of data points. The following forms of data interpolation are supported:

Name	Type (enum)	MinPoints	Description
Linear	Linear	2	Linear interpolation approximates a curve by concatinating line segments between known data points. This results in a continuous curve with a discontinuities of the derivative at the known data points. This interpolation type uses the GSL library routines.
Polynomial	Polynomial	3	Computes a polynomial interpolation. This method should only be used for interpolating small numbers of points because polynomial interpolation introduces large oscillations, even for well-behaved datasets. The number of terms in the interpolating polynomial is equal to the number of points. This interpolation type uses the GSL library routines.
Neville's Algorithm for Polynomial Interpolation	PolynomialNeville	3	Polynomial interpolation using Neville's algorithm. This first fits a polynomial of degree 0 through each point. On the second iteration, the polynomial of adjoining indices are combined to fit through pairs of points. This process repeats until a pyramid of approximations is reached. This is based on the Newton form of the interpolating polynomial and the recursion relation of the divided differences.
Natural Cubic Spline	CubicNatural	3	Cubic spline with natural boundary conditions. The resulting curve is piecewise cubic on each interval with matching first and second derivatives at the supplied data points. The second derivative is chosen to be zero at the endpoints, i.e. the first and last points of the data set. This is also refered to as free boundary condtions. This interpolation type uses the GSL library routines.
Clamped Cubic Spline	CubicClamped	3	Cubic Spline interpolation with clamped boundary conditions. The resulting curve is piecewise cubic on each interval. For this type of boundary condition to hold, it is necessary to have either the values of the derivative at the endpoints or an accurate approximation to those values. In general, clamped boundary conditions lead to more accurate approximations since they include more information about the function.
Periodic Natural Cubic Spline	CubicNatPeriodic	2	Cubic spline with periodic boundary conditions. The resulting curve is piecewise cubic on each interval with matching first and second derivatives at the supplied data points. The derivatives at the first and last points are also matched. Note that the last point in the data must have the same y-value as the first point, otherwise the resulting periodic interpolation will have a discontinuity at the boundary. This interpolation type uses the GSL library routines.
Cubic Spline about 4-point Neighborhood	CubicNeighborhood	4	Cubic Spline interpolation using 4-pt Neighborhoods with natural boundary conditions. This interpolation type is adapted from the IDL interpol.pro application using the "/spline" keyword on an irregular grid. This type of cubic spline fits a natural cubic spline to the 4-point neighborhood of known data points surrounding the x value at which we wish to evaluate. For example, suppose {x₀, x₁, ..., x_n} is the array of known domain values in the data set and \( x_i \leq u < x_{i+1} \) for some i such that \( 0 \leq i \leq n \), then to evaluate the y value at u, this method of interpolation evaluates the ordinary cubic spline consisting of the data set {x_i-1, x_i, x_i+1, x_i+2} at u.
Cubic Spline using Hermite cubic polynomial	CubicHermite	2	Cubic Spline interpolation using the Hermite cubic polynomial (also called the unique Hermite interpolating fundamental polynomial of degree 3). This method requires the input of the slopes of the tangent lines of the curve (i.e. velocities, or first derivatives) at each of the data points. This ensures that the approximation has the same "shape" as the curve at the known data points. This interpolation type uses piecewise Hermite cubic polynomials for each interval, (x₀, x₁), using the known data points, (x₀,f(x₀)) and (x₁,f(x₁)), and their derivatives, f '(x₀) and f '(x₁). The Hermite cubic polynomial is defined \[ H_3(x) = f(x_0)h_0(x)+f(x_1)h_1(x) +f\prime(x_0)\hat{h}_0(x)+f\prime(x_1)\hat{h}_1(x) \] where \( h_k(x) = [1-2(x-x_k)L\prime_k(x_k)](L_k(x))^2 \) and \( \hat{h}_k(x) = (x-x_k)(L_k(x))^2 \) for the kth-Lagrange coefficient polynomials of degree n = 1, \( L_0(x) = \frac{x- x_1}{x_0-x_1}\) and \( L_1(x) = \frac{x- x_0}{x_1-x_0}\).
Akima	Akima	5	Non-rounded Akima spline with natural boundary conditions. This method uses non-rounded corner algorithm of Wodicka. This interpolation type uses the GSL library routines.
Periodic Akima	AkimaPeriodic	5	Non-rounded Akima spline with periodic boundary conditions. This method uses non-rounded corner algorithm of Wodicka. This interpolation type uses the GSL library routines.

Numerical analysis approximation methods for differentiating and integrating unknown functions represented by a data set are implemented by using the interpolations on the data set created with a NumericalApproximation object. The Lagrange polynomial,

\[ L(x) = \sum_{k=0}^{n} \left(f(x_k) \prod_{i=0,i \neq k}^{n} \frac{x- x_i}{x_k-x_i}\right) \]

is used to determine formulas for numerical differentiation and integration.
Numerical Differentiation
The differentiation methods use difference formulas to approximate first and second derivatives at a domain value for a given a set of known (x,y) data points. Once all of the data points are added, a Isis::NumericalApproximation interpolation is used on the dataset to estimate the function, \(f:x \to y\), mapping x values to corresponding y values. Then, the derivative of f evaluated at x₀ is approximated. To do so, a uniform step size of h is chosen to determine the distance between x_i and x_i+1.

Differentiation methods require the parameters a (domain value at which the derivative will be evaluated), n (number of points used in the difference formula), and h (step size of points). Note: a is denoted x₀ in the above formulas.
Numerical Differentiation Type	Difference Formulas	Methods Available	Description
Backward Difference	2-point backward difference. \[ f\prime(x_0) \approx \frac{1}{h}[f(x_0) - f(x_0 - h)] \] 3-point backward difference. \[ f\prime(x_0) \approx \frac{1}{2h}[3f(x_0) - 4f(x_0 - h) + f(x_0 - 2h)] \] 3-point backward second difference. \[ f\prime\prime(x_0) \approx \frac{1}{h^2}[f(x_0) - 2f(x_0 - h) + f(x_0 - 2h)] \]	BackwardFirstDifference() BackwardSecondDifference()	Backward difference formulas use a uniform step-size moving in the negative direction from the given x₀. These formulas are derived by differentiating the Lagrange polynomials for x_i = x₀ - ih and evaluating them atx₀.
Forward Difference	2-point forward difference. \[ f\prime(x_0) \approx \frac{1}{h}[f(x_0 + h) - f(x_0)] \] 3-point forward difference. \[ f\prime(x_0) \approx \frac{1}{2h}[-f(x_0 + 2h) + 4f(x_0 + h) - 3f(x_0)] \] 3-point forward second difference. \[ f\prime\prime(x_0) \approx \frac{1}{h^2}[f(x_0 + 2h) - 2f(x_0 + h) + f(x_0)] \]	ForwardFirstDifference() ForwardSecondDifference()	Forward difference formulas use a uniform step-size moving in the positive direction from x₀. These formulas are derived by differentiating the Lagrange polynomials for x_i = x₀ + ih and evaluating them atx₀.
Center Difference	3-point center difference. \[ f\prime(x_0) \approx \frac{1}{2h}[f(x_0 + h) - f(x_0 - h)] \] 5-point center difference. \[ f\prime(x_0) \approx \frac{1}{12h}[-f(x_0 + 2h) + 8f(x_0 +h) - 8f(x_0 - h) + f(x_0 - 2h)] \] 3-point center second difference. \[ f\prime\prime(x_0) \approx \frac{1}{h^2}[f(x_0 + h) - 2f(x_0) + f(x_0 - h)] \] 5-point center second difference. \[ f\prime\prime(x_0) \approx \frac{1}{12h^2}[-f(x_0 + 2h) + 16f(x_0 +h) - 30f(x_0) + 16f(x_0 - h) - f(x_0 - 2h)] \]	CenterFirstDifference() CenterSecondDifference()	Center difference formulas use a uniform step-size moving in both directions from x₀ so this point stays centered. These formulas are derived by differentiating the Lagrange polynomials for x_i = x₀ (i - (n - 1)/2)h and evaluating them at x₀.
GSL Differentiation	Unknown	GslFirstDerivative() GslSecondDerivative()	No documentation was found about the algorithms used for these methods. They may only be called when interpolation type is one of the following: Linear, Polynomial, CubicNatural, CubicNatPeriodic, Akima, or AkimaPeriodic.

Numerical Integration
The integration methods were derived using Newton-Cotes, or quadrature, formulas for approximating integrals given a set of known (x,y) data points. The x values may be irregularly spaced, but must be unique and added to the dataset in ascending order. Once all of the data points are added, a Isis::NumericalApproximation interpolation is used on the dataset to estimate the function, \(f:x \to y\), mapping x values to corresponding y values. Then, the integral of f on the interval from a to b is approximated. To do so, an algorithm for creating a composite formula by applying the original formula to segments that share a boundary point is used. The NumericalApproximation::InterpType chosen for interpolation computation seems to have little affect on the error between the actual integral and the return values of the integration methods. The BoolesRule() method varies the most in error. Although errors are not high for any of the interpolation types, CubicNatural spline interpolations seem to have the smallest error most often with BoolesRule(). For any other numerical integration method, the errors appear to be identical for any NumericalApproximation::InterpType except Polynomial, which usually has a slightly larger error than the other interpolation types. Note: A portion of this algorithm is derived from the IDL function int_tabulated for Boole's Method.

Integration methods require the parameters a, b (interval of domain over which to integrate).
Numerical Integration Type	Integration Formulas	Methods Available	Description
Trapezoidal Rule	2-point Newton-Cotes trapezoidal rule: \[ \int_{a}^b f(x)dx \approx \frac{h}{2}[f(a) + f(b)] \] where h = b - a.	TrapezoidalRule()	The 2-point closed rule, known as the trapezoidal rule, uses straight line segments between known data points to estimate the area under the curve (integral). This Newton-Cotes formula uses a uniform step-size of h = b - a and is derived by integrating the Lagrange polynomials over the closed interval [a, a+h] for x_i = x₀ + ih.
Simpson's Rule, or Simpson's 3-Point Rule	3-point Newton-Cotes, Simpson's Rule \[ \int_{a}^b f(x)dx \approx \frac{h}{3}[f(a) + 4f(a+h) + f(a+2h)] \] where h = (b - a)/2.	Simpsons3PointRule()	The 3-point closed rule, known as Simpson's Rule or Simpson's 3-Point Rule, uses parabolic arcs between known data points to estimate the area under the curve (integral). This Newton-Cotes formula uses a uniform step-size of h = (b - a)/2 and is derived by integrating the Lagrange polynomials over the closed interval [a, a+2h] for x_i = x₀ + ih.
Simpson's 3/8 Rule, or Simpson's 4-Point Rule	4-point Newton-Cotes, Simpson's 3/8 Rule \[ \int_{a}^b f(x)dx \approx \frac{3h}{8}[f(a) + 3f(a+h) + 3f(a+2h) + f(a+3h)] \] where h = (b - a)/3.	Simpsons4PointRule()	The 4-point closed rule, known as Simpson's 3/8 Rule or Simpson's 4-Point Rule, uses cubic curves between known data points to estimate the area under the curve (integral). This Newton-Cotes formula uses a uniform step-size of h = (b - a)/3 and is derived by integrating the Lagrange polynomials over the closed interval [a, a+3h] for x_i = x₀ + ih.
Boole's Rule	5-point Newton-Cotes, Boole's Rule \[ \int_{a}^b f(x)dx \approx \frac{2h}{45}[7f(a) + 32f(a+h) + 12f(a+2h) + 32f(a+3h) + 7f(a+4h)] \] where h = (b - a)/4.	BoolesRule()	The 5-point closed rule, known as Boole's Rule, uses quartic curves between known data points to estimate the area under the curve (integral). This Newton-Cotes formula uses a uniform step-size of h = (b - a)/4 and is derived by integrating the Lagrange polynomials over the closed interval [a, a+4h] for x_i = x₀ + ih.
Refinements of Extended Trapezoidal Rule	Extended closed trapezoidal rule \[ \int_{a}^b f(x)dx \approx h[\frac12 f(x_0) + f(x_1) + ... + f(x_{n-2} + \frac12 f(x_{n-1})] \] where h = (b - a)/4, x₀ = a, and x_n-1 = b.	RefineExtendedTrap()	The extended (or composite) trapezoidal rule can be used to with a series of refinements to approximate the integral. The first stage of refinement returns the ordinary trapezoidal estimate of the integral. Subsequent stages will improve the accuracy, where, for the n^th stage 2^n-2 interior points are added.
Romberg's Method	Romberg's Method for Integration	RombergsMethod()	Romberg's Method is a potent numerical integration tool. It uses the extended trapezoidal rule and Neville's algorithm for polynomial interpolation.
GSL Integration	Unknown	GslIntegral()	No documentation was found about the algorithm used for this methods. They may only be called when interpolation type is one of the following: Linear, Polynomial, CubicNatural, CubicNatPeriodic, Akima, or AkimaPeriodic.

Below is an example demonstating the use of this class:

inline double f(double x) { return (x + cos ((x * x))); }
NumericalApproximation
spline(NumericalApproximation::CubicClamped);
for (int x = 0; x < 200 ; x++) {
  spline.addPoint(x, f(x));
}
spline.SetCubicClampedEndptDeriv(0,0);
double yinterp = spline.Evaluate(50.7);
double yextrap =
    spline.Evaluate(201,NumericalApproximation::Extrapolate);
double deriv = spline.CenterFirstDifference(10);
double integ =
    spline.RombergsMethod(spline.DomainMinimum(),spline.DomainMaximum());

To compute the same data set using the Akima spline, just use the following:

spline.Reset(NumericalApproximation::Akima);
double yinterp = spline.Evaluate(50.7);
double yextrap =
    spline.Evaluate(201,NumericalApproximation::NearestEndpoint);
double deriv = spline.CenterFirstDifference(10);
double integ =
    spline.RombergsMethod(spline.DomainMinimum(),spline.DomainMaximum());

Caveats

When using this class, there are some important details that require consideration. Several interpolation types of this class use the GSL library. In addition, the GSL library default error handling scheme has implications when used in C++ objects. The default behavior is to terminate the application when an error occurs. Options for developers are to turn off error trapping within the GSL, which means users of the library must do their own error checking, or a specific handler can be specified to replace the default mode. Neither option is well suited in object-oriented implementations because they apply globally, to all users of the library. The approach used here is to turn off error handling and require users to handle their own error checking. This is not entirely safe as any user that does not follow this strategy could change this behavior at runtime. Other options should be explored. An additional option is that the default behavior can be altered prior to building GSL but this is also problematic since this behavior cannot be guaranteed universally.

For these types, interpolation is strictly adhered to and extrapolation is not handled well. If the input to be evaluated is outside of the minimum and maximum x values of the original data points, then the y value corresponding to the x value that is nearest to that input is returned. However, for sufficiently close values, the clamped cubic and polynomial Neville's types are fairly accurate in their extrapolation techniques. All differentiation and integration methods throw an error if the value passed is outside of the domain.

Author: 2008-11-05 Janet Barrett, Kris Becker, K Teal Thompson, Jeannie Walldren

Member Typedef Documentation

◆ FunctorConstIter

typedef FunctorList::const_iterator Isis::NumericalApproximation::FunctorConstIter

protected

GSL Iterator.

◆ FunctorList

typedef map<InterpType, InterpFunctor> Isis::NumericalApproximation::FunctorList

protected

Set up a std::map of GSL interpolator functors. List of function types.

◆ InterpFunctor

typedef const gsl_interp_type* Isis::NumericalApproximation::InterpFunctor

protected

GSL Interpolation specs.

Member Enumeration Documentation

◆ ExtrapType

enum Isis::NumericalApproximation::ExtrapType

This enum defines the manner in which a value outside of the domain should be handled if passed to the Evaluate() method.

Enumerator
ThrowError	Evaluate() throws an error if a is outside of the domain.
Extrapolate	Evaluate() attempts to extrapolate if a is outside of the domain. This is only valid for NumericalApproximation::InterpType CubicClamped or PolynomialNeville and the result will be accurate only if sufficiently close to the domain boundary.
NearestEndpoint	Evaluate() returns the y-value of the nearest endpoint if a is outside of the domain.

◆ InterpType

enum Isis::NumericalApproximation::InterpType

This enum defines the types of interpolation supported in this class.

Enumerator
Linear	Linear interpolation.
Polynomial	Polynomial interpolation.
PolynomialNeville	Polynomial interpolation using Neville's algorithm.
CubicNatural	Cubic Spline interpolation with natural boundary conditions.
CubicClamped	Cubic Spline interpolation with clamped boundary conditions.
CubicNatPeriodic	Cubic Spline interpolation with periodic boundary conditions.
CubicNeighborhood	Cubic Spline interpolation using 4-pt Neighborhoods with natural boundary conditions.
CubicHermite	Cubic Spline interpolation using the Hermite cubic polynomial.
Akima	Non-rounded Akima Spline interpolation with natural boundary conditions.
AkimaPeriodic	Non-rounded Akima Spline interpolation with periodic boundary conditions.

Constructor & Destructor Documentation

◆ NumericalApproximation() [1/4]

Isis::NumericalApproximation::NumericalApproximation ( const NumericalApproximation::InterpType & itype = CubicNatural )

Default constructor creates NumericalApproximation object.

Sets the NumericalApproximation::InterpType to the enumerated value of itype. Default is CubicNatural.

Parameters

itype NumericalApproximation::InterpType enum value to be assigned to this object (Default: CubicNatural)

Exceptions

Isis::IException::Programmer "Unable to construct NumericalApproximation object"

References _FILEINFO_, and Isis::IException::errorType().

◆ NumericalApproximation() [2/4]

Isis::NumericalApproximation::NumericalApproximation	(	unsigned int	n,
		double *	x,
		double *	y,
		const NumericalApproximation::InterpType &	itype = `CubicNatural`
	)

Constructs NumericalApproximation object, sets interpolation type, populates the data set with pointer values.

This constructor verifies the NumericalApproximation::InterpType passed in as a parameter. If it passes, it assigns this type to the object and x and y are added to the data set and validated. If the interpolation type is CubicClamped, first derivatives of the endpoints are required before Evaluate() may be called. If no value for itype is given, the default is CubicNatural.

Parameters

n	Number of data points to be used.
x	Array of domain values to add to data set
y	Array of range values corresponding to each domain value in x
itype	NumericalApproximation::InterpType enum value to be assigned to this object (Default: CubicNatural)

Exceptions

Isis::IException::Programmer "Unable to construct NumericalApproximation object"

References _FILEINFO_, and Isis::IException::errorType().

◆ NumericalApproximation() [3/4]

Isis::NumericalApproximation::NumericalApproximation	(	const vector< double > &	x,
		const vector< double > &	y,
		const NumericalApproximation::InterpType &	itype = `CubicNatural`
	)

Constructs NumericalApproximation object, sets interpolation type, populates the data set with vector values.

This constructor verifies the interpolation type and checks that the x and y vectors have the same number of elements. If so, the object is assigned the NumericalApproximation::InterpType passed in as a parameter and x and y are added to the data set and validated. If the interpolation type is CubicClamped, first derivatives of the endpoints are required before evaluation is allowed. If no value for itype is given, the default is CubicNatural.

Parameters

x	Vector of domain values to add to data set
y	Vector of range values corresponding to each domain value in x
itype	NumericalApproximation::InterpType enum value to be assigned to this object (Default: CubicNatural)

Exceptions

Isis::IException::Programmer "Unable to construct NumericalApproximation object"

References _FILEINFO_, and Isis::IException::errorType().

◆ NumericalApproximation() [4/4]

Isis::NumericalApproximation::NumericalApproximation ( const NumericalApproximation & oldObject )

Copy constructor creates NumericalApproximation object and sets the copies the class variable values and state of input NumericalApproximation object to the new object.

Copying of class objects must be handled explicitly in order for it to be completed properly. The data, interpolator type and other class variables are copied from the supplied object.

Parameters

oldObject NumericalApproximation object to gleen data from for creation of this object

Exceptions

Isis::IException::Programmer "Unable to construct NumericalApproximation object"

References _FILEINFO_, Isis::IException::errorType(), p_clampedComputed, p_clampedDerivFirstPt, p_clampedDerivLastPt, p_clampedEndptsSet, p_clampedSecondDerivs, p_dataValidated, p_fprimeOfx, p_itype, p_polyNevError, p_x, and p_y.

◆ ~NumericalApproximation()

Isis::NumericalApproximation::~NumericalApproximation ( )

virtual

Destructor deallocates memory being used.

Explicit operations are required to free resources of the interpolation

Member Function Documentation

◆ AddCubicHermiteDeriv() [1/3]

void Isis::NumericalApproximation::AddCubicHermiteDeriv	(	unsigned int	n,
		double *	fprimeOfx
	)

Adds values for the first derivatives of the data points.

This method can only be called for cubic Hermite splines, i.e. if NumericalApproximation::InterpType is CubicHermite. These values should be entered in the order of the corresponding data points.

Parameters

n	Number of derivative values to be added.
fprimeOfx	Array of derivative values to be added.

Exceptions

Isis::IException::Programmer "Method only used for cspline-Hermite interpolation type"

References _FILEINFO_.

Referenced by Isis::SpicePosition::HermiteIndices(), and Isis::SpicePosition::SetEphemerisTimeHermiteCache().

◆ AddCubicHermiteDeriv() [2/3]

void Isis::NumericalApproximation::AddCubicHermiteDeriv ( const vector< double > & fprimeOfx )

Adds values for the first derivatives of the data points.

This method can only be called for cubic Hermite splines, i.e. if NumericalApproximation::InterpType is CubicHermite. These values should be entered in the order of the corresponding data points.

Parameters

fprimeOfx Vector of derivative values to be added.

Exceptions

Isis::IException::Programmer "Method only used for cspline-Hermite interpolation type"

References _FILEINFO_.

◆ AddCubicHermiteDeriv() [3/3]

void Isis::NumericalApproximation::AddCubicHermiteDeriv ( const double fprimeOfx )

Adds value of a first derivative of a data point.

This method can only be called for cubic Hermite splines, i.e. if NumericalApproximation::InterpType is CubicHermite. Values should be entered in the order of the corresponding data points.

Parameters

fprimeOfx Derivative value to be added.

Exceptions

Isis::IException::Programmer "Method only used for cspline-Hermite interpolation type"

References _FILEINFO_.

◆ AddData() [1/3]

void Isis::NumericalApproximation::AddData	(	const double	x,
		const double	y
	)

Add a datapoint to the set.

This method allows the user to add a point to the set of data in preparation for interpolation over an interval.

Note: All data sets must have unique x values. If the interpolation type is not PolynomialNeville, x values must be sorted in ascending order. If CubicNatPeriodic interpolation type is used, the first and last points added to the data set must have the same y value.

Parameters

x	Domain value to add to data set
y	Range value corresponding to domain value x

Referenced by EvaluateCubicNeighborhood(), Isis::SpicePosition::HermiteIndices(), Isis::LunarLambertEmpirical::LunarLambertEmpirical(), Isis::MinnaertEmpirical::MinnaertEmpirical(), Isis::rebin(), Isis::NumericalAtmosApprox::RombergsMethod(), RombergsMethod(), and Isis::SpicePosition::SetEphemerisTimeHermiteCache().

◆ AddData() [2/3]

void Isis::NumericalApproximation::AddData	(	unsigned int	n,
		double *	x,
		double *	y
	)

Add set of points to the data set using pointer arrays.

This method allows the user to add a set of points to the set of data in preparation for interpolation over an interval. This method does not overwrite previously added data. Rather, it adds the new data to the end of the existing data set.

Note: Behavior is undefined if the domain is not sorted in ascending or descending order.

Parameters

n	Number of data points to be used.
x	Array of domain values to add to data set
y	Array of range values corresponding to each domain value in x

◆ AddData() [3/3]

void Isis::NumericalApproximation::AddData	(	const vector< double > &	x,
		const vector< double > &	y
	)

Add set of points to the data set using vectors.

This method allows the user to add a set of points to the set of data in preparation for interpolation over an interval. This method does not overwrite previously added data. Rather, it adds the new data to the end of the existing data set.

Note: Behavior is undefined if the domain is not sorted in ascending or descending order.

Parameters

x	Vector of domain values to add to data set
y	Vector of range values corresponding to each domain value in x

Exceptions

Isis::IException::Programmer "Invalid arguments. The number of x-values does not equal the number of y-values."

References _FILEINFO_.

◆ BackwardFirstDifference()

double Isis::NumericalApproximation::BackwardFirstDifference	(	const double	a,
		const unsigned int	n = `3`,
		const double	h = `0.1`
	)

Uses an n point backward first difference formula to approximate the first derivative evaluated at a given domain value.

This method uses backward first difference formulas to return an approximation of the first derivative of the interpolated data set function evaluated at given a valid domain value, a. Backward difference formulas use n points, with the largest x value at a, for numerical differentiation approximation. This method uses one of the following formulas:

2-point backward difference.
\[ f\prime(a) \approx \frac{1}{h}[f(a) - f(a - h)] \]
3-point backward difference.
\[ f\prime(a) \approx \frac{1}{2h}[3f(a) - 4f(a - h) + f(a - 2h)] \]

Parameters

a	Domain value at which first deriviative is evaluated.
n	The number of points used in the formula
h	Distance between nearest points in the formula

Returns: double First derivative approximation for the given domain value

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain."
Isis::IException::Programmer	"Formula steps outside of domain." If a-(n-1)h is less than domain min.
Isis::IException::Programmer	"Invalid argument. There is no n-point backward difference formula in use." @throws Isis::IException::Programmer "Unable to calculate backward first difference for given (a,n,h)"

References _FILEINFO_, and Isis::IException::errorType().

◆ BackwardSecondDifference()

double Isis::NumericalApproximation::BackwardSecondDifference	(	const double	a,
		const unsigned int	n = `3`,
		const double	h = `0.1`
	)

Uses an n point backward second difference formula to approximate the second derivative evaluated at a given domain value.

This method uses backward second difference formulas to return an approximation of the second derivative of the interpolated data set function evaluated at given a valid domain value, a. Backward second difference formulas use n points, with the largest x value at a, for numerical differentiation approximation. This method uses the following formula:

3-point backward second difference.
\[ f\prime\prime(a) \approx \frac{1}{h^2}[f(a) - 2f(a - h) + f(a - 2h)] \]

Parameters

a	Domain value at which second deriviative is evaluated.
n	The number of points used in the formula.
h	Distance between nearest points in the formula.

Returns: double Second derivative approximation for the given domain value

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain."
Isis::IException::Programmer	"Formula steps outside of domain." If a-(n-1)h is less than domain min.
Isis::IException::Programmer	"Invalid argument. There is no n-point backward difference formula in use."
Isis::IException::Programmer

References _FILEINFO_, and Isis::IException::errorType().

◆ BoolesRule()

double Isis::NumericalApproximation::BoolesRule	(	const double	a,
		const double	b
	)

Uses Boole's Rule to approximate the integral of the interpolated data set function on the interval (a,b).

The Boole's Rule for integration approximation uses a 5-point Newton-Cote formula. This rule states:

\[ \int_{a}^b f(x)dx \approx \frac{2h}{45}[7f(a) + 32f(a+h) + 12f(a+2h) + 32f(a+3h) + 7f(a+4h)] \]

where h = (b - a)/4.

This method uses a composite, or extended, Boole's rule formula to approximate the integral.

Note: The method uses an algorithm that is adapted from the IDL function int_tabulated.pro.

Parameters

a	Lower bound of interval.
b	Upper bound of interval.

Returns: double Approximate integral of data set function from a to b.

Exceptions

Isis::IException::Programmer "Unable to calculate the integral on the interval (a,b) using Boole's rule"

References _FILEINFO_, and Isis::IException::errorType().

◆ CenterFirstDifference()

double Isis::NumericalApproximation::CenterFirstDifference	(	const double	a,
		const unsigned int	n = `5`,
		const double	h = `0.1`
	)

Uses an n point center first difference formula to approximate the first derivative evaluated at a given domain value.

This method uses center first difference formulas to return an approximation of the first derivative of the interpolated data set function evaluated at given a valid domain value, a. Center difference formulas use n points, centered at a, for numerical differentiation approximation. This method uses one of the following formulas:

3-point center difference.
\[ f\prime(a) \approx \frac{1}{2h}[f(a + h) - f(a - h)] \]
5-point center difference.
\[ f\prime(a) \approx \frac{1}{12h}[-f(a + 2h) + 8f(a +h) - 8f(a - h) + f(a - 2h)] \]

Parameters

a	Domain value at which first deriviative is evaluated.
n	The number of points used in the formula.
h	Distance between nearest points in the formula.

Returns: double First derivative approximation for the given domain value

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain."
Isis::IException::Programmer	"Formula steps outside of domain." If a+(n-1)h is greater than domain max or a-(n-1)h is less than domain min.
Isis::IException::Programmer	"Invalid argument. There is no n-point center difference formula in use."
Isis::IException::Programmer	"Unable to calculate center first difference for (a,n,h)"

References _FILEINFO_, and Isis::IException::errorType().

◆ CenterSecondDifference()

double Isis::NumericalApproximation::CenterSecondDifference	(	const double	a,
		const unsigned int	n = `5`,
		const double	h = `0.1`
	)

Uses an n point center second difference formula to approximate the second derivative evaluated at a given domain value.

This method uses center second difference formulas to return an approximation of the second derivative of the interpolated data set function evaluated at given a valid domain value, a. Center second difference formulas use n points, centered at a, for numerical differentiation approximation. This method uses one of the following formulas:

3-point center second difference.
\[ f\prime\prime(a) \approx \frac{1}{h^2}[f(a + h) - 2f(a) + f(a - h)] \]
5-point center second difference.
\[ f\prime\prime(a) \approx \frac{1}{12h^2}[-f(a + 2h) + 16f(a +h) - 30f(a) + 16f(a - h) - f(a - 2h)] \]

Parameters

a	Domain value at which second deriviative is evaluated.
n	The number of points used in the formula.
h	Distance between nearest points in the formula.

Returns: double Second derivative approximation for the given domain value

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain."
Isis::IException::Programmer	"Formula steps outside of domain." If a+(n-1)h is greater than domain max or a-(n-1)h is less than domain min.
Isis::IException::Programmer	"Invalid argument. There is no n-point center difference formula in use."
Isis::IException::Programmer	"Unable to calculate center second difference for (a,n,h)"

References _FILEINFO_, and Isis::IException::errorType().

◆ ComputeCubicClamped()

void Isis::NumericalApproximation::ComputeCubicClamped ( )

protected

Computes the cubic clamped interpolation for a set of (x,y) data points, given the first derivatives of the endpoints of the data set.

This protected method is called only if the object is assigned a cubic clamped interpolation type and if it has not already been computed for the given data set. It calculates the second derivatives for each p_x value of the known data set and stores these values in p_clampedSecondDerivs so that the EvaluateCubicClamped() method may be called to interpolate the set using clamped boundary conditions, if possible. This method must be called when all the data points have been added to the object for the data set and after SetCubicClampedEndptDeriv() has been called. If the endpoint derivatives are greater than or equal to 1 x 10³⁰, the routine is signaled to set the corresponding boundary condition for a natural spline, with zero second derivative on that boundary.

Exceptions

Issis::IException::Programmer "Must use SetCubicClampedEndptDeriv() before computing cubic spline with clamped boundary" @throws Issis::IException::Programmer "Unable to compute cubic clamped interpolation"

See also: SetCubicClampedEndptDeriv(); EvaluateCubicClamped()

References _FILEINFO_, and Isis::IException::errorType().

◆ ComputeGsl()

void Isis::NumericalApproximation::ComputeGsl ( )

protected

Computes the GSL interpolation for a set of (x,y) data points.

This protected method is called only if the object is assigned a GSL interpolation type and if it has not already been computed on the given data set. It will compute the interval of interpolated range values over the given domain. A copy of this data is maintained in the object so the data points do not need to be interpolated on each evaluation of a point unless the data set changes.

Exceptions

Isis::IException::Programmer "Unable to compute GSL interpolation"

References _FILEINFO_, and Isis::IException::errorType().

◆ Contains()

bool Isis::NumericalApproximation::Contains ( double x )

Returns whether the passed value is an element of the set of x-values in the data set.

This method uses a binary search of the set of x-values to determine whether the input is contained in this set.

Parameters

x	Value to search for in the data set.

Returns: bool Whether the passed value is contained in the x-values of the data set.

◆ CubicClampedSecondDerivatives()

vector< double > Isis::NumericalApproximation::CubicClampedSecondDerivatives ( )

Retrieves the second derivatives of the data set.

This method returns a vector of the same size as the data set. Each component is the second derivative of the corresponding p_x value , as estimated by the ComputeCubicClamped() method.

Returns: vector <double> List of second derivatives for each p_x value in the data set.

Exceptions

Isis::IException::Programmer	"Method only used for cspline-clamped interpolation type"
Isis::IException::Programmer	"Unable to calculate the second derivatives of the data set for a clamped cubic spline."

See also: ComputeCubicClamped()

References _FILEINFO_, and Isis::IException::errorType().

◆ DomainMaximum()

double Isis::NumericalApproximation::DomainMaximum ( )

Input data domain maximum value.

This method validates the data set for the assigned interpolation type and returns the largest value in p_x of the data set.

Returns: double Maximum domain value

Exceptions

Isis::IException::Programmer "Unable to calculate the domain minimum for the data set."

References _FILEINFO_, and Isis::IException::errorType().

◆ DomainMinimum()

double Isis::NumericalApproximation::DomainMinimum ( )

Input data domain minimum value.

This method validates the data set for the assigned interpolation type and returns the smallest value in p_x of the data set.

Returns: double Minimum domain value

Exceptions

Isis::IException::Programmer "Unable to calculate the domain maximum for the data set."

References _FILEINFO_, and Isis::IException::errorType().

◆ Evaluate() [1/2]

double Isis::NumericalApproximation::Evaluate	(	const double	a,
		const ExtrapType &	etype = `ThrowError`
	)

Calculates interpolated or extrapolated value of tabulated data set for given domain value.

This method returns the approximate value for f(a), where a is an element near the domain and f is the approximated function for the given data set. If the given value, a, falls outside of the domain, then the extrapoltion type is examined to determine the result. CubicNeighborhood and GSL interpolation types can not extrapolate, so the user must choose to throw an error or return f evaluated at the nearest domain boundary. CubicClamped, CubicHermite, and PolynomialNeville interpolation types can extrapolate a value with accuracy only if a is near enough to the domain boundary. Default NumericalApproximation::ExtrapType is ThrowError.

Parameters

a	Domain value from which to interpolate a corresponding range value
etype	NumericalApproximation::ExtrapType enum value indicates how to evaluate if a falls outside of the domain. (Default: ThrowError)

Returns: double Range value interpolated or extrapolated for the given domain value

Exceptions

Isis::IException::Programmer "Unable to evaluate the function at the point a"

References _FILEINFO_, and Isis::IException::errorType().

Referenced by Isis::HapkeAtm1::AtmosModelAlgorithm(), Isis::HapkeAtm2::AtmosModelAlgorithm(), EvaluateCubicNeighborhood(), Isis::SpicePosition::HermiteIndices(), Isis::LunarLambertEmpirical::PhotoModelAlgorithm(), Isis::MinnaertEmpirical::PhotoModelAlgorithm(), Isis::rebin(), Isis::NumericalAtmosApprox::RombergsMethod(), RombergsMethod(), and Isis::SpicePosition::SetEphemerisTimeHermiteCache().

◆ Evaluate() [2/2]

vector< double > Isis::NumericalApproximation::Evaluate	(	const vector< double > &	a,
		const ExtrapType &	etype = `ThrowError`
	)

Calculates interpolated value for given set of domain values.

This method returns the approximate values for f(a_i), where a_i is an element in the domain and f is the interpolated function for the given data set. If any of the given values in a fall outside of the domain, then the extrapoltion type is examined to determine the result. CubicNeighborhood and GSL interpolation types can not extrapolate, so the user must choose to throw an error or return f evaluated at the nearest domain boundary. CubicClamped and PolynomialNeville interpolation types can extrapolate a value with accuracy only if a_i is near enough to the domain boundary.

Parameters

a	Vector of domain values from which to interpolate a vector of corresponding range values
etype	NumericalApproximation::ExtrapType enum value indicates how to evaluate any value of a that falls outside of the domain. (Default: ThrowError)

Returns: vector <double> Set of interpolated range values corresponding to the elements of the vector of domain values given

Exceptions

Isis::IException::Programmer "Unable to evaluate the function at the vector of points, a."

References _FILEINFO_, and Isis::IException::errorType().

◆ EvaluateCubicClamped()

double Isis::NumericalApproximation::EvaluateCubicClamped ( const double a )

protected

Performs cubic spline interpolation with clamped boundary conditions, if possible.

This is a protected method called by Evaluate() if the NumericalApproximation::InterpType is CubicClamped and SetCubicClampedEndptDeriv() has been called. It uses the second derivative vector, p_clampedSecondDerivs, to interpolate the value for f(a) using a clamped cubic spline formula,

Note: If the given value, a, falls outside of the domain, then extrapolation is attempted and the result is accurate only if a is near enough to the domain boundary.

Parameters

a	Domain value from which to interpolate a corresponding range value

Returns: double Value returned from interpolation or extrapolation.

See also: Evaluate(); ComputeCubicClamped(); SetCubicClampedEndptDeriv()

◆ EvaluateCubicHermite()

double Isis::NumericalApproximation::EvaluateCubicHermite ( const double a )

protected

Performs interpolation using the Hermite cubic polynomial.

This is a protected method called by Evaluate() if the NumericalApproximation::InterpType is CubicHermite. It returns an approximate value for f(a) by using the Hermite cubic polynomial, which uses Lagrange coefficient polynomials. The data points and each corresponding first derivative should have been already added.

Parameters

a	Domain value from which to interpolate a corresponding range value

Returns: double Value returned from interpolation or extrapolation

Exceptions

Isis::IException::User "Invalid arguments. The size of the first derivative vector does not match the number of (x,y) data points."

See also: http://mathworld.wolfram.com/HermitesInterpolatingPolynomial.html; http://en.wikipedia.org/wiki/Lagrange_Polynomial; Evaluate(); AddCubicHermiteDeriv()

References _FILEINFO_.

◆ EvaluateCubicHermiteFirstDeriv()

double Isis::NumericalApproximation::EvaluateCubicHermiteFirstDeriv ( const double a )

Approximates the first derivative of the data set function evaluated at the given domain value for CubicHermite interpolation type.

This method returns an approximation of the first derivative evaluated at given a valid domain value, a. It is able to extrapolate for values not far outside of the domain

References _FILEINFO_.

Referenced by Isis::SpicePosition::SetEphemerisTimeHermiteCache().

◆ EvaluateCubicHermiteSecDeriv()

double Isis::NumericalApproximation::EvaluateCubicHermiteSecDeriv ( const double a )

Approximates the second derivative of the data set function evaluated at the given domain value for CubicHermite interpolation type.

This method returns an approximation of the second derivative evaluated at given a valid domain value, a. It is able to extrapolate for values not far outside of the domain

References _FILEINFO_.

◆ EvaluateCubicNeighborhood() [1/2]

double Isis::NumericalApproximation::EvaluateCubicNeighborhood ( const double a )

protected

Performs cubic spline interpolation for a neighborhood about a.

This is a protected method called by Evaluate() if the NumericalApproximation::InterpType is CubicNeighborhood. It uses an algorithm that is adapted from the IDL interpol.pro application using the "/spline" keyword on an irregular grid. This type of cubic spline fits a natural cubic spline to the 4-point neighborhood of known data points surrounding a. For example, suppose {x₀, x₁, ..., x_n} is the array of known domain values in the data set and \( x_i \leq a < x_{i+1}\) for some i such that \( 0 \leq i \leq n \), then f(a) is evaluated by interpolating the natural cubic spline consisting of the data set {x_i-1, x_i, x_i+1, x_i+2} at a.

Note: If the given value, a, falls outside of the domain, then f evaluated at the nearest domain boundary is returned.

Parameters

a	Domain value from which to interpolate a corresponding range value.

Returns: double Result of interpolated value of f(a).

Exceptions

Isis::IException::Programmer "Unable to evaluate cubic neighborhood interpolation at a"

See also: Evaluate()

References _FILEINFO_, AddData(), Isis::IException::errorType(), and Evaluate().

◆ EvaluateCubicNeighborhood() [2/2]

vector< double > Isis::NumericalApproximation::EvaluateCubicNeighborhood	(	const vector< double > &	a,
		const ExtrapType &	etype
	)

protected

Performs cubic spline interpolations for neighborhoods about each value of a.

This is a protected method called by Evaluate() if the NumericalApproximation::InterpType is CubicNeighborhood. It uses an algorithm that is adapted from the IDL interpol.pro application using the "/spline" keyword on an irregular grid. For each component of a, this method fits a natural cubic spline using the 4-point neighborhood of known data points surrounding that component. For example, suppose {x₀, x₁, ..., x_n} is the array of known domain values in the data set, then for each component of a, a_k, in the domain, there is an i such that \( 0 \leq i \leq n \) and \( x_i \leq a_k < x_{i+1}\). Then, f(a_k) is evaluated by interpolating the natural cubic spline consisting of the data set {x_i-1, x_i, x_i+1, x_i+2} at a_k.

Note: If any given value, a_i, falls outside of the domain, then f is evaluated at the nearest domain boundary.

Parameters

a	Vector of domain values from which to interpolate a vector of corresponding range values
etype	NumericalApproximation::ExtrapType enum value indicates how to evaluate if a falls outside of the domain. (Default: ThrowError)

Returns: vector <double> Result of interpolated value of f at each value of a.

Exceptions

Isis::IException::Programmer "Unable to evaluate cubic neighborhood interpolation at the values in the vector, a"

See also: Evaluate()

References _FILEINFO_, AddData(), Isis::IException::errorType(), Evaluate(), and Reset().

◆ EvaluateForIntegration()

vector< double > Isis::NumericalApproximation::EvaluateForIntegration	(	const double	a,
		const double	b,
		const unsigned int	n
	)

protected

Evaluates data set in order to have enough data points to approximate the function to be integrated.

Parameters

a	Lower bound of interval.
b	Upper bound of interval.
n	Number of points used in Newton-Cotes formula.

Returns: vector <double> Array of y values to be used in numerical integration

Exceptions

Isis::IException::Programmer	(When a > b) "Invalid interval entered"
Isis::IException::Programmer	"Invalid arguments. Interval entered is not contained within domain"
Isis::IException::Programmer	"Unable to evaluate the data set for integration"

References _FILEINFO_, and Isis::IException::errorType().

◆ EvaluatePolynomialNeville()

double Isis::NumericalApproximation::EvaluatePolynomialNeville ( const double a )

protected

Performs polynomial interpolation using Neville's algorithm.

This is a protected method called by Evaluate() if the NumericalApproximation::InterpType is PolynomialNeville. It uses Neville's algorithm for Lagrange polynomials. It returns a value f(a) and sets the error estimate p_polyNevError. After this method is called, the user may access this error estimate by calling PolynomialNevilleErrorEstimate().

Note: If the given value, a, falls outside of the domain, then extrapolation is attempted and the result is accurate only if a is near enough to the domain boundary.

Parameters

a	Domain value from which to interpolate a corresponding range value

Returns: double Value returned from interpolation or extrapolation

See also: http://mathworld.wolfram.com/NevillesAlgorithm.html; Evaluate(); PolynomialNevilleErrorEstimate()

◆ FindIntervalLowerIndex()

int Isis::NumericalApproximation::FindIntervalLowerIndex ( const double a )

protected

Find the index of the x-value in the data set that is just below the input value, a.

This method is used by EvaluateCubicHermite(), EvaluateCubFirstDeriv() and EvaluateCubicHermiteSecDeriv() to determine in which interval of x-values a lies. It returns the index of the lower endpoint of the interval. If a is below the domain minimum, the method returns 0 as the lower index. If a is above the domain maximum, it returns the second to last index of the data set, Size()-2, as the lower index.

Parameters

a	Domain value around which the interval lies

Returns: int Index of the x-value that is the lower endpoint of the interval of data points that surrounds a. Returns 0 if a is below domain min and Size()-2 if a is above domain max.

◆ ForwardFirstDifference()

double Isis::NumericalApproximation::ForwardFirstDifference	(	const double	a,
		const unsigned int	n = `3`,
		const double	h = `0.1`
	)

Uses an n point forward first difference formula to approximate the first derivative evaluated at a given domain value.

This method uses forward first difference formulas to return an approximation of the first derivative of the interpolated data set function evaluated at given a valid domain value, a. Forward difference formulas use n points, with the smallest x value at a, for numerical differentiation approximation. This method uses one of the following formulas:

2-point forward difference.
\[ f\prime(a) \approx \frac{1}{h}[f(a + h) - f(a)] \]
3-point forward difference.
\[ f\prime(a) \approx \frac{1}{2h}[-f(a + 2h) + 4f(a + h) - 3f(a)] \]

Parameters

a	Domain value at which first deriviative is evaluated.
n	The number of points used in the formula.
h	Distance between nearest points in the formula.

Returns: double First derivative approximation for the given domain value

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain."
Isis::IException::Programmer	"Formula steps outside of domain." If a+(n-1)h is greater than domain max.
Isis::IException::Programmer	"Invalid argument. There is no n-point forward difference formula in use."
Isis::IException::Programmer	"Unable to calculate forward first difference for (a,n,h)"

References _FILEINFO_, and Isis::IException::errorType().

◆ ForwardSecondDifference()

double Isis::NumericalApproximation::ForwardSecondDifference	(	const double	a,
		const unsigned int	n = `3`,
		const double	h = `0.1`
	)

Uses an n point forward second difference formula to approximate the second derivative evaluated at a given domain value.

This method uses forward second difference formulas to return an approximation of the second derivative of the interpolated data set function evaluated at given a valid domain value, a. Forward second difference formulas use n points, with the smallest x value at a, for numerical differentiation approximation. This method uses the following formula:

3-point forward second difference.
\[ f\prime\prime(a) \approx \frac{1}{h^2}[f(a + 2h) - 2f(a + h) + f(a)] \]

Parameters

a	Domain value at which second deriviative is evaluated.
n	The number of points used in the formula.
h	Distance between nearest points in the formula.

Returns: double Second derivative approximation for the given domain value

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain."
Isis::IException::Programmer	"Formula steps outside of domain." If a+(n-1)h is greater than domain max.
Isis::IException::Programmer	"Invalid argument. There is no n-point forward difference formula in use."
Isis::IException::Programmer	"Unable to calculate forward second difference for (a,n,h)"

References _FILEINFO_, and Isis::IException::errorType().

◆ GslAllocation()

void Isis::NumericalApproximation::GslAllocation ( unsigned int npoints )

protected

Allocates GSL interpolation functions.

This method is called within Compute to allocate pointers to a GSL interpolation object and to a GSL accelerator object (for interpolation look-ups). This is only called for the GSL interpolation types. If it is deemed invalid, an exception will be thrown.

Parameters

npoints Number of points to allocate for the GSL interpolator

◆ GslComputed()

bool Isis::NumericalApproximation::GslComputed ( ) const

protected

Returns whether a GSL interpolation computation of the data set has been performed.

This method is only applicable to GSL interpolation types.

Returns: bool True if GSL interpolation has been computed.

Exceptions

Isis::IException::Programmer "Method only valid for GSL interpolation types"

See also: ComputeGsl()

References _FILEINFO_.

◆ GslDeallocation()

void Isis::NumericalApproximation::GslDeallocation ( )

protected

Deallocate GSL interpolator resources, if used.

If a GSL interpolator function has been allocated, this routine will free its resources and reset internal pointers to reflect this state and provide a mechanism to test its state. Otherwise, this method sets the internal GSL pointers to their default, 0.

◆ GslFirstDerivative()

double Isis::NumericalApproximation::GslFirstDerivative ( const double a )

Approximates the first derivative of the data set function evaluated at the given domain value for GSL supported interpolation types.

This method returns an approximation of the first derivative evaluated at given a valid domain value, a. It is a wrapper for the GSL subroutine gsl_spline_eval_deriv_e(). If the NumericalApproximation::InterpType is not a GSL type, then this method throws an error. No documentation was found concerning the algorithm used by this method.

Parameters

a	Domain value at which first deriviative is evaluated.

Returns: double First derivative approximation for the given domain value, if valid.

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain."
Isis::IException::Programmer	"Cannot use this method for interpolation type" (If the interpolation type is not GSL)
Isis::IException::Programmer	"Unable to compute the first derivative at a using the GSL interpolation" @throws Isis::IException::Programmer "Unable to compute the first derivative at a. GSL integrity check failed"

References _FILEINFO_, and Isis::IException::errorType().

◆ GslFunctor()

NumericalApproximation::InterpFunctor Isis::NumericalApproximation::GslFunctor ( NumericalApproximation::InterpType itype ) const

protected

Search for a GSL interpolation function.

This method searches the supported GSL options table for a given interpolation function as requested by the caller. If it is not found, an exception will be thrown indicating the erroneous request.

Parameters

itype Type of GSL interpolator to find

Returns: NumericalApproximation::InterpFunctor Pointer to the GSL spline interpolator construct.

Exceptions

Isis::IException::Programmer "Invalid argument. Unable to find GSL interpolator"

References _FILEINFO_.

Referenced by MinPoints().

◆ GslIntegral()

double Isis::NumericalApproximation::GslIntegral	(	const double	a,
		const double	b
	)

Approximates the integral of the data set function evaluated on the given interval for GSL supported interpolation types.

This method returns an approximation of the integral evaluated on a given valid domain interval, (a,b). It is a wrapper for the GSL subroutine gsl_spline_eval_integ_e(). If the NumericalApproximation::InterpType is not a GSL type, then this method throws an error. No documentation was found concerning the algorithm used by this method.

Parameters

a	Lower endpoint at which integral is evaluated.
b	Upper endpoint at which integral is evaluated.

Returns: double Integral approximation for the given interval, if valid.

Exceptions

Isis::IException::Programmer	(When a > b) "Invalid interval entered"
Isis::IException::Programmer	"Invalid arguments. Interval entered is not contained within domain"
Isis::IException::Programmer	"Cannot use this method for interpolation type" (If the interpolation type is not GSL)
Isis::IException::Programmer	"Unable to compute the integral on the interval (a,b) using GSL interpolation" @throws Isis::IException::Programmer "Unable to compute the integral on the interval (a,b). GSL integrity check failed."

References _FILEINFO_, and Isis::IException::errorType().

◆ GslIntegrityCheck()

void Isis::NumericalApproximation::GslIntegrityCheck	(	int	gsl_status,
		const char *	src,
		int	line
	)

protected

Checks the status of the GSL interpolation operations.

This method takes a return status from a GSL call and determines if it is completed successfully. This implementation currently allows the GSL_DOMAIN error to propagate through sucessfully as the domain can be checked by the caller if they deem this necessary.

Parameters

gsl_status	Return status of a GSL function call
src	Name of the calling source invoking the check. This allows more precise determination of the error.
line	Line of the calling source that invoked the check

Exceptions

Isis::IException::Programmer "GSL error occured"

◆ GslInterpType()

bool Isis::NumericalApproximation::GslInterpType ( NumericalApproximation::InterpType itype ) const

protected

Returns whether an interpolation type is adapted from the GSL library.

GSL interpolation types include the following:

Linear = 0
Polynomial = 1
CubicNatural = 3
CubicNatPeriodic = 5
Akima = 7
AkimaPeriodic = 8

Parameters

itype Interpolation type to be compared to GSL list

Returns: bool True if interpolation type is GSL

◆ GslSecondDerivative()

double Isis::NumericalApproximation::GslSecondDerivative ( const double a )

Approximates the second derivative of the interpolated data set function evaluated at the given domain value for GSL supported interpolation types.

This method returns an approximation of the second derivative evaluated at given a valid domain value, a. It is a wrapper for the GSL subroutine gsl_spline_eval_deriv2_e(). If the NumericalApproximation::InterpType is not a GSL type, then this method throws an error. No documentation was found concerning the algorithm used by this method.

Parameters

a	Domain value at which second deriviative is evaluated.

Returns: double Second derivative approximation for the given domain value, if valid.

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain."
Isis::IException::Programmer	"Cannot use this method for interpolation type" (If the interpolation type is not GSL)
Isis::IException::Programmer	"Unable to compute the second derivative at a using GSL interpolation"
Isis::IException::Programmer	"Unable to compute the second derivative at a. GSL integrity check failed"

References _FILEINFO_, and Isis::IException::errorType().

◆ Init()

void Isis::NumericalApproximation::Init ( NumericalApproximation::InterpType itype )

protected

Initializes the object upon instantiation.

This method sets up the initial state of the object, typically at instantiation. It populates the interpolation function table identifying which options are available to the users of this class.

GSL error handling is turned off - upon repeated instantiation of this object. The GSL default error handling, termination of the application via an abort() is unacceptable. This calls adopts an alternative policy provided by the GSL whereby error checking must be done by the calling environment. This has an unfortunate drawback in that it is not enforceable in an object oriented environment that utilizes the GSL in disjoint classes.

Parameters

itype NumericalApproximation::InterpType enum value to be assigned to this object

Exceptions

Isis::IException::Programmer "Unable to initialize NumericalApproximation object"

References _FILEINFO_, and Isis::IException::errorType().

◆ InsideDomain()

bool Isis::NumericalApproximation::InsideDomain ( const double a )

protected

Returns whether the passed value is greater than or equal to the domain minimum and less than or equal to the domain maximum.

Parameters

a	Value to be verified as valid domain value.

Returns: bool True if passed parameter is within the domain.

Exceptions

Isis::IException::Programmer "Unable to compute domain boundaries"

References _FILEINFO_, and Isis::IException::errorType().

◆ InterpolationType()

InterpType Isis::NumericalApproximation::InterpolationType ( )

inline

Returns the enumerated type of interpolation chosen.

This method can be selected after all the points are added in the interpolation by using Compute() method. Note that this prints out as an integer representaion of the enumerated type:

Linear = 0
Polynomial = 1
PolynomialNeville = 2
CubicNatural = 3
CubicClamped = 4
CubicNatPeriodic = 5
CubicNeighborhood = 6
CubicHermite = 7
Akima = 8
AkimaPeriodic = 9

Returns: NumericalApproximation::InterpType Currently assigned interpolation type

◆ MinPoints() [1/2]

int Isis::NumericalApproximation::MinPoints ( )

Minimum number of points required by interpolating function.

This method returns the minimum number of points that are required by the interpolating function in order for it to be applied to a data set. It returns the number of the of the currently selected/active interpolation function.

Returns: int Minimum number of data points required for the interpolation function

Exceptions

Isis::IException::Programmer "Unable to calculate minimum points."

References _FILEINFO_, and Isis::IException::errorType().

◆ MinPoints() [2/2]

int Isis::NumericalApproximation::MinPoints ( NumericalApproximation::InterpType itype )

Minimum number of points required by interpolating function.

This method returns the minimum number of points that are required by the interpolating function in order for it to be applied to a data set. It returns the minimum number for the specifed interpolation function as specified by the caller.

Parameters

itype Type of interpolation function for which to return minimum points

Returns: int Minimum number of data points required for the interpolation function

Exceptions

Isis::IException::Programmer	"Invalid argument. Unknown interpolation type"
Isis::IException::Programmer	"Unable to calculate minimum points"

References _FILEINFO_, Isis::IException::errorType(), and GslFunctor().

◆ Name()

string Isis::NumericalApproximation::Name ( ) const

Get name of interpolating function assigned to object.

This method returns the name of the interpolation type that is currently assigned to this object as a std::string. This may be called without adding any points. If called before computation, the result reflects the name of the function chosen at instantiation.

Returns: std::string Name of the interpolation function used/to be used.

Exceptions

Isis::IException::Programmer "Unable to retrieve numerical approximation name",

References _FILEINFO_, and Isis::IException::errorType().

◆ operator=()

NumericalApproximation & Isis::NumericalApproximation::operator= ( const NumericalApproximation & oldObject )

NumericalApproximation assigmment operator sets this object "equal to" another.

Assigning one object to the other requires a deallocation of the current state of this object and reinitialization of data from the right hand object. This may be a bit costly if the number of values is large.

Parameters

oldObject NumericalApproximation object to copy data from for initialization

Exceptions

Isis::IException::Programmer "Unable to copy NumericalApproximation object"

References _FILEINFO_, Isis::IException::errorType(), p_clampedDerivFirstPt, p_clampedDerivLastPt, p_clampedSecondDerivs, p_dataValidated, p_fprimeOfx, p_itype, p_polyNevError, p_x, and p_y.

◆ PolynomialNevilleErrorEstimate()

vector< double > Isis::NumericalApproximation::PolynomialNevilleErrorEstimate ( )

Retrieves the error estimate for the Neville's polynomial interpolation type.

This method must be called after the Evaluate() method has been invoked. If the Evaluate() method is passed a vector, this error will contain an error estimate for each of the values of the passed in vector. Otherwise, it will contain a single value.

Returns: vector <double> Estimation of difference between actual value of f(a) and the polynomial Neville interpolated return value of Evaluate(), where f is the function that defines the given data set.

Exceptions

Isis::IException::Programmer	"Method only used for polynomial-Neville's interpolation type"
Isis::IException::Programmer	"Error not calculated"

References _FILEINFO_.

Referenced by Isis::NumericalAtmosApprox::RombergsMethod(), and RombergsMethod().

◆ RefineExtendedTrap()

double Isis::NumericalApproximation::RefineExtendedTrap	(	double	a,
		double	b,
		double	s,
		unsigned int	n
	)

Calculates refinements extended trapezoidal rule to approximate the integral of the interpolated data set function on the interval (a,b).

This method calculates the n^th stage of refinement of an extended trapezoidal rule. When called with n = 1, the method returns the non-composite trapezoidal estimate of the integral. Subsequent calls with n = 2, 3, ... (in sequential order) will improve the accuracy by adding 2^n-2 additional interior points. This method can be used to integrate by the extended trapeziodal rule if you know the number of steps you want to take. For example, if you want 2^M + 1, use the following code:

double result;
for(int j = 1; j <= M+1; j++){
  result = RefineExtendedTrap(a,b,result,j);
}

Note: Although this method may be used to approximate an integral, as described above, it is most often used by RombergsMethod() to integrate.

Parameters

a	Lower limit of integration
b	Upper limit of integration
s	Previous value of refinement
n	Number of partitions to use when integrating

Returns: double Integral (refined) approximation of the function on the interval (a, b)

Exceptions

Isis::IException::Programmer "Unable to calculate the integral on the interval (a,b) using the extended trapezoidal rule"

See also: RombergsMethod()

References _FILEINFO_, and Isis::IException::errorType().

◆ ReportException()

void Isis::NumericalApproximation::ReportException	(	IException::ErrorType	type,
		const string &	methodName,
		const string &	message,
		const char *	filesrc,
		int	lineno
	)		const

protected

Generalized error report generator.

This method is used throughout this class to standardize error reporting as a convenience to its implementor.

Parameters

type	Type of Isis::IException
methodName	Name of method where exception originated
message	Error context string provided by the caller
filesrc	Name of the file the error occured in.
lineno	Line number of the calling source where the error occured.

Exceptions

Isis::IException::errType equal to type and error message equal to (methodName + " - " + message)

◆ Reset() [1/2]

void Isis::NumericalApproximation::Reset ( )

Resets the state of the object.

This method deallocates the internal state of the object and clears the data set and class variables. The object is returned to its original state, so data points must be entered before computing again. This does not clear or reset the interpolation type.

Referenced by Isis::HapkeAtm1::AtmosModelAlgorithm(), Isis::HapkeAtm2::AtmosModelAlgorithm(), EvaluateCubicNeighborhood(), Isis::AtmosModel::GenerateHahgTables(), Isis::AtmosModel::GenerateHahgTablesShadow(), Isis::LunarLambertEmpirical::LunarLambertEmpirical(), Isis::MinnaertEmpirical::MinnaertEmpirical(), Isis::NumericalAtmosApprox::RombergsMethod(), RombergsMethod(), Isis::LunarLambertEmpirical::~LunarLambertEmpirical(), and Isis::MinnaertEmpirical::~MinnaertEmpirical().

◆ Reset() [2/2]

void Isis::NumericalApproximation::Reset ( NumericalApproximation::InterpType itype )

Resets the state of the object and resets interpolation type.

This method will clear the data set, reset the validation status of the data to false, reset the interpolation type, clear class variables and deallocate (inactivate) the internal state of the object. The object is returned to its original state, so data points must be entered before computing again.

Parameters

itype NumericalApproximation::InterpType enum value to be assigned to this object

Exceptions

Isis::IException::Programmer "Unable to reset interpolation type"

References _FILEINFO_, and Isis::IException::errorType().

◆ RombergsMethod()

double Isis::NumericalApproximation::RombergsMethod	(	double	a,
		double	b
	)

Uses Romberg's method to approximate the integral of the interpolated data set function on the interval (a,b).

This method returns the integral of the function from a to b using Romberg's method for Numerical Integration of order 2K, where, e.g., K=2 is simpson's rule. This is a generalization of the trapezoidal rule. Romberg Integration uses a series of refinements on the extended (or composite) trapezoidal rule to reduce error terms. This method makes use of Neville's algorithm for polynomial interpolation to extrapolate successive refinements.

Parameters

a	Lower limit of integration
b	Upper limit of integration

Returns: double Integral approximation of the function on the interval (a, b)

Exceptions

Isis::IException::Programmer	"Failed to converge."
Isis::IException::Programmer	"Unable to calculate the integral on the interval (a,b) using Romberg's method"

See also: http://mathworld.wolfram.com/RombergIntegration.html; RefineExtendedTrap()

References _FILEINFO_, AddData(), Isis::IException::errorType(), Evaluate(), PolynomialNevilleErrorEstimate(), and Reset().

◆ SetCubicClampedEndptDeriv()

void Isis::NumericalApproximation::SetCubicClampedEndptDeriv	(	const double	yp1,
		const double	ypn
	)

Sets the values for the first derivatives of the endpoints of the data set.

This method can only be called for cubic splines with clamped boundary conditions, i.e. if NumericalApproximation::InterpType is CubicClamped.

Parameters

yp1	First derivative of the function evaluated at the domain minimum.
ypn	First derivative of the function evaluated at the domain maximum.

Exceptions

Isis::IException::Programmer "Method only used for cspline-clamped interpolation type"

References _FILEINFO_.

Referenced by Isis::LunarLambertEmpirical::LunarLambertEmpirical(), and Isis::MinnaertEmpirical::MinnaertEmpirical().

◆ SetInterpType()

void Isis::NumericalApproximation::SetInterpType ( NumericalApproximation::InterpType itype )

Sets interpolation type.

This method will reset the interpolation type to that of the input parameter and deallocate memory. Unlike the Reset() method, this method does NOT discard the data set. However, since interpolations have differing requirements for valid data sets, the data set stored is not yet validated for the new interpolation type. Other class variables are cleared if they are interpolation type dependent.

Parameters

itype NumericalApproximation::InterpType enum value to be assigned to this object

Exceptions

Isis::IException::Programmer	"Invalid argument. Unknown interpolation type"
Isis::IException::Programmer	"Unable to set interpolation type"

See also: Reset()

References _FILEINFO_, and Isis::IException::errorType().

Referenced by Isis::LunarLambertEmpirical::LunarLambertEmpirical(), and Isis::MinnaertEmpirical::MinnaertEmpirical().

◆ Simpsons3PointRule()

double Isis::NumericalApproximation::Simpsons3PointRule	(	const double	a,
		const double	b
	)

Uses Simpson's 3-point rule to approximate the integral of the interpolated data set function on the interval (a,b).

The Simpson's 3-Point Rule for numerical integration uses a 3-point Newton-Cote formula. This rule states:

\[ \int_{a}^b f(x)dx \approx \frac{h}{3}[f(a) + f(a+h) + 4*f(a+2h)] \]

where h = (b - a)/2. This method uses a composite, or extended, Simpson's rule algorithm to approximate the integral.

Parameters

a	Lower bound of interval.
b	Upper bound of interval.

Returns: double Approximate integral of data set function from a to b.

Exceptions

Isis::IException::Programmer "Unable to calculate the integral on the interval (a,b) using Simpson's 3 point rule"

References _FILEINFO_, and Isis::IException::errorType().

◆ Simpsons4PointRule()

double Isis::NumericalApproximation::Simpsons4PointRule	(	const double	a,
		const double	b
	)

Uses Simpson's 4-point rule to approximate the integral of the interpolated data set function on the interval (a,b).

The Simpson's 4-point Rule for numerical integration uses a 4-point Newton-Cote formula and is sometimes called the Simpson's 3/8 Rule. This rule states:

\[ \int_{a}^b f(x)dx \approx \frac{3h}{8}[f(a) + 3f(a+h) + 3f(a+2h) + f(a+3h)] \]

where h = (b - a)/3. This method uses a composite, or extended, Simpson's 3/8 rule algorithm to approximate the integral.

Parameters

a	Lower bound of interval.
b	Upper bound of interval.

Returns: double Approximate integral of data set function from a to b.

Exceptions

Isis::IException::Programmer "Unable to calculate the integral on the interval (a,b) using Simpson's 4 point rule"

References _FILEINFO_, and Isis::IException::errorType().

◆ Size()

unsigned int Isis::NumericalApproximation::Size ( )

inline

Returns the number of the coordinates added to the data set.

Returns: unsigned int Size of data set.

◆ TrapezoidalRule()

double Isis::NumericalApproximation::TrapezoidalRule	(	const double	a,
		const double	b
	)

Uses the trapezoidal rule to approximate the integral of the interpolated data set function on the interval (a, b).

The trapeziod rule for integration approximation uses a 2-point Newton-Cote formula. This rule states:

\[ \int_{a}^b f(x)dx \approx \frac{h}{2}[f(a) + f(b)] \]

where h = b - a. This method uses a composite, or extended, trapeziodal rule algorithm to approximate the integral.

Parameters

a	Lower bound of interval.
b	Upper bound of interval.

Returns: double Approximate integral of data set function from a to b.

Exceptions

Isis::IException::Programmer "Unable to calculate the integral on the interval (a,b) using the trapezoidal rule"

References _FILEINFO_, and Isis::IException::errorType().

◆ ValidateDataSet()

void Isis::NumericalApproximation::ValidateDataSet ( )

protected

Validates the data set before computing interpolation.

This method is called from the ComputeCubicClamped() and ComputeGsl() methods to verify that the data set contains the minimum number of required points and that the components of the vector of domain values are unique. For all interpolation types other than polynomial-Neville's, the method verifies that the vector of domain values are also sorted in ascending order. For CubicNatPeriodic interpolation type, this method verifies that the first and last p_y values are equal, i.e. \( f(x_0) = f(x_{n-1}) \).

Exceptions

Isis::IException::Programmer	"Interpolation requires a minimim of data points"
Isis::IException::Programmer	"Invalid data set, x-values must be unique"
Isis::IException::Programmer	"Invalid data set, x-values must be in ascending order" (if interpolation type is not polynomial-Neville's)
Isis::IException::Programmer	"First and last points of the data set must have the same y-value" (if interpolation type is cubic-periodic)

References _FILEINFO_.

◆ ValueToExtrapolate()

double Isis::NumericalApproximation::ValueToExtrapolate	(	const double	a,
		const ExtrapType &	etype
	)

protected

Returns the domain value at which to evaluate.

This protected method is called by Evaluate() if a falls outside of the domain of p_x values in the data set. The return value is determined by the NumericalApproximation::ExtrapType. If it is ThrowError, an error is thrown indicating that a is out of the domain. If it is NearestEndpoint, then the nearest domain boundary value is returned. Otherwise, i.e. Extrapolate, a is returned as long as the NumericalApproximation::InterpType is not GSL or cubic neighborhood.

Parameters

a	Value passed into Evaluate() that falls outside of domain.
etype	NumericalApproximation::ExtrapType enum value indicates how to evaluate if a falls outside of the domain. (Default: ThrowError)

Returns: double Value returned to Evaluate() to be extrapolated.

Exceptions

Isis::IException::Programmer	"Invalid argument. Value entered is outside of domain
Isis::IException::Programmer	"Invalid argument. Cannot extrapolate for interpolation type" (GSL or CubicNeighborhood)

See also: Evaluate()

References _FILEINFO_.

Member Data Documentation

◆ p_acc

gsl_interp_accel* Isis::NumericalApproximation::p_acc

protected

Lookup accelorator.

◆ p_clampedComputed

bool Isis::NumericalApproximation::p_clampedComputed

protected

Flag variable to determine whether ComputeCubicClamped() has been called.

Referenced by NumericalApproximation().

◆ p_clampedDerivFirstPt

double Isis::NumericalApproximation::p_clampedDerivFirstPt

protected

First derivative of first x-value, p_x[0]. This is only used for the CubicClamped interpolation type.

Referenced by NumericalApproximation(), and operator=().

◆ p_clampedDerivLastPt

double Isis::NumericalApproximation::p_clampedDerivLastPt

protected

First derivative of last x-value, p_x[n-1]. This is only used for the CubicClamped interpolation type.

Referenced by NumericalApproximation(), and operator=().

◆ p_clampedEndptsSet

bool Isis::NumericalApproximation::p_clampedEndptsSet

protected

Flag variable to determine whether SetCubicClampedEndptDeriv() has been called after all data was added for CubicClamped interpolation.

Referenced by NumericalApproximation().

◆ p_clampedSecondDerivs

vector<double> Isis::NumericalApproximation::p_clampedSecondDerivs

protected

List of second derivatives evaluated at p_x values. This is only used for the CubicClamped interpolation type.

Referenced by NumericalApproximation(), and operator=().

◆ p_dataValidated

bool Isis::NumericalApproximation::p_dataValidated

protected

Flag variable to determine whether ValidateDataSet() has been called.

Referenced by NumericalApproximation(), and operator=().

◆ p_fprimeOfx

vector<double> Isis::NumericalApproximation::p_fprimeOfx

protected

List of first derivatives corresponding to each x value in the data set (i.e. each value in p_x)

Referenced by NumericalApproximation(), and operator=().

◆ p_interp

gsl_spline* Isis::NumericalApproximation::p_interp

protected

Currently active interpolator.

◆ p_interpFunctors

NumericalApproximation::FunctorList Isis::NumericalApproximation::p_interpFunctors

staticprotected

Maintains list of interpolator options.

◆ p_itype

InterpType Isis::NumericalApproximation::p_itype

protected

Interpolation type.

Referenced by NumericalApproximation(), and operator=().

◆ p_polyNevError

vector<double> Isis::NumericalApproximation::p_polyNevError

protected

Estimate of error for interpolation evaluated at x. This is only used for the PolynomialNeville interpolation type. 91 taken from AtmosModel.

Referenced by NumericalApproximation(), and operator=().

◆ p_x

vector<double> Isis::NumericalApproximation::p_x

protected

List of X values.

Referenced by NumericalApproximation(), and operator=().

◆ p_y

vector<double> Isis::NumericalApproximation::p_y

protected

List of Y values.

Referenced by NumericalApproximation(), and operator=().

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

/usgs/pkgs/isis3.9.0_docs/ISIS3/isis/src/base/objs/NumericalApproximation/NumericalApproximation.h
/usgs/pkgs/isis3.9.0_docs/ISIS3/isis/src/base/objs/NumericalApproximation/NumericalApproximation.cpp

Public Types

Public Member Functions

Protected Types

Protected Member Functions

Protected Attributes

Static Protected Attributes

Detailed Description

Caveats

Member Typedef Documentation

◆ FunctorConstIter

◆ FunctorList

◆ InterpFunctor

Member Enumeration Documentation

◆ ExtrapType

◆ InterpType

Constructor & Destructor Documentation

◆ NumericalApproximation() [1/4]

◆ NumericalApproximation() [2/4]

◆ NumericalApproximation() [3/4]

◆ NumericalApproximation() [4/4]

◆ ~NumericalApproximation()

Member Function Documentation

◆ AddCubicHermiteDeriv() [1/3]

◆ AddCubicHermiteDeriv() [2/3]

◆ AddCubicHermiteDeriv() [3/3]

◆ AddData() [1/3]

◆ AddData() [2/3]

◆ AddData() [3/3]

◆ BackwardFirstDifference()

◆ BackwardSecondDifference()

◆ BoolesRule()

◆ CenterFirstDifference()

◆ CenterSecondDifference()

◆ ComputeCubicClamped()

◆ ComputeGsl()

◆ Contains()

◆ CubicClampedSecondDerivatives()

◆ DomainMaximum()

◆ DomainMinimum()

◆ Evaluate() [1/2]

◆ Evaluate() [2/2]

◆ EvaluateCubicClamped()

◆ EvaluateCubicHermite()

◆ EvaluateCubicHermiteFirstDeriv()

◆ EvaluateCubicHermiteSecDeriv()

◆ EvaluateCubicNeighborhood() [1/2]

◆ EvaluateCubicNeighborhood() [2/2]

◆ EvaluateForIntegration()

◆ EvaluatePolynomialNeville()

◆ FindIntervalLowerIndex()

◆ ForwardFirstDifference()

◆ ForwardSecondDifference()

◆ GslAllocation()

◆ GslComputed()

◆ GslDeallocation()

◆ GslFirstDerivative()

◆ GslFunctor()

◆ GslIntegral()

◆ GslIntegrityCheck()

◆ GslInterpType()

◆ GslSecondDerivative()

◆ Init()

◆ InsideDomain()

◆ InterpolationType()

◆ MinPoints() [1/2]

◆ MinPoints() [2/2]

◆ Name()

◆ operator=()

◆ PolynomialNevilleErrorEstimate()

◆ RefineExtendedTrap()

◆ ReportException()

◆ Reset() [1/2]

◆ Reset() [2/2]

◆ RombergsMethod()

◆ SetCubicClampedEndptDeriv()

◆ SetInterpType()

◆ Simpsons3PointRule()

◆ Simpsons4PointRule()

◆ Size()

◆ TrapezoidalRule()