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Student[NumericalAnalysis]

  

InterpolantRemainderTerm

  

return the interpolating polynomial and remainder term from an interpolation structure

 

Calling Sequence

Parameters

Options

Description

Notes

Examples

Calling Sequence

InterpolantRemainderTerm(p, opts)

Parameters

p

-

a POLYINTERP structure

opts

-

(optional) equations of the form keyword=value where keyword is one of errorboundvar, independentvar, showapproximatepoly, showremainder; options for returning the interpolant and remainder term

Options

• 

errorboundvar = name

  

The name to assign to the independent variable in the remainder term. By default, the errorboundvar given when the POLYINTERP structure was created is used.

• 

independentvar = name

  

The name to assign to the independent variable in the approximated polynomial. By default, the independentvar given when the POLYINTERP structure was created is used.

• 

showapproximatepoly = true or false

  

Whether to return the approximated polynomial. By default this is set to true.

• 

showremainder = true or false

  

Whether to return the remainder term. By default, this is set to true.

Description

• 

The InterpolantRemainderTerm command returns the approximate polynomial and remainder term from a POLYINTERP structure.

• 

The interpolant and remainder term are returned in an expression sequence of the form Pn, Rn, where Pn is the interpolant and Rn is the remainder term.

• 

The POLYINTERP structure is created using the PolynomialInterpolation command or the CubicSpline command.

• 

If the POLYINTERP structure p was created using the CubicSpline command then the InterpolantRemainderTerm command can only return the approximate polynomial and therefore showremainder must be set to false.

• 

In order for the remainder term to exist, the POLYINTERP structure p must have an associated exact function that has been given.

Notes

• 

The remainder term is also called an error term.

• 

The interpolant is also called the approximating polynomial or interpolating polynomial.

Examples

withStudent[NumericalAnalysis]:

xy0,4.0,0.5,0,1.0,2.0,1.5,0,2.0,1.0,2.5,0,3.0,0.5

xy:=0,4.0,0.5,0,1.0,2.0,1.5,0,2.0,1.0,2.5,0,3.0,0.5

(1)

p1PolynomialInterpolationxy,function=22xcosπx,method=lagrange,extrapolate=0.25,0.75,1.25,errorboundvar='ξ':

InterpolantRemainderTermp1

0.3555555556x0.5x1.0x1.5x2.0x2.5x3.02.666666667xx0.5x1.5x2.0x2.5x3.0+1.333333333xx0.5x1.0x1.5x2.5x3.00.04444444444xx0.5x1.0x1.5x2.0x2.5,1504022ξln27cosπξ722ξln26πsinπξ+2122ξln25π2cosπξ+3522ξln24π3sinπξ3522ξln23π4cosπξ2122ξln22π5sinπξ+722ξln2π6cosπξ+22ξπ7sinπξxx0.5x1.0x1.5x2.0x2.5x3.0 &where 0.ξandξ3.0

(2)

See Also

Student[NumericalAnalysis]

Student[NumericalAnalysis][ComputationOverview]

Student[NumericalAnalysis][CubicSpline]

Student[NumericalAnalysis][Interpolant]

Student[NumericalAnalysis][PolynomialInterpolation]

Student[NumericalAnalysis][RemainderTerm]

Student[NumericalAnalysis][UpperBoundOfRemainderTerm]

 


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