InterpolatingPolynomial

InterpolatingPolynomial[{f1,f2,},x]

constructs an interpolating polynomial in x which reproduces the function values f_(i) at successive integer values 1, 2, of x.

InterpolatingPolynomial[{{x1,f1},{x2,f2},},x]

constructs an interpolating polynomial for the function values f_(i) corresponding to x values x_(i).

InterpolatingPolynomial[{{{x1,y1,},f1},{{x2,y2,},f2},},{x,y,}]

constructs a multidimensional interpolating polynomial in the variables x, y, .

InterpolatingPolynomial[{{{x1,},f1,df1,},},{x,}]

constructs an interpolating polynomial that reproduces derivatives as well as function values.

Details and Options

  • The function values f_(i) and sample points x_(i), etc. can be arbitrary real or complex numbers, and in 1D can be arbitrary symbolic expressions.
  • With a 1D list of data of length , InterpolatingPolynomial gives a polynomial of degree .
  • With any given specified set of data, there are infinitely many possible interpolating polynomials; InterpolatingPolynomial always tries to find the one with lowest total degree.
  • InterpolatingPolynomial gives the interpolating polynomial in a Horner form, suitable for numerical evaluation.
  • Different elements in the data can have different numbers of derivatives specified.
  • For multidimensional data, the n^(th) derivative can be given as a tensor with a structure corresponding to D[f,{{x,y,},n}]. »
  • InterpolatingPolynomial allows any function value or derivative to be given as Automatic, in which case it will attempt to fill in the necessary information from derivatives or other function values. »
  • The option setting Modulus->n specifies that the interpolating polynomial should be found modulo n. »

Examples

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Basic Examples  (2)

Construct an interpolating polynomial for the squares:

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Check the result:

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Construct an interpolating polynomial through three points:

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Check the result at a single point:

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Scope  (3)

Generalizations & Extensions  (3)

Options  (1)

Applications  (5)

Properties & Relations  (2)

Possible Issues  (3)

See Also

Interpolation  FunctionInterpolation  Fit  Roots  InterpolatingFunction  FindSequenceFunction  FindGeneratingFunction

Tutorials

Introduced in 1991
(2.0)
| Updated in 2007
(6.0)