InterpolatingPolynomial

InterpolatingPolynomial[{f1, f2, ...}, x]
constructs an interpolating polynomial in x which reproduces the function values at successive integer values 1, 2, ... of .

InterpolatingPolynomial[{{x1, f1}, {x2, f2}, ...}, x]
constructs an interpolating polynomial for the function values corresponding to values .

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 OptionsDetails and Options

  • The function values and sample points , 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 ^(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 . »
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