# 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 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 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 . »

# Examples

open allclose all

## Basic Examples(2)

Construct an interpolating polynomial for the squares:

 In[1]:=
 Out[1]=

Check the result:

 In[2]:=
 Out[2]=

Construct an interpolating polynomial through three points:

 In[1]:=
 Out[1]=

Check the result at a single point:

 In[2]:=
 Out[2]=