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InterpolatingPolynomial

InterpolatingPolynomial
constructs an interpolating polynomial in x which reproduces the function values at successive integer values 1, 2, ... of .
InterpolatingPolynomial
constructs an interpolating polynomial for the function values corresponding to values .
InterpolatingPolynomial
constructs a multidimensional interpolating polynomial in the variables x, y, ....
InterpolatingPolynomial
constructs an interpolating polynomial that reproduces derivatives as well as function values.
MORE INFORMATION
  • The function values and sample points , etc. can be arbitrary real or complex numbers, and in 1D can be arbitrary symbolic expressions.
  • 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.
  • 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. »
  • 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 . »
EXAMPLESCLOSE ALL
Basic Examples   (2)
Construct an interpolating polynomial for the squares:
Check the result:
Construct an interpolating polynomial through three points:
Check the result at a single point:
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:
In[1]:=
Click for copyable input
Out[1]=
Check the result at a single point:
In[2]:=
Click for copyable input
Out[2]=
Scope   (3)
Make the polynomial have derivative 0 when it has value 8:
Interpolate values depending on 2 variables:
Generalizations & Extensions   (3)
Make the polynomial have zero derivative at and without specifying the values there:
Specify some of the partial derivatives in 2 dimensions:
Interpolate values and a gradient in 3 variables:
Options   (1)
Find a polynomial interpolating the given points in arithmetic mod 47:
The polynomial takes on the specified values mod 47:
Applications   (5)
Construct a polynomial with roots , , and :
Newton-Cotes integration formulas with points:
Centered finite difference formula of order for approximating the first derivative:
Interpolate to find the characteristic polynomial of a matrix:
Create a tensor product interpolation:
Create an interpolating polynomial for each fixed value:
Show the interpolation curves in the direction:
Interpolate between the curves in the direction:
Show the interpolating surface and data points:
Properties & Relations   (2)
The interpolating polynomial always goes through the data points:
ListInterpolation creates a tensor product interpolation:
Create a numerical InterpolatingFunction object:
Create a symbolic polynomial by interpolating in each dimension separately:
Verify that results agree with random data points:
Possible Issues   (3)
Runge's function:
Sampling at evenly spaced intervals in the interval from to :
The polynomial that interpolates these points has large oscillations:
Interpolation uses a lower-order piecewise polynomial that does not have this problem:
When derivatives are specified without function values an interpolant may not be found:
There is no quadratic polynomial that satisfies the interpolation conditions:
Points with abscissas lying on a line in 2 dimensions:
In multiple dimensions an interpolant may not be found for some arrangements of points:
This polynomial interpolates the data above, but has total degree 2:
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