Construct an interpolating polynomial for the squares:

Check the result:

Construct an interpolating polynomial through three points:

Check the result at a single point:

Make the polynomial have derivative 0 when it has value 8:

Interpolate values depending on 2 variables:

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:

Construct a polynomial with roots a, b, and c:

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:

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:

Choose some points to be interpolated:

Use VandermondeMatrix in LinearSolve to obtain the coefficients of the interpolating polynomial:

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: