LAGRANGE_INTERP_1D
Polynomial Interpolation in 1D using Lagrange Polynomials

LAGRANGE_INTERP_1D is a MATLAB library which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).

LAGRANGE_INTERP_1D needs the R8LIB library. The test uses the TEST_INTERP_1D library.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

LAGRANGE_INTERP_1D is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BARYCENTRIC_INTERP_1D, a MATLAB library which defines and evaluates the barycentric Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i). The barycentric approach means that very high degree polynomials can safely be used.

CHEBYSHEV_INTERP_1D, a MATLAB library which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

DIVDIF, a MATLAB library which uses divided differences to compute the polynomial interpolant to a given set of data.

HERMITE, a MATLAB library which computes the Hermite interpolant, a polynomial that matches function values and derivatives.

LAGRANGE_APPROX_1D, a MATLAB library which defines and evaluates the Lagrange polynomial p(x) of degree m which approximates a set of nd data points (x(i),y(i)).

LAGRANGE_BASIS_DISPLAY, a MATLAB library which displays the basis functions associated with a given set of nodes used with the Lagrange interpolation scheme.

LAGRANGE_INTERP_2D, a MATLAB library which defines and evaluates the Lagrange polynomial p(x,y) which interpolates a set of data depending on a 2D argument that was evaluated on a product grid, so that p(x(i),y(j)) = z(i,j).

LAGRANGE_INTERP_ND, a MATLAB library which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data depending on a multidimensional argument x that was evaluated on a product grid, so that p(x(i)) = z(i).

NEAREST_INTERP_1D, a MATLAB library which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

PWL_INTERP_1D, a MATLAB library which interpolates a set of data using a piecewise linear interpolant.

R8LIB, a MATLAB library which contains many utility routines using double precision real (R8) arithmetic.

RBF_INTERP_1D, a MATLAB library which defines and evaluates radial basis function (RBF) interpolants to 1D data.

SHEPARD_INTERP_1D, a MATLAB library which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.

SPLINE, a MATLAB library which constructs and evaluates spline interpolants and approximants.

TEST_INTERP_1D, a MATLAB library which defines test problems for interpolation of data y(x), depending on a 2D argument.

VANDERMONDE_INTERP_1D, a MATLAB library which finds a polynomial interpolant to a function of 1D data by setting up and solving a linear system for the polynomial coefficients, involving the Vandermonde matrix.

Reference:

1. Kendall Atkinson,
   An Introduction to Numerical Analysis,
   Prentice Hall, 1989,
   ISBN: 0471624896,
   LC: QA297.A94.1989.
2. Philip Davis,
   Interpolation and Approximation,
   Dover, 1975,
   ISBN: 0-486-62495-1,
   LC: QA221.D33
3. David Kahaner, Cleve Moler, Steven Nash,
   Numerical Methods and Software,
   Prentice Hall, 1989,
   ISBN: 0-13-627258-4,
   LC: TA345.K34.

Source Code:

• lagrange_basis_1d.m, evaluates the Lagrange basis polynomials.
• lagrange_value_1d.m, evaluates the Lagrange interpolant.

Examples and Tests:

The test code requires the test_interp library for test01, and test_interp_1d for test02 and test03. If these libraries are available in a separate folder at the same "level", then a Matlab command such as "addpath ( '../test_interp')" will make them accessible for a run of the test program.

• lagrange_interp_1d_test.m, calls all the tests;
• lagrange_interp_1d_test_output.txt, the output file.
• lagrange_interp_1d_test02.m, samples a function at ND evenly-spaced points, plots the piecewise linear interpolant to this data, and the Lagrange polynomial interpolant.
• lagrange_interp_1d_test03.m, samples a function at ND Chebyshev-spaced points, plots the piecewise linear interpolant to this data, and the Lagrange polynomial interpolant.

lagrange_interp_1d_test02() plots the data and Lagrange interpolant for even spacing.

Images for Problem 1:

• p01_dataeven_04.png, a plot of the 4 node even data for problem p01;
• p01_lageven_04.png, a plot of the Lagrange/even 4 node polynomial interpolant for problem p01;
• p01_dataeven_08.png, a plot of the 8 node even data for problem p01;
• p01_lageven_08.png, a plot of the Lagrange/even 8 node polynomial interpolant for problem p01;
• p01_dataeven_16.png, a plot of the 16 node even data for problem p01;
• p01_lageven_16.png, a plot of the Lagrange/even 16 node polynomial interpolant for problem p01;
• p01_dataeven_32.png, a plot of the 32 node even data for problem p01;
• p01_lageven_32.png, a plot of the Lagrange/even 32 node polynomial interpolant for problem p01;
• p01_dataeven_64.png, a plot of the 64 node even data for problem p01;
• p01_lageven_64.png, a plot of the Lagrange/even 64 node polynomial interpolant for problem p01;

Images for Problem p02:

• p02_dataeven_04.png, a plot of the 4 node even data for problem p02;
• p02_lageven_04.png, a plot of the Lagrange/even 4 node polynomial interpolant for problem p02;
• p02_dataeven_08.png, a plot of the 8 node even data for problem p02;
• p02_lageven_08.png, a plot of the Lagrange/even 8 node polynomial interpolant for problem p02;
• p02_dataeven_16.png, a plot of the 16 node even data for problem p02;
• p02_lageven_16.png, a plot of the Lagrange/even 16 node polynomial interpolant for problem p02;
• p02_dataeven_32.png, a plot of the 32 node even data for problem p02;
• p02_lageven_32.png, a plot of the Lagrange/even 32 node polynomial interpolant for problem p02;
• p02_dataeven_64.png, a plot of the 64 node even data for problem p02;
• p02_lageven_64.png, a plot of the Lagrange/even 64 node polynomial interpolant for problem p02;

Images for problem p03:

• p03_dataeven_04.png, a plot of the 4 node even data for problem p03;
• p03_lageven_04.png, a plot of the Lagrange/even 4 node polynomial interpolant for problem p03;
• p03_dataeven_08.png, a plot of the 8 node even data for problem p03;
• p03_lageven_08.png, a plot of the Lagrange/even 8 node polynomial interpolant for problem p03;
• p03_dataeven_16.png, a plot of the 16 node even data for problem p03;
• p03_lageven_16.png, a plot of the Lagrange/even 16 node polynomial interpolant for problem p03;
• p03_dataeven_32.png, a plot of the 32 node even data for problem p03;
• p03_lageven_32.png, a plot of the Lagrange/even 32 node polynomial interpolant for problem p03;
• p03_dataeven_64.png, a plot of the 64 node even data for problem p03;
• p03_lageven_64.png, a plot of the Lagrange/even 64 node polynomial interpolant for problem p03;

Images for problem p04:

• p04_dataeven_04.png, a plot of the 4 node even data for problem p04;
• p04_lageven_04.png, a plot of the Lagrange/even 4 node polynomial interpolant for problem p04;
• p04_dataeven_08.png, a plot of the 8 node even data for problem p04;
• p04_lageven_08.png, a plot of the Lagrange/even 8 node polynomial interpolant for problem p04;
• p04_dataeven_16.png, a plot of the 16 node even data for problem p04;
• p04_lageven_16.png, a plot of the Lagrange/even 16 node polynomial interpolant for problem p04;
• p04_dataeven_32.png, a plot of the 32 node even data for problem p04;
• p04_lageven_32.png, a plot of the Lagrange/even 32 node polynomial interpolant for problem p04;
• p04_dataeven_64.png, a plot of the 64 node even data for problem p04;
• p04_lageven_64.png, a plot of the Lagrange/even 64 node polynomial interpolant for problem p04;

Images for problem p05:

• p05_dataeven_04.png, a plot of the 4 node even data for problem p05;
• p05_lageven_04.png, a plot of the Lagrange/even 4 node polynomial interpolant for problem p05;
• p05_dataeven_08.png, a plot of the 8 node even data for problem p05;
• p05_lageven_08.png, a plot of the Lagrange/even 8 node polynomial interpolant for problem p05;
• p05_dataeven_16.png, a plot of the 16 node even data for problem p05;
• p05_lageven_16.png, a plot of the Lagrange/even 16 node polynomial interpolant for problem p05;
• p05_dataeven_32.png, a plot of the 32 node even data for problem p05;
• p05_lageven_32.png, a plot of the Lagrange/even 32 node polynomial interpolant for problem p05;
• p05_dataeven_64.png, a plot of the 64 node even data for problem p05;
• p05_lageven_64.png, a plot of the Lagrange/even 64 node polynomial interpolant for problem p05;

Images for problem p06:

• p06_dataeven_04.png, a plot of the 4 node even data for problem p06;
• p06_lageven_04.png, a plot of the Lagrange/even 4 node polynomial interpolant for problem p06;
• p06_dataeven_08.png, a plot of the 8 node even data for problem p06;
• p06_lageven_08.png, a plot of the Lagrange/even 8 node polynomial interpolant for problem p06;
• p06_dataeven_16.png, a plot of the 16 node even data for problem p06;
• p06_lageven_16.png, a plot of the Lagrange/even 16 node polynomial interpolant for problem p06;
• p06_dataeven_32.png, a plot of the 32 node even data for problem p06;
• p06_lageven_32.png, a plot of the Lagrange/even 32 node polynomial interpolant for problem p06;
• p06_dataeven_64.png, a plot of the 64 node even data for problem p06;
• p06_lageven_64.png, a plot of the Lagrange/even 64 node polynomial interpolant for problem p06;

Images for problem p07:

• p07_dataeven_04.png, a plot of the 4 node even data for problem p07;
• p07_lageven_04.png, a plot of the Lagrange/even 4 node polynomial interpolant for problem p07;
• p07_dataeven_08.png, a plot of the 8 node even data for problem p07;
• p07_lageven_08.png, a plot of the Lagrange/even 8 node polynomial interpolant for problem p07;
• p07_dataeven_16.png, a plot of the 16 node even data for problem p07;
• p07_lageven_16.png, a plot of the Lagrange/even 16 node polynomial interpolant for problem p07;
• p07_dataeven_32.png, a plot of the 32 node even data for problem p07;
• p07_lageven_32.png, a plot of the Lagrange/even 32 node polynomial interpolant for problem p07;
• p07_dataeven_64.png, a plot of the 64 node even data for problem p07;
• p07_lageven_64.png, a plot of the Lagrange/even 64 node polynomial interpolant for problem p07;

Images for problem p08:

• p08_dataeven_04.png, a plot of the 4 node even data for problem p08;
• p08_lageven_04.png, a plot of the Lagrange/even 4 node polynomial interpolant for problem p08;
• p08_dataeven_08.png, a plot of the 8 node even data for problem p08;
• p08_lageven_08.png, a plot of the Lagrange/even 8 node polynomial interpolant for problem p08;
• p08_dataeven_16.png, a plot of the 16 node even data for problem p08;
• p08_lageven_16.png, a plot of the Lagrange/even 16 node polynomial interpolant for problem p08;
• p08_dataeven_32.png, a plot of the 32 node even data for problem p08;
• p08_lageven_32.png, a plot of the Lagrange/even 32 node polynomial interpolant for problem p08;
• p08_dataeven_64.png, a plot of the 64 node even data for problem p08;
• p08_lageven_64.png, a plot of the Lagrange/even 64 node polynomial interpolant for problem p08;

lagrange_interp_1d_test03() plots the data and Lagrange interpolant for Chebyshev spacing.

• p01_datacheby_04.png, a plot of the 4 node Chebyshev data for problem p01;
• p01_lagcheby_04.png, a plot of the Lagrange/Chebyshev 4 node polynomial interpolant for problem p01;
• p01_datacheby_08.png, a plot of the 8 node Chebyshev data for problem p01;
• p01_lagcheby_08.png, a plot of the Lagrange/Chebyshev 8 node polynomial interpolant for problem p01;
• p01_datacheby_16.png, a plot of the 16 node Chebyshev data for problem p01;
• p01_lagcheby_16.png, a plot of the Lagrange/Chebyshev 16 node polynomial interpolant for problem p01;
• p01_datacheby_32.png, a plot of the 32 node Chebyshev data for problem p01;
• p01_lagcheby_32.png, a plot of the Lagrange/Chebyshev 32 node polynomial interpolant for problem p01;
• p01_datacheby_64.png, a plot of the 64 node Chebyshev data for problem p01;
• p01_lagcheby_64.png, a plot of the Lagrange/Chebyshev 64 node polynomial interpolant for problem p01;

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