# test_interp

test_interp, a FORTRAN90 code which defines data that may be used to test interpolation algorithms.

The following sets of data are available:

1. p01_plot.png, 18 data points, 2 dimensions. This example is due to Hans-Joerg Wenz. It is an example of good data, which is dense enough in areas where the expected curvature of the interpolant is large. Good results can be expected with almost any reasonable interpolation method.
2. p02_plot.png, 18 data points, 2 dimensions. This example is due to ETY Lee of Boeing. Data near the corners is more dense than in regions of small curvature. A local interpolation method will produce a more plausible interpolant than a nonlocal interpolation method, such as cubic splines.
3. p03_plot.png, 11 data points, 2 dimensions. This example is due to Fred Fritsch and Ralph Carlson. This data can cause problems for interpolation methods. There are sudden changes in direction, and at the same time, sparsely-placed data. This can cause an interpolant to overshoot the data in a way that seems implausible.
4. p04_plot.png, 8 data points, 2 dimensions. This example is due to Larry Irvine, Samuel Marin and Philip Smith. This data can cause problems for interpolation methods. There are sudden changes in direction, and at the same time, sparsely-placed data. This can cause an interpolant to overshoot the data in a way that seems implausible.
5. p05_plot.png, 9 data points, 2 dimensions. This example is due to Larry Irvine, Samuel Marin and Philip Smith. This data can cause problems for interpolation methods. There are sudden changes in direction, and at the same time, sparsely-placed data. This can cause an interpolant to overshoot the data in a way that seems implausible.
6. p06_plot.png, 49 data points, 2 dimensions. The data is due to deBoor and Rice. The data represents a temperature dependent property of titanium. The data has been used extensively as an example in spline approximation with variably-spaced knots. DeBoor considers two sets of knots: (595,675,755,835,915,995,1075) and (595,725,850,910,975,1040,1075).
7. p07_plot.png, 4 data points, 2 dimensions. The data is a simple symmetric set of 4 points, for which it is interesting to develop the Shepard interpolants for varying values of the exponent p.
8. p08_plot.png, 12 data points, 2 dimensions. This is equally spaced data for y = x^2, except for one extra point whose x value is close to another, but whose y value is not so close. A small disagreement in nearby data can become a disaster.

The code requires access to a compiled copy of the R8LIB code.

### Languages:

test_interp is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

DIVDIF, a FORTRAN90 code which includes many routines to construct and evaluate divided difference interpolants.

HERMITE, a FORTRAN90 code which computes the Hermite interpolant, a polynomial that matches function values and derivatives.

INTERP, a FORTRAN90 code which can compute interpolants to data.

INTERPOLATION, a dataset directory which contains datasets to be interpolated.

LAGRANGE_INTERP_1D, a FORTRAN90 code which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).

NEAREST_INTERP_1D, a FORTRAN90 code which interpolates a set of data using a piecewise constant interpolant defined by the nearest neighbor criterion.

PWL_INTERP_1D, a FORTRAN90 code which interpolates a set of data using a piecewise linear function.

PPPACK, a FORTRAN90 code which implements Carl de Boor's piecewise polynomial functions, including, particularly, cubic splines.

R8LIB, a FORTRAN90 code which contains many utility routines using double precision real (R8) arithmetic.

RBF_INTERP, a FORTRAN90 code which defines and evaluates radial basis interpolants to multidimensional data.

SPLINE, a FORTRAN90 code which includes many routines to construct and evaluate spline interpolants and approximants.

TEST_APPROX, a FORTRAN90 code which defines tests for approximation and interpolation algorithms.

VANDERMONDE_INTERP_1D, a FORTRAN90 code which finds a polynomial interpolant to data y(x) of a 1D argument, by setting up and solving a linear system for the polynomial coefficients, involving the Vandermonde matrix.

### Reference:

1. Carl DeBoor, John Rice,
Least-squares cubic spline approximation II - variable knots.
Technical Report CSD TR 21,
Purdue University, Lafayette, Indiana, 1968.
2. Carl DeBoor,
A Practical Guide to Splines,
Springer, 2001,
ISBN: 0387953663,
LC: QA1.A647.v27.
3. Fred Fritsch, Ralph Carlson,
Monotone Piecewise Cubic Interpolation,
SIAM Journal on Numerical Analysis,
Volume 17, Number 2, April 1980, pages 238-246.
4. Larry Irvine, Samuel Marin, Philip Smith,
Constrained Interpolation and Smoothing,
Constructive Approximation,
Volume 2, Number 1, December 1986, pages 129-151.
5. ETY Lee,
Choosing Nodes in Parametric Curve Interpolation,
Computer-Aided Design,
Volume 21, Number 6, July/August 1989, pages 363-370.
6. Hans-Joerg Wenz,
Interpolation of Curve Data by Blended Generalized Circles,
Computer Aided Geometric Design,
Volume 13, Number 8, November 1996, pages 673-680.

### Source Code:

Last revised on 08 June 2020.