test_approx

test_approx, a C++ code which provides sets of test data for approximation algorithms.

The code contains a number of vectors of data (X(1:N),Y(1:N)) for which no underlying functional relationship is given.

The task of interpolation software is to find, from some given class of functions, the function G(X) which exactly matches the known data values. That is, G(X(1:N)) = Y(1:N).

The task of approximation software is to find, from some given class of functions, the function H(X) for which some approximation error is minimized. There are many forms of error measurement. For instance, the error might simply be the sum of the differences of the function and the data at the data abscissas:

l1(X) = sum ( 1 <= I <= N ) abs ( H(X(I)) - Y(I) )

l2(X) = sqrt ( sum ( 1 <= I <= N ) ( H(X(I)) - Y(I) )^2 )

l_inf(X) = max ( abs ( H(X(I)) - Y(I) ) )

L1(X,A,B) = integral ( A <= X <= B ) abs ( H(X) - F(X) ) dx

The problems available include:

1. p01_data.png: DeBoor example, Mars position data
2. p02_data.png: DeBoor example, roughly linear data
3. p03_data.png: The pulse data, 0 0 0 0 0 1 0 0 0 0 0;
4. p04_data.png: The jump data, 0 0 0 0 0 1/2 1 1 1 1 1;
5. p05_data.png: DeBoor's Titanium Property data;
6. p06_data.png: The Sawtooth data;
7. p07_data.png: Concavity test data;
8. p08_data.png: Extrapolation test data;
9. p09_data.png: Sunspot data, 1700-1960;
10. p10_data.png: 100 samples of y=2+5x+10*N(0,1), where N(0,1) is a random normal value with 0 mean and unit variance;

Licensing:

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

Languages:

test_approx is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a C++ code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV, a C++ code which computes the Chebyshev interpolant/approximant to a given function over an interval.

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

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

TEST_APPROX, a dataset directory which contains sets of data (x,y) for which an approximating formula is desired.

test_approx, a C++ code which defines test problems for approximation, provided as a set of (x,y) data.

test_approx_test

TEST_INTERP_1D, a C++ code which defines test problems for interpolation of data y(x), depending on a 1D argument.

Reference:

1. Samuel Conte, Carl deBoor,
   Elementary Numerical Analysis,
   Second Edition,
   McGraw Hill, 1972,
   ISBN: 07-012446-4,
   LC: QA297.C65.
2. Carl deBoor,
   A Practical Guide to Splines,
   Springer, 2001,
   ISBN: 0387953663,
   LC: QA1.A647.v27.
3. Max Waldmeier,
   The Sunspot-Activity in the Years 1610-1960,
   Shulthess, 1961,
   LC: QB525.W34.

Source Code:

• test_approx.cpp, the source code;
• test_approx.sh, compiles the source code;
• test_approx.hpp, the include file;

• test_approx_test.cpp, the calling program;
• test_approx_test.sh, runs all the tests.
• test_approx_test.txt, the sample output.

TEST07 creates data files for an Overhauser spline interpolant to data set 7.

• test07_data.txt the data.
• test07_approx.txt the interpolant values.
• test07.png a PNG image of the data and interpolant.

TEST08 creates data files for a cubic spline interpolant to data set 7.

• test08_data.txt the data.
• test08_approx.txt the interpolant values.
• test08.png a PNG image of the data and interpolant.

TEST10 creates data files for a B-spline approximant to data set 7.

• test10_data.txt the data.
• test10_approx.txt the approximant values.
• test10.png a PNG image of the data and approximant.

TEST11 creates data files for a beta spline approximant to data set 7:

• test11_data.txt the data.
• test11_approx.txt the approximant values.
• test11.png a PNG image of the data and approximant.

TEST12 creates data files for a Bernstein approximant to data set 5.

• test12_data.txt the data.
• test12_approx.txt the interpolant values.
• test12.png a PNG image of the data and approximant.

TEST13 creates data files for a cubic spline interpolant to data set 5.

• test13_data.txt the data.
• test13_approx.txt the interpolant values.
• test13.png a PNG image of the data and interpolant.