SPLINE
Interpolation and Approximation of Data
SPLINE,
a C++ code which
constructs and evaluates spline functions.
These spline functions are typically used to

interpolate data exactly at a set of points;

approximate data at many points, or over an interval.
The most common use of this software is for situations where
a set of (X,Y) data points is known, and it is desired to
determine a smooth function which passes exactly through
those points, and which can be evaluated everywhere.
Thus, it is possible to get a formula that allows you to
"connect the dots".
Of course, you could could just connect the dots with
straight lines, but that would look ugly, and if there really
is some function that explains your data, you'd expect it to
curve around rather than make sudden angular turns. The
functions in SPLINE offer a variety of choices for
slinky curves that will make pleasing interpolants of your data.
There are a variety of types of approximation curves
available, including:

least squares polynomials,

divided difference polynomials,

piecewise polynomials,

B splines,

Bernstein splines,

beta splines,

Bezier splines,

Hermite splines,

Overhauser (or CatmullRom) splines.
Also included are a set of routines that return the local "basis matrix",
which allows the evaluation of the spline in terms of local function
data.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
SPLINE 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
uses divided differences to interpolate data.
HERMITE_CUBIC,
a C++ code which
can compute the value, derivatives or integral of a Hermite cubic polynomial,
or manipulate an interpolating function made up of piecewise Hermite cubic
polynomials.
LAGRANGE_INTERP_1D,
a C++ code which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of data, so that p(x(i)) = y(i).
spline_test
TEST_APPROX,
a C++ code which
defines test problems for approximation,
provided as a set of (x,y) data.
TEST_INTERP_1D,
a C++ code which
defines test problems for interpolation of data y(x),
depending on a 1D argument.
Reference:

JA Brewer, DC Anderson,
Visual Interaction with Overhauser Curves and Surfaces,
SIGGRAPH 77,
in Proceedings of the 4th Annual Conference on Computer Graphics
and Interactive Techniques,
ASME, July 1977, pages 132137.

Edwin Catmull, Raphael Rom,
A Class of Local Interpolating Splines,
in Computer Aided Geometric Design,
edited by Robert Barnhill, Richard Reisenfeld,
Academic Press, 1974,
ISBN: 0120790505.

Samuel Conte, Carl deBoor,
Elementary Numerical Analysis,
Second Edition,
McGraw Hill, 1972,
ISBN: 070124464.

Alan Davies, Philip Samuels,
An Introduction to Computational Geometry for Curves and Surfaces,
Clarendon Press, 1996,
ISBN: 0198514786,
LC: QA448.D38.

Carl deBoor,
A Practical Guide to Splines,
Springer, 2001,
ISBN: 0387953663.

Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
LINPACK User's Guide,
SIAM, 1979,
ISBN13: 9780898711721.

Gisela EngelnMuellges, Frank Uhlig,
Numerical Algorithms with C,
Springer, 1996,
ISBN: 3540605304.

James Foley, Andries vanDam, Steven Feiner, John Hughes,
Computer Graphics, Principles and Practice,
Second Edition,
Addison Wesley, 1995,
ISBN: 0201848406,
LC: T385.C5735.

Fred Fritsch, Judy Butland,
A Method for Constructing Local Monotone Piecewise
Cubic Interpolants,
SIAM Journal on Scientific and Statistical Computing,
Volume 5, Number 2, 1984, pages 300304.

Fred Fritsch, Ralph Carlson,
Monotone Piecewise Cubic Interpolation,
SIAM Journal on Numerical Analysis,
Volume 17, Number 2, April 1980, pages 238246.

David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0136272584,
LC: TA345.K34.

David Rogers, Alan Adams,
Mathematical Elements of Computer Graphics,
Second Edition,
McGraw Hill, 1989,
ISBN: 0070535299.
Source Code:
Last revised on 17 April 2020.