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<html>
<head>
<title>
SPLINE - Interpolation and Approximation of Data
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPLINE <br> Interpolation and Approximation of Data
</h1>
<hr>
<p>
<b>SPLINE</b>
is a FORTRAN90 library which
defines and evaluates spline functions.
</p>
<p>
These spline functions are typically used to
<ul>
<li>
interpolate data exactly at a set of points;
</li>
<li>
approximate data at many points, or over an interval.
</li>
</ul>
</p>
<p>
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".
</p>
<p>
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 <b>SPLINE</b> offer a variety of choices for
slinky curves that will make pleasing interpolants of your data.
</p>
<p>
There are a variety of types of approximation curves
available, including:
<ul>
<li>
least squares polynomials,
</li>
<li>
divided difference polynomials,
</li>
<li>
piecewise polynomials,
</li>
<li>
B splines,
</li>
<li>
Bernstein splines,
</li>
<li>
beta splines,
</li>
<li>
Bezier splines,
</li>
<li>
Hermite splines,
</li>
<li>
Overhauser (or Catmull-Rom) splines.
</li>
</ul>
</p>
<p>
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.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPLINE</b> is available in
<a href = "../../c_src/spline/spline.html">a C version</a> and
<a href = "../../cpp_src/spline/spline.html">a C++ version</a> and
<a href = "../../f77_src/spline/spline.html">a FORTRAN77 version</a> and
<a href = "../../f_src/spline/spline.html">a FORTRAN90 version</a> and
<a href = "../../m_src/spline/spline.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/bernstein/bernstein.html">
BERNSTEIN</a>,
a FORTRAN90 library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../f_src/chebyshev/chebyshev.html">
CHEBYSHEV</a>,
a FORTRAN90 library which
computes the Chebyshev interpolant/approximant to a given function
over an interval.
</p>
<p>
<a href = "../../f_src/divdif/divdif.html">
DIVDIF</a>,
a FORTRAN90 library which
uses divided differences to interpolate data.
</p>
<p>
<a href = "../../f_src/hermite/hermite.html">
HERMITE</a>,
a FORTRAN90 library which
computes the Hermite interpolant, a polynomial that matches function values
and derivatives.
</p>
<p>
<a href = "../../f_src/hermite_cubic/hermite_cubic.html">
HERMITE_CUBIC</a>,
a FORTRAN90 library 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.
</p>
<p>
<a href = "../../f_src/interp/interp.html">
INTERP</a>,
a FORTRAN90 library which
can be used for parameterizing and interpolating data;
</p>
<p>
<a href = "../../f_src/lagrange_approx_1d/lagrange_approx_1d.html">
LAGRANGE_APPROX_1D</a>,
a FORTRAN90 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)).
</p>
<p>
<a href = "../../f_src/lagrange_interp_1d/lagrange_interp_1d.html">
LAGRANGE_INTERP_1D</a>,
a FORTRAN90 library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of data, so that p(x(i)) = y(i).
</p>
<p>
<a href = "../../f_src/nms/nms.html">
NMS</a>,
a FORTRAN90 library which
includes a wide variety of numerical software, including
solvers for linear systems of equations, interpolation of data,
numerical quadrature, linear least squares data fitting,
the solution of nonlinear equations, ordinary differential equations,
optimization and nonlinear least squares, simulation and random numbers,
trigonometric approximation and Fast Fourier Transforms.
</p>
<p>
<a href = "../../f_src/pppack/pppack.html">
PPPACK</a>,
a FORTRAN90 library which
implements piecewise polynomial functions,
including, in particular, cubic splines,
by Carl deBoor.
</p>
<p>
<a href = "../../f_src/test_approx/test_approx.html">
TEST_APPROX</a>,
a FORTRAN90 library which
defines a number of test problems for approximation and interpolation.
</p>
<p>
<a href = "../../f_src/test_interp_1d/test_interp_1d.html">
TEST_INTERP_1D</a>,
a FORTRAN90 library which
defines test problems for interpolation of data y(x),
depending on a 1D argument.
</p>
<p>
<a href = "../../f_src/toms446/toms446.html">
TOMS446</a>,
a FORTRAN90 library which
manipulates Chebyshev series for interpolation and approximation;<br>
this is a version of ACM TOMS algorithm 446,
by Roger Broucke.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
JA Brewer, DC Anderson,<br>
Visual Interaction with Overhauser Curves and Surfaces,<br>
SIGGRAPH 77,<br>
in Proceedings of the 4th Annual Conference on Computer Graphics
and Interactive Techniques,<br>
ASME, July 1977, pages 132-137.
</li>
<li>
Edwin Catmull, Raphael Rom,<br>
A Class of Local Interpolating Splines,<br>
in Computer Aided Geometric Design,<br>
edited by Robert Barnhill, Richard Reisenfeld,<br>
Academic Press, 1974,<br>
ISBN: 0120790505,<br>
LC: QA464.I58.
</li>
<li>
Samuel Conte, Carl deBoor,<br>
Elementary Numerical Analysis,<br>
Second Edition,<br>
McGraw Hill, 1972,<br>
ISBN: 07-012446-4,<br>
LC: QA297.C65.
</li>
<li>
Alan Davies, Philip Samuels,<br>
An Introduction to Computational Geometry for Curves and Surfaces,<br>
Clarendon Press, 1996,<br>
ISBN: 0-19-851478-6,<br>
LC: QA448.D38.
</li>
<li>
Carl deBoor,<br>
A Practical Guide to Splines,<br>
Springer, 2001,<br>
ISBN: 0387953663,<br>
LC: QA1.A647.v27.
</li>
<li>
Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,<br>
LINPACK User's Guide,<br>
SIAM, 1979,<br>
ISBN13: 978-0-898711-72-1,<br>
LC: QA214.L56.
</li>
<li>
Gisela Engeln-Muellges, Frank Uhlig,<br>
Numerical Algorithms with C,<br>
Springer, 1996,<br>
ISBN: 3-540-60530-4,<br>
LC: QA297.E56213.
</li>
<li>
James Foley, Andries vanDam, Steven Feiner, John Hughes,<br>
Computer Graphics, Principles and Practice,<br>
Second Edition,<br>
Addison Wesley, 1995,<br>
ISBN: 0201848406,<br>
LC: T385.C5735.
</li>
<li>
Fred Fritsch, Judy Butland,<br>
A Method for Constructing Local Monotone Piecewise
Cubic Interpolants,<br>
SIAM Journal on Scientific and Statistical Computing,<br>
Volume 5, Number 2, 1984, pages 300-304.
</li>
<li>
Fred Fritsch, Ralph Carlson,<br>
Monotone Piecewise Cubic Interpolation,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 17, Number 2, April 1980, pages 238-246.
</li>
<li>
David Kahaner, Cleve Moler, Steven Nash,<br>
Numerical Methods and Software,<br>
Prentice Hall, 1989,<br>
ISBN: 0-13-627258-4,<br>
LC: TA345.K34.
</li>
<li>
David Rogers, Alan Adams,<br>
Mathematical Elements for Computer Graphics,<br>
Second Edition,<br>
McGraw Hill, 1989,<br>
ISBN: 0070535299,<br>
LC: T385.R6.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "spline.f90">spline.f90</a>, the source code;
</li>
<li>
<a href = "spline.sh">spline.sh</a>, commands to compile
the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "spline_prb.f90">spline_prb.f90</a>, the test program;
</li>
<li>
<a href = "spline_prb.sh">spline_prb.sh</a>, commands
to compile, link and run the test program;
</li>
<li>
<a href = "spline_prb_output.txt">spline_prb_output.txt</a>, the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BASIS_FUNCTION_B_VAL</b> evaluates the B spline basis function.
</li>
<li>
<b>BASIS_FUNCTION_BETA_VAL</b> evaluates the beta spline basis function.
</li>
<li>
<b>BASIS_MATRIX_B_UNI</b> sets up the uniform B spline basis matrix.
</li>
<li>
<b>BASIS_MATRIX_BETA_UNI</b> sets up the uniform beta spline basis matrix.
</li>
<li>
<b>BASIS_MATRIX_BEZIER</b> sets up the cubic Bezier spline basis matrix.
</li>
<li>
<b>BASIS_MATRIX_HERMITE</b> sets up the Hermite spline basis matrix.
</li>
<li>
<b>BASIS_MATRIX_OVERHAUSER_NONUNI:</b> nonuniform Overhauser spline basis matrix.
</li>
<li>
<b>BASIS_MATRIX_OVERHAUSER_NUL</b> sets the nonuniform left Overhauser basis matrix.
</li>
<li>
<b>BASIS_MATRIX_OVERHAUSER_NUR:</b> the nonuniform right Overhauser basis matrix.
</li>
<li>
<b>BASIS_MATRIX_OVERHAUSER_UNI</b> sets the uniform Overhauser spline basis matrix.
</li>
<li>
<b>BASIS_MATRIX_OVERHAUSER_UNI_L</b> sets the left uniform Overhauser basis matrix.
</li>
<li>
<b>BASIS_MATRIX_OVERHAUSER_UNI_R</b> sets the right uniform Overhauser basis matrix.
</li>
<li>
<b>BASIS_MATRIX_TMP</b> computes Q = T * MBASIS * P
</li>
<li>
<b>BC_VAL</b> evaluates a parameterized N-th degree Bezier curve in 2D.
</li>
<li>
<b>BEZ_VAL</b> evaluates an N-th degree Bezier function at a point.
</li>
<li>
<b>BP_APPROX</b> evaluates the Bernstein polynomial approximant to F(X) on [A,B].
</li>
<li>
<b>BP01</b> evaluates the Bernstein basis polynomials for [0,1] at a point.
</li>
<li>
<b>BPAB</b> evaluates the Bernstein basis polynomials for [A,B] at a point.
</li>
<li>
<b>CHFEV</b> evaluates a cubic polynomial given in Hermite form.
</li>
<li>
<b>D3_MXV</b> multiplies a D3 matrix times a vector.
</li>
<li>
<b>D3_NP_FS</b> factors and solves an D3 system.
</li>
<li>
<b>D3_UNIFORM</b> randomizes a D3 matrix.
</li>
<li>
<b>DATA_TO_DIF</b> sets up a divided difference table from raw data.
</li>
<li>
<b>DIF_VAL</b> evaluates a divided difference polynomial at a point.
</li>
<li>
<b>LEAST_SET_OLD</b> constructs the least squares polynomial approximation to data.
</li>
<li>
<b>LEAST_VAL_OLD</b> evaluates a least squares polynomial defined by LEAST_SET_OLD.
</li>
<li>
<b>LEAST_SET</b> defines a least squares polynomial for given data.
</li>
<li>
<b>LEAST_VAL</b> evaluates a least squares polynomial defined by LEAST_SET.
</li>
<li>
<b>LEAST_VAL2</b> evaluates a least squares polynomial defined by LEAST_SET.
</li>
<li>
<b>PARABOLA_VAL2</b> evaluates a parabolic interpolant through tabular data.
</li>
<li>
<b>PCHST:</b> PCHIP sign-testing routine.
</li>
<li>
<b>R8_SWAP</b> swaps two real values.
</li>
<li>
<b>R8_UNIFORM_01</b> is a portable pseudorandom number generator.
</li>
<li>
<b>R8VEC_BRACKET</b> searches a sorted R8VEC for successive brackets of a value.
</li>
<li>
<b>R8VEC_BRACKET3</b> finds the interval containing or nearest a given value.
</li>
<li>
<b>R8VEC_DISTINCT</b> is true if the entries in an R8VEC are distinct.
</li>
<li>
<b>R8VEC_EVEN</b> returns N real values, evenly spaced between ALO and AHI.
</li>
<li>
<b>R8VEC_INDICATOR</b> sets an R8VEC to the indicator vector.
</li>
<li>
<b>R8VEC_ORDER_TYPE</b> determines the order type of an R8VEC.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_SORT_BUBBLE_A</b> ascending sorts an R8VEC using bubble sort.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> sets a double precision vector to unit pseudorandom numbers.
</li>
<li>
<b>R8VEC_UNIQUE_COUNT</b> counts the unique elements in an unsorted R8VEC.
</li>
<li>
<b>SPLINE_B_VAL</b> evaluates a cubic B spline approximant.
</li>
<li>
<b>SPLINE_BETA_VAL</b> evaluates a cubic beta spline approximant.
</li>
<li>
<b>SPLINE_BEZIER_VAL</b> evaluates a cubic Bezier spline.
</li>
<li>
<b>SPLINE_CONSTANT_VAL</b> evaluates a piecewise constant spline at a point.
</li>
<li>
<b>SPLINE_CUBIC_SET</b> computes the second derivatives of a piecewise cubic spline.
</li>
<li>
<b>SPLINE_CUBIC_VAL</b> evaluates a piecewise cubic spline at a point.
</li>
<li>
<b>SPLINE_CUBIC_VAL2</b> evaluates a piecewise cubic spline at a point.
</li>
<li>
<b>SPLINE_HERMITE_SET</b> sets up a piecewise cubic Hermite interpolant.
</li>
<li>
<b>SPLINE_HERMITE_VAL</b> evaluates a piecewise cubic Hermite interpolant.
</li>
<li>
<b>SPLINE_LINEAR_INT</b> evaluates the integral of a piecewise linear spline.
</li>
<li>
<b>SPLINE_LINEAR_INTSET</b> sets a piecewise linear spline with given integral properties.
</li>
<li>
<b>SPLINE_LINEAR_VAL</b> evaluates a piecewise linear spline at a point.
</li>
<li>
<b>SPLINE_OVERHAUSER_NONUNI_VAL</b> evaluates the nonuniform Overhauser spline.
</li>
<li>
<b>SPLINE_OVERHAUSER_UNI_VAL</b> evaluates the uniform Overhauser spline.
</li>
<li>
<b>SPLINE_OVERHAUSER_VAL</b> evaluates an Overhauser spline.
</li>
<li>
<b>SPLINE_PCHIP_SET</b> sets derivatives for a piecewise cubic Hermite interpolant.
</li>
<li>
<b>SPLINE_PCHIP_VAL</b> evaluates a piecewise cubic Hermite function.
</li>
<li>
<b>SPLINE_QUADRATIC_VAL</b> evaluates a piecewise quadratic spline at a point.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 29 January 2007.
</i>
<!-- John Burkardt -->
</body>
</html>