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<html>
<head>
<title>
HERMITE_CUBIC - Hermite Cubic Polynomial Evaluation, Interpolation, Integration, Splines
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
HERMITE_CUBIC <br> Hermite Cubic Polynomial Evaluation, Interpolation, Integration, Splines
</h1>
<hr>
<p>
<b>HERMITE_CUBIC</b>
is a FORTRAN90 library which
demonstrates the use of cubic polynomials in the Hermite form.
</p>
<h3 align = "center">
The Hermite Cubic
</h3>
<p>
The Hermite form of a cubic polynomial defines the polynomial <b>p(x)</b>
by specifying two distinct points <b>x1</b> and <b>x2</b>, and
providing values for the following four items:
<pre>
f1 = p(x1)
d1 = p'(x1)
f2 = p(x2)
d2 = p'(x2)
</pre>
The locations of the abscissas and the four data values are enough
to uniquely define a cubic polynomial, known as the Hermite cubic.
</p>
<p>
From the Hermite cubic representation, it is possible to determine
the standard power series form:
<pre>
p(x) = c<sub>0</sub> + c<sub>1</sub> * x + c<sub>2</sub> * x<sup>2</sup> + c<sub>3</sub> * x<sup>3</sup>
</pre>
</p>
<p>
It is possible, given any value of the argument <b>x</b> and the
data values that define the Hermite cubic polynomial, to determine the
value of <b>p(x)</b>, as well as the values of the first, second and third
derivatives.
</p>
<p>
It is possible, given two values of the argument <b>x3</b> and
<b>x4</b>, and the data values that define the Hermite cubic polynomial,
to determine the value of the integral of <b>p(x)</b> over the interval
[x3,x4].
</p>
<h3 align = "center">
Hermite Cubic Splines:
</h3>
<p>
A sequence of Hermite cubic polynomials can be used to produce a
piecewise cubic Hermite interpolant, if we are given a strictly
increasing sequence of <b>n</b> nodes <b>x(1:n)</b>, and corresponding
data vectors <b>f(1:n)</b> and <b>d(1:n)</b>. This is done by
defining <b>n-1</b> cubic Hermite polynomials, with the <b>i</b>-th
polynomial defined using the data at nodes <b>x(i)</b> and <b>x(i+1)</b>.
The resulting function interpolates the value and derivative data,
and is continuous and continuously differentiable everywhere,
and in particular, at the nodes.
</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>HERMITE_CUBIC</b> is available in
<a href = "../../c_src/hermite_cubic/hermite_cubic.html">a C version</a> and
<a href = "../../cpp_src/hermite_cubic/hermite_cubic.html">a C++ version</a> and
<a href = "../../f77_src/hermite_cubic/hermite_cubic.html">a FORTRAN77 version</a> and
<a href = "../../f_src/hermite_cubic/hermite_cubic.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hermite_cubic/hermite_cubic.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
computes divided difference polynomials from 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/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/spline/spline.html">
SPLINE</a>,
a FORTRAN90 library which
includes many routines to construct
and evaluate spline interpolants and approximants.
</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>
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>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "hermite_cubic.f90">hermite_cubic.f90</a>, the source code.
</li>
<li>
<a href = "hermite_cubic.sh">hermite_cubic.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "hermite_cubic_prb.f90">hermite_cubic_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "hermite_cubic_prb.sh">hermite_cubic_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "hermite_cubic_prb_output.txt">hermite_cubic_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>HERMITE_CUBIC_INTEGRAL</b> returns the integral of a Hermite cubic polynomial.
</li>
<li>
<b>HERMITE_CUBIC_INTEGRATE</b> integrates a Hermite cubic polynomial from A to B.
</li>
<li>
<b>HERMITE_CUBIC_LAGRANGE_INTEGRAL:</b> Hermite cubic Lagrange integrals.
</li>
<li>
<b>HERMITE_CUBIC_LAGRANGE_INTEGRATE</b> integrates Hermite cubic Lagrange polynomials.
</li>
<li>
<b>HERMITE_CUBIC_LAGRANGE_VALUE</b> evaluates the Hermite cubic Lagrange polynomials.
</li>
<li>
<b>HERMITE_CUBIC_SPLINE_INTEGRAL:</b> Hermite cubic spline integral.
</li>
<li>
<b>HERMITE_CUBIC_SPLINE_INTEGRATE</b> integrates a Hermite cubic spline over [A,B].
</li>
<li>
<b>HERMITE_CUBIC_SPLINE_VALUE</b> evaluates a Hermite cubic spline.
</li>
<li>
<b>HERMITE_CUBIC_TO_POWER_CUBIC</b> converts a Hermite cubic to power form.
</li>
<li>
<b>HERMITE_CUBIC_VALUE</b> evaluates a Hermite cubic polynomial.
</li>
<li>
<b>POWER_CUBIC_TO_HERMITE_CUBIC</b> converts a power cubic to Hermite form.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8VEC_BRACKET3</b> finds the interval containing or nearest a given value.
</li>
<li>
<b>R8VEC_EVEN</b> returns an R8VEC of evenly spaced values.
</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 modified on 10 February 2011.
</i>
<!-- John Burkardt -->
</body>
</html>