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<html>
<head>
<title>
BERNSTEIN - The Bernstein Polynomials
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
BERNSTEIN <br> The Bernstein Polynomials
</h1>
<hr>
<p>
<b>BERNSTEIN</b>
is a FORTRAN90 library which
evaluates the Bernstein polynomials.
</p>
<p>
The k-th Bernstein basis polynomial of degree n is defined by
<pre>
B(n,k)(x) = C(n,k) * (1-x)^(n-k) * x^k
</pre>
for k = 0 to n and C(n,k) is the combinatorial function "N choose K"
defined by
<pre>
C(n,k) = n! / k! / ( n - k )!
</pre>
</p>
<p>
For an arbitrary value of n, the set B(n,k) forms a basis
for the space of polynomials of degree n or less.
</p>
<p>
Every basis polynomial B(n,k) is nonnegative in [0,1], and may be zero
only at the endpoints.
</p>
<p>
Except for the case n = 0, the basis polynomial B(n,k)(x) has a
unique maximum value at
<pre>
x = k/n.
</pre>
</p>
<p>
For any point x, (including points outside [0,1]), the basis polynomials
for an arbitrary value of n sum to 1:
<pre>
sum ( 1 <= k <= n ) B(n,k)(x) = 1
</pre>
</p>
<p>
For 0 < n, the Bernstein basis polynomial can be written as a combination
of two lower degree basis polynomials:
<pre>
B(n,k)(x) = ( 1 - x ) * B(n-1,k)(x) + x * B(n-1,k-1)(x) +
</pre>
where, if k is 0, the factor B(n-1,k-1)(x) is taken to be 0,
and if k is n, the factor B(n-1,k)(x) is taken to be 0.
</p>
<p>
A Bernstein basis polynomial can be written as a combination
of two higher degree basis polynomials:
<pre>
B(n,k)(x) = ( (n+1-k) * B(n+1,k)(x) + (k+1) * B(n+1,k+1)(x) ) / ( n + 1 )
</pre>
</p>
<p>
The derivative of B(n,k)(x) can be written as:
<pre>
d/dx B(n,k)(x) = n * B(n-1,k-1)(x) - B(n-1,k)(x)
</pre>
</p>
<p>
A Bernstein polynomial can be written in terms of the standard power basis:
<pre>
B(n,k)(x) = sum ( k <= i <= n ) (-1)^(i-k) * C(n,k) * C(i,k) * x^i
</pre>
</p>
<p>
A power basis monomial can be written in terms of the Bernstein basis
of degree n where k <= n:
<pre>
x^k = sum ( k-1 <= i <= n-1 ) C(i,k) * B(n,k)(x) / C(n,k)
</pre>
</p>
<p>
Over the interval [0,1], the n-th degree Bernstein approximation polynomial to
a function f(x) is defined by
<pre>
BA(n,f)(x) = sum ( 0 <= k <= n ) f(k/n) * B(n,k)(x)
</pre>
As a function of n, the Bernstein approximation polynomials form a sequence
that slowly, but uniformly, converges to f(x) over [0,1].
</p>
<p>
By a simple linear process, the Bernstein basis polynomials can be shifted
to an arbitrary interval [a,b], retaining their properties.
</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>BERNSTEIN</b> is available in
<a href = "../../c_src/bernstein/bernstein.html">a C version</a> and
<a href = "../../cpp_src/bernstein/bernstein.html">a C++ version</a> and
<a href = "../../f77_src/bernstein/bernstein.html">a FORTRAN77 version</a> and
<a href = "../../f_src/bernstein/bernstein.html">a FORTRAN90 version</a> and
<a href = "../../m_src/bernstein/bernstein.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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/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/spline/spline.html">
SPLINE</a>,
a FORTRAN90 library which
constructs and evaluates spline interpolants and approximants.
</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>
Kenneth Joy,<br>
"Bernstein Polynomials",<br>
On-Line Geometric Modeling Notes,<br>
idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf
</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>
Josef Reinkenhof,<br>
Differentiation and integration using Bernstein's polynomials,<br>
International Journal of Numerical Methods in Engineering,<br>
Volume 11, Number 10, 1977, pages 1627-1630.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "bernstein.f90">bernstein.f90</a>, the source code.
</li>
<li>
<a href = "bernstein.sh">bernstein.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "bernstein_prb.f90">bernstein_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "bernstein_prb.sh">bernstein_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "bernstein_prb_output.txt">bernstein_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BERNSTEIN_MATRIX</b> returns the Bernstein matrix.
</li>
<li>
<b>BERNSTEIN_MATRIX_INVERSE</b> returns the inverse Bernstein matrix.
</li>
<li>
<b>BERNSTEIN_POLY</b> evaluates the Bernstein polynomials at a point X.
</li>
<li>
<b>BERNSTEIN_POLY_VALUES</b> returns some values of the Bernstein polynomials.
</li>
<li>
<b>BPAB</b> evaluates at X the Bernstein polynomials based in [A,B].
</li>
<li>
<b>BPAB_APPROX</b> evaluates the Bernstein polynomial approximant to F(X) on [A,B].
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_MOP</b> returns the I-th power of -1 as an R8.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8MAT_IS_IDENTITY</b> determines if an R8MAT is the identity.
</li>
<li>
<b>R8MAT_NORM_FRO</b> returns the Frobenius norm of an M by N R8MAT.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly 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 revised on 10 July 2011.
</i>
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