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lagrange_basis_display.html
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<html>
<head>
<title>
LAGRANGE_BASIS_DISPLAY - Display Basis Functions for Lagrange Interpolation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LAGRANGE_BASIS_DISPLAY <br> Display Basis Functions for Lagrange Interpolation
</h1>
<hr>
<p>
<b>LAGRANGE_BASIS_DISPLAY</b>
is a MATLAB program which
displays the basis functions associated with any set of interpolation
points to be used for Lagrange interpolation.
</p>
<p>
The Lagrange interpolating polynomial to a set of m+1 data pairs (xi,yi)
can be represented as
<pre>
p(x) = sum ( 1 <= i <= m + 1 ) yi * l(i,x)
</pre>
Each function l(i,x) is a Lagrange basis function associated with the
set of x data values. Each l(i,x) is a polynomial of degree m, which
is 1 at node xi and zero at the other nodes. Moreover, there is an
explicit formula:
<pre>
l(i,x) = product ( 1 <= j <= m + 1, j /= i ) ( x - xj )
/ product ( 1 <= j <= m + 1, j /= i ) ( xi - xj )
</pre>
Thus the interpolating polynomial can be represented as a linear combination
of the Lagrange basis functions, and the coefficients are simply the
data values yi.
</p>
<p>
For a given set of m+1 data pairs (xi,yi), you may also define the same
interpolating polynomial using a Vandermonde matrix; this approach
essentially uses the monomials 1, x, x^2, ..., x^m as the basis functions.
The unknown polynomial coefficients c must be determined by forming
and solving the Vandermonde system; not only is this method more costly,
but this linear system is numerically ill-conditioned, so that the resulting
answers can be unreliable.
</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>LAGRANGE_BASIS_DISPLAY</b> is available in
<a href = "../../m_src/lagrange_basis_display/lagrange_basis_display.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/lagrange_interp_1d/lagrange_interp_1d.html">
LAGRANGE_INTERP_1D</a>,
a MATLAB 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 = "../../m_src/vandermonde_interp_1d/vandermonde_interp_1d.html">
VANDERMONDE_INTERP_1D</a>,
a MATLAB library which
finds a polynomial interpolant to data y(x) of a 1D argument,
by setting up and solving a linear system for the polynomial coefficients,
involving the Vandermonde matrix.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Kendall Atkinson,<br>
An Introduction to Numerical Analysis,<br>
Prentice Hall, 1989,<br>
ISBN: 0471624896,<br>
LC: QA297.A94.1989.
</li>
<li>
Philip Davis,<br>
Interpolation and Approximation,<br>
Dover, 1975,<br>
ISBN: 0-486-62495-1,<br>
LC: QA221.D33
</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>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lagrange_basis_display.m">lagrange_basis_display.m</a>,
displays Lagrange basis functions over [A,B], of degree M,
for given nodes XD.
</li>
<li>
<a href = "standard_basis_display.m">standard_basis_display.m</a>,
displays standard monomial basis functions over [A,B], up to degree M.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the YMDHMS date as a timestamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lagrange_basis_display_test.m">lagrange_basis_display_test.m</a>,
tests the software.
</li>
</ul>
</p>
<p>
Plots made by the test include:
<ul>
<li>
<a href = "lagrange_10_chebyshev.png">lagrange_10_chebyshev.png</a>,
the Lagrange basis functions of degree 10, for the Chebyshev points
on [-1,+1].
</li>
<li>
<a href = "lagrange_10_even.png">lagrange_10_evem.png</a>,
the Lagrange basis functions of degree 10, for evenly-spaced points
on [0,+1].
</li>
<li>
<a href = "standard_10.png">standard_10.png</a>,
the standard monomial basis functions up to degree 10, for evenly-spaced points
on [0,+1].
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 06 August 2012.
</i>
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</body>
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</html>