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<html>
<head>
<title>
CHEBYSHEV_POLYNOMIAL - Chebyshev Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
CHEBYSHEV_POLYNOMIAL <br> Chebyshev Polynomials
</h1>
<hr>
<p>
<b>CHEBYSHEV_POLYNOMIAL</b>
is a FORTRAN90 library which
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
The Chebyshev polynomial T(n,x), or Chebyshev polynomial of the first kind,
may be defined, for 0 <= n, and -1 <= x <= +1 by:
<pre>
cos ( t ) = x
T(n,x) = cos ( n * t )
</pre>
For any value of x, T(n,x) may be evaluated by a three
term recurrence:
<pre>
T(0,x) = 1
T(1,x) = x
T(n+1,x) = 2x T(n,x) - T(n-1,x)
</pre>
</p>
<p>
The Chebyshev polynomial U(n,x), or Chebyshev polynomial of the second kind,
may be defined, for 0 <= n, and -1 <= x <= +1 by:
<pre>
cos ( t ) = x
U(n,x) = sin ( ( n + 1 ) t ) / sin ( t )
</pre>
For any value of x, U(n,x) may be evaluated by a three
term recurrence:
<pre>
U(0,x) = 1
U(1,x) = 2x
U(n+1,x) = 2x U(n,x) - U(n-1,x)
</pre>
</p>
<p>
The Chebyshev polynomial V(n,x), or Chebyshev polynomial of the third kind,
may be defined, for 0 <= n, and -1 <= x <= +1 by:
<pre>
cos ( t ) = x
V(n,x) = cos ( (2n+1)*t/2) / cos ( t/2)
</pre>
For any value of x, V(n,x) may be evaluated by a three
term recurrence:
<pre>
V(0,x) = 1
V(1,x) = 2x-1
V(n+1,x) = 2x V(n,x) - V(n-1,x)
</pre>
</p>
<p>
The Chebyshev polynomial W(n,x), or Chebyshev polynomial of the fourth kind,
may be defined, for 0 <= n, and -1 <= x <= +1 by:
<pre>
cos ( t ) = x
W(n,x) = sin((2*n+1)*t/2)/sin(t/2)
</pre>
For any value of x, W(n,x) may be evaluated by a three
term recurrence:
<pre>
W(0,x) = 1
W(1,x) = 2x+1
W(n+1,x) = 2x W(n,x) - W(n-1,x)
</pre>
</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>CHEBYSHEV_POLYNOMIAL</b> is available in
<a href = "../../cpp_src/chebyshev_polynomial/chebyshev_polynomial.html">a C++ version</a> and
<a href = "../../f_src/chebyshev_polynomial/chebyshev_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/chebyshev_polynomial/chebyshev_polynomial.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/chebyshev1_rule/chebyshev1_rule.html">
CHEBYSHEV1_RULE</a>,
a FORTRAN90 program which
computes and prints a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../f_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a FORTRAN90 program which
compute and print a Gauss-Chebyshev type 2 quadrature rule.
</p>
<p>
<a href = "../../f_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../f_src/int_exactness_chebyshev1/int_exactness_chebyshev1.html">
INT_EXACTNESS_CHEBYSHEV1</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_chebyshev2/int_exactness_chebyshev2.html">
INT_EXACTNESS_CHEBYSHEV2</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.
</p>
<p>
<a href = "../../f_src/jacobi_polynomial/jacobi_polynomial.html">
JACOBI_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Jacobi polynomial and associated functions.
</p>
<p>
<a href = "../../f_src/laguerre_polynomial/laguerre_polynomial.html">
LAGUERRE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomial,
and the Laguerre function.
</p>
<p>
<a href = "../../f_src/legendre_polynomial/legendre_polynomial.html">
LEGENDRE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Legendre polynomial and associated functions.
</p>
<p>
<a href = "../../f_src/polpak/polpak.html">
POLPAK</a>,
a FORTRAN90 library which
evaluates a variety of mathematical functions.
</p>
<p>
<a href = "../../f_src/test_values/test_values.html">
TEST_VALUES</a>,
a FORTRAN90 library which
supplies test values of various mathematical functions.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Theodore Chihara,<br>
An Introduction to Orthogonal Polynomials,<br>
Gordon and Breach, 1978,<br>
ISBN: 0677041500,<br>
LC: QA404.5 C44.
</li>
<li>
Walter Gautschi,<br>
Orthogonal Polynomials: Computation and Approximation,<br>
Oxford, 2004,<br>
ISBN: 0-19-850672-4,<br>
LC: QA404.5 G3555.
</li>
<li>
John Mason, David Handscomb,<br>
Chebyshev Polynomials,<br>
CRC Press, 2002,<br>
ISBN: 0-8493-035509,<br>
LC: QA404.5.M37.
</li>
<li>
Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,<br>
NIST Handbook of Mathematical Functions,<br>
Cambridge University Press, 2010,<br>
ISBN: 978-0521192255,<br>
LC: QA331.N57.
</li>
<li>
Gabor Szego,<br>
Orthogonal Polynomials,<br>
American Mathematical Society, 1992,<br>
ISBN: 0821810235,<br>
LC: QA3.A5.v23.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "chebyshev_polynomial.f90">chebyshev_polynomial.f90</a>, the source code.
</li>
<li>
<a href = "chebyshev_polynomial.sh">chebyshev_polynomial.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "chebyshev_polynomial_prb.f90">chebyshev_polynomial_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "chebyshev_polynomial_prb.sh">chebyshev_polynomial_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "chebyshev_polynomial_prb_output.txt">chebyshev_polynomial_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>DAXPY</b> computes constant times a vector plus a vector.
</li>
<li>
<b>DDOT</b> forms the dot product of two vectors.
</li>
<li>
<b>DNRM2</b> returns the euclidean norm of a vector.
</li>
<li>
<b>DROT</b> applies a plane rotation.
</li>
<li>
<b>DROTG</b> constructs a Givens plane rotation.
</li>
<li>
<b>DSCAL</b> scales a vector by a constant.
</li>
<li>
<b>DSVDC</b> computes the singular value decomposition of a real rectangular matrix.
</li>
<li>
<b>DSWAP</b> interchanges two vectors.
</li>
<li>
<b>I4_UNIFORM</b> returns a scaled pseudorandom I4.
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_SIGN</b> returns the sign of an R8.
</li>
<li>
<b>R8VEC_IN_AB</b> is TRUE if the entries of an R8VEC are in the range [A,B].
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>R8VEC2_PRINT</b> prints an R8VEC2.
</li>
<li>
<b>SVD_SOLVE</b> solves a linear system in the least squares sense.
</li>
<li>
<b>T_DOUBLE_PRODUCT_INTEGRAL:</b> integral (-1<=x<=1) T(i,x)*T(j,x)/sqrt(1-x^2) dx
</li>
<li>
<b>T_INTEGRAL:</b> integral ( -1 <= x <= +1 ) x^e dx / sqrt ( 1 - x^2 ).
</li>
<li>
<b>T_POLYNOMIAL</b> evaluates Chebyshev polynomials T(n,x).
</li>
<li>
<b>T_POLYNOMIAL_AB:</b> evaluates Chebyshev polynomials T(n,x) in [A,B].
</li>
<li>
<b>T_POLYNOMIAL_COEFFICIENTS:</b> coefficients of the Chebyshev polynomial T(n,x).
</li>
<li>
<b>T_POLYNOMIAL_VALUE:</b> returns the single value T(n,x).
</li>
<li>
<b>T_POLYNOMIAL_VALUES</b> returns values of Chebyshev polynomials T(n,x).
</li>
<li>
<b>T_POLYNOMIAL_ZEROS</b> returns zeroes of the Chebyshev polynomial T(n,x).
</li>
<li>
<b>T_PROJECT_COEFFICIENTS:</b> function projected onto Chebyshev polynomials T(n,x).
</li>
<li>
<b>T_PROJECT_COEFFICIENTS_AB:</b> function projected onto T(n,x) over [a,b].
</li>
<li>
<b>T_PROJECT_COEFFICIENTS_DATA:</b> project data onto Chebyshev polynomials T(n,x).
</li>
<li>
<b>T_PROJECT_VALUE</b> evaluates an expansion in Chebyshev polynomials T(n,x).
</li>
<li>
<b>T_PROJECT_VALUE_AB</b> evaluates an expansion in Chebyshev polynomials T(n,x).
</li>
<li>
<b>T_QUADRATURE_RULE:</b> quadrature rule for T(n,x).
</li>
<li>
<b>T_TRIPLE_PRODUCT_INTEGRAL:</b> integral (-1<=x<=1) T(i,x)*T(j,x)*T(k,x)/sqrt(1-x^2) dx
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>U_DOUBLE_PRODUCT_INTEGRAL:</b> integral (-1<=x<=1) U(i,x)*U(j,x)*sqrt(1-x^2) dx
</li>
<li>
<b>U_INTEGRAL:</b> integral ( -1 <= x <= +1 ) x^e sqrt ( 1 - x^2 ) dx.
</li>
<li>
<b>U_POLYNOMIAL</b> evaluates Chebyshev polynomials U(n,x).
</li>
<li>
<b>U_POLYNOMIAL_AB:</b> evaluates Chebyshev polynomials U(n,x) in [A,B].
</li>
<li>
<b>U_POLYNOMIAL_COEFFICIENTS:</b> coefficients of Chebyshev polynomials U(n,x).
</li>
<li>
<b>U_POLYNOMIAL_VALUES</b> returns values of Chebyshev polynomials U(n,x).
</li>
<li>
<b>U_POLYNOMIAL_ZEROS</b> returns zeroes of Chebyshev polynomials U(n,x).
</li>
<li>
<b>U_QUADRATURE_RULE:</b> quadrature rule for U(n,x).
</li>
<li>
<b>V_DOUBLE_PRODUCT_INTEGRAL:</b> integral (-1<=x<=1) V(i,x)*V(j,x)*sqrt(1+x)/sqrt(1-x) dx
</li>
<li>
<b>V_POLYNOMIAL</b> evaluates Chebyshev polynomials V(n,x).
</li>
<li>
<b>V_POLYNOMIAL_VALUES</b> returns values of Chebyshev polynomials V(n,x).
</li>
<li>
<b>V_POLYNOMIAL_ZEROS</b> returns zeroes of Chebyshev polynomials V(n,x).
</li>
<li>
<b>W_DOUBLE_PRODUCT_INTEGRAL:</b> integral (-1<=x<=1) W(i,x)*W(j,x)*sqrt(1-x)/sqrt(1+x) dx
</li>
<li>
<b>W_POLYNOMIAL</b> evaluates Chebyshev polynomials W(n,x).
</li>
<li>
<b>W_POLYNOMIAL_VALUES</b> returns values of Chebyshev polynomials W(n,x).
</li>
<li>
<b>W_POLYNOMIAL_ZEROS</b> returns zeroes of Chebyshev polynomials W(n,x).
</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 26 April 2012.
</i>
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