LOBATTO_POLYNOMIAL
Lobatto Polynomials

LOBATTO_POLYNOMIAL is a FORTRAN77 library which evaluates the completed Lobatto polynomial and associated functions.

The completed Lobatto polynomial Lo(n,x) can be defined by:

        Lo(n,x) = n * ( P(n-1,x) - x * P(n,x) )
      

The completed Lobatto polynomial Lo(n,x) has degree n+1, and is zero at x = -1 and x = +1.

GNUPLOT is used to create images of some of the functions.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

LOBATTO_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a FORTRAN77 library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a FORTRAN77 library which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

GNUPLOT, FORTRAN77 programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

HERMITE_POLYNOMIAL, a FORTRAN77 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

JACOBI_POLYNOMIAL, a FORTRAN77 library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Legendre polynomial and associated functions.

POLPAK, a FORTRAN77 library which evaluates a variety of mathematical functions.

TEST_VALUES, a FORTRAN77 library which supplies test values of various mathematical functions.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Larry Andrews,
   Special Functions of Mathematics for Engineers,
   Second Edition,
   Oxford University Press, 1998,
   ISBN: 0819426164,
   LC: QA351.A75.
3. Daniel Zwillinger, editor,
   CRC Standard Mathematical Tables and Formulae,
   30th Edition,
   CRC Press, 1996.

Source Code:

• lobatto_polynomial.f, the source code.
• lobatto_polynomial.sh, BASH commands to compile the source code.

Examples and Tests:

• lobatto_polynomial_prb.f, a sample calling program.
• lobatto_polynomial_prb.sh, BASH commands to compile and run the sample program.
• lobatto_polynomial_prb_output.txt, the output file.

The example program creates data files that can be used by GNUPLOT to create plots.

• test_value_commands.txt
• test_value_data.txt
• test_value.png
• test_derivative_commands.txt
• test_derivative_data.txt
• test_derivative.png

List of Routines:

• GET_UNIT returns a free FORTRAN unit number.
• LOBATTO_POLYNOMIAL_DERIVATIVE: derivative of completed Lobatto polynomial.
• LOBATTO_POLYNOMIAL_DERIVATIVES: derivatives of completed Lobatto polynomials.
• LOBATTO_POLYNOMIAL_PLOT plots one or more completed Lobatto polynomials.
• LOBATTO_POLYNOMIAL_VALUE evaluates the completed Lobatto polynomials Lo(n,x).
• LOBATTO_POLYNOMIAL_VALUES: values of the completed Lobatto polynomials.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN77 source codes.