# LOBATTO_POLYNOMIAL Lobatto Polynomials

LOBATTO_POLYNOMIAL is a C++ 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) )
```
where n is a positive integer called the order, x is a real value between -1 and +1, and P(n,x) is the Legendre polynomial.

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.

### Languages:

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

### Related Data and Programs:

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

CHEBYSHEV_POLYNOMIAL, a C++ 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.

GEGENBAUER_POLYNOMIAL, a C++ library which evaluates the Gegenbauer polynomial and associated functions.

GNUPLOT, C++ 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 C++ library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

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

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

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

LEGENDRE_SHIFTED_POLYNOMIAL, a C++ library which evaluates the shifted Legendre polynomial, with domain [0,1].

POLPAK, a C++ library which evaluates a variety of mathematical functions.

TEST_VALUES, a C++ 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.

### Examples and Tests:

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

### List of Routines:

• 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 C++ source codes.

Last revised on 20 November 2014.