# lobatto_polynomial

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

CHEBYSHEV_POLYNOMIAL, a C++ code 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++ code which evaluates the Gegenbauer polynomial and associated functions.

HERMITE_POLYNOMIAL, a C++ code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

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

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

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

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

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

TEST_VALUES, a C++ code 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:

Last revised on 26 March 2020.