LEGENDRE_POLYNOMIAL
Legendre Polynomials

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

The Legendre polynomial P(n,x) can be defined by:

        P(0,x) = 1
        P(1,x) = x
        P(n,x) = (2*n-1)/n * x * P(n-1,x) - (n-1)/n * P(n-2,x)
      

The N zeroes of P(n,x) are the abscissas used for Gauss-Legendre quadrature of the integral of a function F(X) with weight function 1 over the interval [-1,1].

The Legendre polynomials are orthogonal under the inner product defined as integration from -1 to 1:

        Integral ( -1 <= x <= 1 ) P(i,x) * P(j,x) dx 
          = 0 if i =/= j
          = 2 / ( 2*i+1 ) if i = j.
      

Licensing:

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

Languages:

LEGENDRE_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python 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.

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

INT_EXACTNESS_LEGENDRE, a C++ program which tests the polynomial exactness of Gauss-Legendre quadrature rules.

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_PRODUCT_POLYNOMIAL, a C++ library which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.

LEGENDRE_RULE, a C++ program which computes a 1D Gauss-Legendre quadrature rule.

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

LOBATTO_POLYNOMIAL, a C++ library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

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. Theodore Chihara,
   An Introduction to Orthogonal Polynomials,
   Gordon and Breach, 1978,
   ISBN: 0677041500,
   LC: QA404.5 C44.
2. Walter Gautschi,
   Orthogonal Polynomials: Computation and Approximation,
   Oxford, 2004,
   ISBN: 0-19-850672-4,
   LC: QA404.5 G3555.
3. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
   NIST Handbook of Mathematical Functions,
   Cambridge University Press, 2010,
   ISBN: 978-0521192255,
   LC: QA331.N57.
4. Gabor Szego,
   Orthogonal Polynomials,
   American Mathematical Society, 1992,
   ISBN: 0821810235,
   LC: QA3.A5.v23.

Source Code:

• legendre_polynomial.cpp, the source code.
• legendre_polynomial.hpp, the include file.

Examples and Tests:

• legendre_polynomial_test.cpp, a sample calling program.
• legendre_polynomial_test.txt, the output file.

List of Routines:

• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• P_EXPONENTIAL_PRODUCT: exponential products for P(n,x).
• P_INTEGRAL evaluates a monomial integral associated with P(n,x).
• P_POLYNOMIAL evaluates the Legendre polynomials P(n,x).
• P_POLYNOMIAL_COEFFICIENTS: coefficients of Legendre polynomials P(n,x).
• P_POLYNOMIAL_PRIME evaluates the derivative of Legendre polynomials P'(n,x).
• P_POLYNOMIAL_VALUES returns values of the Legendre polynomials P(n,x).
• P_POLYNOMIAL_ZEROS: zeros of Legendre function P(n,x).
• P_POWER_PRODUCT: power products for Legendre polynomial P(n,x).
• P_QUADRATURE_RULE: quadrature for Legendre function P(n,x).
• PM_POLYNOMIAL evaluates the Legendre polynomials Pm(n,m,x).
• PM_POLYNOMIAL_VALUES returns values of Legendre polynomials Pm(n,m,x).
• LEGENDRE_ASSOCIATED_NORMALIZED_VALUES: normalized associated Legendre.
• LEGENDRE_FUNCTION_Q_VALUES returns values of the Legendre Q function.
• LEGENDRE_ASSOCIATED_NORMALIZED: normalized associated Legendre functions.
• LEGENDRE_FUNCTION_Q evaluates the Legendre Q functions.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8VEC_PRINT prints an R8VEC.
• R8VEC2_PRINT prints an R8VEC2.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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