jacobi_polynomial
jacobi_polynomial,
an Octave code which
evaluates the Jacobi polynomial.
For a given choice of the parameters a and b, both greater than -1,
the Jacobi polynomials are a set of polynomials which are
pairwise orthogonal with respect to the integral:
integral (-1<=x<=+1) J(i,a,b,x) J(j,a,b,x) (1-x)^a (1+x)^b dx
That is, this integral is 0 unless i = j. J(i,a,b,x) indicates the
Jacobi polynomial of degree i.
The standard Jacobi polynomials can be defined by a three term
recurrence formula that is a bit too ugly to quote here.
It is worth noting that the definition of the Jacobi polynomials
is general enough that it includes some familiar families as
special cases:
-
if a = b = 0, we have the Legendre polynomials, P(n,x);
-
if a = b = -1/2, we have the Chebyshev polynomials of the first kind, T(n,x);
-
if a = b = 1/2, we have the Chebyshev polynomials of the second kind, U(n,x);
-
if a = b, we have the Gegenbauer polynomials;
Licensing:
The computer code and data files made available on this web page
are distributed under
the MIT license
Languages:
jacobi_polynomial is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version and
an Octave version.
Related Data and Programs:
jacobi_polynomial_test
polpak,
an Octave code which
evaluates a variety of mathematical functions.
test_values,
an Octave code which
supplies test values of various mathematical functions.
Reference:
-
Theodore Chihara,
An Introduction to Orthogonal Polynomials,
Gordon and Breach, 1978,
ISBN: 0677041500,
LC: QA404.5 C44.
-
Walter Gautschi,
Orthogonal Polynomials: Computation and Approximation,
Oxford, 2004,
ISBN: 0-19-850672-4,
LC: QA404.5 G3555.
-
Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
NIST Handbook of Mathematical Functions,
Cambridge University Press, 2010,
ISBN: 978-0521192255,
LC: QA331.N57.
-
Gabor Szego,
Orthogonal Polynomials,
American Mathematical Society, 1992,
ISBN: 0821810235,
LC: QA3.A5.v23.
Source Code:
-
imtqlx.m,
diagonalizes a symmetric tridiagonal matrix;
-
j_double_product_integral.m,
integral of J(i,x)*J(j,x)*(1-x)^a*(1+x)^b.
-
j_integral.m,
evaluates a monomial Jacobi integral for J(n,a,b,x).
-
j_polynomial.m,
evaluates the Jacobi polynomial J(n,a,b,x).
-
j_polynomial_plot.m,
plots one or more Jacobi polynomials J(n,a,b,x) over [-1,+1].
-
j_polynomial_values.m,
a few tabulated values of the Jacobi polynomial J(n,a,b,x).
-
j_polynomial_zeros.m,
returns zeros of the Jacobi polynomial J(n,a,b,x).
-
j_quadrature_rule.m,
returns quadrature rules associated with the Jacobi polynomial J(n,a,b,x).
-
r8_factorial.m,
computes the factorial function;
-
r8_sign.m,
returns the sign of an R8.
-
r8mat_print.m,
prints an R8MAT;
-
r8mat_print_some.m,
prints some of an R8MAT;
-
r8vec_print.m,
prints an R8VEC;
-
r8vec2_print.m,
prints a pair of R8VEC's;
Last modified on 25 September 2022.