chebyshev_polynomial, a Python 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.
The Chebyshev polynomial T(n,x), or Chebyshev polynomial of the first kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:
cos ( t ) = x
T(n,x) = cos ( n * t )
For any value of x, T(n,x) may be evaluated by a three
term recurrence:
T(0,x) = 1
T(1,x) = x
T(n+1,x) = 2x T(n,x) - T(n-1,x)
The Chebyshev polynomial U(n,x), or Chebyshev polynomial of the second kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:
cos ( t ) = x
U(n,x) = sin ( ( n + 1 ) t ) / sin ( t )
For any value of x, U(n,x) may be evaluated by a three
term recurrence:
U(0,x) = 1
U(1,x) = 2x
U(n+1,x) = 2x U(n,x) - U(n-1,x)
The Chebyshev polynomial V(n,x), or Chebyshev polynomial of the third kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:
cos ( t ) = x
V(n,x) = cos ( (2n+1)*t/2) / cos ( t/2)
For any value of x, V(n,x) may be evaluated by a three
term recurrence:
V(0,x) = 1
V(1,x) = 2x-1
V(n+1,x) = 2x V(n,x) - V(n-1,x)
The Chebyshev polynomial W(n,x), or Chebyshev polynomial of the fourth kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:
cos ( t ) = x
W(n,x) = sin((2*n+1)*t/2)/sin(t/2)
For any value of x, W(n,x) may be evaluated by a three
term recurrence:
W(0,x) = 1
W(1,x) = 2x+1
W(n+1,x) = 2x W(n,x) - W(n-1,x)
The computer code and data files described and made available on this web page are distributed under the MIT license
chebyshev_polynomial is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
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