function c = u_polynomial_coefficients ( n ) %*****************************************************************************80 % %% U_POLYNOMIAL_COEFFICIENTS: coefficients of Chebyshev polynomials U(n,x). % % First terms: % % N/K 0 1 2 3 4 5 6 7 8 9 10 % % 0 1 % 1 0 2 % 2 -1 0 4 % 3 0 -4 0 8 % 4 1 0 -12 0 16 % 5 0 6 0 -32 0 32 % 6 -1 0 24 0 -80 0 64 % 7 0 -8 0 80 0 -192 0 128 % % Recursion: % % U(0)(X) = 1, % U(1)(X) = 2*X, % U(N)(X) = 2 * X * U(N-1)(X) - U(N-2)(X) % % Licensing: % % This code is distributed under the GNU LGPL license. % % Modified: % % 01 August 2004 % % Author: % % John Burkardt % % Reference: % % Milton Abramowitz and Irene Stegun, % Handbook of Mathematical Functions, % US Department of Commerce, 1964. % % Parameters: % % Input, integer N, the highest order polynomial to compute. % Note that polynomials 0 through N will be computed. % % Output, real C(1:N+1,1:N+1), the coefficients of the Chebyshev U % polynomials. % if ( n < 0 ) c = []; return end c(1:n+1,1:n+1) = 0.0; c(1,1) = 1.0; if ( n == 0 ) return end c(2,2) = 2.0; for i = 2 : n c(i+1,1) = - c(i-1,1); c(i+1,2:i-1) = 2.0 * c(i,1:i-2) - c(i-1,2:i-1); c(i+1, i ) = 2.0 * c(i, i-1); c(i+1, i+1) = 2.0 * c(i, i ); end return end