DipoleAntennacode.zip_dipole antenna_dipole current_massagejh2_s

% quadr.m - Gauss-Legendre quadrature weights and evaluation points % % Usage: [w,x] = quadr(a,b,N) % [w,x] = quadr(a,b) (equivalent to N=16) % % a,b = integration interval [a,b] % N = number of weights in quadrature formula (default N=16, even N avoids x=0) % % w = length-N column vector of (symmetric) weights % x = length-N column vector of shifted/scaled Legendre evaluation points % % notes: J = \int_a^b f(t) dt is approximated by J = w'*f(x), where f(x) is % the column vector of values of f(t) evaluated at the Legendre points x. % The weights w have been pre-multiplied by the scale factor (b-a)/2 % % for double integration of f(t1,t2) over the intervals [a1,b1] and [a2,b2], use % [w1,x1] = quadr(a1,b1,N1), [w2,x2] = quadr(a2,b2,N2), J = w1'*f(x1,x2)*w2 % where f(x1,x2) is an N1xN2 matrix of values of f(t1,t2) % % construction method: use LEGENDRE to calculate the coefficients of the % order-N Legendre polynomial, then find its N roots z, and construct x % by shifting and scaling, x = (z*(b-a) + a+b)/2, or, z = (2*x-a-b)/(b-a), % and finally, solve for w by matching the quadrature integration formula % exactly at the polynomial powers f(t) = t^k, k=0,1,...,N % % example: f(t) = e^t + 1/t has J0 = \int_1^2 f(t)dt = e^2-e^1+ln(2) = 5.36392145 % percent error of different methods, 100*abs(J-J0)/J0, is as follows: % method: quad(f,1,2) quad8(f,1,2) quadr(1,2,5) quadr(1,2,15) % error: 2.5492e-004 3.0302e-012 4.2336e-007 1.6558e-014 % % see also QUADRS for splitting the integration interval into subintervals % S. J. Orfanidis - 1999 - www.ece.rutgers.edu/~orfanidi/ewa function [w,x] = quadr(a,b,N) if nargin==0, help quadr; return; end if nargin==2, N=16; end P = legendre(N,0); % evaluate Legendre functions at x=0 m = (0:N)'; % coefficient index P = (-1).^m .* P ./ gamma(m+1); % order-N Legendre polynomial coefficients z = sort(roots(flip(P))); % sort roots in increasing magnitude x = (z*(b-a) + a + b)/2; % shifted Legendre roots k = (0:N-1)'; c = (1 + (-1).^k) ./ (k+1); % this is \int_{-1}^1 t^k dt A = []; % coefficient matrix of the system A*w = c for m=1:N A = [A, z(m).^k]; % build the columns of A end w = A\c * (b-a)/2; % The weights w can also be computed more simply, but less accurately, as follows: % % P = legendre(N,z); % Legendre functions evaluated at the roots z % w = (b-a) ./ (P(2,:)').^2; % 2nd row of P is (1-z.^2).^(1/2) .* P'_N(z) % % the unscaled weights are w(m) = 2/[(1-z(m)^2)*P'_N(z(m))^2], m=0,1,...,N, % where P'_N(z(m)) are the derivatives of the order-N Legendre polynomial, evaluated % at the Legendre zeros. The second row of LEGENDRE(N,z) contains these derivatives.