% LANCZOS Lanczos algorithm.
%
%    Given the discrete inner product whose nodes are contained 
%    in the first column, and whose weights are contained in the 
%    second column, of the nx2 array xw, the call ab=LANCZOS(n,xw)
%    generates the first n recurrence coefficients ab of the 
%    corresponding discrete orthogonal polynomials. The n alpha-
%    coefficients are stored in the first column, the n beta-
%    coefficients in the second column, of the nx2 array ab.
%
%    The script is adapted from the routine RKPW in
%    W.B. Gragg and W.J. Harrod, ``The numerically stable 
%    reconstruction of Jacobi matrices from spectral data'', 
%    Numer. Math. 44 (1984), 317-335.
%
function ab=lanczos(N,xw)
Ncap=size(xw,1);
if(N<=0 | N>Ncap), error('N out of range'), end
p0=xw(:,1); p1=zeros(Ncap,1); p1(1)=xw(1,2);
for n=1:Ncap-1
  pn=xw(n+1,2); gam=1; sig=0; t=0; xlam=xw(n+1,1);
  for k=1:n+1
    rho=p1(k)+pn; tmp=gam*rho; tsig=sig;
    if rho<=0
      gam=1; sig=0;
    else
      gam=p1(k)/rho;
      sig=pn/rho;
    end
    tk=sig*(p0(k)-xlam)-gam*t;
    p0(k)=p0(k)-(tk-t); t=tk;
    if sig<=0
      pn=tsig*p1(k);
    else
      pn=(t^2)/sig;
    end
    tsig=sig; p1(k)=tmp;
  end
end
ab=[p0(1:N) p1(1:N)];