% MCDIS Multiple-component discretization procedure.
%
%    This routine performs a sequence of discretizations of the 
%    given weight function (or measure), each discretization being
%    followed by an application of the Stieltjes, or Lanczos, 
%    procedure to produce approximations to the desired recurrence
%    coefficients. The fineness of the discretization is 
%    characterized by a discretization parameter N. The support of
%    the continuous part of the weight function is decomposed into
%    a given number mc of subintervals (some or all of which may 
%    be identical). The routine then applies to each subinterval 
%    an N-point quadrature rule to discretize the weight function 
%    on that subinterval. The discrete part of the weight function
%    (if there is any) is added on to the discretized continuous 
%    weight function. The sequence of discretizations, if chosen
%    judiciously, leads to convergence of the recurrence
%    coefficients for the discretized measures to those of the
%    given measure. If convergence to within a prescribed accuracy
%    eps0 occurs before N reaches its maximum allowed value Nmax,
%    then the value of N that yields convergence is output as 
%    Ncap, and so is the number of iterations, kount. If there is 
%    no convergence, the routine displays the message "Ncap
%    exceeds Nmax in mcdis" prior to exiting.
%
%    The choice between the Stieltjes and the Lanczos procedure is
%    made by setting the parameter irout equal to 1 in the former,
%    and different from 1, in the latter case.
%        
%    The details of the discretization are to be specified prior 
%    to calling the procedure. They are embodied in the following 
%    global parameters:
%
%    mc     = the number of component intervals
%    mp     = the number of points in the discrete part of the 
%             measure (mp=0 if there is none)
%    iq     = a parameter to be set equal to 1, if the user
%             provides his or her own quadrature routine, and 
%             different from 1 otherwise
%    idelta = a parameter whose default value is 1, but is 
%             preferably set equal to 2, if iq=1 and the user
%             provides Gauss-type quadrature routines
% 
%    The component intervals have to be specified (in the order 
%    left to right) by a global mcx2 array AB=[[a1 b1];[a2 b2];
%    ...;[amc bmc]],  where for infinite extreme intervals a1=-Inf
%    resp. bmc=Inf. The discrete spectrum (if mp>0) is similarly
%    specified by a global mpx2 array DM=[[x1 y1];[x2 y2];...;
%    ;[xmp ymp]] containing the abscissae and jumps.
%
%    If the user provides his or her own quadrature routine 
%    "quadown", the routine mcdis must be called with the input 
%    parameter "quad" replaced by "@quadown", otherwise with
%    "quad" replaced by "@quadgp", a general-purpose routine
%    provided in the package. The quadrature routine must have 
%    the form
%
%                             function xw=quad(N,i)
%
%    where N is the number of nodes and i identifies the interval
%    to which the routine is to be applied.
%
%    The routine mcdis also applies to measures given originally 
%    in multi-component form.
%
function [ab,Ncap,kount]=mcdis(N,eps0,quad,Nmax)
global mc mp iq idelta irout DM uv AB
f='Ncap exceeds Nmax in mcdis with irout=%2.0f\n';
if N<1, error('Input variable N out of range'), end
iNcap=1; kount=-1;
ab(:,2)=zeros(N,1); b=ones(N,1);
Ncap=floor((2*N-1)/idelta);
while any(abs(ab(:,2)-b)>eps0*abs(ab(:,2)))
  b=ab(:,2);
  kount=kount+1;
  if kount>1, iNcap=2^(floor(kount/5))*N; end
  Ncap=Ncap+iNcap;
  if Ncap>Nmax 
    fprintf(f,irout)
    return
  end
  mtNcap=mc*Ncap;
  if iq~=1, uv=fejer(Ncap); end 
  for i=1:mc
    im1tn=(i-1)*Ncap;
    xw=feval(quad,Ncap,i);
    xwm(im1tn+1:im1tn+Ncap,:)=xw;
  end
  if mp~=0, xwm(mtNcap+1:mtNcap+mp,:)=DM; end
  if irout==1
    ab=stieltjes(N,xwm);
  else
    ab=lanczos(N,xwm);
  end 
end
% Ncap, kount