function demo1d
% *** demo1b.m with better integral equation      ***
% *** and inner product preserving discretization ***
  close all
  format long
  format compact
%
% *** user specified quantities **********
  lambda=0.999   % parameter lambda
  theta=pi/2     % corner opening angle
  npan=10;       % number of coarse panels
  evec=1;        % a unit vector 
% ****************************************
  T16=Tinit16;
  W16=Winit16;
%
  for nsub=1:100
%
% *** panel breakpoints and interval lengths in parameter ***
  npanfin=npan+2*nsub;
  sinterfin=zeros(npanfin+1,1);
  sinterfin(1:npan+1)=linspace(0,1,npan+1);
  sinterfindiff=ones(npan+2*nsub,1)/npan;
  for k=1:nsub
    sinterfin(3:end)=sinterfin(2:end-1);
    sinterfin(2)=(sinterfin(1)+sinterfin(2))/2;   
    sinterfindiff(3:end)=sinterfindiff(2:end-1);
    sinterfindiff(2)=sinterfindiff(1)/2;
    sinterfindiff(1)=sinterfindiff(1)/2;   
  end
  sinterfin(end-nsub:end)=1-flipud(sinterfin(1:nsub+1));
  sinterfindiff(end-nsub-1:end)=flipud(sinterfindiff(1:nsub+2));
%
% *** discretization points and weights ***  
  npfin=16*npanfin
  sfin=zeros(npfin,1);
  wfin=zeros(npfin,1);
  for k=1:npanfin
    myind=k*16-15:k*16;
    sdif=sinterfindiff(k)/2;
    sfin(myind)=(sinterfin(k)+sinterfin(k+1))/2+sdif*T16;
    wfin(myind)=W16*sdif;
  end
  zfin=zfunc(sfin,theta) ;
  zpfin=zpfunc(sfin,theta);
  zppfin=zppfunc(sfin,theta);
% *** some extra presicion gained from symmetry ***
  zfin(npfin/2+1:npfin)=conj(flipud(zfin(1:npfin/2)));
  zpfin(npfin/2+1:npfin)=-conj(flipud(zpfin(1:npfin/2)));
  zppfin(npfin/2+1:npfin)=conj(flipud(zppfin(1:npfin/2)));
% *************************************************
  nzfin=-1i*zpfin./abs(zpfin);
  wzpfin=wfin.*zpfin;
  arclength=sum(abs(wzpfin))
  area=0.5*imag(conj(zfin).'*wzpfin)
%
% *** The inner product preserving K_fin matrix is set up ***
  sqrtwfin=sqrt(wfin);
  wzpfinPres=sqrtwfin.*zpfin;
  Kfin=M1AinitPres(zfin,zpfin,zppfin,nzfin,wfin,sqrtwfin,wzpfinPres,npfin);
%
% *** Solving main linear system ***
  g=2*sqrtwfin.*real(conj(evec)*nzfin);     
  MAT=lambda*Kfin+sqrtwfin*abs(wzpfinPres)';
%
  [rhofin,it]=myGMRES(MAT,lambda*g,npfin,100,eps);
  GMRES_iter=it
%
% *** Post processing ***
  qfin=rhofin.'*real(conj(evec)*zfin.*abs(wzpfinPres))

    qref=1.1300163213105365;
    myerr=abs(qref-qfin)/abs(qref);
%
    set(0,'DefaultAxesFontSize',16)
    set(0,'DefaultAxesLineWidth',1) 
%
    figure(1)
    semilogy(nsub,myerr,'*')
    axis([0 100 1e-16 1e0])
    title('Convergence of {\itq} with mesh refinement','FontSize',16)
    xlabel('Number of subdivisions {\itn}_{sub}','FontSize',16)
    ylabel('Estimated relative error in {\itq}','FontSize',16)  
    axis square
    grid on  
    hold on

    figure(2)
    semilogy(nsub,it,'*')
    axis([0 100 1 100])
    title('GMRES iterations for full convergence','FontSize',16)
    xlabel('Number of subdivisions {\itn}_{sub}','FontSize',16)
    ylabel('Number of iterations ','FontSize',16)  
    axis square
    grid on  
    hold on  
  
  end  
  
  
  function [x,it]=myGMRES(A,b,n,m,tol)
% *** GMRES with low-threshold stagnation control ***
  V=zeros(n,m+1);
  H=zeros(m);
  cs=zeros(m,1);
  sn=zeros(m,1);
  bnrm2=norm(b);
  V(:,1)=b/bnrm2;
  s=bnrm2*eye(m+1,1);
  for it = 1:m                                  
    it1=it+1;                                   
    w=A*V(:,it);
    for k = 1:it
      H(k,it)=V(:,k)'*w;
      w=w-H(k,it)*V(:,k);
    end
    H(it,it)=H(it,it)+1;
    wnrm2=norm(w);
    V(:,it1)=w/wnrm2;
    for k=1:it-1                                
      temp     = cs(k)*H(k,it)+sn(k)*H(k+1,it);
      H(k+1,it)=-sn(k)*H(k,it)+cs(k)*H(k+1,it);
      H(k,it)  = temp;
    end
    [cs(it),sn(it)]=rotmat(H(it,it),wnrm2);     
    H(it,it)= cs(it)*H(it,it)+sn(it)*wnrm2;
    s(it1) =-sn(it)*s(it);                      
    s(it)  = cs(it)*s(it);                         
    myerr=abs(s(it1))/bnrm2;
    if (myerr<=tol)|(it==m)                     
      predicted_residual=myerr
      y=triu(H(1:it,1:it))\s(1:it);             
      x=fliplr(V(:,1:it))*flipud(y);
      true_residual=norm(x+A*x-b)/bnrm2
      break;
    end
  end

  function [c,s]=rotmat(a,b)
  if  b==0
    c=1;
    s=0;
  elseif abs(b)>abs(a)
    temp=a/b;
    s=1/sqrt(1+temp^2);
    c=temp*s;
  else
    temp=b/a;
    c=1/sqrt(1+temp^2);
    s=temp*c;
  end

  function M1=M1AinitPres(z,zp,zpp,nz,w,sqrtw,wzp,N)
% *** inner product preserving adjoint of double layer potential ***   
  M1=zeros(N);
  for m=1:N
    M1(:,m)=sqrtw*abs(wzp(m)).*real(nz./(z(m)-z));
    M1(m,m)=-w(m)*imag(zpp(m)/zp(m))/2;      
  end
  M1=M1/pi;
    
  function zout=zfunc(s,theta)
  zout=sin(pi*s).*exp(1i*theta*(s-0.5));

  function zpout=zpfunc(s,theta)
  zpout=(pi*cos(pi*s)+1i*theta*sin(pi*s)).*exp(1i*theta*(s-0.5));
  
  function zppout=zppfunc(s,theta)
  zppout=(2i*pi*theta*cos(pi*s)-(theta^2+pi^2)*sin(pi*s)).* ...
	 exp(1i*theta*(s-0.5));  

  function T=Tinit16
% *** 16-point Gauss-Legendre nodes ***  
  T=zeros(16,1);
  T( 1)=-0.989400934991649932596154173450332627;
  T( 2)=-0.944575023073232576077988415534608345;
  T( 3)=-0.865631202387831743880467897712393132;
  T( 4)=-0.755404408355003033895101194847442268;
  T( 5)=-0.617876244402643748446671764048791019;
  T( 6)=-0.458016777657227386342419442983577574;
  T( 7)=-0.281603550779258913230460501460496106;
  T( 8)=-0.095012509837637440185319335424958063;
  T( 9)= 0.095012509837637440185319335424958063;
  T(10)= 0.281603550779258913230460501460496106;
  T(11)= 0.458016777657227386342419442983577574;
  T(12)= 0.617876244402643748446671764048791019;
  T(13)= 0.755404408355003033895101194847442268;
  T(14)= 0.865631202387831743880467897712393132;
  T(15)= 0.944575023073232576077988415534608345;
  T(16)= 0.989400934991649932596154173450332627;

  function W=Winit16
% *** 16-point Gauss-Legendre weights ***  
  W=zeros(16,1); 
  W( 1)= 0.027152459411754094851780572456018104;
  W( 2)= 0.062253523938647892862843836994377694;
  W( 3)= 0.095158511682492784809925107602246226;
  W( 4)= 0.124628971255533872052476282192016420;
  W( 5)= 0.149595988816576732081501730547478549;
  W( 6)= 0.169156519395002538189312079030359962;
  W( 7)= 0.182603415044923588866763667969219939;
  W( 8)= 0.189450610455068496285396723208283105;
  W( 9)= 0.189450610455068496285396723208283105;
  W(10)= 0.182603415044923588866763667969219939;
  W(11)= 0.169156519395002538189312079030359962;
  W(12)= 0.149595988816576732081501730547478549;
  W(13)= 0.124628971255533872052476282192016420;
  W(14)= 0.095158511682492784809925107602246226;
  W(15)= 0.062253523938647892862843836994377694;
  W(16)= 0.027152459411754094851780572456018104;