//  ***  kraw.sci  ***
//  Krawczyk operator
//  by Shin'ichi OISHI
//  1999.12.20
//  x is an [n,1]-mat-tlist expressing approximate
//  solution of func(x)=0.
//  name is a function name, for instance,  name = 'func'. 
//  T=||B(0,d1)|| and H=||K(x+T)-x||.
//  Thus if T > H, then an exact solution exists 
//  in x+B(0,d1). 
function [T,H]=kraw(x, name)
   ss=x('dim');
   dim=ss(1);
   e1=0.5*10^{-15};
   h=i(-e1,e1);
   ix=x;
   for i=1:dim;
      ix(i+2) = x(i+2) + h;
   end;
   s = name + '(ix)';
   z = evstr(s);
   v = x;
   for i=1:dim,
      v(i+2)=z(i+2);
   end;
   y=init_dif(x);
   s=name + '(y)';
   zz=evstr(s);
   j=jacobi(zz);
   vv=val(zz);
   d=j\vv;
   rad=2*norm(d,'inf');
   R=inv(j);
   d1=max(10^{-13},abs((rad)));
   hh=i(-d1,d1);
   ix=x;
   for i=1:dim,
      ix(i+2) = x(i+2) + hh;
   end;
   y=init_dif(ix);
   F = evstr(s);
   Fc = eye(dim,dim);
   Fw = eye(dim,dim);
   fc = ones(dim,1);
   fw = ones(dim,1);
   up();
   for i=1:dim,
      fc(i) = (v(i+2)(2)+v(i+2)(3))/2;
      fw(i) = fc(i)-v(i+2)(2);
   end;
   for i=1:dim,
      for j=1:dim,
         Fc(i,j)=(F(i+2)(4)(j+2)(2)+F(i+2)(4)(j+2)(3))/2;
         Fw(i,j)=Fc(i,j)-F(i+2)(4)(j+2)(2);
      end;
   end;
   aR=abs(R);
   ru = (-R)*fc + aR * fw;
   Mu = eye(dim,dim) + (-R)*Fc + aR * Fw;
   down();
   rd = (-R)*fc + (-aR)*fw;
   Md = eye(dim,dim) + (-R)*Fc + (-aR) * Fw;
   Mm = max(abs(Md),abs(Mu));
   up();
   q = d1*Mm*ones(dim,1);
   Hu = ru+q;
   down();
   Hd = rd+(-q);
   H=max(norm(Hu,'inf'),norm(Hd,'inf'));
   T=d1;
endfunction