Sun Aug 18 18:00:52 2002 run on Windows
%***********************************************************************
%*****                                                             *****
%***** Package  F I D E - Test Examples Ver. 1.1.2 May 29,1995     *****
%*****                                                             *****
%***********************************************************************
%***********************************************************************
%*****                                                             *****
%*****    T e s t     Examples   ---   Module    E X P R E S       *****
%*****                                                             *****
%***********************************************************************

let cos th**2=1 - sin th**2,
    cos fi**2=1 - sin fi**2;


factor df;


on rat;


for all x,y let diff(x,y)=df(x,y);


depend u,r,th,fi;


depend v,r,th,fi;


depend f,r,th,fi;


depend w,r,th,fi;


% Spherical coordinate system
scalefactors 3,r*sin th*cos fi,r*sin th*sin fi,r*cos th,r,th,fi;


tensor a1,a2,a3,a4,a5;


vectors u,v;


dyads w;


a1:=grad f;

a1:= ( df(f,r) , 

        df(f,th)
       ---------- , 
           r

        df(f,fi)
       ----------- ) 
        sin(th)*r


a2:=div u;

     df(u(3),fi)     df(u(2),th)                  cos(th)*u(2) + 2*sin(th)*u(1)
a2:=------------- + ------------- + df(u(1),r) + -------------------------------
      sin(th)*r           r                                 sin(th)*r


a3:=curl v;

        df(v(3),th)      - df(v(2),fi)     cos(th)*v(3)
a3:= ( ------------- + ---------------- + -------------- , 
             r            sin(th)*r         sin(th)*r

                        df(v(1),fi)      - v(3)
        - df(v(3),r) + ------------- + --------- , 
                         sin(th)*r         r

                      - df(v(1),th)     v(2)
       df(v(2),r) + ---------------- + ------ ) 
                           r             r


a4:=lapl v;

         - 2*df(v(3),fi)      - 2*df(v(2),th)     df(v(1),fi,2)
a4:= ( ------------------ + ------------------ + --------------- + df(v(1),r,2)
                    2                2                    2  2
           sin(th)*r                r              sin(th) *r

           2*df(v(1),r)     df(v(1),th,2)     df(v(1),th)*cos(th)
        + -------------- + --------------- + ---------------------
                r                 2                        2
                                 r                sin(th)*r

           2*(cos(th)*v(2) + sin(th)*v(1))
        - --------------------------------- , 
                              2
                     sin(th)*r

         - 2*df(v(3),fi)*cos(th)     df(v(2),fi,2)
       -------------------------- + --------------- + df(v(2),r,2)
                     2  2                    2  2
              sin(th) *r              sin(th) *r

           2*df(v(2),r)     df(v(2),th,2)     df(v(2),th)*cos(th)
        + -------------- + --------------- + ---------------------
                r                 2                        2
                                 r                sin(th)*r

           2*df(v(1),th)        - v(2)
        + --------------- + ------------- , 
                 2                  2  2
                r            sin(th) *r

        df(v(3),fi,2)                    2*df(v(3),r)     df(v(3),th,2)
       --------------- + df(v(3),r,2) + -------------- + ---------------
                2  2                          r                 2
         sin(th) *r                                            r

           df(v(3),th)*cos(th)     2*df(v(2),fi)*cos(th)     2*df(v(1),fi)
        + --------------------- + ----------------------- + ---------------
                        2                      2  2                    2
               sin(th)*r                sin(th) *r            sin(th)*r

              - v(3)
        + ------------- ) 
                  2  2
           sin(th) *r


a3:=2*a3+a4;

         - 2*df(v(3),fi)     2*df(v(3),th)      - 2*df(v(2),fi)
a3:= ( ------------------ + --------------- + ------------------
                    2              r              sin(th)*r
           sin(th)*r

            - 2*df(v(2),th)     df(v(1),fi,2)                    2*df(v(1),r)
        + ------------------ + --------------- + df(v(1),r,2) + --------------
                   2                    2  2                          r
                  r              sin(th) *r

           df(v(1),th,2)     df(v(1),th)*cos(th)
        + --------------- + ---------------------
                 2                        2
                r                sin(th)*r

           2*(cos(th)*v(3)*r - cos(th)*v(2) - sin(th)*v(1))
        + -------------------------------------------------- , 
                                       2
                              sin(th)*r

         - 2*df(v(3),fi)*cos(th)                    df(v(2),fi,2)
       -------------------------- - 2*df(v(3),r) + ---------------
                     2  2                                   2  2
              sin(th) *r                             sin(th) *r

                          2*df(v(2),r)     df(v(2),th,2)
        + df(v(2),r,2) + -------------- + ---------------
                               r                 2
                                                r

           df(v(2),th)*cos(th)     2*df(v(1),fi)     2*df(v(1),th)
        + --------------------- + --------------- + ---------------
                        2            sin(th)*r             2
               sin(th)*r                                  r

                       2
            - 2*sin(th) *v(3)*r - v(2)
        + ----------------------------- , 
                          2  2
                   sin(th) *r

        df(v(3),fi,2)                    2*df(v(3),r)     df(v(3),th,2)
       --------------- + df(v(3),r,2) + -------------- + ---------------
                2  2                          r                 2
         sin(th) *r                                            r

           df(v(3),th)*cos(th)     2*df(v(2),fi)*cos(th)
        + --------------------- + ----------------------- + 2*df(v(2),r)
                        2                      2  2
               sin(th)*r                sin(th) *r

                                                           2
           2*df(v(1),fi)      - 2*df(v(1),th)     2*sin(th) *v(2)*r - v(3)
        + --------------- + ------------------ + -------------------------- ) 
                     2              r                          2  2
            sin(th)*r                                   sin(th) *r


a5:=lapl f;

     df(f,fi,2)                  2*df(f,r)     df(f,th,2)     df(f,th)*cos(th)
a5:=------------- + df(f,r,2) + ----------- + ------------ + ------------------
            2  2                     r              2                     2
     sin(th) *r                                    r             sin(th)*r


a1:=a1+div w;

        df(w(3,1),fi)     df(w(2,1),th)
a1:= ( --------------- + --------------- + df(w(1,1),r) + df(f,r)
          sin(th)*r             r

           cos(th)*w(2,1) - sin(th)*w(3,3) - sin(th)*w(2,2) + 2*sin(th)*w(1,1)
        + ---------------------------------------------------------------------
                                        sin(th)*r

        , 

        df(w(3,2),fi)     df(w(2,2),th)                    df(f,th)
       --------------- + --------------- + df(w(1,2),r) + ---------- + 
          sin(th)*r             r                             r

         - cos(th)*w(3,3) + cos(th)*w(2,2) + sin(th)*w(2,1) + 2*sin(th)*w(1,2)
       ------------------------------------------------------------------------
                                      sin(th)*r

        , 

        df(w(3,3),fi)     df(w(2,3),th)                    df(f,fi)
       --------------- + --------------- + df(w(1,3),r) + -----------
          sin(th)*r             r                          sin(th)*r

           cos(th)*w(3,2) + cos(th)*w(2,3) + sin(th)*w(3,1) + 2*sin(th)*w(1,3)
        + ---------------------------------------------------------------------
                                        sin(th)*r

     ) 


a1:=u.dyad((a,0,1),(1,b,3),(0,c,d));

a1:= ( u(2) + u(1)*a , 

       u(3)*c + u(2)*b , 

       u(3)*d + 3*u(2) + u(1) ) 


a2:=vect(a,b,c);

a2:= ( a , 

       b , 

       c ) 


a1.a2;

                                     2                    2
u(3)*b*c + u(3)*c*d + u(2)*a + u(2)*b  + 3*u(2)*c + u(1)*a  + u(1)*c


%  Scalar product
u.v;

u(3)*v(3) + u(2)*v(2) + u(1)*v(1)


%  Vector product
u?v;

 (  - u(3)*v(2) + u(2)*v(3) , 

   u(3)*v(1) - u(1)*v(3) , 

    - u(2)*v(1) + u(1)*v(2) ) 


%  Dyadic
u&v;

 (  ( u(1)*v(1) , 

      u(1)*v(2) , 

      u(1)*v(3) )  , 

    ( u(2)*v(1) , 

      u(2)*v(2) , 

      u(2)*v(3) )  , 

    ( u(3)*v(1) , 

      u(3)*v(2) , 

      u(3)*v(3) )  ) 


%  Directional derivative
dirdf(u,v);

    df(v(1),fi)*u(3)                       df(v(1),th)*u(2)
 ( ------------------ + df(v(1),r)*u(1) + ------------------
       sin(th)*r                                  r

       u(3)*v(3) + u(2)*v(2)
    - ----------------------- , 
                 r

    df(v(2),fi)*u(3)                       df(v(2),th)*u(2)
   ------------------ + df(v(2),r)*u(1) + ------------------
       sin(th)*r                                  r

        - cos(th)*u(3)*v(3) + sin(th)*u(2)*v(1)
    + ------------------------------------------ , 
                      sin(th)*r

    df(v(3),fi)*u(3)                       df(v(3),th)*u(2)
   ------------------ + df(v(3),r)*u(1) + ------------------
       sin(th)*r                                  r

       u(3)*(cos(th)*v(2) + sin(th)*v(1))
    + ------------------------------------ ) 
                   sin(th)*r


clear a1,a2,a3,a4,a5,u,v,w;


for all x,y clear diff(x,y);


clear cos th**2,
      cos fi**2;


remfac df;


off rat;


scalefactors 3,x,y,z,x,y,z;



%***********************************************************************
%*****                                                             *****
%*****      T e s t     Examples   ---   Module    I I M E T       *****
%*****                                                             *****
%***********************************************************************

% Example I.1 - 1-D Lagrangian Hydrodynamics

off exp;


factor diff;


on rat,eqfu;



% Declare which indexes will be given to coordinates
coordinates x,t into j,m;



% Declares uniform grid in x coordinate
grid uniform,x;



% Declares dependencies of functions on coordinates
dependence eta(t,x),v(t,x),eps(t,x),p(t,x);



% Declares p as known function
given p;



same eta,v,p;



iim a, eta,diff(eta,t)-eta*diff(v,x)=0,
    v,diff(v,t)+eta/ro*diff(p,x)=0,
    eps,diff(eps,t)+eta*p/ro*diff(v,x)=0;


*****************************
*****      Program      *****          IIMET Ver 1.1.2
*****************************

      Partial Differential Equations
      ==============================

diff(eta,t) - diff(v,x)*eta    =    0

 diff(p,x)*eta
--------------- + diff(v,t)    =    0
      ro

               diff(v,x)*eta*p
diff(eps,t) + -----------------    =    0
                     ro


 Backtracking needed in grid optimalization
0 interpolations are needed in x coordinate
  Equation for eta variable is integrated in half grid point
  Equation for v variable is integrated in half grid point
  Equation for eps variable is integrated in half grid point
0 interpolations are needed in t coordinate
  Equation for eta variable is integrated in half grid point
  Equation for v variable is integrated in half grid point
  Equation for eps variable is integrated in half grid point

         Equations after Discretization Using IIM :
         ==========================================

(4*(eta(j,m + 1) - eta(j,m) - eta(j + 1,m) + eta(j + 1,m + 1))*hx - (

    (eta(j + 1,m + 1) + eta(j,m + 1))*(v(j + 1,m + 1) - v(j,m + 1))

     + (eta(j + 1,m) + eta(j,m))*(v(j + 1,m) - v(j,m)))*(ht(m + 1) + ht(m)))/(4

   *(ht(m + 1) + ht(m))*hx)   =   0


(4*(v(j,m + 1) - v(j,m) - v(j + 1,m) + v(j + 1,m + 1))*hx*ro + (

    (eta(j + 1,m + 1) + eta(j,m + 1))*(p(j + 1,m + 1) - p(j,m + 1))

     + (eta(j + 1,m) + eta(j,m))*(p(j + 1,m) - p(j,m)))*(ht(m + 1) + ht(m)))/(4

   *(ht(m + 1) + ht(m))*hx*ro)   =   0


(4*(eps(j,m + 1) - eps(j,m) - eps(j + 1,m) + eps(j + 1,m + 1))*hx*ro + (

    (eta(j + 1,m + 1)*p(j + 1,m + 1) + eta(j,m + 1)*p(j,m + 1))

    *(v(j + 1,m + 1) - v(j,m + 1))

     + (eta(j + 1,m)*p(j + 1,m) + eta(j,m)*p(j,m))*(v(j + 1,m) - v(j,m)))

 *(ht(m + 1) + ht(m)))/(4*(ht(m + 1) + ht(m))*hx*ro)   =   0




clear a;


clearsame;


cleargiven;



%***********************************************************************

% Example  I.2 - How other functions (here sin, cos) can be used in
%                discretized terms

diffunc sin,cos;


difmatch all,diff(u*sin x,x),u=one,2,(u(i+1)*sin x(i+1)-u(i-1)
                   *sin x(i-1))/(dim1+dip1),
                 u=half,0,(u(i+1/2)*sin x(i+1/2)-u(i-1/2)*sin x(i-1/2))
                   /di;


difmatch all,cos x*diff(u,x,2),u=one,0,cos x i*(u(i+1)-2*u(i)+u(i-1))
                   /di^2,
                 u=half,3,(u(i+3/2)-u(i+1/2))/dip2/2 -
                          (u(i-1/2)-u(i-3/2))/dim2/2;


off exp;


coordinates x,t into j,m;


grid uniform,x,t;


dependence u(x,t),v(x,t);


iim a,u,diff(u,t)+diff(u,x)+cos x*diff(v,x,2)=0,
      v,diff(v,t)+diff(sin x*u,x)=0;


*****************************
*****      Program      *****          IIMET Ver 1.1.2
*****************************

      Partial Differential Equations
      ==============================

diff(u,t) + diff(u,x) + diff(v,x,2)*cos(x)    =    0

diff(sin(x)*u,x) + diff(v,t)    =    0


0 interpolations are needed in x coordinate
  Equation for u variable is integrated in half grid point
  Equation for v variable is integrated in half grid point
0 interpolations are needed in t coordinate
  Equation for u variable is integrated in half grid point
  Equation for v variable is integrated in half grid point

         Equations after Discretization Using IIM :
         ==========================================

           2*j + 1              2*j + 1           2*j + 3
 - ((2*(v(---------,m + 1) + v(---------,m)) - v(---------,m)
              2                    2                 2

           2*j + 3              2*j - 1          2*j - 1
      - v(---------,m + 1) - v(---------,m) - v(---------,m + 1))
              2                    2                2

            2*j + 1
    *cos(x(---------))*ht + (
               2

       (u(j,m + 1) + u(j,m) - u(j + 1,m) - u(j + 1,m + 1))*ht

                                                                              2
        - (u(j,m + 1) - u(j,m) - u(j + 1,m) + u(j + 1,m + 1))*hx)*hx)/(2*ht*hx )

   =   0


( - ((u(j,m + 1) + u(j,m))*sin(x(j))*ht

              2*j + 1              2*j + 1
      - 2*(v(---------,m + 1) - v(---------,m))*hx)
                 2                    2

  + (u(j + 1,m + 1) + u(j + 1,m))*sin(x(j + 1))*ht)/(2*ht*hx)   =   0



clear a;



%***********************************************************************

% Example I.3 - Schrodinger equation

factor diff;


coordinates t,x into m,j;


grid uniform,x,t;


dependence ur(x,t),ui(x,t);


same ui,ur;


iim a,ur,-diff(ui,t)+1/2*diff(ur,x,2)+(ur**2+ui**2)*ur=0,
       ui,diff(ur,t)+1/2*diff(ui,x,2)+(ur**2+ui**2)*ui=0;


*****************************
*****      Program      *****          IIMET Ver 1.1.2
*****************************

      Partial Differential Equations
      ==============================

                 diff(ur,x,2)                   2     2
 - diff(ui,t) + --------------    =     - ur*(ui  + ur )
                      2

 diff(ui,x,2)                                2     2
-------------- + diff(ur,t)    =     - ui*(ui  + ur )
      2


0 interpolations are needed in t coordinate
  Equation for ur variable is integrated in half grid point
  Equation for ui variable is integrated in half grid point
0 interpolations are needed in x coordinate
  Equation for ur variable is integrated in one grid point
  Equation for ui variable is integrated in one grid point

         Equations after Discretization Using IIM :
         ==========================================

((ur(m,j + 1) - 2*ur(m,j) + ur(m,j - 1) - 2*ur(m + 1,j) + ur(m + 1,j + 1)

                                                       2          2
   + ur(m + 1,j - 1))*ht - 4*(ui(m + 1,j) - ui(m,j))*hx )/(4*ht*hx )   =   

               3          3          2                      2
    ur(m + 1,j)  + ur(m,j)  + ui(m,j) *ur(m,j) + ui(m + 1,j) *ur(m + 1,j)
 - -----------------------------------------------------------------------
                                      2


((ui(m,j + 1) - 2*ui(m,j) + ui(m,j - 1) - 2*ui(m + 1,j) + ui(m + 1,j + 1)

                                                       2          2
   + ui(m + 1,j - 1))*ht + 4*(ur(m + 1,j) - ur(m,j))*hx )/(4*ht*hx )   =   

                2              2                        2          2
    (ui(m + 1,j)  + ur(m + 1,j) )*ui(m + 1,j) + (ui(m,j)  + ur(m,j) )*ui(m,j)
 - ---------------------------------------------------------------------------
                                        2



clear a;


clearsame;



%***********************************************************************

% Example I.4 - Vector calculus in p.d.e. input
%               cooperation with expres module
%               2-D hydrodynamics

scalefactors 2,x,y,x,y;


vectors u;


off exp,twogrid;


on eqfu;


factor diff,ht,hx,hy;


coordinates x,y,t into j,i,m;


grid uniform,x,y,t;


dependence n(t,x,y),u(t,x,y),p(t,x,y);


iim a,n,diff(n,t)+u.grad n+n*div u=0,
     u,m*n*(diff(u,t)+u.grad u)+grad p=vect(0,0),
     p,3/2*(diff(p,t)+u.grad p)+5/2*p*div u=0;


*****************************
*****      Program      *****          IIMET Ver 1.1.2
*****************************

      Partial Differential Equations
      ==============================

diff(n,t) + diff(n,x)*u1 + diff(n,y)*u2 + diff(u1,x)*n + diff(u2,y)*n    =    0

diff(p,x) + diff(u1,t)*m*n + diff(u1,x)*m*n*u1 + diff(u1,y)*m*n*u2    =    0

diff(p,y) + diff(u2,t)*m*n + diff(u2,x)*m*n*u1 + diff(u2,y)*m*n*u2    =    0

 3*diff(p,t)     3*diff(p,x)*u1     3*diff(p,y)*u2     5*diff(u1,x)*p
------------- + ---------------- + ---------------- + ----------------
      2                2                  2                  2

    5*diff(u2,y)*p
 + ----------------    =    0
          2


0 interpolations are needed in x coordinate
  Equation for n variable is integrated in half grid point
  Equation for u1 variable is integrated in half grid point
  Equation for u2 variable is integrated in half grid point
  Equation for p variable is integrated in half grid point
0 interpolations are needed in y coordinate
  Equation for n variable is integrated in half grid point
  Equation for u1 variable is integrated in half grid point
  Equation for u2 variable is integrated in half grid point
  Equation for p variable is integrated in half grid point
0 interpolations are needed in t coordinate
  Equation for n variable is integrated in half grid point
  Equation for u1 variable is integrated in half grid point
  Equation for u2 variable is integrated in half grid point
  Equation for p variable is integrated in half grid point

         Equations after Discretization Using IIM :
         ==========================================

   -1   -1
(hy  *hx  *(n(j + 1,i + 1,m + 1)*u1(j + 1,i + 1,m + 1)*hy

             + n(j + 1,i + 1,m + 1)*u2(j + 1,i + 1,m + 1)*hx

             + n(j + 1,i + 1,m)*u1(j + 1,i + 1,m)*hy

             + n(j + 1,i + 1,m)*u2(j + 1,i + 1,m)*hx

             + n(j + 1,i,m + 1)*u1(j + 1,i,m + 1)*hy

             - n(j + 1,i,m + 1)*u2(j + 1,i,m + 1)*hx

             + n(j + 1,i,m)*u1(j + 1,i,m)*hy - n(j + 1,i,m)*u2(j + 1,i,m)*hx

             - n(j,i + 1,m + 1)*u1(j,i + 1,m + 1)*hy

             + n(j,i + 1,m + 1)*u2(j,i + 1,m + 1)*hx

             - n(j,i + 1,m)*u1(j,i + 1,m)*hy + n(j,i + 1,m)*u2(j,i + 1,m)*hx

             - n(j,i,m + 1)*u1(j,i,m + 1)*hy - n(j,i,m + 1)*u2(j,i,m + 1)*hx

                                                                      -1
             - n(j,i,m)*u1(j,i,m)*hy - n(j,i,m)*u2(j,i,m)*hx))/4 + (ht  *(

      n(j,i,m + 1) - n(j,i,m) - n(j,i + 1,m) + n(j,i + 1,m + 1) - n(j + 1,i,m)

       + n(j + 1,i,m + 1) - n(j + 1,i + 1,m) + n(j + 1,i + 1,m + 1)))/4   =   0


   -1
(hx  *((n(j + 1,i,m + 1)*u1(j + 1,i,m + 1) + n(j,i,m + 1)*u1(j,i,m + 1))

       *(u1(j + 1,i,m + 1) - u1(j,i,m + 1)) + 

       (n(j + 1,i,m)*u1(j + 1,i,m) + n(j,i,m)*u1(j,i,m))

       *(u1(j + 1,i,m) - u1(j,i,m)) + 

       (n(j + 1,i + 1,m)*u1(j + 1,i + 1,m) + n(j,i + 1,m)*u1(j,i + 1,m))

       *(u1(j + 1,i + 1,m) - u1(j,i + 1,m)) + (

          n(j + 1,i + 1,m + 1)*u1(j + 1,i + 1,m + 1)

           + n(j,i + 1,m + 1)*u1(j,i + 1,m + 1))

                                                               -1   -1   -1
       *(u1(j + 1,i + 1,m + 1) - u1(j,i + 1,m + 1)))*m)/8 + (hy  *hx  *ht  *(((

            (n(j,i + 1,m + 1) + n(j,i + 1,m))

            *(u1(j,i + 1,m + 1) - u1(j,i + 1,m))

             + (n(j,i,m + 1) + n(j,i,m))*(u1(j,i,m + 1) - u1(j,i,m)) + 

            (n(j + 1,i,m + 1) + n(j + 1,i,m))

            *(u1(j + 1,i,m + 1) - u1(j + 1,i,m)) + 

            (n(j + 1,i + 1,m + 1) + n(j + 1,i + 1,m))

            *(u1(j + 1,i + 1,m + 1) - u1(j + 1,i + 1,m)))*hx*m + 2*(

            p(j + 1,i,m + 1) + p(j + 1,i,m) + p(j + 1,i + 1,m)

             + p(j + 1,i + 1,m + 1)

             - (p(j,i,m + 1) + p(j,i,m) + p(j,i + 1,m) + p(j,i + 1,m + 1)))*ht)

      *hy + ((n(j,i + 1,m + 1)*u2(j,i + 1,m + 1) + n(j,i,m + 1)*u2(j,i,m + 1))

             *(u1(j,i + 1,m + 1) - u1(j,i,m + 1)) + 

             (n(j,i + 1,m)*u2(j,i + 1,m) + n(j,i,m)*u2(j,i,m))

             *(u1(j,i + 1,m) - u1(j,i,m)) + 

             (n(j + 1,i + 1,m)*u2(j + 1,i + 1,m) + n(j + 1,i,m)*u2(j + 1,i,m))

             *(u1(j + 1,i + 1,m) - u1(j + 1,i,m)) + (

                n(j + 1,i + 1,m + 1)*u2(j + 1,i + 1,m + 1)

                 + n(j + 1,i,m + 1)*u2(j + 1,i,m + 1))

             *(u1(j + 1,i + 1,m + 1) - u1(j + 1,i,m + 1)))*ht*hx*m))/8   =   0


   -1   -1
(hy  *hx  *(((n(j + 1,i,m + 1)*u1(j + 1,i,m + 1) + n(j,i,m + 1)*u1(j,i,m + 1))

             *(u2(j + 1,i,m + 1) - u2(j,i,m + 1)) + 

             (n(j + 1,i,m)*u1(j + 1,i,m) + n(j,i,m)*u1(j,i,m))

             *(u2(j + 1,i,m) - u2(j,i,m)) + 

             (n(j + 1,i + 1,m)*u1(j + 1,i + 1,m) + n(j,i + 1,m)*u1(j,i + 1,m))

             *(u2(j + 1,i + 1,m) - u2(j,i + 1,m)) + (

                n(j + 1,i + 1,m + 1)*u1(j + 1,i + 1,m + 1)

                 + n(j,i + 1,m + 1)*u1(j,i + 1,m + 1))

             *(u2(j + 1,i + 1,m + 1) - u2(j,i + 1,m + 1)))*hy + (

               (n(j,i + 1,m + 1)*u2(j,i + 1,m + 1) + n(j,i,m + 1)*u2(j,i,m + 1))

               *(u2(j,i + 1,m + 1) - u2(j,i,m + 1)) + 

               (n(j,i + 1,m)*u2(j,i + 1,m) + n(j,i,m)*u2(j,i,m))

               *(u2(j,i + 1,m) - u2(j,i,m)) + 

               (n(j + 1,i + 1,m)*u2(j + 1,i + 1,m) + n(j + 1,i,m)*u2(j + 1,i,m))

               *(u2(j + 1,i + 1,m) - u2(j + 1,i,m)) + (

                  n(j + 1,i + 1,m + 1)*u2(j + 1,i + 1,m + 1)

                   + n(j + 1,i,m + 1)*u2(j + 1,i,m + 1))

                                                                              -1
               *(u2(j + 1,i + 1,m + 1) - u2(j + 1,i,m + 1)))*hx)*m)/8 + ( - hy

      -1
   *ht  *(2*(p(j,i,m + 1) + p(j,i,m) - p(j,i + 1,m) - p(j,i + 1,m + 1)

              + p(j + 1,i,m) + p(j + 1,i,m + 1) - p(j + 1,i + 1,m)

              - p(j + 1,i + 1,m + 1))*ht - ((n(j,i + 1,m + 1) + n(j,i + 1,m))

             *(u2(j,i + 1,m + 1) - u2(j,i + 1,m))

              + (n(j,i,m + 1) + n(j,i,m))*(u2(j,i,m + 1) - u2(j,i,m)) + 

             (n(j + 1,i,m + 1) + n(j + 1,i,m))

             *(u2(j + 1,i,m + 1) - u2(j + 1,i,m)) + 

             (n(j + 1,i + 1,m + 1) + n(j + 1,i + 1,m))

             *(u2(j + 1,i + 1,m + 1) - u2(j + 1,i + 1,m)))*hy*m))/8   =   0


   -1   -1
(hy  *hx  *(3*((p(j + 1,i,m + 1) - p(j,i,m + 1))

               *(u1(j + 1,i,m + 1) + u1(j,i,m + 1))

                + (p(j + 1,i,m) - p(j,i,m))*(u1(j + 1,i,m) + u1(j,i,m)) + 

               (p(j + 1,i + 1,m) - p(j,i + 1,m))

               *(u1(j + 1,i + 1,m) + u1(j,i + 1,m)) + 

               (p(j + 1,i + 1,m + 1) - p(j,i + 1,m + 1))

               *(u1(j + 1,i + 1,m + 1) + u1(j,i + 1,m + 1)))*hy + 2*(

               4*p(j + 1,i + 1,m + 1)*u2(j + 1,i + 1,m + 1)

                - p(j + 1,i + 1,m + 1)*u2(j + 1,i,m + 1)

                + 4*p(j + 1,i + 1,m)*u2(j + 1,i + 1,m)

                - p(j + 1,i + 1,m)*u2(j + 1,i,m)

                + p(j + 1,i,m + 1)*u2(j + 1,i + 1,m + 1)

                - 4*p(j + 1,i,m + 1)*u2(j + 1,i,m + 1)

                + p(j + 1,i,m)*u2(j + 1,i + 1,m) - 4*p(j + 1,i,m)*u2(j + 1,i,m)

                + 4*p(j,i + 1,m + 1)*u2(j,i + 1,m + 1)

                - p(j,i + 1,m + 1)*u2(j,i,m + 1) + 4*p(j,i + 1,m)*u2(j,i + 1,m)

                - p(j,i + 1,m)*u2(j,i,m) + p(j,i,m + 1)*u2(j,i + 1,m + 1)

                - 4*p(j,i,m + 1)*u2(j,i,m + 1) + p(j,i,m)*u2(j,i + 1,m)

                - 4*p(j,i,m)*u2(j,i,m))*hx + 5*(

               (p(j + 1,i,m + 1) + p(j,i,m + 1))

               *(u1(j + 1,i,m + 1) - u1(j,i,m + 1))

                + (p(j + 1,i,m) + p(j,i,m))*(u1(j + 1,i,m) - u1(j,i,m)) + 

               (p(j + 1,i + 1,m) + p(j,i + 1,m))

               *(u1(j + 1,i + 1,m) - u1(j,i + 1,m)) + 

               (p(j + 1,i + 1,m + 1) + p(j,i + 1,m + 1))

                                                                            -1
               *(u1(j + 1,i + 1,m + 1) - u1(j,i + 1,m + 1)))*hy))/16 + (3*ht  *(

      p(j,i,m + 1) - p(j,i,m) - p(j,i + 1,m) + p(j,i + 1,m + 1) - p(j + 1,i,m)

       + p(j + 1,i,m + 1) - p(j + 1,i + 1,m) + p(j + 1,i + 1,m + 1)))/8   =   0



clear a,u;



%***********************************************************************

% Example I.5 - 1-D hydrodynamics up to 3-rd moments (heat flow)

coordinates x,t into j,m;


grid uniform,x,t;


dependence n(x,t),u(x,t),tt(x,t),p(x,t),q(x,t);


iim a, n,diff(n,t)+u*diff(n,x)+diff(u,x)=0,
    u,n*m*(diff(u,t)+u*diff(u,x))+k*diff(n*tt,x)+diff(p,x)=0,
    tt,3/2*k*n*(diff(tt,t)+u*diff(tt,x))+n*k*tt*diff(u,x)+1/2*p
       *diff(u,x)+diff(q,x)=0,
    p,diff(p,t)+u*diff(p,x)+p*diff(u,x)+n*k*tt*diff(u,x)+2/5*diff(q,x)
       =0,
    q,diff(q,t)+u*diff(q,x)+q*diff(u,x)+5/2*n*k**2*tt/m*diff(tt,x)+n*k
       *tt*diff(p,x)-p*diff(p,x)=0;


*****************************
*****      Program      *****          IIMET Ver 1.1.2
*****************************

      Partial Differential Equations
      ==============================

diff(n,t) + diff(n,x)*u + diff(u,x)    =    0

diff(n*tt,x)*k + diff(p,x) + diff(u,t)*m*n + diff(u,x)*m*n*u    =    0

             3*diff(tt,t)*k*n     3*diff(tt,x)*k*n*u
diff(q,x) + ------------------ + --------------------
                    2                     2

    diff(u,x)*(2*k*n*tt + p)
 + --------------------------    =    0
               2

                           2*diff(q,x)
diff(p,t) + diff(p,x)*u + ------------- + diff(u,x)*(k*n*tt + p)    =    0
                                5

                                                                  2
                                                    5*diff(tt,x)*k *n*tt
diff(p,x)*(k*n*tt - p) + diff(q,t) + diff(q,x)*u + ----------------------
                                                            2*m

 + diff(u,x)*q    =    0


0 interpolations are needed in x coordinate
  Equation for n variable is integrated in half grid point
  Equation for u variable is integrated in half grid point
  Equation for tt variable is integrated in half grid point
  Equation for p variable is integrated in half grid point
  Equation for q variable is integrated in half grid point
0 interpolations are needed in t coordinate
  Equation for n variable is integrated in half grid point
  Equation for u variable is integrated in half grid point
  Equation for tt variable is integrated in half grid point
  Equation for p variable is integrated in half grid point
  Equation for q variable is integrated in half grid point

         Equations after Discretization Using IIM :
         ==========================================

   -1
(hx  *((n(j + 1,m + 1) - n(j,m + 1))*(u(j + 1,m + 1) + u(j,m + 1))

        + (n(j + 1,m) - n(j,m))*(u(j + 1,m) + u(j,m))

        + 2*(u(j + 1,m + 1) + u(j + 1,m) - (u(j,m + 1) + u(j,m)))))/4

      -1
    ht  *(n(j,m + 1) - n(j,m) - n(j + 1,m) + n(j + 1,m + 1))
 + ----------------------------------------------------------   =   0
                               2


      -1
( - hx  *(n(j,m + 1)*tt(j,m + 1) + n(j,m)*tt(j,m) - n(j + 1,m)*tt(j + 1,m)

                                                       -1   -1
           - n(j + 1,m + 1)*tt(j + 1,m + 1))*k)/2 + (hx  *ht  *((

         (n(j + 1,m + 1) + n(j + 1,m))*(u(j + 1,m + 1) - u(j + 1,m))

          + (n(j,m + 1) + n(j,m))*(u(j,m + 1) - u(j,m)))*hx*m

       + 2*(p(j + 1,m + 1) + p(j + 1,m) - (p(j,m + 1) + p(j,m)))*ht + (

         (n(j + 1,m + 1)*u(j + 1,m + 1) + n(j,m + 1)*u(j,m + 1))

         *(u(j + 1,m + 1) - u(j,m + 1))

          + (n(j + 1,m)*u(j + 1,m) + n(j,m)*u(j,m))*(u(j + 1,m) - u(j,m)))*ht*m)

   )/4   =   0


   -1
(hx  *((n(j + 1,m + 1)*tt(j + 1,m + 1) + n(j,m + 1)*tt(j,m + 1))

       *(u(j + 1,m + 1) - u(j,m + 1))

        + (n(j + 1,m)*tt(j + 1,m) + n(j,m)*tt(j,m))*(u(j + 1,m) - u(j,m)))*k)/4 

     -1   -1
+ (hx  *ht  *(((p(j + 1,m + 1) + p(j,m + 1))*(u(j + 1,m + 1) - u(j,m + 1))

                + (p(j + 1,m) + p(j,m))*(u(j + 1,m) - u(j,m))

                + 4*(q(j + 1,m + 1) + q(j + 1,m) - (q(j,m + 1) + q(j,m))))*ht + 

              3*((n(j + 1,m + 1) + n(j + 1,m))*(tt(j + 1,m + 1) - tt(j + 1,m))

                  + (n(j,m + 1) + n(j,m))*(tt(j,m + 1) - tt(j,m)))*hx*k + 3*(

                 (n(j + 1,m + 1)*u(j + 1,m + 1) + n(j,m + 1)*u(j,m + 1))

                 *(tt(j + 1,m + 1) - tt(j,m + 1)) + 

                 (n(j + 1,m)*u(j + 1,m) + n(j,m)*u(j,m))*(tt(j + 1,m) - tt(j,m))

                 )*ht*k))/8   =   0


   -1
(hx  *(5*((n(j + 1,m + 1)*tt(j + 1,m + 1) + n(j,m + 1)*tt(j,m + 1))

          *(u(j + 1,m + 1) - u(j,m + 1))

           + (n(j + 1,m)*tt(j + 1,m) + n(j,m)*tt(j,m))*(u(j + 1,m) - u(j,m)))*k 

       + 2*(5*p(j + 1,m + 1)*u(j + 1,m + 1) + 5*p(j + 1,m)*u(j + 1,m)

             - 5*p(j,m + 1)*u(j,m + 1) - 5*p(j,m)*u(j,m) + 2*q(j + 1,m + 1)

             + 2*q(j + 1,m) - 2*q(j,m + 1) - 2*q(j,m))))/20

      -1
    ht  *(p(j,m + 1) - p(j,m) - p(j + 1,m) + p(j + 1,m + 1))
 + ----------------------------------------------------------   =   0
                               2


      -1
( - hx  *(2*((p(j + 1,m + 1) + p(j,m + 1))*(p(j + 1,m + 1) - p(j,m + 1))

              + (p(j + 1,m) + p(j,m))*(p(j + 1,m) - p(j,m)) - 2*(

                q(j + 1,m + 1)*u(j + 1,m + 1) + q(j + 1,m)*u(j + 1,m)

                 - q(j,m + 1)*u(j,m + 1) - q(j,m)*u(j,m)))*m - 5*(

             (n(j + 1,m + 1)*tt(j + 1,m + 1) + n(j,m + 1)*tt(j,m + 1))

             *(tt(j + 1,m + 1) - tt(j,m + 1)) + 

             (n(j + 1,m)*tt(j + 1,m) + n(j,m)*tt(j,m))*(tt(j + 1,m) - tt(j,m)))

            2
          *k  - 2*((n(j + 1,m + 1)*tt(j + 1,m + 1) + n(j,m + 1)*tt(j,m + 1))

             *(p(j + 1,m + 1) - p(j,m + 1))

              + (n(j + 1,m)*tt(j + 1,m) + n(j,m)*tt(j,m))*(p(j + 1,m) - p(j,m)))

          *k*m))/(8*m)

      -1
    ht  *(q(j,m + 1) - q(j,m) - q(j + 1,m) + q(j + 1,m + 1))
 + ----------------------------------------------------------   =   0
                               2



clear a;



remfac diff,ht,hx,hy;


on exp;


off rat;



%***********************************************************************
%*****                                                             *****
%*****     T e s t     Examples   ---   Module    A P P R O X      *****
%*****                                                             *****
%***********************************************************************

% Example A.1

coordinates x,t into j,n;


maxorder t=2,x=3;


functions u,v;


approx( (u(n+1/2)-u(n-1/2))/ht=(v(n+1/2,j+1/2)-v(n+1/2,j-1/2)
                               +v(n-1/2,j+1/2)-v(n-1/2,j-1/2))/(2*hx) );


 Difference scheme approximates differential equation df(u,t)=df(v,x)

 with orders of approximation:

  2
hx

  2
ht


% Example A.2

maxorder t=3,x=3;


approx( (u(n+1)-u(n))/ht=(u(n+1,j+1/2)-u(n+1,j-1/2)
                               +u(n,j+1/2)-u(n,j-1/2))/(2*hx) );


 Difference scheme approximates differential equation df(u,t)=df(u,x)

 with orders of approximation:

  2
hx

ht


% Example A.3

maxorder t=2,x=3;


center t=1/2;


approx( (u(n+1)-u(n))/ht=(v(n+1,j+1/2)-v(n+1,j-1/2)
                               +v(n,j+1/2)-v(n,j-1/2))/(2*hx) );


 Difference scheme approximates differential equation df(u,t)=df(v,x)

 with orders of approximation:

  2
hx

  2
ht


% Example A.4

approx( u(n+1)/ht=(v(n+1,j+1/2)-v(n+1,j-1/2)
                               +v(n,j+1/2)-v(n,j-1/2))/(2*hx) );


 Reformulate difference scheme, grid steps remain in denominators

 Difference scheme approximates differential equation 0=df(v,x)

 with orders of approximation:

  2
hx

  -1
ht


% Example A.5

maxorder t=3,x=3;


approx( (u(n+1)-u(n))/ht=(u(n+1,j+1/2)-u(n+1,j-1/2))/hx);


 Difference scheme approximates differential equation df(u,t)=df(u,x)

 with orders of approximation:

  2
hx

ht


% Example A.6

approx( (u(n+1)-u(n))/ht=(u(n+1/2,j+1/2)-u(n+1/2,j-1/2))/hx);


 Difference scheme approximates differential equation df(u,t)=df(u,x)

 with orders of approximation:

  2
hx

  2
ht


% Example A.7;

maxorder x=4;


approx((u(n+1)-u(n))/ht=(u(n+1/2,j+1)-2*u(n+1/2,j)+u(n+1/2,j-1))/hx**2);


 Difference scheme approximates differential equation df(u,t)=df(u,x,2)

 with orders of approximation:

  2
hx

  2
ht


%***********************************************************************
%*****                                                             *****
%*****     T e s t     Examples   ---   Module    C H A R P O L    *****
%*****                                                             *****
%***********************************************************************

% Example C.1

coordinates t,x into i,j;


grid uniform,t,x;


let cos ax**2=1-sin ax**2;


unfunc u,v;


matrix aa(1,2),bb(2,2);


aa(1,1):=(u(i+1)-u(i))/ht+(v(j+1)-v(j))/hx$


aa(1,2):=(v(i+1)-v(i))/ht+(u(j+1/2)-u(j-1/2))/hx$


bb:=ampmat aa;


       kx*hx
ax := -------
         2


      [hx  0 ]
h1 := [      ]
      [0   hx]


      [       hx          2*sin(ax)*ht*( - i*cos(ax) + sin(ax))]
h0 := [                                                        ]
      [ - 2*i*sin(ax)*ht                   hx                  ]


      [                      2*sin(ax)*ht*( - i*cos(ax) + sin(ax)) ]
      [         1           ---------------------------------------]
      [                                       hx                   ]
bb := [                                                            ]
      [  - 2*i*sin(ax)*ht                                          ]
      [-------------------                     1                   ]
      [        hx                                                  ]


bb:=denotemat bb;


      [  1     ai12*i + ar12]
bb := [                     ]
      [ai21*i        1      ]


factor lam;


pol:=charpol bb;


          2
pol := lam  - 2*lam + ai12*ai21 - i*ai21*ar12 + 1

prdenot;


                  2
         2*sin(ax) *ht
ar12 := ---------------
              hx

          - 2*cos(ax)*sin(ax)*ht
ai12 := -------------------------
                   hx

          - 2*sin(ax)*ht
ai21 := -----------------
               hx

cleardenot;


clear aa,bb,pol;



%***********************************************************************

% Example C.2 : Reprint Vorozcov, Ganza, Mazurik: Simvolno-cislennyj
%               interfejs.  v zadacach ..., Novosibirsk 1986,  p.47.

unfunc u;


matrix aa(1,1),bb(1,1);


aa(1,1):=(u(i+1)-u(i))/ht+a*(u(j)-u(j-1))/hx$


bb:=ampmat aa;


ax := kx*hx


h1 := [hx]


h0 := [cos(ax)*a*ht - i*sin(ax)*a*ht - a*ht + hx]


      [ cos(ax)*a*ht - i*sin(ax)*a*ht - a*ht + hx ]
bb := [-------------------------------------------]
      [                    hx                     ]


bb:=denotemat bb;


bb := [ai11*i + ar11]


pol:=charpol bb;


pol := lam - i*ai11 - ar11

prdenot;


         cos(ax)*a*ht - a*ht + hx
ar11 := --------------------------
                    hx

          - sin(ax)*a*ht
ai11 := -----------------
               hx

cleardenot;


clear aa,bb,pol;



%***********************************************************************

% Example C.3 : Reprint Vorozcov, Ganza, Mazurik: Simvolno-cislennyj
%               interfejs.  v zadacach ..., Novosibirsk 1986,  p.52.

coordinates t,x into m,j;


unfunc u,r;


matrix aa(1,2),bb(2,2);


aa(1,1):=(r(m+1)-r(m))/ht+u0*(r(m+1,j+1)-r(m+1,j-1))/2/hx
                         +r0*(u(m+1,j+1)-u(m+1,j-1))/2/hx$


aa(1,2):=(u(m+1)-u(m))/ht+u0*(u(m+1,j+1)-u(m+1,j-1))/2/hx
                         +c0**2/r0*(r(m,j+1)-u(m,j-1))/2/hx$


bb:=ampmat aa;


ax := kx*hx


      [      i*sin(ax)*ht*r0        i*sin(ax)*ht*u0 + hx]
h1 := [                                                 ]
      [2*r0*(i*sin(ax)*ht*u0 + hx)           0          ]


h0 := mat((0,hx),

                     2                  2
          (cos(ax)*c0 *ht - i*sin(ax)*c0 *ht + 2*hx*r0,

             2
           c0 *ht*( - cos(ax) - i*sin(ax))))


                           2                2
             - i*cos(ax)*c0 *ht - sin(ax)*c0 *ht - 2*i*hx*r0
bb := mat((--------------------------------------------------,
                      2*r0*(sin(ax)*ht*u0 - i*hx)

              2
            c0 *ht*(i*cos(ax) - sin(ax))
           ------------------------------),
            2*r0*(sin(ax)*ht*u0 - i*hx)

                                    2                2
            sin(ax)*ht*(i*cos(ax)*c0 *ht + sin(ax)*c0 *ht + 2*i*hx*r0)
          (------------------------------------------------------------,(
                          2   2   2                            2
                2*(sin(ax) *ht *u0  - 2*i*sin(ax)*ht*hx*u0 - hx )

                                     2   2          2   2   2
               - i*cos(ax)*sin(ax)*c0 *ht  + sin(ax) *c0 *ht

                                            2
               - 2*i*sin(ax)*ht*hx*u0 - 2*hx )/(2

                       2   2   2                            2
              *(sin(ax) *ht *u0  - 2*i*sin(ax)*ht*hx*u0 - hx ))))


bb:=denotemat bb;


      [ai11*i + ar11  ai12*i + ar12]
bb := [                            ]
      [ai21*i + ar21  ai22*i + ar22]


pol:=charpol bb;


          2
pol := lam  + lam*( - i*ai11 - i*ai22 - ar11 - ar22) - ai11*ai22 + i*ai11*ar22

        + ai12*ai21 - i*ai12*ar21 - i*ai21*ar12 + i*ai22*ar11 + ar11*ar22

        - ar12*ar21

prdenot;


              2
          - c0
ar11 := ---------
         2*r0*u0

                      2           2
          - cos(ax)*c0 *ht*u0 - c0 *hx - 2*hx*r0*u0
ai11 := --------------------------------------------
               2*r0*u0*(sin(ax)*ht*u0 - hx*i)

              2
          - c0
ar12 := ---------
         2*r0*u0

              2
            c0 *(cos(ax)*ht*u0 - hx)
ai12 := --------------------------------
         2*r0*u0*(sin(ax)*ht*u0 - hx*i)

                     2   2   2
              sin(ax) *c0 *ht
ar21 := ----------------------------
                   2   2   2     2
         2*(sin(ax) *ht *u0  - hx )

                                    2   2   3   2             2      2
ai21 := (sin(ax)*ht*(cos(ax)*sin(ax) *c0 *ht *u0  - cos(ax)*c0 *ht*hx

                        2   2   2                  2   2         2       3
             + 2*sin(ax) *c0 *ht *hx*u0 + 2*sin(ax) *ht *hx*r0*u0  - 2*hx *r0))/

                   4   4   4            3   3        3            2   2   2   2
        (2*(sin(ax) *ht *u0  - 2*sin(ax) *ht *hx*i*u0  - 2*sin(ax) *ht *hx *u0

                              3          4
             + 2*sin(ax)*ht*hx *i*u0 + hx ))

                 2   2   2       2
          sin(ax) *c0 *ht  - 2*hx
ar22 := ----------------------------
                   2   2   2     2
         2*(sin(ax) *ht *u0  - hx )

                                       2   2   3   2             2      2
ai22 := (sin(ax)*ht*( - cos(ax)*sin(ax) *c0 *ht *u0  + cos(ax)*c0 *ht*hx

                        2   2   2                  2   2      3       3
             + 2*sin(ax) *c0 *ht *hx*u0 - 2*sin(ax) *ht *hx*u0  - 2*hx *u0))/(2*

                   4   4   4            3   3        3            2   2   2   2
           (sin(ax) *ht *u0  - 2*sin(ax) *ht *hx*i*u0  - 2*sin(ax) *ht *hx *u0

                              3          4
             + 2*sin(ax)*ht*hx *i*u0 + hx ))

cleardenot;


clear aa,bb,pol;



%***********************************************************************

% Example C.4 : Richtmyer, Morton: Difference methods for initial value
%               problems, &10.3.  p.262

coordinates t,x into n,j;


unfunc v,w;


matrix aa(1,2),bb(2,2);


aa(1,1):=(v(n+1)-v(n))/ht-c*(w(j+1/2)-w(j-1/2)+
                             w(n+1,j+1/2)-w(n+1,j-1/2))/(2*hx)$


aa(1,2):=(w(n+1,j-1/2)-w(n,j-1/2))/ht-c*(v(n+1,j)-v(n+1,j-1)+
                                         v(j)-v(j-1))/(2*hx)$


bb:=ampmat aa;


       kx*hx
ax := -------
         2


      [                 hx                        - i*sin(ax)*c*ht    ]
h1 := [                                                               ]
      [sin(ax)*c*ht*( - i*cos(ax) - sin(ax))  hx*(cos(ax) - i*sin(ax))]


      [                hx                       i*sin(ax)*c*ht     ]
h0 := [                                                            ]
      [sin(ax)*c*ht*(i*cos(ax) + sin(ax))  hx*(cos(ax) - i*sin(ax))]


      [           2  2   2     2                             ]
      [  - sin(ax) *c *ht  + hx       2*i*sin(ax)*c*ht*hx    ]
      [--------------------------   -----------------------  ]
      [         2  2   2     2              2  2   2     2   ]
      [  sin(ax) *c *ht  + hx        sin(ax) *c *ht  + hx    ]
bb := [                                                      ]
      [                                       2  2   2     2 ]
      [   2*i*sin(ax)*c*ht*hx        - sin(ax) *c *ht  + hx  ]
      [ -----------------------    --------------------------]
      [         2  2   2     2              2  2   2     2   ]
      [  sin(ax) *c *ht  + hx        sin(ax) *c *ht  + hx    ]


bb:=denotemat bb;


      [ ar11   ai12*i]
bb := [              ]
      [ai21*i   ar22 ]


pol:=charpol bb;


          2
pol := lam  - lam*(ar11 + ar22) + ai12*ai21 + ar11*ar22

prdenot;


                   2  2   2     2
          - sin(ax) *c *ht  + hx
ar11 := --------------------------
                 2  2   2     2
          sin(ax) *c *ht  + hx

           2*sin(ax)*c*ht*hx
ai12 := -----------------------
                2  2   2     2
         sin(ax) *c *ht  + hx

           2*sin(ax)*c*ht*hx
ai21 := -----------------------
                2  2   2     2
         sin(ax) *c *ht  + hx

                   2  2   2     2
          - sin(ax) *c *ht  + hx
ar22 := --------------------------
                 2  2   2     2
          sin(ax) *c *ht  + hx

cleardenot;


clear aa,bb,pol;



%***********************************************************************

% Example C.5: Mazurik: Algoritmy resenia zadaci..., Preprint no.24-85,
%               AN USSR SO, Inst. teor. i prikl. mechaniky, p.34

coordinates t,x,y into n,m,k;


grid uniform,t,x,y;


unfunc u1,u2,u3;


matrix aa(1,3),bb(3,3);


aa(1,1):=(u1(n+1)-u1(n))/ht+c/2*((-u1(m-1)+2*u1(m)-u1(m+1))/hx +
         (u2(m+1)-u2(m-1))/hx - (u1(k-1)-2*u1(k)+u1(k+1))/hy +
         (u3(k+1)-u3(k-1))/hy)$


aa(1,2):=(u2(n+1)-u2(n))/ht+c/2*((u1(m+1)-u1(m-1))/hx -
         (u2(m-1)-2*u2(m)+u2(m+1))/hx)$


aa(1,3):=(u3(n+1)-u3(n))/ht + c/2*((u1(k+1)-u1(k-1))/hy -
         (u3(k-1)-2*u3(k)+u3(k+1))/hy)$


off prfourmat;


bb:=ampmat aa;


ax := kx*hx


ay := ky*hy


            cos(ax)*c*ht*hy + cos(ay)*c*ht*hx - c*ht*hx - c*ht*hy + hx*hy
bb := mat((---------------------------------------------------------------,
                                        hx*hy

             - i*sin(ax)*c*ht    - i*sin(ay)*c*ht
           -------------------,-------------------),
                   hx                  hy

             - i*sin(ax)*c*ht   cos(ax)*c*ht - c*ht + hx
          (-------------------,--------------------------,0),
                   hx                      hx

             - i*sin(ay)*c*ht     cos(ay)*c*ht - c*ht + hy
          (-------------------,0,--------------------------))
                   hy                        hy


pol:=charpol bb;


           3   2   2      2
pol := (lam *hx *hy  + lam *hx*hy*( - 2*cos(ax)*c*ht*hy - 2*cos(ay)*c*ht*hx

            + 2*c*ht*hx + 2*c*ht*hy - 3*hx*hy) + lam*(

                              2   2                    2   2
           3*cos(ax)*cos(ay)*c *ht *hx*hy - 3*cos(ax)*c *ht *hx*hy

                         2   2   2                       2          2  2   2   2
            - 2*cos(ax)*c *ht *hy  + 4*cos(ax)*c*ht*hx*hy  + cos(ay) *c *ht *hx

                         2   2   2              2   2
            - 2*cos(ay)*c *ht *hx  - 3*cos(ay)*c *ht *hx*hy

                               2             2  2   2   2    2   2   2
            + 4*cos(ay)*c*ht*hx *hy + sin(ay) *c *ht *hx  + c *ht *hx

                 2   2            2   2   2            2                  2
            + 3*c *ht *hx*hy + 2*c *ht *hy  - 4*c*ht*hx *hy - 4*c*ht*hx*hy

                  2   2                   2  3   3
            + 3*hx *hy ) - cos(ax)*cos(ay) *c *ht *hx

                              3   3                         3   3
         + 2*cos(ax)*cos(ay)*c *ht *hx + 2*cos(ax)*cos(ay)*c *ht *hy

                              2   2                        2  3   3
         - 3*cos(ax)*cos(ay)*c *ht *hx*hy - cos(ax)*sin(ay) *c *ht *hx

                    3   3                 3   3                 2   2
         - cos(ax)*c *ht *hx - 2*cos(ax)*c *ht *hy + 3*cos(ax)*c *ht *hx*hy

                      2   2   2                       2          2  3   3
         + 2*cos(ax)*c *ht *hy  - 2*cos(ax)*c*ht*hx*hy  + cos(ay) *c *ht *hx

                  2  2   2   2              3   3                 3   3
         - cos(ay) *c *ht *hx  - 2*cos(ay)*c *ht *hx - 2*cos(ay)*c *ht *hy

                      2   2   2              2   2                          2
         + 2*cos(ay)*c *ht *hx  + 3*cos(ay)*c *ht *hx*hy - 2*cos(ay)*c*ht*hx *hy

                  2  3   3             2  2   2   2    3   3         3   3
         + sin(ay) *c *ht *hx - sin(ay) *c *ht *hx  + c *ht *hx + 2*c *ht *hy

            2   2   2      2   2            2   2   2            2
         - c *ht *hx  - 3*c *ht *hx*hy - 2*c *ht *hy  + 2*c*ht*hx *hy

                       2     2   2     2   2
         + 2*c*ht*hx*hy  - hx *hy )/(hx *hy )

let
  cos ax=cos ax2**2-sin ax2**2,
  cos ay=cos ay2**2-sin ay2**2,
  sin ax=2*sin ax2*cos ax2,
  sin ay=2*sin ay2*cos ay2,
  cos ax2**2=1-sin ax2**2,
  cos ay2**2=1-sin ay2**2,
  sin ax2=s1,
  sin ay2=s2,
  hx=c*ht/cap1,
  hy=c*ht/cap2;


order s1,s2;


pol:=pol;


          3      2      2            2                       2   2
pol := lam  + lam *(4*s1 *cap1 + 4*s2 *cap2 - 3) + lam*(12*s1 *s2 *cap1*cap2

                 2     2       2            2     2       2
           + 4*s1 *cap1  - 8*s1 *cap1 + 4*s2 *cap2  - 8*s2 *cap2 + 3)

              2   2     2            2   2          2        2   2
        + 8*s1 *s2 *cap1 *cap2 + 8*s1 *s2 *cap1*cap2  - 12*s1 *s2 *cap1*cap2

              2     2       2            2     2       2
        - 4*s1 *cap1  + 4*s1 *cap1 - 4*s2 *cap2  + 4*s2 *cap2 - 1

clear cos ax,cos ay,sin ax,sin ay,cos ax2**2,cos ay2**2,sin ax2,sin ay2,
      hx,hy;


pol:=complexpol pol;


         2   2
 If  8*s1 *s2 *cap1*cap2*(cap1 + cap2)  =  0  and  0

  =  0  , a root of the polynomial is equal to 1.

          3      2      2            2                       2   2
pol := lam  + lam *(4*s1 *cap1 + 4*s2 *cap2 - 3) + lam*(12*s1 *s2 *cap1*cap2

                 2     2       2            2     2       2
           + 4*s1 *cap1  - 8*s1 *cap1 + 4*s2 *cap2  - 8*s2 *cap2 + 3)

              2   2     2            2   2          2        2   2
        + 8*s1 *s2 *cap1 *cap2 + 8*s1 *s2 *cap1*cap2  - 12*s1 *s2 *cap1*cap2

              2     2       2            2     2       2
        - 4*s1 *cap1  + 4*s1 *cap1 - 4*s2 *cap2  + 4*s2 *cap2 - 1

pol1:=hurw pol;


             3   2   2                                2         2   2     2
pol1 := 8*lam *s1 *s2 *cap1*cap2*(cap1 + cap2) + 8*lam *( - 3*s1 *s2 *cap1 *cap2

                  2   2          2       2   2               2     2     2     2
            - 3*s1 *s2 *cap1*cap2  + 3*s1 *s2 *cap1*cap2 + s1 *cap1  + s2 *cap2

                          2   2     2            2   2          2
           ) + 8*lam*(3*s1 *s2 *cap1 *cap2 + 3*s1 *s2 *cap1*cap2

                  2   2                 2     2       2            2     2
            - 6*s1 *s2 *cap1*cap2 - 2*s1 *cap1  + 2*s1 *cap1 - 2*s2 *cap2

                  2                 2   2     2          2   2          2
            + 2*s2 *cap2) + 8*( - s1 *s2 *cap1 *cap2 - s1 *s2 *cap1*cap2

                  2   2               2     2       2          2     2
            + 3*s1 *s2 *cap1*cap2 + s1 *cap1  - 2*s1 *cap1 + s2 *cap2

                  2
            - 2*s2 *cap2 + 1)

denotid cp;


pol:=denotepol pol;


          3      2
pol := lam  + lam *cpr02 + lam*cpr01 + cpr00

prdenot;


             2   2     2            2   2          2        2   2
cpr00 := 8*s1 *s2 *cap1 *cap2 + 8*s1 *s2 *cap1*cap2  - 12*s1 *s2 *cap1*cap2

                2     2       2            2     2       2
          - 4*s1 *cap1  + 4*s1 *cap1 - 4*s2 *cap2  + 4*s2 *cap2 - 1

cpr01 := 

     2   2                 2     2       2            2     2       2
12*s1 *s2 *cap1*cap2 + 4*s1 *cap1  - 8*s1 *cap1 + 4*s2 *cap2  - 8*s2 *cap2 + 3

             2            2
cpr02 := 4*s1 *cap1 + 4*s2 *cap2 - 3

cleardenot;


clear aa,bb,pol,pol1;



%***********************************************************************

% Example C.6 : Lax-Wendrov (V. Ganzha)

coordinates t,x,y into n,m,k;


grid uniform,t,x,y;


let cos ax**2=1-sin ax**2,
    cos ay**2=1-sin ay**2;


unfunc u1,u2,u3,u4;


matrix aa(1,4),bb(4,4);


aa(1,1):=4*(u1(n+1)-u1(n))/ht+
         (w*(u1(m+2)-u1(m-2)+u1(m+1,k+1)+u1(m+1,k-1)-
         u1(m-1,k+1)-u1(m-1,k-1))+p*(u2(m+2)-u2(m-2)+u2(m+1,k+1)+
         u2(m+1,k-1)-u2(m-1,k+1)-u2(m-1,k-1))+
         v*(u1(m+1,k+1)+u1(m-1,k+1)-
         u1(m+1,k-1)-u1(m-1,k-1)+u1(k+2)-u1(k-2))+p*(u3(m+1,k+1)+
         u3(m-1,k+1)-u3(m+1,k-1)-u3(m-1,k-1)+u3(k+2)-u3(k-2)))/hx+ht*
         (2*w**2*(-u1(m+2)+2*u1(m)-u1(m-2))+4*w*p*(-u2(m+2)+2*u2(m)-
         u2(m-2))+2*(-u4(m+2)+2*u4(m)-u4(m-2))+2*v**2*(-u1(k+2)+
         2*u1(k)-u1(k-2))+4*v*p*(u3(k+2)+2*u3(k)-u3(k-2))+2*(-u4(k+2)+
         2*u4(k)-u4(k-2))+4*w*v*(-u1(m+1,k+1)+u1(m+1,k-1)+u1(m-1,k+1)-
         u1(m-1,k-1))+4*p*v*(-u2(m+1,k+1)+u2(m+1,k-1)+u2(m-1,k+1)-
         u2(m-1,k-1))+4*w*p*(-u3(m+1,k+1)+u3(m+1,k-1)+u3(m-1,k+1)-
         u3(m-1,k-1)))/hx/hx$


aa(1,2):=4*p*(u2(n+1)-u2(n))/ht+
       (w*p*(u2(m+2)-u2(m-2)+u2(m+1,k+1)+
         u2(m+1,k-1)-u2(m-1,k+1)-u2(m-1,k-1))+u4(m+2)-u4(m-2)+
         u4(m+1,k+1)+
         u4(m+1,k-1)-u4(m-1,k+1)-u4(m-1,k-1)+
         p*v*(u2(m+1,k+1)+u2(m-1,k+1)+
         u2(k+2)-u2(k-2)-u2(m+1,k-1)-u2(m-1,k-1)))/hx+ht*(2*w**2*p*
         (-u2(m+2)+2*u2(m)-u2(m-2))+2*p*c**2*(-u2(m+2)+2*u2(m)-u2(m-2))
         +4*w*(-u4(m+2)+2*u4(m)-u4(m-2))+2*p*v**2*(-u2(k+2)+2*u2(k)-
         u2(k-2))+4*w*p*v*(-u2(m+1,k+1)+u2(m+1,k-1)+u2(m-1,k+1)-
         u2(m-1,k-1))+2*p*c**2*(-u3(m+1,k+1)+u3(m+1,k-1)+u3(m-1,k+1)
         -u3(m-1,k-1))+4*v*(-u4(m+1,k+1)+u4(m+1,k-1)+u4(m-1,k+1)-
         u4(m-1,k-1)))/hx/hx$


aa(1,3):=4*p*(u3(n+1)-u3(n))/ht+(w*p*(u3(m+2)-u3(m-2)+u3(m+1,k+1)+
         u3(m+1,k-1)-u3(m-1,k+1)-u3(m-1,k-1))+u4(k+2)-u4(k-2)+
         u4(m+1,k+1)-u4(m+1,k-1)+u4(m-1,k+1)-u4(m-1,k-1)+
         p*v*(u3(m+1,k+1)+u3(m-1,k+1)+u3(k+2)-u3(k-2)-u3(m+1,k-1)-
         u3(m-1,k-1)))/hx+ht*(2*w**2*p*(-u3(m+2)+2*u3(m)-u3(m-2))+
         2*p*c**2*(-u3(k+2)+2*u3(k)-u3(k-2))+4*v*(-u4(k+2)+
         2*u4(k)-u4(k-2))+2*p*v**2*(-u3(k+2)+2*u3(k)-u3(k-2))+
         4*w*p*v*(-u3(m+1,k+1)+u3(m+1,k-1)+u3(m-1,k+1)-
         u3(m-1,k-1))+2*p*c**2*(-u2(m+1,k+1)+u2(m+1,k-1)+
         u2(m-1,k+1)-u2(m-1,k-1))+4*w*(u4(m+1,k+1)+u4(m+1,k-1)+
         u4(m-1,k+1)-u4(m-1,k-1)))/hx/hx$


aa(1,4):=4*(u4(n+1)-u4(n))/ht+(p*c**2*(u2(m+2)-u2(m-2)+u2(m+1,k+1)+
         u2(m+1,k-1)-u2(m-1,k+1)-u2(m-1,k-1))+w*(u4(m+2)-
         u4(m-2)+u4(m+1,k+1)+u4(m+1,k-1)-u4(m-1,k+1)-u4(m-1,k-1))+
         +p*c**2*(u3(m+1,k+1)+u3(m-1,k+1)-u3(m+1,k-1)-
         u3(m-1,k-1)+u3(k+2)-u3(k-2))+v*(u4(m+1,k+1)+u4(m-1,k+1)-
         u4(m+1,k-1)-u4(m-1,k-1)+u4(k+2)-u4(k-2)))/hx+ht*
         (2*w**2*(-u4(m+2)+2*u4(m)-u4(m-2))+4*w*p*c**2*(-u2(m+2)+
         2*u2(m)-u2(m-2))+2*c**2*(-u4(m+2)+2*u4(m)-u4(m-2))+
         4*p*v*c**2*(-u3(k+2)+2*u3(k)-u3(k-2))+2*c**2*(-u4(k+2)+
         2*u4(k)-u4(k-2))+2*v**2*(-u4(k+2)+2*u4(k)-u4(k-2))+
         4*p*v*c**2*(-u2(m+1,k+1)+u2(m+1,k-1)+u2(m-1,k+1)-
         u2(m-1,k-1))+4*w*p*c**2*(-u3(m+1,k+1)+u3(m+1,k-1)+
         u3(m-1,k+1)-u3(m-1,k-1))+4*w*v*(-u4(m+1,k+1)+
         u4(m+1,k-1)+u4(m-1,k+1)-u4(m-1,k-1)))/hx/hx$


bb:=ampmat aa;


ax := kx*hx


ay := ky*hy


bb := mat((( - i*cos(ax)*sin(ax)*ht*hx*w - i*cos(ax)*sin(ay)*ht*hx*v

             - i*cos(ay)*sin(ax)*ht*hx*w - i*cos(ay)*sin(ay)*ht*hx*v

                        2   2  2                       2                2   2  2
             - 2*sin(ax) *ht *w  - 4*sin(ax)*sin(ay)*ht *v*w - 2*sin(ay) *ht *v

                 2    2
             + hx )/hx ,(sin(ax)*ht*p*( - i*cos(ax)*hx - i*cos(ay)*hx

                                                        2
                  - 4*sin(ax)*ht*w - 4*sin(ay)*ht*v))/hx ,(ht*p*(

                  - i*cos(ax)*sin(ay)*hx - 4*i*cos(ay)*sin(ay)*ht*v

                                                                               2
                  - i*cos(ay)*sin(ay)*hx - 4*sin(ax)*sin(ay)*ht*w - 2*ht*v))/hx

                    2         2          2
              - 2*ht *(sin(ax)  + sin(ay) )
           ,--------------------------------),
                            2
                          hx

          (0,( - i*cos(ax)*sin(ax)*ht*hx*w - i*cos(ax)*sin(ay)*ht*hx*v

               - i*cos(ay)*sin(ax)*ht*hx*w - i*cos(ay)*sin(ay)*ht*hx*v

                          2  2   2            2   2  2
               - 2*sin(ax) *c *ht  - 2*sin(ax) *ht *w

                                     2                2   2  2     2    2
               - 4*sin(ax)*sin(ay)*ht *v*w - 2*sin(ay) *ht *v  + hx )/hx ,

                                  2   2
             - 2*sin(ax)*sin(ay)*c *ht
           -----------------------------,(sin(ax)*ht*( - i*cos(ax)*hx
                          2
                        hx

                                                                        2
                  - i*cos(ay)*hx - 4*sin(ax)*ht*w - 4*sin(ay)*ht*v))/(hx *p)),

                                    2   2
               - 2*sin(ax)*sin(ay)*c *ht
          (0,-----------------------------,( - i*cos(ax)*sin(ax)*ht*hx*w
                            2
                          hx

               - i*cos(ax)*sin(ay)*ht*hx*v - i*cos(ay)*sin(ax)*ht*hx*w

                                                      2   2  2
               - i*cos(ay)*sin(ay)*ht*hx*v - 2*sin(ax) *ht *w

                                     2                2  2   2
               - 4*sin(ax)*sin(ay)*ht *v*w - 2*sin(ay) *c *ht

                          2   2  2     2    2
               - 2*sin(ay) *ht *v  + hx )/hx ,(ht*( - 2*cos(ax)*cos(ay)*ht*w

                  - 2*i*cos(ax)*sin(ay)*ht*w - i*cos(ax)*sin(ay)*hx

                  - 2*i*cos(ay)*sin(ax)*ht*w - i*cos(ay)*sin(ay)*hx

                                                      2           2
                  - 2*sin(ax)*sin(ay)*ht*w - 4*sin(ay) *ht*v))/(hx *p)),

                       2
          (0,(sin(ax)*c *ht*p*( - i*cos(ax)*hx - i*cos(ay)*hx - 4*sin(ax)*ht*w

                                       2           2
                  - 4*sin(ay)*ht*v))/hx ,(sin(ay)*c *ht*p*( - i*cos(ax)*hx

                                                                       2
                  - i*cos(ay)*hx - 4*sin(ax)*ht*w - 4*sin(ay)*ht*v))/hx ,(

               - i*cos(ax)*sin(ax)*ht*hx*w - i*cos(ax)*sin(ay)*ht*hx*v

               - i*cos(ay)*sin(ax)*ht*hx*w - i*cos(ay)*sin(ay)*ht*hx*v

                          2  2   2            2   2  2
               - 2*sin(ax) *c *ht  - 2*sin(ax) *ht *w

                                     2                2  2   2
               - 4*sin(ax)*sin(ay)*ht *v*w - 2*sin(ay) *c *ht

                          2   2  2     2    2
               - 2*sin(ay) *ht *v  + hx )/hx ))


let
  sin(ax)=s1,
  cos(ax)=c1,
  sin(ay)=s2,
  cos(ay)=c2,
  w=k1*hx/ht,
  v=k2*hx/ht,
  c=k3*hx/ht,
  ht=r1*hx;


denotid a;


bb:=denotemat bb;


      [ai11*i + ar11  ai12*i + ar12  ai13*i + ar13      ar14     ]
      [                                                          ]
      [      0        ai22*i + ar22      ar23       ai24*i + ar24]
bb := [                                                          ]
      [      0            ar32       ai33*i + ar33  ai34*i + ar34]
      [                                                          ]
      [      0        ai42*i + ar42  ai43*i + ar43  ai44*i + ar44]


clear sin ax,cos ax,sin ay,cos ay,w,v,c,ht;


pol:=charpol bb;


          4      3
pol := lam  + lam

       *( - i*ai11 - i*ai22 - i*ai33 - i*ai44 - ar11 - ar22 - ar33 - ar44) + 

          2
       lam *( - ai11*ai22 - ai11*ai33 - ai11*ai44 + i*ai11*ar22 + i*ai11*ar33

              + i*ai11*ar44 - ai22*ai33 - ai22*ai44 + i*ai22*ar11 + i*ai22*ar33

              + i*ai22*ar44 + ai24*ai42 - i*ai24*ar42 - ai33*ai44 + i*ai33*ar11

              + i*ai33*ar22 + i*ai33*ar44 + ai34*ai43 - i*ai34*ar43

              - i*ai42*ar24 - i*ai43*ar34 + i*ai44*ar11 + i*ai44*ar22

              + i*ai44*ar33 + ar11*ar22 + ar11*ar33 + ar11*ar44 + ar22*ar33

              + ar22*ar44 - ar23*ar32 - ar24*ar42 + ar33*ar44 - ar34*ar43) + lam

       *(i*ai11*ai22*ai33 + i*ai11*ai22*ai44 + ai11*ai22*ar33 + ai11*ai22*ar44

          - i*ai11*ai24*ai42 - ai11*ai24*ar42 + i*ai11*ai33*ai44

          + ai11*ai33*ar22 + ai11*ai33*ar44 - i*ai11*ai34*ai43 - ai11*ai34*ar43

          - ai11*ai42*ar24 - ai11*ai43*ar34 + ai11*ai44*ar22 + ai11*ai44*ar33

          - i*ai11*ar22*ar33 - i*ai11*ar22*ar44 + i*ai11*ar23*ar32

          + i*ai11*ar24*ar42 - i*ai11*ar33*ar44 + i*ai11*ar34*ar43

          + i*ai22*ai33*ai44 + ai22*ai33*ar11 + ai22*ai33*ar44

          - i*ai22*ai34*ai43 - ai22*ai34*ar43 - ai22*ai43*ar34 + ai22*ai44*ar11

          + ai22*ai44*ar33 - i*ai22*ar11*ar33 - i*ai22*ar11*ar44

          - i*ai22*ar33*ar44 + i*ai22*ar34*ar43 - i*ai24*ai33*ai42

          - ai24*ai33*ar42 - ai24*ai42*ar11 - ai24*ai42*ar33 + ai24*ai43*ar32

          + i*ai24*ar11*ar42 - i*ai24*ar32*ar43 + i*ai24*ar33*ar42

          - ai33*ai42*ar24 + ai33*ai44*ar11 + ai33*ai44*ar22 - i*ai33*ar11*ar22

          - i*ai33*ar11*ar44 - i*ai33*ar22*ar44 + i*ai33*ar24*ar42

          + ai34*ai42*ar23 - ai34*ai43*ar11 - ai34*ai43*ar22 + i*ai34*ar11*ar43

          + i*ai34*ar22*ar43 - i*ai34*ar23*ar42 + i*ai42*ar11*ar24

          - i*ai42*ar23*ar34 + i*ai42*ar24*ar33 + i*ai43*ar11*ar34

          + i*ai43*ar22*ar34 - i*ai43*ar24*ar32 - i*ai44*ar11*ar22

          - i*ai44*ar11*ar33 - i*ai44*ar22*ar33 + i*ai44*ar23*ar32

          - ar11*ar22*ar33 - ar11*ar22*ar44 + ar11*ar23*ar32 + ar11*ar24*ar42

          - ar11*ar33*ar44 + ar11*ar34*ar43 - ar22*ar33*ar44 + ar22*ar34*ar43

          + ar23*ar32*ar44 - ar23*ar34*ar42 - ar24*ar32*ar43 + ar24*ar33*ar42)

        + ai11*ai22*ai33*ai44 - i*ai11*ai22*ai33*ar44 - ai11*ai22*ai34*ai43

        + i*ai11*ai22*ai34*ar43 + i*ai11*ai22*ai43*ar34 - i*ai11*ai22*ai44*ar33

        - ai11*ai22*ar33*ar44 + ai11*ai22*ar34*ar43 - ai11*ai24*ai33*ai42

        + i*ai11*ai24*ai33*ar42 + i*ai11*ai24*ai42*ar33 - i*ai11*ai24*ai43*ar32

        - ai11*ai24*ar32*ar43 + ai11*ai24*ar33*ar42 + i*ai11*ai33*ai42*ar24

        - i*ai11*ai33*ai44*ar22 - ai11*ai33*ar22*ar44 + ai11*ai33*ar24*ar42

        - i*ai11*ai34*ai42*ar23 + i*ai11*ai34*ai43*ar22 + ai11*ai34*ar22*ar43

        - ai11*ai34*ar23*ar42 - ai11*ai42*ar23*ar34 + ai11*ai42*ar24*ar33

        + ai11*ai43*ar22*ar34 - ai11*ai43*ar24*ar32 - ai11*ai44*ar22*ar33

        + ai11*ai44*ar23*ar32 + i*ai11*ar22*ar33*ar44 - i*ai11*ar22*ar34*ar43

        - i*ai11*ar23*ar32*ar44 + i*ai11*ar23*ar34*ar42 + i*ai11*ar24*ar32*ar43

        - i*ai11*ar24*ar33*ar42 - i*ai22*ai33*ai44*ar11 - ai22*ai33*ar11*ar44

        + i*ai22*ai34*ai43*ar11 + ai22*ai34*ar11*ar43 + ai22*ai43*ar11*ar34

        - ai22*ai44*ar11*ar33 + i*ai22*ar11*ar33*ar44 - i*ai22*ar11*ar34*ar43

        + i*ai24*ai33*ai42*ar11 + ai24*ai33*ar11*ar42 + ai24*ai42*ar11*ar33

        - ai24*ai43*ar11*ar32 + i*ai24*ar11*ar32*ar43 - i*ai24*ar11*ar33*ar42

        + ai33*ai42*ar11*ar24 - ai33*ai44*ar11*ar22 + i*ai33*ar11*ar22*ar44

        - i*ai33*ar11*ar24*ar42 - ai34*ai42*ar11*ar23 + ai34*ai43*ar11*ar22

        - i*ai34*ar11*ar22*ar43 + i*ai34*ar11*ar23*ar42 + i*ai42*ar11*ar23*ar34

        - i*ai42*ar11*ar24*ar33 - i*ai43*ar11*ar22*ar34 + i*ai43*ar11*ar24*ar32

        + i*ai44*ar11*ar22*ar33 - i*ai44*ar11*ar23*ar32 + ar11*ar22*ar33*ar44

        - ar11*ar22*ar34*ar43 - ar11*ar23*ar32*ar44 + ar11*ar23*ar34*ar42

        + ar11*ar24*ar32*ar43 - ar11*ar24*ar33*ar42

denotid cp;


pol:=denotepol pol;


          4      3                        2
pol := lam  + lam *(cpi03*i + cpr03) + lam *(cpi02*i + cpr02)

        + lam*(cpi01*i + cpr01) + cpi00*i + cpr00

pol:=complexpol pol;


 If  cpr00 + cpr01 + cpr02 + cpr03 + 1  =  0  and  cpi00 + cpi01 + cpi02 + cpi03

  =  0  , a root of the polynomial is equal to 1.

          8        7            6       2                  2
pol := lam  + 2*lam *cpr03 + lam *(cpi03  + 2*cpr02 + cpr03 )

               5
        + 2*lam *(cpi02*cpi03 + cpr01 + cpr02*cpr03)

             4                       2                                  2
        + lam *(2*cpi01*cpi03 + cpi02  + 2*cpr00 + 2*cpr01*cpr03 + cpr02 )

               3
        + 2*lam *(cpi00*cpi03 + cpi01*cpi02 + cpr00*cpr03 + cpr01*cpr02)

             2                       2                        2
        + lam *(2*cpi00*cpi02 + cpi01  + 2*cpr00*cpr02 + cpr01 )

                                                   2        2
        + 2*lam*(cpi00*cpi01 + cpr00*cpr01) + cpi00  + cpr00

denotid rp;


pol:=denotepol pol;


          8      7            6            5            4            3
pol := lam  + lam *rpr07 + lam *rpr06 + lam *rpr05 + lam *rpr04 + lam *rpr03

             2
        + lam *rpr02 + lam*rpr01 + rpr00

prdenot;


               2   2                       2   2
ar11 :=  - 2*s1 *k1  - 4*s1*s2*k1*k2 - 2*s2 *k2  + 1

ai11 :=  - (s1*c1*k1 + s1*c2*k1 + s2*c1*k2 + s2*c2*k2)

ar12 :=  - 4*s1*p*r1*(s1*k1 + s2*k2)

ai12 :=  - s1*p*r1*(c1 + c2)

ar13 := 2*p*r1*( - 2*s1*s2*k1 - k2)

ai13 := s2*p*r1*( - c1 - 4*c2*k2 - c2)

               2    2     2
ar14 :=  - 2*r1 *(s1  + s2 )

               2   2       2   2                       2   2
ar22 :=  - 2*s1 *k1  - 2*s1 *k3  - 4*s1*s2*k1*k2 - 2*s2 *k2  + 1

ai22 :=  - (s1*c1*k1 + s1*c2*k1 + s2*c1*k2 + s2*c2*k2)

                     2
ar23 :=  - 2*s1*s2*k3

          - 4*s1*r1*(s1*k1 + s2*k2)
ar24 := ----------------------------
                     p

          - s1*r1*(c1 + c2)
ai24 := --------------------
                 p

                     2
ar32 :=  - 2*s1*s2*k3

               2   2                       2   2       2   2
ar33 :=  - 2*s1 *k1  - 4*s1*s2*k1*k2 - 2*s2 *k2  - 2*s2 *k3  + 1

ai33 :=  - (s1*c1*k1 + s1*c2*k1 + s2*c1*k2 + s2*c2*k2)

                                 2
         2*r1*( - s1*s2*k1 - 2*s2 *k2 - c1*c2*k1)
ar34 := ------------------------------------------
                            p

         r1*( - 2*s1*c2*k1 - 2*s2*c1*k1 - s2*c1 - s2*c2)
ai34 := -------------------------------------------------
                                p

                   2
          - 4*s1*k3 *p*(s1*k1 + s2*k2)
ar42 := -------------------------------
                      r1

                 2
          - s1*k3 *p*(c1 + c2)
ai42 := -----------------------
                  r1

                   2
          - 4*s2*k3 *p*(s1*k1 + s2*k2)
ar43 := -------------------------------
                      r1

                 2
          - s2*k3 *p*(c1 + c2)
ai43 := -----------------------
                  r1

               2   2       2   2                       2   2       2   2
ar44 :=  - 2*s1 *k1  - 2*s1 *k3  - 4*s1*s2*k1*k2 - 2*s2 *k2  - 2*s2 *k3  + 1

ai44 :=  - (s1*c1*k1 + s1*c2*k1 + s2*c1*k2 + s2*c2*k2)

cpr00 := ai11*ai22*ai33*ai44 - ai11*ai22*ai34*ai43 - ai11*ai22*ar33*ar44

          + ai11*ai22*ar34*ar43 - ai11*ai24*ai33*ai42 - ai11*ai24*ar32*ar43

          + ai11*ai24*ar33*ar42 - ai11*ai33*ar22*ar44 + ai11*ai33*ar24*ar42

          + ai11*ai34*ar22*ar43 - ai11*ai34*ar23*ar42 - ai11*ai42*ar23*ar34

          + ai11*ai42*ar24*ar33 + ai11*ai43*ar22*ar34 - ai11*ai43*ar24*ar32

          - ai11*ai44*ar22*ar33 + ai11*ai44*ar23*ar32 - ai22*ai33*ar11*ar44

          + ai22*ai34*ar11*ar43 + ai22*ai43*ar11*ar34 - ai22*ai44*ar11*ar33

          + ai24*ai33*ar11*ar42 + ai24*ai42*ar11*ar33 - ai24*ai43*ar11*ar32

          + ai33*ai42*ar11*ar24 - ai33*ai44*ar11*ar22 - ai34*ai42*ar11*ar23

          + ai34*ai43*ar11*ar22 + ar11*ar22*ar33*ar44 - ar11*ar22*ar34*ar43

          - ar11*ar23*ar32*ar44 + ar11*ar23*ar34*ar42 + ar11*ar24*ar32*ar43

          - ar11*ar24*ar33*ar42

cpi00 :=  - ai11*ai22*ai33*ar44 + ai11*ai22*ai34*ar43 + ai11*ai22*ai43*ar34

          - ai11*ai22*ai44*ar33 + ai11*ai24*ai33*ar42 + ai11*ai24*ai42*ar33

          - ai11*ai24*ai43*ar32 + ai11*ai33*ai42*ar24 - ai11*ai33*ai44*ar22

          - ai11*ai34*ai42*ar23 + ai11*ai34*ai43*ar22 + ai11*ar22*ar33*ar44

          - ai11*ar22*ar34*ar43 - ai11*ar23*ar32*ar44 + ai11*ar23*ar34*ar42

          + ai11*ar24*ar32*ar43 - ai11*ar24*ar33*ar42 - ai22*ai33*ai44*ar11

          + ai22*ai34*ai43*ar11 + ai22*ar11*ar33*ar44 - ai22*ar11*ar34*ar43

          + ai24*ai33*ai42*ar11 + ai24*ar11*ar32*ar43 - ai24*ar11*ar33*ar42

          + ai33*ar11*ar22*ar44 - ai33*ar11*ar24*ar42 - ai34*ar11*ar22*ar43

          + ai34*ar11*ar23*ar42 + ai42*ar11*ar23*ar34 - ai42*ar11*ar24*ar33

          - ai43*ar11*ar22*ar34 + ai43*ar11*ar24*ar32 + ai44*ar11*ar22*ar33

          - ai44*ar11*ar23*ar32

cpr01 := ai11*ai22*ar33 + ai11*ai22*ar44 - ai11*ai24*ar42 + ai11*ai33*ar22

          + ai11*ai33*ar44 - ai11*ai34*ar43 - ai11*ai42*ar24 - ai11*ai43*ar34

          + ai11*ai44*ar22 + ai11*ai44*ar33 + ai22*ai33*ar11 + ai22*ai33*ar44

          - ai22*ai34*ar43 - ai22*ai43*ar34 + ai22*ai44*ar11 + ai22*ai44*ar33

          - ai24*ai33*ar42 - ai24*ai42*ar11 - ai24*ai42*ar33 + ai24*ai43*ar32

          - ai33*ai42*ar24 + ai33*ai44*ar11 + ai33*ai44*ar22 + ai34*ai42*ar23

          - ai34*ai43*ar11 - ai34*ai43*ar22 - ar11*ar22*ar33 - ar11*ar22*ar44

          + ar11*ar23*ar32 + ar11*ar24*ar42 - ar11*ar33*ar44 + ar11*ar34*ar43

          - ar22*ar33*ar44 + ar22*ar34*ar43 + ar23*ar32*ar44 - ar23*ar34*ar42

          - ar24*ar32*ar43 + ar24*ar33*ar42

cpi01 := ai11*ai22*ai33 + ai11*ai22*ai44 - ai11*ai24*ai42 + ai11*ai33*ai44

          - ai11*ai34*ai43 - ai11*ar22*ar33 - ai11*ar22*ar44 + ai11*ar23*ar32

          + ai11*ar24*ar42 - ai11*ar33*ar44 + ai11*ar34*ar43 + ai22*ai33*ai44

          - ai22*ai34*ai43 - ai22*ar11*ar33 - ai22*ar11*ar44 - ai22*ar33*ar44

          + ai22*ar34*ar43 - ai24*ai33*ai42 + ai24*ar11*ar42 - ai24*ar32*ar43

          + ai24*ar33*ar42 - ai33*ar11*ar22 - ai33*ar11*ar44 - ai33*ar22*ar44

          + ai33*ar24*ar42 + ai34*ar11*ar43 + ai34*ar22*ar43 - ai34*ar23*ar42

          + ai42*ar11*ar24 - ai42*ar23*ar34 + ai42*ar24*ar33 + ai43*ar11*ar34

          + ai43*ar22*ar34 - ai43*ar24*ar32 - ai44*ar11*ar22 - ai44*ar11*ar33

          - ai44*ar22*ar33 + ai44*ar23*ar32

cpr02 :=  - ai11*ai22 - ai11*ai33 - ai11*ai44 - ai22*ai33 - ai22*ai44

          + ai24*ai42 - ai33*ai44 + ai34*ai43 + ar11*ar22 + ar11*ar33

          + ar11*ar44 + ar22*ar33 + ar22*ar44 - ar23*ar32 - ar24*ar42

          + ar33*ar44 - ar34*ar43

cpi02 := ai11*ar22 + ai11*ar33 + ai11*ar44 + ai22*ar11 + ai22*ar33 + ai22*ar44

          - ai24*ar42 + ai33*ar11 + ai33*ar22 + ai33*ar44 - ai34*ar43

          - ai42*ar24 - ai43*ar34 + ai44*ar11 + ai44*ar22 + ai44*ar33

cpr03 :=  - (ar11 + ar22 + ar33 + ar44)

cpi03 :=  - (ai11 + ai22 + ai33 + ai44)

              2        2
rpr00 := cpi00  + cpr00

rpr01 := 2*(cpi00*cpi01 + cpr00*cpr01)

                              2                        2
rpr02 := 2*cpi00*cpi02 + cpi01  + 2*cpr00*cpr02 + cpr01

rpr03 := 2*(cpi00*cpi03 + cpi01*cpi02 + cpr00*cpr03 + cpr01*cpr02)

                              2                                  2
rpr04 := 2*cpi01*cpi03 + cpi02  + 2*cpr00 + 2*cpr01*cpr03 + cpr02

rpr05 := 2*(cpi02*cpi03 + cpr01 + cpr02*cpr03)

              2                  2
rpr06 := cpi03  + 2*cpr02 + cpr03

rpr07 := 2*cpr03

cleardenot;


clear aa,bb,pol;



%***********************************************************************
%*****                                                             *****
%*****      T e s t     Examples   ---   Module    H U R W P       *****
%*****                                                             *****
%***********************************************************************

% Example H.1

x0:=lam-1;


x0 := lam - 1

x1:=lam-(ar+i*ai);


x1 := lam - (ai*i + ar)

x2:=lam-(br+i*bi);


x2 := lam - (bi*i + br)

x3:=lam-(cr+i*ci);


x3 := lam - (ci*i + cr)

hurwitzp x1;

Necessary and sufficient conditions are:
 - ar  >  0


cond


% Example H.2

x:=hurw(x0*x1);


x := 2*lam*( - ai*i - ar + 1) + 2*(ai*i + ar + 1)

hurwitzp x;

Necessary and sufficient conditions are:
        2     2
4*( - ai  - ar  + 1)  >  0


cond


% Example H.3

x:=(x1*x2);


        2
x := lam  - lam*(ai*i + ar + bi*i + br) - ai*bi + ai*br*i + ar*bi*i + ar*br

hurwitzp x;

Necessary and sufficient conditions are:
 - (ar + br)  >  0

         2               2               2     2
ar*br*(ai  - 2*ai*bi + ar  + 2*ar*br + bi  + br )  >  0


cond


% Example H.4

x:=(x1*x2*x3);


        3      2
x := lam  - lam *(ai*i + ar + bi*i + br + ci*i + cr) + lam*( - ai*bi + ai*br*i

         - ai*ci + ai*cr*i + ar*bi*i + ar*br + ar*ci*i + ar*cr - bi*ci + bi*cr*i

         + br*ci*i + br*cr) + ai*bi*ci*i + ai*bi*cr + ai*br*ci - ai*br*cr*i

      + ar*bi*ci - ar*bi*cr*i - ar*br*ci*i - ar*br*cr

hurwitzp x;

Necessary and sufficient conditions are:
 - (ar + br + cr)  >  0

  2           2                                           3        3
ai *ar*br + ai *ar*cr - 2*ai*ar*bi*br - 2*ai*ar*ci*cr + ar *br + ar *cr

       2   2       2             2   2        2           3          2
 + 2*ar *br  + 4*ar *br*cr + 2*ar *cr  + ar*bi *br + ar*br  + 4*ar*br *cr

             2        2           3     2                           3
 + 4*ar*br*cr  + ar*ci *cr + ar*cr  + bi *br*cr - 2*bi*br*ci*cr + br *cr

       2   2        2           3
 + 2*br *cr  + br*ci *cr + br*cr   >  0

               4   2       4           4   2       4           4   2     4   2
ar*br*cr*( - ai *bi  + 2*ai *bi*ci - ai *br  - 2*ai *br*cr - ai *ci  - ai *cr

                 3   3       3   2          3      2       3
           + 2*ai *bi  - 2*ai *bi *ci + 2*ai *bi*br  + 4*ai *bi*br*cr

                 3      2       3      2       3   2          3
           - 2*ai *bi*ci  + 2*ai *bi*cr  + 2*ai *br *ci + 4*ai *br*ci*cr

                 3   3       3      2       2   2   2       2   2
           + 2*ai *ci  + 2*ai *ci*cr  - 2*ai *ar *bi  + 4*ai *ar *bi*ci

                 2   2   2       2   2             2   2   2       2   2   2
           - 2*ai *ar *br  - 4*ai *ar *br*cr - 2*ai *ar *ci  - 2*ai *ar *cr

                 2      2          2      2          2
           - 2*ai *ar*bi *br - 2*ai *ar*bi *cr + 4*ai *ar*bi*br*ci

                 2                   2      3       2      2
           + 4*ai *ar*bi*ci*cr - 2*ai *ar*br  - 6*ai *ar*br *cr

                 2         2       2         2       2      2          2      3
           - 2*ai *ar*br*ci  - 6*ai *ar*br*cr  - 2*ai *ar*ci *cr - 2*ai *ar*cr

               2   4       2   3          2   2   2       2   2
           - ai *bi  - 2*ai *bi *ci - 2*ai *bi *br  - 2*ai *bi *br*cr

                 2   2   2       2   2   2       2      2          2
           + 6*ai *bi *ci  - 2*ai *bi *cr  - 2*ai *bi*br *ci - 8*ai *bi*br*ci*cr

                 2      3       2         2     2   4       2   3
           - 2*ai *bi*ci  - 2*ai *bi*ci*cr  - ai *br  - 2*ai *br *cr

                 2   2   2       2   2   2       2      2          2      3
           - 2*ai *br *ci  - 2*ai *br *cr  - 2*ai *br*ci *cr - 2*ai *br*cr

               2   4       2   2   2     2   4          2   3          2   2
           - ai *ci  - 2*ai *ci *cr  - ai *cr  + 2*ai*ar *bi  - 2*ai*ar *bi *ci

                    2      2          2                   2      2
           + 2*ai*ar *bi*br  + 4*ai*ar *bi*br*cr - 2*ai*ar *bi*ci

                    2      2          2   2             2
           + 2*ai*ar *bi*cr  + 2*ai*ar *br *ci + 4*ai*ar *br*ci*cr

                    2   3          2      2             3                2
           + 2*ai*ar *ci  + 2*ai*ar *ci*cr  + 4*ai*ar*bi *cr + 4*ai*ar*bi *br*ci

                       2                      2                      2
           - 8*ai*ar*bi *ci*cr + 4*ai*ar*bi*br *cr - 8*ai*ar*bi*br*ci

                             2                2                   3
           + 8*ai*ar*bi*br*cr  + 4*ai*ar*bi*ci *cr + 4*ai*ar*bi*cr

                       3                2                      3
           + 4*ai*ar*br *ci + 8*ai*ar*br *ci*cr + 4*ai*ar*br*ci

                             2          4             3   2          3   2
           + 4*ai*ar*br*ci*cr  + 2*ai*bi *ci - 2*ai*bi *ci  + 2*ai*bi *cr

                    2   2             2                   2   3
           + 4*ai*bi *br *ci + 4*ai*bi *br*ci*cr - 2*ai*bi *ci

                    2      2             2   2             2   2
           - 2*ai*bi *ci*cr  - 2*ai*bi*br *ci  + 2*ai*bi*br *cr

                          2                   3             4             2   2
           + 4*ai*bi*br*ci *cr + 4*ai*bi*br*cr  + 2*ai*bi*ci  + 4*ai*bi*ci *cr

                       4          4             3                2   3
           + 2*ai*bi*cr  + 2*ai*br *ci + 4*ai*br *ci*cr + 2*ai*br *ci

                    2      2     4   2       4           4   2       4
           + 2*ai*br *ci*cr  - ar *bi  + 2*ar *bi*ci - ar *br  - 2*ar *br*cr

               4   2     4   2       3   2          3   2          3
           - ar *ci  - ar *cr  - 2*ar *bi *br - 2*ar *bi *cr + 4*ar *bi*br*ci

                 3                3   3       3   2          3      2
           + 4*ar *bi*ci*cr - 2*ar *br  - 6*ar *br *cr - 2*ar *br*ci

                 3      2       3   2          3   3     2   4       2   3
           - 6*ar *br*cr  - 2*ar *ci *cr - 2*ar *cr  - ar *bi  + 2*ar *bi *ci

                 2   2   2       2   2             2   2   2       2   2   2
           - 2*ar *bi *br  - 6*ar *bi *br*cr - 2*ar *bi *ci  - 2*ar *bi *cr

                 2      2          2                   2      3
           + 2*ar *bi*br *ci + 8*ar *bi*br*ci*cr + 2*ar *bi*ci

                 2         2     2   4       2   3          2   2   2
           + 2*ar *bi*ci*cr  - ar *br  - 6*ar *br *cr - 2*ar *br *ci

                  2   2   2       2      2          2      3     2   4
           - 10*ar *br *cr  - 6*ar *br*ci *cr - 6*ar *br*cr  - ar *ci

                 2   2   2     2   4          4             3
           - 2*ar *ci *cr  - ar *cr  - 2*ar*bi *cr + 4*ar*bi *ci*cr

                    2   2             2      2          2      2
           - 4*ar*bi *br *cr - 2*ar*bi *br*ci  - 6*ar*bi *br*cr

                    2   2             2   3             2                      3
           - 2*ar*bi *ci *cr - 2*ar*bi *cr  + 4*ar*bi*br *ci*cr + 4*ar*bi*br*ci

                             2          4             3   2          3   2
           + 4*ar*bi*br*ci*cr  - 2*ar*br *cr - 2*ar*br *ci  - 6*ar*br *cr

                    2   2             2   3             4             2   2
           - 6*ar*br *ci *cr - 6*ar*br *cr  - 2*ar*br*ci  - 4*ar*br*ci *cr

                       4     4   2     4   2       3   3       3      2
           - 2*ar*br*cr  - bi *ci  - bi *cr  + 2*bi *ci  + 2*bi *ci*cr

                 2   2   2       2   2   2       2      2          2      3
           - 2*bi *br *ci  - 2*bi *br *cr  - 2*bi *br*ci *cr - 2*bi *br*cr

               2   4       2   2   2     2   4          2   3          2      2
           - bi *ci  - 2*bi *ci *cr  - bi *cr  + 2*bi*br *ci  + 2*bi*br *ci*cr

               4   2     4   2       3   2          3   3     2   4
           - br *ci  - br *cr  - 2*br *ci *cr - 2*br *cr  - br *ci

                 2   2   2     2   4
           - 2*br *ci *cr  - br *cr )  >  0


cond

clear x,x0,x1,x2,x3;



%***********************************************************************
%*****                                                             *****
%*****    T e s t     Examples   ---   Module    L I N B A N D     *****
%*****                                                             *****
%***********************************************************************

on evallhseqp;

  % So both sides of equations evaluate.

% Example L.1

operator v;


off echo;

      dimension u(200),v(200),acof(200,3),arhs(200),xl(200,3)
      dx=5.0e-2
      x=0.1
      do 25001 i=1,101
          v(i)=x**2/2.0
          x=x+dx
25001 continue
      iacof=200
      iarhs=200
      n=1
      ad(n)=1.0
      aucd(n)=0.0
      arhs(n)=v(1)
      n=n+1
      alcd(n)=1.0
      ad(n)=-2.0
      aucd(n)=1.0
      arhs(n)=v(3)-(2.0*v(2))+v(1)
      do 25002 k=3,99,1
          n=n+1
          alcd(n)=1.0
          ad(n)=-2.0
          aucd(n)=1.0
          arhs(n)=v(k-1)+v(k+1)-(2.0*v(k))
25002 continue
      n=n+1
      alcd(n)=1.0
      ad(n)=-2.0
      aucd(n)=1.0
      arhs(n)=v(101)-(2.0*v(100))+v(99)
      n=n+1
      alcd(n)=0.0
      ad(n)=1.0
      arhs(n)=v(101)
      call dgtsl(n,alcd,ad,aucd,arhs,ier)
c  n is number of equations
c  alcd,ad,aucd,arhs are arrays of dimension at least (n)
c  if (ier.ne.0) matrix acof is algorithmically singular
      if(ier.ne.0) call errout
      n=1
      u(1)=arhs(n)
      n=n+1
      u(2)=arhs(n)
      do 25003 k=3,99,1
          n=n+1
          u(k)=arhs(n)
25003 continue
      n=n+1
      u(100)=arhs(n)
      n=n+1
      u(101)=arhs(n)
      amer=0.0
      arer=0.0
      do 25004 i=1,101
          am=abs(real(u(i)-v(i)))
          ar=am/v(i)
          if(am.gt.amer) amer=am
          if(ar.gt.arer) arer=ar
25004 continue
      write(*,100)amer,arer
      stop
100     format(' max. abs. error = ',e12.2,' max. rel. error = ',e12.2)
      end


%***********************************************************************

% Example L.2

on nag;


off echo;

      dimension u(200),v(200),acof(200,3),arhs(200),xl(200,3)
      dx=5.0e-2
      x=0.1
      do 25005 i=1,101
          v(i)=x**2/2.0
          x=x+dx
25005 continue
      iacof=200
      iarhs=200
      n=1
      ad(n)=1.0
      aucd(n+1)=0.0
      arhs(n)=v(1)
      n=n+1
      alcd(n)=1.0
      ad(n)=-2.0
      aucd(n+1)=1.0
      arhs(n)=v(3)-(2.0*v(2))+v(1)
      do 25006 k=3,99,1
          n=n+1
          alcd(n)=1.0
          ad(n)=-2.0
          aucd(n+1)=1.0
          arhs(n)=v(k-1)+v(k+1)-(2.0*v(k))
25006 continue
      n=n+1
      alcd(n)=1.0
      ad(n)=-2.0
      aucd(n+1)=1.0
      arhs(n)=v(101)-(2.0*v(100))+v(99)
      n=n+1
      alcd(n)=0.0
      ad(n)=1.0
      arhs(n)=v(101)
      ier=0
      call f01lef(n,ad,0.,aucd,alcd,1.e-10,au2cd,in,ier)
c  n is number of equations
c  alcd,ad,aucd,au2cd,arhs are arrays of dimension at least (n)
c  in is integer array of dimension at least (n)
      if(ier.ne.0 .or. in(n).ne.0) call errout
      call f04lef(1,n,ad,aucd,alcd,au2cd,in,arhs,0.,ier)
      if(ier.ne.0) call errout
      n=1
      u(1)=arhs(n)
      n=n+1
      u(2)=arhs(n)
      do 25007 k=3,99,1
          n=n+1
          u(k)=arhs(n)
25007 continue
      n=n+1
      u(100)=arhs(n)
      n=n+1
      u(101)=arhs(n)
      amer=0.0
      arer=0.0
      do 25008 i=1,101
          am=abs(real(u(i)-v(i)))
          ar=am/v(i)
          if(am.gt.amer) amer=am
          if(ar.gt.arer) arer=ar
25008 continue
      write(*,100)amer,arer
      stop
100     format(' max. abs. error = ',e12.2,' max. rel. error = ',e12.2)
      end


%***********************************************************************

% Example L.3

on imsl;


off echo,nag;

      dimension u(200),v(200),acof(200,3),arhs(200),xl(200,3)
      dx=5.0e-2
      x=0.1
      do 25009 i=1,101
          v(i)=x**2/2.0
          x=x+dx
25009 continue
      iacof=200
      iarhs=200
      n=1
      acof(n,1)=0.0
      acof(n,2)=1.0
      acof(n,3)=0.0
      arhs(n)=v(1)
      n=n+1
      acof(n,1)=1.0
      acof(n,2)=-2.0
      acof(n,3)=1.0
      arhs(n)=v(3)-(2.0*v(2))+v(1)
      do 25010 k=3,99,1
          n=n+1
          acof(n,1)=1.0
          acof(n,2)=-2.0
          acof(n,3)=1.0
          arhs(n)=v(k-1)+v(k+1)-(2.0*v(k))
25010 continue
      n=n+1
      acof(n,1)=1.0
      acof(n,2)=-2.0
      acof(n,3)=1.0
      arhs(n)=v(101)-(2.0*v(100))+v(99)
      n=n+1
      acof(n,1)=0.0
      acof(n,2)=1.0
      acof(n,3)=0.0
      arhs(n)=v(101)
      call leqt1b(acof,n,1,1,iacof,arhs,1,iarhs,0,xl,ier)
c  iacof is actual 1-st dimension of the acof array
c  iarhs is actual 1-st dimension of the arhs array
c  xl is working array with size n*(nlc+1)
c  where n is number of equations nlc number of lower
c  codiagonals
c  if ier=129( .ne.0) matrix acof is algorithmically singular
      if(ier.ne.0) call errout
      n=1
      u(1)=arhs(n)
      n=n+1
      u(2)=arhs(n)
      do 25011 k=3,99,1
          n=n+1
          u(k)=arhs(n)
25011 continue
      n=n+1
      u(100)=arhs(n)
      n=n+1
      u(101)=arhs(n)
      amer=0.0
      arer=0.0
      do 25012 i=1,101
          am=abs(real(u(i)-v(i)))
          ar=am/v(i)
          if(am.gt.amer) amer=am
          if(ar.gt.arer) arer=ar
25012 continue
      write(*,100)amer,arer
      stop
100     format(' max. abs. error = ',e12.2,' max. rel. error = ',e12.2)
      end


%***********************************************************************

% Example L.4

on essl;


off echo,imsl;

      dimension u(200),v(200),acof(200,3),arhs(200),xl(200,3)
      dx=5.0e-2
      x=0.1
      do 25013 i=1,101
          v(i)=x**2/2.0
          x=x+dx
25013 continue
      iacof=200
      iarhs=200
      n=1
      ad(n)=1.0
      aucd(n)=0.0
      arhs(n)=v(1)
      n=n+1
      alcd(n)=1.0
      ad(n)=-2.0
      aucd(n)=1.0
      arhs(n)=v(3)-(2.0*v(2))+v(1)
      do 25014 k=3,99,1
          n=n+1
          alcd(n)=1.0
          ad(n)=-2.0
          aucd(n)=1.0
          arhs(n)=v(k-1)+v(k+1)-(2.0*v(k))
25014 continue
      n=n+1
      alcd(n)=1.0
      ad(n)=-2.0
      aucd(n)=1.0
      arhs(n)=v(101)-(2.0*v(100))+v(99)
      n=n+1
      alcd(n)=0.0
      ad(n)=1.0
      arhs(n)=v(101)
      call dgtf(n,alcd,ad,aucd,af,ipvt)
c  n is number of equations
c  alcd,ad,aucd,af,arhs are arrays of dimension at least (n)
c  these arrays are double precision type
c  for single precision change dgtf to sgtf and dgts to sgts
c  ipvt is integer array of dimension at least (n+3)/8
      call dgts(n,alcd,ad,aucd,af,ipvt,arhs)
      n=1
      u(1)=arhs(n)
      n=n+1
      u(2)=arhs(n)
      do 25015 k=3,99,1
          n=n+1
          u(k)=arhs(n)
25015 continue
      n=n+1
      u(100)=arhs(n)
      n=n+1
      u(101)=arhs(n)
      amer=0.0
      arer=0.0
      do 25016 i=1,101
          am=abs(real(u(i)-v(i)))
          ar=am/v(i)
          if(am.gt.amer) amer=am
          if(ar.gt.arer) arer=ar
25016 continue
      write(*,100)amer,arer
      stop
100     format(' max. abs. error = ',e12.2,' max. rel. error = ',e12.2)
      end

off essl;



%***********************************************************************
%*****                                                             *****
%*****      T e s t     Complex Examples   ---   More Modules      *****
%*****                                                             *****
%***********************************************************************

% Example M.1

off exp;


coordinates t,x into n,j;


grid uniform,x,t;


dependence v(t,x),w(t,x);


isgrid v(x..one),w(x..half);


iim aa, v, diff(v,t)=c*diff(w,x),
        w, diff(w,t)=c*diff(v,x);


*****************************
*****      Program      *****          IIMET Ver 1.1.2
*****************************

      Partial Differential Equations
      ==============================

diff(v,t) - diff(w,x)*c    =    0

 - diff(v,x)*c + diff(w,t)    =    0


0 interpolations are needed in t coordinate
  Equation for v variable is integrated in half grid point
  Equation for w variable is integrated in half grid point
0 interpolations are needed in x coordinate
  Equation for v variable is integrated in one grid point
  Equation for w variable is integrated in half grid point

         Equations after Discretization Using IIM :
         ==========================================

       2*j - 1          2*j + 1              2*j + 1              2*j - 1
((w(n,---------) - w(n,---------) - w(n + 1,---------) + w(n + 1,---------))*c
          2                2                    2                    2

 *ht + 2*(v(n + 1,j) - v(n,j))*hx)/(2*ht*hx)   =   0


( - (v(n,j + 1) - v(n,j) - v(n + 1,j) + v(n + 1,j + 1))*c*ht

                2*j + 1          2*j + 1
  + 2*(w(n + 1,---------) - w(n,---------))*hx)/(2*ht*hx)   =   0
                   2                2



on exp;


center t=1/2;


functions v,w;


approx( aa(0,0)=aa(0,1));


 Difference scheme approximates differential equation df(v,t) - df(w,x)*c=0

 with orders of approximation:

  2
ht

  2
hx

center x=1/2;


approx( aa(1,0)=aa(1,1));


 Difference scheme approximates differential equation  - df(v,x)*c + df(w,t)=0

 with orders of approximation:

  2
ht

  2
hx

let cos ax**2=1-sin ax**2;


unfunc v,w;


matrix a(1,2),b(2,2),bt(2,2);


a(1,1):=aa(0,0);


                                                    2*j - 1
a(1,1) := (2*v(n + 1,j)*hx - 2*v(n,j)*hx + w(n + 1,---------)*c*ht
                                                       2

                       2*j + 1               2*j - 1
            - w(n + 1,---------)*c*ht + w(n,---------)*c*ht
                          2                     2

                   2*j + 1
            - w(n,---------)*c*ht)/(2*ht*hx)
                      2

a(1,2):=aa(1,0);


a(1,2) := ( - v(n + 1,j + 1)*c*ht + v(n + 1,j)*c*ht - v(n,j + 1)*c*ht

                                       2*j + 1               2*j + 1
            + v(n,j)*c*ht + 2*w(n + 1,---------)*hx - 2*w(n,---------)*hx)/(2*ht
                                          2                     2

             *hx)

off prfourmat;


b:=ampmat a;


       kx*hx
ax := -------
         2


     [           2  2   2     2                             ]
     [  - sin(ax) *c *ht  + hx       2*i*sin(ax)*c*ht*hx    ]
     [--------------------------   -----------------------  ]
     [         2  2   2     2              2  2   2     2   ]
     [  sin(ax) *c *ht  + hx        sin(ax) *c *ht  + hx    ]
b := [                                                      ]
     [                                       2  2   2     2 ]
     [   2*i*sin(ax)*c*ht*hx        - sin(ax) *c *ht  + hx  ]
     [ -----------------------    --------------------------]
     [         2  2   2     2              2  2   2     2   ]
     [  sin(ax) *c *ht  + hx        sin(ax) *c *ht  + hx    ]


clear a,aa;


factor lam;


pol:=charpol b;


           2         4  4   4            2  2   2   2     4
pol := (lam *(sin(ax) *c *ht  + 2*sin(ax) *c *ht *hx  + hx )

                         4  4   4     4           4  4   4
         + 2*lam*(sin(ax) *c *ht  - hx ) + sin(ax) *c *ht

                    2  2   2   2     4          4  4   4            2  2   2   2
         + 2*sin(ax) *c *ht *hx  + hx )/(sin(ax) *c *ht  + 2*sin(ax) *c *ht *hx

               4
           + hx )

pol:=troot1 pol;


                 2  2   2
        4*sin(ax) *c *ht
 If  -----------------------  =  0  and  0
             2  2   2     2
      sin(ax) *c *ht  + hx

  =  0  , a root of the polynomial is equal to 1.

           2         4  4   4            2  2   2   2     4
pol := (lam *(sin(ax) *c *ht  + 2*sin(ax) *c *ht *hx  + hx )

                         4  4   4     4           4  4   4
         + 2*lam*(sin(ax) *c *ht  - hx ) + sin(ax) *c *ht

                    2  2   2   2     4          4  4   4            2  2   2   2
         + 2*sin(ax) *c *ht *hx  + hx )/(sin(ax) *c *ht  + 2*sin(ax) *c *ht *hx

               4
           + hx )

pol:=hurw num pol;


pol := 

     2        2  2   2         2  2   2     2        2         2  2   2     2
4*lam *sin(ax) *c *ht *(sin(ax) *c *ht  + hx ) + 4*hx *(sin(ax) *c *ht  + hx )

hurwitzp pol;

Necessary and sufficient conditions are:
         2
       hx
-----------------  >  0
        2  2   2
 sin(ax) *c *ht


cond

bt:=tcon b;


      [           2  2   2     2                             ]
      [  - sin(ax) *c *ht  + hx       - 2*sin(ax)*c*ht*hx*i  ]
      [--------------------------   ------------------------ ]
      [         2  2   2     2              2  2   2     2   ]
      [  sin(ax) *c *ht  + hx        sin(ax) *c *ht  + hx    ]
bt := [                                                      ]
      [                                       2  2   2     2 ]
      [   - 2*sin(ax)*c*ht*hx*i      - sin(ax) *c *ht  + hx  ]
      [ ------------------------   --------------------------]
      [         2  2   2     2              2  2   2     2   ]
      [  sin(ax) *c *ht  + hx        sin(ax) *c *ht  + hx    ]


bt*b;


[1  0]
[    ]
[0  1]


bt*b-b*bt;


[0  0]
[    ]
[0  0]


clear aa,a,b,bt;



%***********************************************************************

% Example M.2 : Richtmyer, Morton: Difference methods for initial value
%               problems, &10.2.  p.261

coordinates t,x into n,j;


grid uniform,t,x;


let cos ax**2=1-sin ax**2;


unfunc v,w;


matrix a(1,2),b(2,2),bt(2,2);


a(1,1):=(v(n+1)-v(n))/ht-c*(w(j+1/2)-w(j-1/2))/hx$


a(1,2):=(w(n+1,j-1/2)-w(n,j-1/2))/ht-c*(v(n+1,j)-v(n+1,j-1))/hx$


off prfourmat;


b:=ampmat a;


       kx*hx
ax := -------
         2


     [                          2*i*sin(ax)*c*ht      ]
     [        1                ------------------     ]
     [                                 hx             ]
     [                                                ]
b := [                                 2  2   2     2 ]
     [ 2*i*sin(ax)*c*ht     - 4*sin(ax) *c *ht  + hx  ]
     [------------------  ----------------------------]
     [        hx                        2             ]
     [                                hx              ]


clear a;


factor lam;


pol:=charpol b;


           2   2                   2  2   2     2      2
        lam *hx  + 2*lam*(2*sin(ax) *c *ht  - hx ) + hx
pol := --------------------------------------------------
                                2
                              hx

pol:=hurw num pol;


            2        2  2   2                2  2   2     2
pol := 4*lam *sin(ax) *c *ht  + 4*( - sin(ax) *c *ht  + hx )

hurwitzp pol;

Necessary and sufficient conditions are:
           2  2   2     2
  - sin(ax) *c *ht  + hx
--------------------------  >  0
            2  2   2
     sin(ax) *c *ht


cond

bt:=tcon b;


      [                            - 2*sin(ax)*c*ht*i     ]
      [          1               ---------------------    ]
      [                                   hx              ]
      [                                                   ]
bt := [                                    2  2   2     2 ]
      [  - 2*sin(ax)*c*ht*i     - 4*sin(ax) *c *ht  + hx  ]
      [---------------------  ----------------------------]
      [         hx                          2             ]
      [                                   hx              ]


bt*b;


[          2  2   2     2                           3  3   3                  ]
[ 4*sin(ax) *c *ht  + hx                   8*sin(ax) *c *ht *i                ]
[-------------------------                ---------------------               ]
[             2                                      3                        ]
[           hx                                     hx                         ]
[                                                                             ]
[             3  3   3                 4  4   4            2  2   2   2     4 ]
[  - 8*sin(ax) *c *ht *i     16*sin(ax) *c *ht  - 4*sin(ax) *c *ht *hx  + hx  ]
[------------------------   --------------------------------------------------]
[            3                                       4                        ]
[          hx                                      hx                         ]


bt*b-b*bt;


[                                      3  3   3   ]
[                            16*sin(ax) *c *ht *i ]
[            0              ----------------------]
[                                      3          ]
[                                    hx           ]
[                                                 ]
[              3  3   3                           ]
[  - 16*sin(ax) *c *ht *i                         ]
[-------------------------            0           ]
[             3                                   ]
[           hx                                    ]


clear a,b,bt;



%***********************************************************************

% Example M.3: Mazurik: Algoritmy resenia zadaci..., preprint no.24-85,
%               AN USSR SO, Inst. teor. i prikl. mechaniky, p.34

operator v1,v2;


matrix a(1,3),b(3,3),bt(3,3);


a(1,1):=(p(n+1)-p(n))/ht+c/2*((-p(m-1)+2*p(m)-p(m+1))/hx +
         (v1(m+1)-v1(m-1))/hx - (p(k-1)-2*p(k)+p(k+1))/hy +
         (v2(k+1)-v2(k-1))/hy)$


a(1,2):=(v1(n+1)-v1(n))/ht+c/2*((p(m+1)-p(m-1))/hx -
         (v1(m-1)-2*v1(m)+v1(m+1))/hx)$


a(1,3):=(v2(n+1)-v2(n))/ht + c/2*((p(k+1)-p(k-1))/hy -
         (v2(k-1)-2*v2(k)+v2(k+1))/hy)$


coordinates t,x,y into n,m,k;


functions p,v1,v2;


for k:=1:3 do approx(a(1,k)=0);


 Difference scheme approximates differential equation 

df(p,t) + df(v1,x)*c + df(v2,y)*c=0

 with orders of approximation:

  2
ht

hx

hy

 Difference scheme approximates differential equation df(p,x)*c + df(v1,t)=0

 with orders of approximation:

  2
ht

hx

1

 Difference scheme approximates differential equation df(p,y)*c + df(v2,t)=0

 with orders of approximation:

  2
ht

1

hy

grid uniform,t,x,y;


unfunc p,v1,v2;


hy:=hx;


hy := hx

off prfourmat;


b:=ampmat a;


ax := kx*hx


ay := ky*hy


           cos(ax)*c*ht + cos(ay)*c*ht - 2*c*ht + hx    - i*sin(ax)*c*ht
b := mat((-------------------------------------------,-------------------,
                              hx                              hx

            - i*sin(ay)*c*ht
          -------------------),
                  hx

            - i*sin(ax)*c*ht   cos(ax)*c*ht - c*ht + hx
         (-------------------,--------------------------,0),
                  hx                      hx

            - i*sin(ay)*c*ht     cos(ay)*c*ht - c*ht + hx
         (-------------------,0,--------------------------))
                  hx                        hx


pol:=charpol b;


           3   3      2   2
pol := (lam *hx  + lam *hx *( - 2*cos(ax)*c*ht - 2*cos(ay)*c*ht + 4*c*ht - 3*hx)

                                      2   2              2   2
         + lam*hx*(3*cos(ax)*cos(ay)*c *ht  - 5*cos(ax)*c *ht

                                             2   2
            + 4*cos(ax)*c*ht*hx - 5*cos(ay)*c *ht  + 4*cos(ay)*c*ht*hx

                 2   2                   2                       3   3
            + 7*c *ht  - 8*c*ht*hx + 3*hx ) + 4*cos(ax)*cos(ay)*c *ht

                              2   2                 3   3              2   2
         - 3*cos(ax)*cos(ay)*c *ht *hx - 4*cos(ax)*c *ht  + 5*cos(ax)*c *ht *hx

                            2              3   3              2   2
         - 2*cos(ax)*c*ht*hx  - 4*cos(ay)*c *ht  + 5*cos(ay)*c *ht *hx

                            2      3   3      2   2               2     3    3
         - 2*cos(ay)*c*ht*hx  + 4*c *ht  - 7*c *ht *hx + 4*c*ht*hx  - hx )/hx

let
  cos ax=cos ax2**2-sin ax2**2,
  cos ay=cos ay2**2-sin ay2**2,
  sin ax=2*sin ax2*cos ax2,
  sin ay=2*sin ay2*cos ay2,
  cos ax2**2=1-sin ax2**2,
  cos ay2**2=1-sin ay2**2,
  sin ax2=s1,
  sin ay2=s2,
  hx=c*ht/cap;


factor lam;


order s1,s2;


pol:=troot1 pol;


          2   2    3
 If  16*s1 *s2 *cap   =  0  and  0

  =  0  , a root of the polynomial is equal to 1.

          3      2      2           2
pol := lam  + lam *(4*s1 *cap + 4*s2 *cap - 3) + lam

              2   2    2       2    2       2           2    2       2
       *(12*s1 *s2 *cap  + 4*s1 *cap  - 8*s1 *cap + 4*s2 *cap  - 8*s2 *cap + 3)

               2   2    3        2   2    2       2    2       2
        + 16*s1 *s2 *cap  - 12*s1 *s2 *cap  - 4*s1 *cap  + 4*s1 *cap

              2    2       2
        - 4*s2 *cap  + 4*s2 *cap - 1

clear cos ax,cos ay,sin ax,sin ay,cos ax2**2,cos ay2**2,sin ax2,sin ay2,
      hx,hy;


pol:=hurw num pol;


             3   2   2    3
pol := 16*lam *s1 *s2 *cap

               2    2         2   2           2   2     2     2
        + 8*lam *cap *( - 6*s1 *s2 *cap + 3*s1 *s2  + s1  + s2 ) + 16*lam*cap

             2   2    2       2   2         2         2     2         2
       *(3*s1 *s2 *cap  - 3*s1 *s2 *cap - s1 *cap + s1  - s2 *cap + s2 ) + 8*(

                 2   2    3       2   2    2     2    2       2         2    2
           - 2*s1 *s2 *cap  + 3*s1 *s2 *cap  + s1 *cap  - 2*s1 *cap + s2 *cap

                 2
           - 2*s2 *cap + 1)

hurwitzp pol;

If we denote:
           2   2    3
 (1)  16*s1 *s2 *cap   >  0

           2         2   2           2   2     2     2
 (2)  8*cap *( - 6*s1 *s2 *cap + 3*s1 *s2  + s1  + s2 )  >  0

                  2   2    2       2   2         2         2     2         2
 (3)  16*cap*(3*s1 *s2 *cap  - 3*s1 *s2 *cap - s1 *cap + s1  - s2 *cap + s2 )

  >  0

                2   2    3       2   2    2     2    2       2         2    2
 (4)  8*( - 2*s1 *s2 *cap  + 3*s1 *s2 *cap  + s1 *cap  - 2*s1 *cap + s2 *cap

                2
          - 2*s2 *cap + 1)  >  0

           2         2   2           2   2     2     2
 (5)  8*cap *( - 6*s1 *s2 *cap + 3*s1 *s2  + s1  + s2 )  >  0

             3          4   4    3        4   4    2       4   4
 (6)  128*cap *( - 16*s1 *s2 *cap  + 24*s1 *s2 *cap  - 9*s1 *s2 *cap

                       4   2    2        4   2           4   2     4         4
                 + 8*s1 *s2 *cap  - 10*s1 *s2 *cap + 3*s1 *s2  - s1 *cap + s1

                       2   4    2        2   4           2   4       2   2
                 + 8*s1 *s2 *cap  - 10*s1 *s2 *cap + 3*s1 *s2  - 2*s1 *s2 *cap

                     2   2     4         4
                 + s1 *s2  - s2 *cap + s2 )  >  0

              3       6   6    6        6   6    5        6   6    4
 (7)  1024*cap *(32*s1 *s2 *cap  - 96*s1 *s2 *cap  + 90*s1 *s2 *cap

                         6   6    3        6   4    5         6   4    4
                  - 27*s1 *s2 *cap  - 32*s1 *s2 *cap  + 100*s1 *s2 *cap

                         6   4    3        6   4    2        6   2    4
                  - 93*s1 *s2 *cap  + 27*s1 *s2 *cap  + 10*s1 *s2 *cap

                         6   2    3        6   2    2       6   2         6    3
                  - 31*s1 *s2 *cap  + 26*s1 *s2 *cap  - 6*s1 *s2 *cap - s1 *cap

                        6    2       6            4   6    5         4   6    4
                  + 3*s1 *cap  - 2*s1 *cap - 32*s1 *s2 *cap  + 100*s1 *s2 *cap

                         4   6    3        4   6    2        4   4    4
                  - 93*s1 *s2 *cap  + 27*s1 *s2 *cap  + 20*s1 *s2 *cap

                         4   4    3        4   4    2        4   4
                  - 76*s1 *s2 *cap  + 73*s1 *s2 *cap  - 21*s1 *s2 *cap

                        4   2    3        4   2    2        4   2
                  - 3*s1 *s2 *cap  + 16*s1 *s2 *cap  - 14*s1 *s2 *cap

                        4   2     4         4        2   6    4
                  + 3*s1 *s2  - s1 *cap + s1  + 10*s1 *s2 *cap

                         2   6    3        2   6    2       2   6
                  - 31*s1 *s2 *cap  + 26*s1 *s2 *cap  - 6*s1 *s2 *cap

                        2   4    3        2   4    2        2   4
                  - 3*s1 *s2 *cap  + 16*s1 *s2 *cap  - 14*s1 *s2 *cap

                        2   4       2   2         2   2     6    3       6    2
                  + 3*s1 *s2  - 2*s1 *s2 *cap + s1 *s2  - s2 *cap  + 3*s2 *cap

                        6         4         4
                  - 2*s2 *cap - s2 *cap + s2 )  >  0

 (c1)   (1) AND (2) AND (4)

 (c2)   (1) AND (3) AND (4)

 (d1)   (5) AND (7)

 (d2)   (6)

Necessary and sufficient conditions are:
  (  (C1)  OR  (C2)  )  AND  (  (D1)  OR  (D2)  )

cond

bt:=tcon b;


            cos(ax)*c*ht + cos(ay)*c*ht - 2*c*ht + hx   sin(ax)*c*ht*i
bt := mat((-------------------------------------------,----------------,
                               hx                             hx

            sin(ay)*c*ht*i
           ----------------),
                  hx

            sin(ax)*c*ht*i   cos(ax)*c*ht - c*ht + hx
          (----------------,--------------------------,0),
                  hx                    hx

            sin(ay)*c*ht*i     cos(ay)*c*ht - c*ht + hx
          (----------------,0,--------------------------))
                  hx                      hx


bt*b;


                         2   2              2   2
mat(((2*cos(ax)*cos(ay)*c *ht  - 4*cos(ax)*c *ht  + 2*cos(ax)*c*ht*hx

                    2   2                          2   2                 2    2
       - 4*cos(ay)*c *ht  + 2*cos(ay)*c*ht*hx + 6*c *ht  - 4*c*ht*hx + hx )/hx ,

               2   2                               2   2
      sin(ax)*c *ht *i*( - cos(ay) + 1)   sin(ay)*c *ht *i*( - cos(ax) + 1)
     -----------------------------------,-----------------------------------),
                       2                                   2
                     hx                                  hx

               2   2
      sin(ax)*c *ht *i*(cos(ay) - 1)
    (--------------------------------,
                     2
                   hx

                    2   2                          2   2                 2
       - 2*cos(ax)*c *ht  + 2*cos(ax)*c*ht*hx + 2*c *ht  - 2*c*ht*hx + hx
     ----------------------------------------------------------------------,
                                        2
                                      hx

                       2   2
      sin(ax)*sin(ay)*c *ht
     ------------------------),
                 2
               hx

               2   2                                    2   2
      sin(ay)*c *ht *i*(cos(ax) - 1)   sin(ax)*sin(ay)*c *ht
    (--------------------------------,------------------------,
                     2                            2
                   hx                           hx

                    2   2                          2   2                 2
       - 2*cos(ay)*c *ht  + 2*cos(ay)*c*ht*hx + 2*c *ht  - 2*c*ht*hx + hx
     ----------------------------------------------------------------------))
                                        2
                                      hx


bt*b-b*bt;


                   2   2
        2*sin(ax)*c *ht *i*( - cos(ay) + 1)
mat((0,-------------------------------------,
                          2
                        hx

                 2   2
      2*sin(ay)*c *ht *i*( - cos(ax) + 1)
     -------------------------------------),
                        2
                      hx

                 2   2
      2*sin(ax)*c *ht *i*(cos(ay) - 1)
    (----------------------------------,0,0),
                      2
                    hx

                 2   2
      2*sin(ay)*c *ht *i*(cos(ax) - 1)
    (----------------------------------,0,0))
                      2
                    hx


clear a,b,bt,pol;



%***********************************************************************

end;


Time for test: 22393 ms
