/* $OpenXM: OpenXM/src/asir-contrib/packages/src/taka_runge_kutta.rr,v 1.20 2013/12/14 04:38:58 takayama Exp $ */
/* From misc/200205/runge-kutta.rr */

/* They have not yet been registered in names.rr */

#define DEVAL(a)  eval(a)
Taka_Runge_kutta_adapted0 = 0$
Taka_Runge_kutta_epsilon = 0.1$
Taka_Runge_kutta_H_Upper_Bound = 0.2$
Taka_Runge_kutta_Make_Larger = 1$ /* Default 1 */

Taka_Runge_kutta_graphic0 = 0$  /* load("glib"); */
Taka_Runge_kutta_yrange = 10$

Taka_Runge_kutta_save_data = 1$
Taka_Runge_kutta_debug = 0$

extern Tk_rk_deep_eval$
Tk_rk_deep_eval=0$

def taka_runge_kutta_2(F,X,Y,X0,Y0,H,X1) {
  extern Taka_Runge_kutta_graphic0, Taka_Runge_kutta_yrange, Taka_Runge_kutta_save_data;
  Alpha =0.5;
  Beta = 0.5;
  P = 1; Q = 1;

  Ans = [];
  if (Taka_Runge_kutta_graphic0) {
     glib_open();
     glib_window(X0,Y0[0]-Taka_Runge_kutta_yrange,X1,Y0[0]+Taka_Runge_kutta_yrange);
  }

  if (type(F) == 5) {
    N = size(F)[0];
  }else{
    N = length(F);
  }
  if (type(Y0) != 5) {
    Y0 = newvect(N,Y0);
  }
  Yk = Y0;
  K1 = newvect(N);
  K2 = newvect(N);
  Yk1 = newvect(N);
  Xk = X0;

  while (Xk < X1) {
    taka_runge_kutta_replace(K1,F,Y,N,X,Xk,Yk);
    taka_runge_kutta_replace(K2,F,Y,N,X,Xk+P*H,Yk+Q*H*K1);
    Yk1 = Yk+H*Alpha*K1+H*Beta*K2;
    if (Taka_Runge_kutta_save_data) {
      Ans = cons(cons(Xk,vtol(Yk)),Ans);
    }
    print([Xk,Yk[0]]);
    if (Taka_Runge_kutta_graphic0) glib_line(Xk,Yk[0],Xk+H,Yk1[0]);
    Xk += H;
    Yk = Yk1;
  }
  return Ans;
}

def taka_runge_kutta_2_test() {
   /* Equation of oscilations */
   F = newvect(2,[y2,-y1]);
   Y = [y1,y2];
   taka_runge_kutta_2(F,x,Y,0,[1,0],0.1,15);
}

def taka_runge_kutta_replace(V,F,Y,N,X,Xk,Rule_vector) {
  for (I=0; I<N; I++) {
    V[I] = subst(F[I],X,Xk);
    for (J=0; J<N; J++) {
      V[I] = subst(V[I],Y[J],Rule_vector[J]);    
    }
    V[I] = eval(V[I]*exp(0));
  }
}

def taka_runge_kutta_abs(V) {
  if (type(V) != 5 && type(V) != 4) { /* not a vector */
    if (ntype(V) == 4) { /* complex number */
       return V*conj(V);
    }else{
      return(V*V);
    }
  }
  if (type(V) == 5) N = size(V)[0];
  if (type(V) == 4) N = length(V);
  S = 0;
  for (I=0; I<N; I++) {
    if (ntype(V[I]) == 4) /* complex number */
      S += V[I]*conj(V[I]);
    else
      S += V[I]*V[I];
  }
  return S;
}

def taka_runge_kutta_4(F,X,Y,X0,Y0,H,X1) {
/* N : rank of the ODE. */
  extern Taka_Runge_kutta_adapted0, Taka_Runge_kutta_epsilon,
         Taka_Runge_kutta_graphic0, Taka_Runge_kutta_yrange,
         Taka_Runge_kutta_save_data, Taka_Runge_kutta_debug;

  OneStep = getopt(onestep);
  if (type(OneStep) <= 0) OneStep = 0; else OneStep = 1;
  if (OneStep) X1=X0+2*H;
  if ((H<0) && (X1-X0)>0) error("taka_runge_kutta_4, X1-X0 should be <0");
  if ((H>0) && (X1-X0)<0) error("taka_runge_kutta_4, X1-X0 should be >0");
  Ans = [];
  if (Taka_Runge_kutta_graphic0) {
     glib_open();
     glib_window(X0,Y0[0]-Taka_Runge_kutta_yrange,X1,Y0[0]+Taka_Runge_kutta_yrange);
  }
  if (X0==X1) return([cons(X0,Y0)]);

  if (type(F) == 5) {
    N = size(F)[0];
  }else{
    N = length(F);
  }
  if (type(Y0) != 5) {
    Y0 = newvect(N,Y0);
  }
  Yk = Y0;
  K1 = newvect(N);
  K2 = newvect(N);
  K3 = newvect(N);
  K4 = newvect(N);
  Yk1 = newvect(N);
  Xk = X0;

  while (H<0? Xk > X1: Xk < X1) {
    taka_runge_kutta_replace(K1,F,Y,N,X,Xk,Yk);
    taka_runge_kutta_replace(K2,F,Y,N,X,Xk+H*(1/2),Yk+K1*(1/2)*H);
    taka_runge_kutta_replace(K3,F,Y,N,X,Xk+H*(1/2),Yk+K2*(1/2)*H);
    taka_runge_kutta_replace(K4,F,Y,N,X,Xk+H,Yk+K3*H);
    Yk1 = Yk+H*(K1/6+K2/3+K3/3+K4/6);
    if (Taka_Runge_kutta_debug) print([Xk,Yk[0]]);
    if (Taka_Runge_kutta_save_data) {
      Ans = cons(cons(Xk,vtol(Yk)),Ans);
    }
    if (OneStep) {
       return([cons(Xk+H,vtol(Yk1)), cons(Xk,vtol(Yk))]);
    }
    if (Taka_Runge_kutta_graphic0) glib_line(Xk,Yk[0],Xk+H,Yk1[0]);
    if (Taka_Runge_kutta_adapted0 && 
        (taka_runge_kutta_abs(Yk1-Yk) > Taka_Runge_kutta_epsilon)) {
      H = H*(1/2);
    }else{
      if (Taka_Runge_kutta_adapted0) H = H*2;
      Xk += H;
      Yk = Yk1;
    }
  }
  return Ans;
}

def taka_runge_kutta_4_adaptive(F,X,Y,X0,Y0,H,X1) {
/* N : rank of the ODE. */
  extern Taka_Runge_kutta_epsilon,
         Taka_Runge_kutta_graphic0, Taka_Runge_kutta_yrange,
         Taka_Runge_kutta_save_data,
         Taka_Runge_kutta_H_Upper_Bound,
         Taka_Runge_kutta_Make_Larger;

  Opt = getopt();
  if (taka_runge_kutta_complex_gt(H,0)) Forward = 1; else Forward = 0;
  while(Opt != []) {
    if (car(Opt)[0] == "forward") {
      Forward = car(Opt)[1];
    }
    Opt = cdr(Opt);
  }

  Ans = [cons(X0,Y0)];
  if (Taka_Runge_kutta_graphic0) {
     glib_open();
     glib_window(X0,Y0[0]-Taka_Runge_kutta_yrange,X1,Y0[0]+Taka_Runge_kutta_yrange);
  }

  if (type(F) == 5) {
    N = size(F)[0];
  }else{
    N = length(F);
  }
  if (type(Y0) != 5) {
    Y0 = newvect(N,Y0);
  }
  Yk = Y0;
  K1 = newvect(N);
  K2 = newvect(N);
  K3 = newvect(N);
  K4 = newvect(N);
  Yk1 = newvect(N);
  Yk2 = newvect(N);
  Yk3 = newvect(N);
  Xk = X0;

  while (true) {
    if (Forward) {
       /* if (Xk > X1) break; */
       if (taka_runge_kutta_complex_gt(Xk,X1)) break;
    } else{
       /* if (Xk < X1) break; */
       if (taka_runge_kutta_complex_gt(X1,Xk)) break;
    }
    /* Regular step */
    taka_runge_kutta_replace(K1,F,Y,N,X,Xk,Yk);
    taka_runge_kutta_replace(K2,F,Y,N,X,Xk+H*(1/2),Yk+K1*(1/2)*H);
    taka_runge_kutta_replace(K3,F,Y,N,X,Xk+H*(1/2),Yk+K2*(1/2)*H);
    taka_runge_kutta_replace(K4,F,Y,N,X,Xk+H,Yk+K3*H);
    Yk1 = Yk+H*(K1/6+K2/3+K3/3+K4/6);
    /* half step */
    H2 = H/2;
    taka_runge_kutta_replace(K1,F,Y,N,X,Xk,Yk);
    taka_runge_kutta_replace(K2,F,Y,N,X,Xk+H2*(1/2),Yk+K1*(1/2)*H2);
    taka_runge_kutta_replace(K3,F,Y,N,X,Xk+H2*(1/2),Yk+K2*(1/2)*H2);
    taka_runge_kutta_replace(K4,F,Y,N,X,Xk+H2,Yk+K3*H2);
    Yk2 = Yk+H2*(K1/6+K2/3+K3/3+K4/6);

    taka_runge_kutta_replace(K1,F,Y,N,X,Xk+H2,Yk2);
    taka_runge_kutta_replace(K2,F,Y,N,X,Xk+H2+H2*(1/2),Yk2+K1*(1/2)*H2);
    taka_runge_kutta_replace(K3,F,Y,N,X,Xk+H2+H2*(1/2),Yk2+K2*(1/2)*H2);
    taka_runge_kutta_replace(K4,F,Y,N,X,Xk+H2+H2,Yk2+K3*H2);
    Yk3 = Yk2+H2*(K1/6+K2/3+K3/3+K4/6);

    /* This is a strategy which you may change. */
    /* WantedPrec = Taka_Runge_kutta_epsilon*taka_runge_kutta_abs(Yk);*/
    WantedPrec = Taka_Runge_kutta_epsilon;

    Delta1 = DEVAL(taka_runge_kutta_abs(Yk3-Yk1));
    if (Delta1 != 0) {
      Habs = DEVAL((WantedPrec/Delta1)^(1/5));
      Habs = (4/5)*Habs; /* 0.8 = (4/5) is the safety factor */
    }else{
      Habs = 2; /* Any large number */
    }
    /* print("Habs="+rtostr(Habs)); */
    if (Habs < 1) { /* Compute again.  */
      H = H*Habs;
      print("Changing to Smaller step size: "+rtostr(H));
      print([Xk,Yk[0]]);
    }else{  /* Go ahead */
      Xk += H;
      Yk = Yk1;
      if ((H<Taka_Runge_kutta_H_Upper_Bound) && Taka_Runge_kutta_Make_Larger) {
        H = (Habs*H > Taka_Runge_kutta_H_Upper_Bound? 
             (H/number_abs(H))*Taka_Runge_kutta_H_Upper_Bound :
             Habs*H);  /* Habs*H2*2 */
        print("Changing to a larger step size: "+rtostr(H));
      }
      print([Xk,Yk[0]]);
      if (Taka_Runge_kutta_save_data) {
        Ans = cons(cons(Xk,vtol(Yk)),Ans);
      }
      if (Taka_Runge_kutta_graphic0) glib_line(Xk,Yk[0],Xk+H,Yk1[0]);
    }
  }
  return Ans;
}

/* load("glib"); load("taka_plot.rr"); to execute the functions below. */
def taka_runge_kutta_4_a_test() {
   /* exponential function */
   F = newvect(1,[y1]);
   Y = [y1];
   T = taka_runge_kutta_4_adaptive(F,x,Y,0,[1],0.1,5);
   taka_plot_auto(T);
   print("Eval by eval(exp(?)) : ",0); print([T[0][0],eval(exp(T[0][0]))]);
}

def taka_runge_kutta_4_a2_test() {
   /* Equation of oscilations */
   F = newvect(2,[y2,-y1]);
   Y = [y1,y2];
   T = taka_runge_kutta_4_adaptive(F,x,Y,0,[1,0],0.1,15);
   taka_plot_auto(T);
   print("Eval by eval((?)) : ",0); print([T[0][0],eval(cos(T[0][0]))]);
}

def taka_runge_kutta_4_test() {
   /* Equation of oscilations */
   F = newvect(2,[y2,-y1]);
   Y = [y1,y2];
   T=taka_runge_kutta_4(F,x,Y,0,[1,0],0.1,15);
   print(T);
   taka_plot_auto(T);
}

def taka_runge_kutta_4_test2() {
   /* Equation of oscilations */
   F = newvect(2,[y2,-y1]);
   Y = [y1,y2];
   T=taka_runge_kutta_4(F,x,Y,15,[1,0],-0.1,0);
   print(T);
   taka_plot_auto(T);
}

def taka_runge_kutta_replace_linear(V,F,Y,N,X,Xk,Rule_vector) {
extern Tk_rk_deep_eval;
  if (Tk_rk_deep_eval) {
    V1=base_replace(F,[[X,Xk]]);
  }else{
    V1=base_replace_n(F,[[X,Xk]]);
  }
  V1=V1*Rule_vector;
  for (I=0; I<N; I++) {
    if (Tk_rk_deep_eval) 
      V[I] = eval(V1[I]*exp(0));
    else
      V[I] = V1[I];
  }
}

/* Y is a dummy */
def taka_runge_kutta_4_linear(F,X,Y,X0,Y0,H,X1) {
/* N : rank of the ODE. */
  extern Taka_Runge_kutta_adapted0, Taka_Runge_kutta_epsilon,
         Taka_Runge_kutta_graphic0, Taka_Runge_kutta_yrange,
         Taka_Runge_kutta_save_data, Taka_Runge_kutta_debug;

  OneStep = getopt(onestep);
  if (type(OneStep) <= 0) OneStep = 0; else OneStep = 1;
  if (OneStep) X1=X0+2*H;
  if ((H<0) && (X1-X0)>0) error("taka_runge_kutta_4_linear, X1-X0 should be <0");
  if ((H>0) && (X1-X0)<0) error("taka_runge_kutta_4_linear, X1-X0 should be >0");
  Ans = [];
  if (Taka_Runge_kutta_graphic0) {
     glib_open();
     glib_window(X0,Y0[0]-Taka_Runge_kutta_yrange,X1,Y0[0]+Taka_Runge_kutta_yrange);
  }
  if (X0==X1) return([cons(X0,Y0)]);

  if (type(F) == 4) {
    F=newmat(length(F),length(F[0]),F);
  }
  N = size(F)[0];

  if (type(Y0) != 5) {
    Y0 = newvect(N,Y0);
  }
  Yk = Y0;
  K1 = newvect(N);
  K2 = newvect(N);
  K3 = newvect(N);
  K4 = newvect(N);
  Yk1 = newvect(N);
  Xk = X0;

  while (H<0? Xk > X1: Xk < X1) {
    taka_runge_kutta_replace_linear(K1,F,Y,N,X,Xk,Yk);
    taka_runge_kutta_replace_linear(K2,F,Y,N,X,Xk+H*(1/2),Yk+K1*(1/2)*H);
    taka_runge_kutta_replace_linear(K3,F,Y,N,X,Xk+H*(1/2),Yk+K2*(1/2)*H);
    taka_runge_kutta_replace_linear(K4,F,Y,N,X,Xk+H,Yk+K3*H);
    Yk1 = Yk+H*(K1/6+K2/3+K3/3+K4/6);
    if (Taka_Runge_kutta_debug) print([Xk,Yk[0]]);
    if (Taka_Runge_kutta_save_data) {
      Ans = cons(cons(Xk,vtol(Yk)),Ans);
    }
    if (OneStep) {
       return([cons(Xk+H,vtol(Yk1)), cons(Xk,vtol(Yk))]);
    }
    if (Taka_Runge_kutta_graphic0) glib_line(Xk,Yk[0],Xk+H,Yk1[0]);
    if (Taka_Runge_kutta_adapted0 && 
        (taka_runge_kutta_abs(Yk1-Yk) > Taka_Runge_kutta_epsilon)) {
      H = H*(1/2);
    }else{
      if (Taka_Runge_kutta_adapted0) H = H*2;
      Xk += H;
      Yk = Yk1;
    }
  }
  return Ans;
}

def taka_runge_kutta_4_linear_test() {
   /* Airy equation y''-x y = 0 
    [evalf(AiryAi(0)),evalf(subs(x=0,diff(AiryAi(x),x)))];
    Y0=[0.3550280540, -0.2588194038]
    evalf(AiryAi(-5)); --> 0.35076
   */
   F = [[0,1],[x,0]];
   Y = [y1,y2]; 
   Y0=[0.3550280540, -0.2588194038];
   T=taka_runge_kutta_4_linear(F,x,Y,0,Y0,-0.1,-5);
   print(T);
   taka_plot_auto(T);
   T2=taka_runge_kutta_4([y2,x*y1],x,Y,0,Y0,-0.1,-5);
   print("AiryAi(-5) --> 0.35076");
   return([T[0],T2[0]]);
}

/*
def base_replace_n(F,R) { return base_replace(F,R); }
*/

/* cf. asir2000/engine/cplx.c  int cmpcplx(a,b),
       which does not compare the real part and imaginary part.
       Instead, it compares NID (number id)
*/
def taka_runge_kutta_complex_gt(A,B) {
  Ar = number_real_part(A); Ai = number_imaginary_part(A);
  Br = number_real_part(B); Bi = number_imaginary_part(B);
  if (Ar > Br) return 1;
  else if (Ar < Br) return 0;
  if (Ai > Bi) return 1;
  else if (Ai < Bi) return 0;
  return 0;
}

Loaded_taka_runge_kutta=1$

/* cf. misc-2003/09/neval/ellip.*  */
/* runge_kutta_4 is still buggy for complex numbers */

module tk_rk;
localf taka_minus;
localf taka_runge_kutta_reverse ;
localf taka_runge_kutta_4a;
localf taka_runge_kutta_4a_linear;
localf test4 ;
localf test4b ;
localf runge_kutta_4;
localf runge_kutta_4_linear;
def taka_minus(Ob) {
  if (type(Ob) != 4) return(-Ob);
  else return map(taka_minus,Ob);
}
def taka_runge_kutta_reverse(A) {
  B=[];
  for (; length(A) != 0; A=cdr(A)) {
     T=car(A);
     B=cons(cons(-T[0],cdr(T)),B);
  }
  return reverse(B);
}
def taka_runge_kutta_4a(FF,X0,Y,S0,Ys,T0,H) {
  if (T0 < S0) {
    /* opposite direction */
    return taka_runge_kutta_reverse(
      taka_runge_kutta_4a(map(taka_minus,base_replace(FF,[[X0,-X0]])),X0,Y,-S0,Ys,-T0,H));
  }
  if (H >= T0-S0) {
    A=taka_runge_kutta_4(FF,X0,Y,S0,Ys,T0-S0,0 | onestep=1);
  }else{
    A=taka_runge_kutta_4(FF,X0,Y,S0,Ys,H,T0);
    T=A[0];
    if (T0-T[0] > 0) {
      B=taka_runge_kutta_4(FF,X0,Y,T[0],cdr(T),T0-T[0],0 | onestep=1);
      T=B[0];
      A=cons(T,A);
    }
 }
 return(A);
}
def runge_kutta_4(FF,X0,Y,S0,Ys,T0,H) {
   return taka_runge_kutta_4a(FF,X0,Y,S0,Ys,T0,H);
}
def runge_kutta_4_linear(FF,X0,Y,S0,Ys,T0,H) {
   return taka_runge_kutta_4a_linear(FF,X0,Y,S0,Ys,T0,H);
}

def taka_runge_kutta_4a_linear(FF,X0,Y,S0,Ys,T0,H) {
  if (T0 < S0) {
    /* opposite direction */
    if (H<0) H=-H;
    return taka_runge_kutta_reverse(
      taka_runge_kutta_4a_linear(map(taka_minus,base_replace(FF,[[X0,-X0]])),X0,Y,-S0,Ys,-T0,H));
  }
  if (H >= T0-S0) {
    A=taka_runge_kutta_4_linear(FF,X0,Y,S0,Ys,T0-S0,0 | onestep=1);
  }else{
    A=taka_runge_kutta_4_linear(FF,X0,Y,S0,Ys,H,T0);
    T=A[0];
    if (T0-T[0] > 0) {
      B=taka_runge_kutta_4_linear(FF,X0,Y,T[0],cdr(T),T0-T[0],0 | onestep=1);
      T=B[0];
      A=cons(T,A);
    }
 }
 return(A);
}

/* equation of oscilation */
def test4() {
  A=runge_kutta_4([y1,-y0],x,[y0,y1],0,[1,0],3.14*2,0.1);
  taka_plot_auto(A);
  return(A);
}

def test4b() {
  A=taka_runge_kutta_4a([y1,-y0],x,[y0,y1],3.14,[-1,0],0,0.1);
  taka_plot_auto(A);
  return(A);
}
endmodule;

import("taka_plot.rr")$
pari(allocatemem,10^7)$
module rktest;
localf re$
localf re2$
localf im$
localf im2$
localf tryA$
localf tryA2$
localf geq$
localf test1$

def re(L) {
  return map(re2,L);
}
def re2(P) {
  return map(number_real_part,P);
}
def im(L) {
  return map(im2,L);
}
def im2(P) {
  return map(number_imaginary_part,P);
}

def geq() {
  L=x*(1-x)*dx^2+(c-(a+b+1)*x)*dx-a*b;
  L=base_replace(L,[[a,1/2],[b,1/2],[c,1]]);

  L2 = -((c-(a+b+1)*x)*y2-a*b*y1)/(x*(1-x));
  L2=base_replace(L2,[[a,1/2],[b,1/2],[c,1]]);
  return [ y2, L2];
}
def tryA() {
  LL = geq();
  A = taka_runge_kutta_4_adaptive(
       LL,
       x,[y1,y2],
       0.5+0.5*@i,[-6.78383-1.28991*@i, -1.51159-1.7935*@i],
       (3-@i)*0.0005, 2.0);
  taka_plot_auto(re(A));
  return A;
}
def tryA2() {
  LL = geq();
  A = taka_runge_kutta_4(
       LL,
       x,[y1,y2],
       0.5+0.5*@i,[-6.78383-1.28991*@i, -1.51159-1.7935*@i],
       (3-@i)*0.0005, 1.0);
  taka_plot_auto(re(A));
  return A;
}

def test1() {
  A=tryA2();
  B=A[100];
  print(B);
/* cf. misc-2008/A2/misc/ellip2.m
  p2 = 0.5+0.5*I --> [-6.78383-1.28991*@i, -1.51159-1.7935*@i].
  p2 = 0.8495+0.3835*I;
Print["-------------------"];
Print[p2];
Print[N[-2*Gamma[1/2]^2*Hypergeometric2F1[1/2,1/2,1,z] /. {z->p2}]]
Print[N[D[-2*Gamma[1/2]^2*Hypergeometric2F1[1/2,1/2,1,z],z] /. {z->p2}]]
*/
 print("math:  [0.8495+0.3835*I, -7.64079 - 2.0799*I, -1.16364 - 4.14709*I] ");
 print("It was Buggy tryA2() and tryA() --> fixed. see log of 1.8");
}

endmodule;

end$