// pde44h_eq.c  homogeneous Biharmonic PDE in four dimensions
// solve uxxxx(x,y,z,t) + uyyyy(x,y,z,t) + uzzzz(x,y,z,t) +
//       utttt(x,y,z,t) + 2*uxx(x,y,z,t) + 2*uyy(x,y,z,t) + 
//       2*uzz(x,y,z,t) + 2*utt(x,y,z,t) +   4*u(x,y,z,t) = 0
//
// boundary conditions computed using u(x,y,z,t)
// analytic solution is u(x,y,z,t) = sin(x+y+z+t)
//
// replace continuous derivatives with discrete derivatives
// solve for Uijkl numerical approximation U(x_i,y_j,z_k,t_l)
// A * U = F, solve simultaneous equations for U
//

// The subscript versions of the derivatives are substituted into the
// equation to be solved. The resulting equation is analytically solved
// for u(i,j,k,l) = other u terms with coefficients and + f(xi,yj,zk,tl)
// The boundaries xmin..xmax, ymin..ymax, zmin..zmax, tmin..tmax are known
// The number of grid points in each dimension is known nx, ny, nz, nt
// hx=(xmax-xmin)/(nx-1), hy=(ymax-ymin)/(ny-1),
// hz=(zmax-zmin)/(nz-1), ht=(tmax-tmin)/(tx-1), 
// thus xi=xmin+i*hx, yj=ymin+j*hy, zk=zmin+k*hz, tl=tmin+l*ht
// then the linear system of equations is built,
// nxyzt=(nx-2)*(ny-2)*(nz-2)*(nt-2) equations because the boundary value
//                                   is known on each end.
// The matrix A is nxyzt rows by nxyzt columns (nxyzt+1) columns including F
// The forcing function vector F is nxyzt elements adjunct to A
// The solution vector U is nxyzt elements.
// The building of the A matix requires 4 nested loops for rows
// i  in 1..nx-1, j  in 1..ny-1,  k  in 1..nz-1, l  in 1..nt-1 for rows of A
// then each term in the differential equation fills in that row
// and cs = (nx-2)*(ny-2)*(nz-2)*(nt-2) the RHS, fills the last column of A
// subscripting functions are used to compute linear subscripts of A, U
// row=(i-1)*(ny-2)*(nz-2)*(nt-2)+(ii-1)*(nz-2)*(nt-2)+(iii-1)*(nt-2)+(iiii-1)
// row*(nxyzt+1)+col for A
//
// care must be taken to move the negative of boundary elements to
// the right hand size of the equation to be solved. These are subtracted
// from the computed value of the forcing function, cs, for that row.
// tests are j==0 || j==nx-1   ...  jjjj==0 || jjjj==nt-1 for boundaries
//

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <time.h>
#include <string.h>
#include "deriv.h"
#include "simeq.h"

#undef  abs
#define abs(x) ((x)<0.0?(-(x)):(x))
#undef  min
#define min(a,b) ((a)<(b)?(a):(b))
#undef  max
#define max(a,b) ((a)>(b)?(a):(b))

// nx=5 0.001 error  nx=6 0.0004 error 
#define nx  6
#define ny  6
#define nz  6
#define nt  6
#define nxyzt (nx-2)*(ny-2)*(nz-2)*(nt-2)
#define nn max(nx, max(ny, max(nz, nt)))

int s(int i, int ii, int iii, int iiii) // for x,y,z,t 
{
  return (i-1)*(ny-2)*(nz-2)*(nt-2) + (ii-1)*(nz-2)*(nt-2) +
         (iii-1)*(nt-2) + (iiii-1);
} // end s 

int sk(int k, int i, int ii, int iii, int iiii) // for x,y,z,t 
{
  return k*(nxyzt+1) + (i-1)*(ny-2)*(nz-2)*(nt-2) + (ii-1)*(nz-2)*(nt-2) +
         (iii-1)*(nt-2) + (iiii-1);

} // end sk 

int ss(int i, int ii, int iii, int iiii, int j, int jj, int jjj, int jjjj)
{
  return s(i,ii,iii,iiii)*(nxyzt+1) + s(j,jj,jjj,jjjj);
} // end ss 

static double A[nxyzt*(nxyzt+1)]; // last column is RHS 
static double U[nxyzt];
static double xg[nx];
static double yg[ny];
static double zg[nz];
static double tg[nt];
static double Ua[nxyzt];
static double xmin = 0.0; // problem parameters 
static double xmax = 1.0;
static double ymin = 0.0;
static double ymax = 1.0;
static double zmin = 0.0;
static double zmax = 1.0;
static double tmin = 0.0;
static double tmax = 1.0;
static double hx, hy, hz, ht;


static double uana(double x, double y, double z, double t)
            //analytic solution for checking code and boundaries
{
  return sin(x+y+z+t);
}

void printA() // optional
{
  int i, ii, iii, iiii, j, jj, jjj, jjjj, cs, k;
  cs = nxyzt; // constant RHS column 
  for(i=1; i<nx-1; i++)
  {
    for(ii=1; ii<ny-1; ii++)
    {
      for(iii=1; iii<nz-1; iii++)
      {
        for(iiii=1; iiii<nt-1; iiii++)
        {
          k = s(i,ii,iii,iii); // row index 
          for(j=1; j<nx-1; j++)
          {
            for(jj=1; jj<ny-1; jj++)
            {
              for(jjj=1; jjj<nz-1; jjj++)
              {
                for(jjjj=1; jjjj<nt-1; jjjj++)
                {
                  printf("A[row(%d,%d,%d,%d),col(%d,%d,%d,%d)] =%f\n",
                         i,ii,iii,iiii,j,jj,jjj,jjjj,
			 A[ss(i,ii,iii,iiii,j,jj,jjj,jjjj)]);
	        } // jjjj 
	      } // jjj 
	    } // jj 
          } // j 
          printf("RHS A[%d,%d]=%f\n\n", k, cs, A[k*(nxyzt+1)+cs]);
          // equivalent A[sk(k,i,ii,iii,iiii+1)] 
        } // iiii 
      } // iii 
    } // ii 
  } // i 
  printf("\n");
} // end printA 

double eval_derivative(int xord, int yord, int zord, int tord,
                       int i, int ii, int iii, int iiii, double u[])
{
  int j, jj, jjj, jjjj;
  double cx[nx];
  double cy[ny];
  double cz[nz];
  double ct[nt];
  double p1[nn];
  double p2[nn];
  double p3[nn];
  double discrete;
  double x, y, z, t;

  rderiv(xord, nx, i, hx, cx);
  x = xg[i];
  rderiv(yord, ny, ii, hy, cy);
  y = yg[ii];
  rderiv(zord, nz, iii, hz, cz);
  z = zg[iii];
  rderiv(tord, nt, iiii, ht, ct);
  t = tg[iiii];
  discrete = 0.0;

  // four cases of one partial x, y, z, t to any order 
  if(xord!=0 && yord==0 && zord==0 && tord==0)
    for(j=0; j<nx; j++)
      if(j==0 || j==nx-1) discrete += cx[j]*uana(xg[j],y,z,t);
      else discrete += cx[j]*u[s(j,ii,iii,iiii)];
  else if(xord==0 && yord!=0 && zord==0 && tord==0)
    for(jj=0; jj<ny; jj++)
      if(jj==0 || jj==ny-1) discrete += cy[jj]*uana(x,yg[jj],z,t);
      else discrete += cy[jj]*u[s(i,jj,iii,iiii)];
  else if(xord==0 && yord==0 && zord!=0 && tord==0)
    for(jjj=0; jjj<nz; jjj++)
      if(jjj==0 || jjj==nz-1) discrete += cz[jjj]*uana(x,y,zg[jjj],t);
      else discrete += cz[jjj]*u[s(i,ii,jjj,iiii)];
  else if(xord==0 && yord==0 && zord==0 && tord!=0)
    for(jjjj=0; jjjj<nt; jjjj++)
      if(jjjj==0 || jjjj==nt-1) discrete += ct[jjjj]*uana(x,y,z,tg[jjjj]);
      else discrete += ct[jjjj]*u[s(i,ii,iii,jjjj)];
  // six cases of two partials xy, xz, xt, yz, yt, zt to any order 
  else if(xord!=0 && yord!=0 && zord==0 && tord==0)
  {
    for(j=0; j<nx; j++)
    {
      p1[j] = 0.0;
      for(jj=0; jj<ny; jj++)
        if(j==0 || j==nx-1 || jj==0 || jj==ny-1)
             p1[j] += cy[jj]*uana(xg[j],yg[jj],zg[iii],tg[iiii]);
        else p1[j] += cy[jj]*u[s(j,jj,iii,iiii)];
      discrete += cx[j]*p1[j];
    }
  }
  else if(xord!=0 && yord==0 && zord!=0 && tord==0)
  {
    for(j=0; j<nx; j++)
    {
      p1[j] = 0.0;
      for(jjj=0; jjj<nz; jjj++)
        if(j==0 || j==nx-1 || jjj==0 || jjj==nz-1)
	     p1[j] += ct[jjj]*uana(xg[j],yg[ii],zg[jjj],tg[iiii]);
        else p1[j] += cz[jjj]*u[s(j,ii,jjj,iiii)];
      discrete += cx[j]*p1[j];
    }
  }
  else if(xord!=0 && yord==0 && zord==0 && tord!=0)
  {
    for(j=0; j<nx; j++)
    {
      p1[j] = 0.0;
      for(jjjj=0; jjjj<nt; jjjj++)
        if(j==0 || j==nx-1 || jjjj==0 || jjjj==nt-1)
             p1[j] += ct[jjjj]*uana(xg[j],yg[ii],zg[iii],tg[jjjj]);
        else p1[j] += ct[jjjj]*u[s(j,ii,iii,jjjj)];
      discrete += cx[j]*p1[j];
    }
  }
  else if(xord==0 && yord!=0 && zord!=0 && tord==0)
  {
    for(jj=0; jj<ny; jj++)
    {
      p1[jj] = 0.0;
      for(jjj=0; jjj<nz; jjj++)
        if(jj==0 || jj==ny-1 || jjj==0 || jjj==nz-1)
             p1[jj] += cz[jjj]*uana(xg[i],yg[jj],zg[jjj],tg[iiii]);
        else p1[jj] += cz[jjj]*u[s(i,jj,jjj,iiii)];
      discrete += cy[jj]*p1[jj];
    }
  }
  else if(xord==0 && yord!=0 && zord==0 && tord!=0)
  {
    for(jj=0; jj<ny; jj++)
    {
      p1[jj] = 0.0;
      for(jjjj=0; jjjj<nt; jjjj++)
        if(jj==0 || jj==ny-1 || jjjj==0 || jjjj==nt-1)
             p1[jj] += ct[jjjj]*uana(xg[i],yg[jj],zg[iii],tg[jjjj]);
        else p1[jj] += ct[jjjj]*u[s(i,jj,iii,jjjj)];
      discrete += cy[jj]*p1[jj];
    }
  }
  else if(xord==0 && yord==0 && zord!=0 && tord!=0)
  {
    for(jjj=0; jjj<nz; jjj++)
    {
      p1[jjj] = 0.0;
      for(jjjj=0; jjjj<nt; jjjj++)
        if(jjj==0 || jjj==nz-1 || jjjj==0 || jjjj==nt-1)
             p1[jjj] += ct[jjjj]*uana(xg[i],yg[ii],zg[jjj],tg[jjjj]);
        else p1[jjj] += ct[jjjj]*u[s(i,ii,jjj,jjjj)];
      discrete += cz[jjj]*p1[jjj];
    }
  }
  // four cases of three partials xyz, xyt, xzt, yzt to any order 
  else if(xord!=0 && yord!=0 && zord!=0 && tord==0)
  {
    for(j=0; j<nx; j++)
    {
      p1[j] = 0.0;
      for(jj=0; jj<ny; jj++)
      {
        p2[jj] = 0.0;
        for(jjj=0; jjj<nz; jjj++)
          if(j==0 || j==nx-1 ||
             jj==0 || jj==ny-1 ||
             jjj==0 || jjj==nz-1)
               p2[jj] += cz[jjj]*uana(xg[j],yg[jj],zg[jjj],tg[iiii]);
          else p2[jj] += cz[jjj]*u[s(j,jj,jjj,iiii)];
        p1[j] += cy[jj]*p2[jj];
      }
      discrete += cx[j]*p1[j];
    }
  }
  else if(xord!=0 && yord!=0 && zord==0 && tord!=0)
  {
    for(j=0; j<nx; j++)
    {
      p1[j] = 0.0;
      for(jj=0; jj<ny; jj++)
      {
        p2[jj] = 0.0;
        for(jjjj=0; jjjj<nt; jjjj++)
          if(j==0 || j==nx-1 ||
             jj==0 || jj==ny-1 ||
             jjjj==0 || jjjj==nt-1)
               p2[jj] += ct[jjjj]*uana(xg[j],yg[jj],zg[iii],tg[jjjj]);
          else p2[jj] += ct[jjjj]*u[s(j,jj,iii,jjjj)];
        p1[j] += cy[jj]*p2[jj];
      }
      discrete += cx[j]*p1[j];
    }
  }
  else if(xord!=0 && yord==0 && zord!=0 && tord!=0)
  {
    for(j=0; j<nx; j++)
    {
      p1[j] = 0.0;
      for(jjj=0; jjj<nz; jjj++)
      {
        p2[jjj] = 0.0;
        for(jjjj=0; jjjj<nt; jjjj++)
          if(j==0 || j==nx-1 ||
             jjj==0 || jjj==nz-1 ||
             jjjj==0 || jjjj==nt-1)
               p2[jjj] += ct[jjjj]*uana(xg[j],yg[ii],zg[jjj],tg[jjjj]);
          else p2[jjj] += ct[jjjj]*u[s(j,ii,jjj,jjjj)];
        p1[j] += cz[jjj]*p2[jjj];
      }
      discrete += cx[j]*p1[j];
    }
  }
  else if(xord==0 && yord!=0 && zord!=0 && tord!=0)
  {
    for(jj=0; jj<ny; jj++)
    {
      p1[jj] = 0.0;
      for(jjj=0; jjj<nz; jjj++)
      {
        p2[jjj] = 0.0;
        for(jjjj=0; jjjj<nt; jjjj++)
          if(jj==0 || jj==ny-1 ||
             jjj==0 || jjj==nz-1 ||
             jjjj==0 || jjjj==nt-1)
               p2[jjj] += ct[jjjj]*uana(xg[i],yg[jj],zg[jjj],tg[jjjj]);
          else p2[jjj] += ct[jjjj]*u[s(i,jj,jjj,jjjj)];
        p1[jj] += cz[jjj]*p2[jjj];
      }
      discrete += cy[jj]*p1[jj];
    }
  }
  // one case of four partials 
  else
  {
    for(j=0; j<nx; j++)
    {
      p1[j] = 0.0;
      for(jj=0; jj<ny; jj++)
      {
        p2[jj] = 0.0;
        for(jjj=0; jjj<nz; jjj++)
        {
          p3[jjj] = 0.0;
          for(jjjj=0; jjjj<nt; jjjj++)
	    if(j==0 || j==nx-1 ||
               jj==0 || jj==ny-1 ||
               jjj==0 || jjj==nz-1 ||
               jjjj==0 || jjjj==nt-1)
               p3[jjj] += ct[jjjj]*uana(xg[j],yg[jj],zg[jjj],tg[jjjj]);
	    else p3[jjj] += ct[jjjj]*u[s(j,jj,jjj,jjjj)];
          p2[jj] += cz[jjj]*p3[jjj];
        } // end jjj 
        p1[j] += cy[jj]*p2[jj];
      } // end jj 
      discrete += cx[j]*p1[j];
    } // end j 
  } // end if 
  return discrete;
} // end eval_deriv 

void check_soln(double u[])
{
  // check uxxxx(x,y,z,t) + uyyyy(x,y,z,t) + uzzzz(x,y,z,t) +
  //       utttt(x,y,z,t) + 2 uxx(x,y,z,t) + 2 uyy(x,y,z,t) +
  //       2 uzz(x,y,z,t) + 2 utt(x,y,z,t) + 4 u(x,y,z,t) = 0
  
  double val, err, smaxerr;
  int i, ii, iii, iiii;

  smaxerr = 0.0;
  for(i=1; i<nx-1; i++)
  {
    for(ii=1; ii<ny-1; ii++)
    {
      for(iii=1; iii<nz-1; iii++)
      {
        for(iiii=1; iiii<nt-1; iiii++)
        {
          val = 0.0;
          // uxxxx(x,y,z,t) 
          val += eval_derivative(4, 0, 0, 0, i, ii, iii, iiii, u);
	  // + uyyyy(x,y,z,t) 
          val += eval_derivative(0, 4, 0, 0, i, ii, iii, iiii, u);
	  // + uzzzz(x,y,z,t) 
          val += eval_derivative(0, 0, 4, 0, i, ii, iii, iiii, u);
          // + utttt(x,y,z,t) 
          val += eval_derivative(0, 0, 0, 4, i, ii, iii, iiii, u);
          // + 2 uxx(x,y,z,t) 
          val += 2.0*eval_derivative(2, 0, 0, 0, i, ii, iii, iiii, u);
	  // + 2 uyy(x,y,z,t) 
          val += 2.0*eval_derivative(0, 2, 0, 0, i, ii, iii, iiii, u);
	  // + 2 uzz(x,y,z,t) 
          val += 2.0*eval_derivative(0, 0, 2, 0, i, ii, iii, iiii, u);
	  // + 2 utt(x,y,z,t) 
          val += 2.0*eval_derivative(0, 0, 0, 2, i, ii, iii, iiii, u);
	  // + 4 u(x,y,z,t) 
          val += 4.0*uana(xg[i],yg[ii],zg[iii],tg[iiii]);
          // -       F(x,y,z,t)  zero 
	  err = abs(val);
          if(err>smaxerr) smaxerr = err;
        } // end iiii 
      } // end iii 
    } // end ii 
  } // end i 
  printf("check_soln maxerr=%g \n",smaxerr);
} // end check_soln 

void write_soln(char tag[], double u[]) // for plot4d_gl 
{
  FILE* outp;
  char name1[32] = "pde44h_eq_c";
  char extn[] = ".dat";
  char * name;
  int i, ii, iii, iiii;

  strcat(name1,tag);
  name = strcat(name1,extn);
  printf("writing solution to %s \n", name);
  outp = fopen(name,"w");
  if(outp==NULL)
  {
    printf("ERROR unable to open %s for writing \n", name);
    exit(1);
  }
  fprintf(outp, "4 %2d %2d %2d %2d \n", nx, ny, nz, nt);
  for(i=1; i<nx-1; i++)
  {
    for(ii=1; ii<ny-1; ii++)
    {
      for(iii=1; iii<nz-1; iii++)
      {
        for(iiii=1; iiii<nt-1; iiii++)
        {
          fprintf(outp, "%f %f %f %f %f \n",
                  xg[i],yg[ii],zg[iii],tg[iiii],U[s(i,ii,iii,iiii)]);
        }
      }
    }
  }
  fclose(outp);
  printf("solution file %s written \n", name);
} // end write_soln 
 

int main(int argc, char *argv[])
{
  double x, y, z, t;
  int i, ii, iii, iiii, j, jj, jjj, jjjj;
  int k, cs;
  // coeficients of terms in first equation 
  // derivative point arrays, dynamic in this version 
  double cxx[nx], cyy[ny], czz[nz], ctt[nt]; 
  double cxxxx[nx], cyyyy[ny], czzzz[nz], ctttt[nt];
  double val, err, avgerr, maxerr;
  double time_start;
  double time_now;
  double time_last;

  printf("pde44h_eq.c running \n");
  printf("Given uxxxx(x,y,z,t) + uyyyy(x,y,z,t) + uzzzz(x,y,z,t) + \n");
  printf("      utttt(x,y,z,t) + 2 uxx(x,y,z,t) + 2 uyy(x,y,z,t) + \n");
  printf("      2 uzz(x,y,z,t) + 2 utt(x,y,z,t) + 4 u(x,y,z,t) = 0 \n");
  printf("\n"); 
  printf("xmin<=x<=xmax ymin<=y<=ymax zmin<=z<=zmax Boundaries \n");
  printf("Analytic solution u(x,y,z,t) =  sin(x+y+z+t) \n");
  printf("\n");

  printf("xmin=%g, xmax=%g, ymin=%g, ymax=%g \n", xmin, xmax, ymin, ymax);
  printf("zmin=%g, zmax=%g, tmin=%g, tmax=%g \n", zmin, zmax, tmin, tmax);
  printf("nx=%d, ny=%d, nz=%d, nt=%d \n", nx, ny, nz, nt);
  time_start = (double)clock()/(double)CLOCKS_PER_SEC;
  printf("time start = %f seconds \n", time_start);
  time_last = time_start;

  printf("x grid and analytic solution at ymin,zmin,tmin \n");
  hx = (xmax-xmin)/(double)(nx-1);
  for(i=0; i<nx; i++)
  {
    xg[i] = xmin+(double)i*hx;
    Ua[i] = uana(xg[i],ymin,zmin,tmin); // temp 
    printf("i=%d, Ua(%6.3f,%6.3f,%6.3f,%6.3f)=%7.4f \n",
            i, xg[i], ymin, zmin, tmin, Ua[i]);
  } // i   
  printf("\n");
  printf("y grid and analytic solution at xmin,zmin,tmin \n");
  hy = (ymax-ymin)/(double)(ny-1);
  for(ii=0; ii<ny; ii++)
  {
    yg[ii] = ymin+(double)ii*hy;
    Ua[ii] = uana(xmin, yg[ii],zmin,tmin); // temp 
    printf("ii=%d, Ua(%6.3f,%6.3f,%6.3f,%6.3f)=%7.4f \n",
            ii, xmin, yg[ii], zmin, tmin, Ua[ii]);
  } // ii 
  printf("\n");
  printf("z grid and analytic solution at xmin,ymin,tmin \n");
  hz = (zmax-zmin)/(double)(nz-1);
  for(iii=0; iii<nz; iii++)
  {
    zg[iii] = zmin+(double)iii*hz;
    Ua[iii] = uana(xmin, ymin, zg[iii],tmin); // temp 
    printf("iii=%d, Ua(%6.3f,%6.3f,%6.3f,%6.3f)=%7.4f \n",
            iii, xmin, ymin, zg[iii], tmin, Ua[iii]);
  } // iii 
  printf("\n");
  printf("t grid and analytic solution at xmin,ymin,zmin \n");
  ht = (tmax-tmin)/(double)(nt-1);
  for(iiii=0; iiii<nt; iiii++)
  {
    tg[iiii] = tmin+(double)iiii*ht;
    Ua[iiii] = uana(xmin, ymin, zmin, tg[iiii]); // temp 
    printf("iiii=%d, Ua(%6.3f,%6.3f,%6.3f,%6.3f)=%7.4f \n",
            iiii, xmin, ymin, zmin, tg[iiii], Ua[iiii]);
  } // iiii 
  printf("\n");

  printf("check rderiv(4,nx,0,hx,cxxxx \n");
  rderiv(4,nx,0,hx,cxxxx);
  for(i=0; i<nx; i++) printf("cxxxx[%d]=%f \n", i, cxxxx[i]);
  printf("check nuderiv(4,nx,0,xg,cxxxx \n");
  nuderiv(4,nx,0,xg,cxxxx);
  for(i=0; i<nx; i++) printf("cxxxx[%d]=%f \n", i, cxxxx[i]);
  printf("\n");
  printf("check rderiv(4,nx,nx-1,hx,cxxxx \n");
  rderiv(4,nx,nx-1,hx,cxxxx);
  for(i=0; i<nx; i++) printf("cxxxx[%d]=%f \n", i, cxxxx[i]);
  printf("check nuderiv(4,nx,nx-1,xg,cxxxx \n");
  nuderiv(4,nx,nx-1,xg,cxxxx);
  for(i=0; i<nx; i++) printf("cxxxx[%d]=%f \n", i, cxxxx[i]);
  printf("\n");
  printf("hx=%f, hy=%f, hz=%f, ht=%f \n", hx, hy, hz, ht);

  printf("put solution in Ua vector \n");
  // put solution in Ua vector 
  for(i=1; i<nx-1; i++)
  {
    x = xg[i];
    for(ii=1; ii<ny-1; ii++)
    {
      y = yg[ii];
      for(iii=1; iii<nz-1; iii++)
      {
        z = zg[iii];
        for(iiii=1; iiii<nt-1; iiii++)
        {
          t = tg[iiii];
          Ua[s(i,ii,iii,iiii)] = uana(x,y,z,t);
          // printf("solution at i=%d,x=%g, ii=%d,y=%g, iii=%d,z=%g,
          //         iiii=%d,t=%g is %g \n",
	  //         i, x, ii, y, iii, z, iiii, t, Ua[s(i,ii,iii,iiii)]); 
        } // iiii 
      } // iii 
    } // ii 
  } // i 
  time_now = (double)clock()/(double)CLOCKS_PER_SEC;
  printf("time now = %f seconds, time for previous section = %f seconds \n",
         time_now, time_now-time_last);
  time_last = time_now;
  printf("\n");

  printf("initialize A \n");
  // initialize A 
  for(i=1; i<nx-1; i++)
  {
    for(ii=1; ii<ny-1; ii++)
    {
      for(iii=1; iii<nz-1; iii++)
      {
        for(iiii=1; iiii<nt-1; iiii++)
        {
          for(j=1; j<nx-1; j++)
          {
            for(jj=1; jj<ny-1; jj++)
            {
              for(jjj=1; jjj<nz-1; jjj++)
              {
                for(jjjj=1; jjjj<nt-1; jjjj++)
                {
                  A[ss(i,ii,iii,iiii,j,jj,jjj,jjjj)] = 0.0;
	        } // jjjj 
	      } // jjj 
	    } // jj 
          } // j 
        } // iiii 
      } // iii 
    } // ii 
  } // i 

  time_now = (double)clock()/(double)CLOCKS_PER_SEC;
  printf("time now = %f seconds, time for previous section = %f seconds \n",
         time_now, time_now-time_last);
  time_last = time_now;
  printf("\n");

  // compute entries in A matrix, inside boundary 
  printf("compute A matrix \n");
  cs = (nx-2)*(ny-2)*(nz-2)*(nt-2); // constant RHS column 
  val = 0.0;
  for(i=1; i<nx-1; i++)
  {
    for(ii=1; ii<ny-1; ii++)
    {
      for(iii=1; iii<nz-1; iii++)
      {
        for(iiii=1; iiii<nt-1; iiii++)
        {
          k = s(i,ii,iii,iiii); // row index 

          // for each term in first equation 

	  // for d^4u/dx^4 
          rderiv(4,nx,i,hx,cxxxx);
          for(j=0; j<nx; j++)
          {
            val = cxxxx[j];
            if(j==0 || j==nx-1)
	         A[k*(nxyzt+1)+cs] -= val*uana(xg[j],yg[ii],zg[iii],tg[iiii]);
            else A[sk(k,j,ii,iii,iiii)] += val;
	  }

	  // for d^4u/dy^4 
          rderiv(4,ny,ii,hy,cyyyy);
          for(jj=0; jj<ny; jj++)
          {
            val = cyyyy[jj];
            if(jj==0 || jj==ny-1)
	         A[k*(nxyzt+1)+cs] -=  val*uana(xg[i],yg[jj],zg[iii],tg[iiii]);
            else A[sk(k,i,jj,iii,iiii)] += val;
	  }

	  // for d^4u/dz^4 
          rderiv(4,nz,iii,hz,czzzz);
          for(jjj=0; jjj<nz; jjj++)
          {
            val = czzzz[jjj];
            if(jjj==0 || jjj==nz-1)
	         A[k*(nxyzt+1)+cs] -= val*uana(xg[i],yg[ii],zg[jjj],tg[iiii]);
            else A[sk(k,i,ii,jjj,iiii)] += val;
	  }

	  // for d^4u/dt^4 
          rderiv(4,nt,iiii,ht,ctttt);
          for(jjjj=0; jjjj<nt; jjjj++)
          {
            val = ctttt[jjjj];
            if(jjjj==0 || jjjj==nt-1)
	         A[k*(nxyzt+1)+cs] -= val*uana(xg[i],yg[ii],zg[iii],tg[jjjj]);
            else A[sk(k,i,ii,iii,jjjj)] += val;
	  }

	  // for 2 d^2u/dx^2 
          rderiv(2,nx,i,hx,cxx);
          for(j=0; j<nx; j++)
          {
            val = 2.0*cxx[j];
            if(j==0 || j==nx-1)
	         A[k*(nxyzt+1)+cs] -= val*uana(xg[j],yg[ii],zg[iii],tg[iiii]);
            else A[sk(k,j,ii,iii,iiii)] += val;
	  }

	  // for 2 d^2u/dy^2 
          rderiv(2,ny,ii,hy,cyy);
          for(jj=0; jj<ny; jj++)
          {
            val = 2.0*cyy[jj];
            if(jj==0 || jj==ny-1)
	         A[k*(nxyzt+1)+cs] -=  val*uana(xg[i],yg[jj],zg[iii],tg[iiii]);
            else A[sk(k,i,jj,iii,iiii)] += val;
	  }

	  // for 2 d^2u/dz^2 
          rderiv(2,nz,iii,hz,czz);
          for(jjj=0; jjj<nz; jjj++)
          {
            val = 2.0*czz[jjj];
            if(jjj==0 || jjj==nz-1)
	         A[k*(nxyzt+1)+cs] -= val*uana(xg[i],yg[ii],zg[jjj],tg[iiii]);
            else A[sk(k,i,ii,jjj,iiii)] += val;
	  }

	  // for 2 d^2u/dt^2 
          rderiv(2,nt,iiii,ht,ctt);
          for(jjjj=0; jjjj<nt; jjjj++)
          {
            val = 2.0*ctt[jjjj];
            if(jjjj==0 || jjjj==nt-1)
	         A[k*(nxyzt+1)+cs] -= val*uana(xg[i],yg[ii],zg[iii],tg[jjjj]);
            else A[sk(k,i,ii,iii,jjjj)] += val;
	  }

	  // for 4 u(x,y,z,t) 
          A[sk(k,i,ii,iii,iiii)] += 4.0;

          // for f(x,y,z,t) RHS zero 

          // end terms 

	} // end iiii loop 
      } // end iii loop 
    } // end ii loop 
  } // end i loop 

  // printA(); // lots of output 

  printf("A computed stiffness matrix \n");
  time_now = (double)clock()/(double)CLOCKS_PER_SEC;
  printf("time now = %f seconds, time for previous section = %f seconds \n",
         time_now, time_now-time_last);
  time_last = time_now;
  printf("\n");

  printf("solve %d equations in %d unknowns \n", nxyzt, nxyzt);

  simeqb(nxyzt, A, U); // A destroyed 

  printf("equations solved \n");
  time_now = (double)clock()/(double)CLOCKS_PER_SEC;
  printf("time now = %f seconds, time for previous section = %f seconds \n",
         time_now, time_now-time_last);
  time_last = time_now;
  printf("\n");
  
  printf("check PDE against known solution\n");
  check_soln(Ua);
  write_soln("1",Ua);

  printf("check PDE against computed solution\n");
  check_soln(U);
  write_soln("2",U);
  printf("\n");

  avgerr = 0.0;
  maxerr = 0.0;
  printf("          U computed, Ua analytic, error \n");
  for(i=1; i<nx-1; i++)
  {
    for(ii=1; ii<ny-1; ii++)
    {
      for(iii=1; iii<nz-1; iii++)
      {
        for(iiii=1; iiii<nt-1; iiii++)
        {
          err = abs(U[s(i,ii,iii,iiii)]-Ua[s(i,ii,iii,iiii)]);
          if(err>maxerr) maxerr = err;
          avgerr = avgerr + err;
          printf("ug[%d,%d,%d,%d]=%9.4f, Ua=%9.4f, err=%g \n",
                  i, ii, iii, iiii, U[s(i,ii,iii,iiii)],
                  Ua[s(i,ii,iii,iiii)], err);
        } // iiii 
      } // iii 
    } // ii 
  } // i 
  time_now = (double)clock()/(double)CLOCKS_PER_SEC;
  printf("total CPU time  = %f seconds \n", time_now-time_start);
  printf("\n");


  printf("                                        maxerr=%g, avgerr=%g \n",
         maxerr, avgerr/(double)(nx*ny*nz*nt));
  printf("\n");
  printf("end pde44h_eq.c \n");
  return 0;
} // end main 

// end pde44h_eq.c