/* pde44e_eq.c  discretize, build linear equations, solve
 * solve uxxxx(x,y,z,t) + uyyyy(x,y,z,t) + uzzzz(x,y,z,t) +
 *       utttt(x,y,z,t) + uxyzt(x,y,z,t) = F(x,y,z,t)
 * F(x,y,z,t) = (1 + 7xyzt + 6xxyyzztt +x^3Y^3z^3t^3)xp(x*y*z*t)
 *              
 * boundary conditions computed using u(x,y,z,t)
 * analytic solution is u(x,y,z,t) = exp(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
 */

/* using rderiv for discretization
 * subscripts in code use x[i], y[ii], z[iii], t[iiii]
 * derivatives are computed with b,hx,a included as in
 * cx[j] for first derivative of x at point j
 * cxxxx[j] for fourth derivative of x at point j
 */

/* 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 "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))

#define nx  5
#define ny  5
#define nz  5
#define nt  5
#define nxyzt (nx-2)*(ny-2)*(nz-2)*(nt-2)

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 f(double x, double y, double z, double t)
{
  return (1.0 + 7.0*x*y*z*t + 6.0*x*x*y*y*z*z*t*t +
          x*x*x*y*y*y*z*z*z*t*t*t)*exp(x*y*z*t); 
}

static double uana(double x, double y, double z, double t)
                                        /*analytic solution and boundaries*/
{
  return exp(x*y*z*t);
}

void printA()
{
  int i, ii, iii, iiii, j, jj, jjj, jjjj, cs, k;
  cs = (nx-2)*(ny-2)*(nz-2)*(nt-2); /* 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 */

int main(int argc, char *argv[])
{
  double x, y, z, t, hx, hy, hz, ht;
  int i, ii, iii, iiii, j, jj, jjj, jjjj;
  int k, cs;
  /* coeficients of terms in first equation */
  double coefdxxxx = 1.0; /* computed below if not constant */
  double coefdyyyy = 1.0;
  double coefdzzzz = 1.0;
  double coefdtttt = 1.0;
  double coefdxyzt = 1.0;
  /* derivative point arrays, dynamic in this version */
  double cx[nx], cy[ny], cz[nz], ct[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("pde44e_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) + uxyzt(x,y,z,t) = F(x,y,z,t) \n");
  printf("  F(x,y,z,t) = (1 + 7xyzt + 6xxyyzztt +x^3Y^3z^3t^3)xp(x*y*z*t)\n");
  printf("xmin<=x<=xmax ymin<=y<=ymax zmin<=z<=zmax Boundaries \n");
  printf("Analytic solution u(x,y,z,t) =  exp(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 rderiv(4,nx,2,hx,cxxxx) \n");
  rderiv(4,nx,2,hx,cxxxx);
  for(i=0; i<nx; i++) printf("cxxxx[%d]=%f \n", i, cxxxx[i]);
  printf("\n");

  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 */
          coefdxxxx = 1.0;
          rderiv(4,nx,i,hx,cxxxx);
          for(j=0; j<nx; j++)
          {
            val = coefdxxxx*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 */
          coefdyyyy = 1.0;
          rderiv(4,ny,ii,hy,cyyyy);
          for(jj=0; jj<ny; jj++)
          {
            val = coefdyyyy*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 */
          coefdzzzz = 1.0;
          rderiv(4,nz,iii,hz,czzzz);
          for(jjj=0; jjj<nz; jjj++)
          {
            val = coefdzzzz*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 */
          coefdtttt = 1.0;
          rderiv(4,nt,iiii,ht,ctttt);
          for(jjjj=0; jjjj<nt; jjjj++)
          {
            val = coefdtttt*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 d^4u/dxdydzdt */
          coefdxyzt  = 1.0;
          rderiv(1,nx,i,hx,cx);
          rderiv(1,ny,ii,hy,cy);
          rderiv(1,nz,iii,hz,cz);
          rderiv(1,nt,iiii,ht,ct);
          for(j=0; j<nx; j++)
	  {
	    for(jj=0; jj<ny; jj++)
	    {
	      for(jjj=0; jjj<nz; jjj++)
	      {
                for(jjjj=0; jjjj<nt; jjjj++)
                {
                  val = coefdxyzt*cx[j]*cy[jj]*cz[jjj]*ct[jjjj];
                  if(j==0 || j==nx-1 || jj==0 || jj==ny-1 ||
                     jjj==0 || jjj==nz-1 || jjjj==0 || jjjj==nt-1)
	               A[k*(nxyzt+1)+cs] -= val*uana(xg[j],yg[jj],zg[jjj],tg[jjjj]);
                  else A[sk(k,j,jj,jjj,jjjj)] += val;
	        } /* end jjjj */
	      } /* end jjj */
	    } /* end jj */
	  } /* end j */

          /* for f(x,y,z,t) RHS constant */
          A[k*(nxyzt+1)+cs] += f(xg[i],yg[ii],zg[iii],tg[iiii]);

          /* 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);

  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");


  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 pde44e_eq.c \n");
  return 0;
} /* end main */

/* end pde44e_eq.c */