/src/variable.cpp

/* ----------------------------------------------------------------------
  LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
   http://lammps.sandia.gov, Sandia National Laboratories
   Steve Plimpton, sjplimp@sandia.gov

   Copyright (2003) Sandia Corporation.  Under the terms of Contract
   DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
   certain rights in this software.  This software is distributed under 
   the GNU General Public License.

   See the README file in the top-level LAMMPS directory.
------------------------------------------------------------------------- */

#include "math.h"
#include "stdlib.h"
#include "string.h"
#include "ctype.h"
#include "unistd.h"
#include "variable.h"
#include "universe.h"
#include "atom.h"
#include "update.h"
#include "group.h"
#include "domain.h"
#include "region.h"
#include "modify.h"
#include "compute.h"
#include "fix.h"
#include "output.h"
#include "thermo.h"
#include "random_mars.h"
#include "memory.h"
#include "error.h"

using namespace LAMMPS_NS;

#define VARDELTA 4
#define MAXLEVEL 4

#define MIN(A,B) ((A) < (B)) ? (A) : (B)
#define MAX(A,B) ((A) > (B)) ? (A) : (B)

#define MYROUND(a) (( a-floor(a) ) >= .5) ? ceil(a) : floor(a)

enum{INDEX,LOOP,WORLD,UNIVERSE,ULOOP,STRING,EQUAL,ATOM};
enum{ARG,OP};

// customize by adding a function

enum{DONE,ADD,SUBTRACT,MULTIPLY,DIVIDE,CARAT,UNARY,
       NOT,EQ,NE,LT,LE,GT,GE,AND,OR,
       SQRT,EXP,LN,LOG,SIN,COS,TAN,ASIN,ACOS,ATAN,ATAN2,
       RANDOM,NORMAL,CEIL,FLOOR,ROUND,RAMP,STAGGER,LOGFREQ,
       VDISPLACE,SWIGGLE,CWIGGLE,GMASK,RMASK,GRMASK,
       VALUE,ATOMARRAY,TYPEARRAY,INTARRAY};

// customize by adding a special function

enum{SUM,XMIN,XMAX,AVE,TRAP};

#define INVOKED_SCALAR 1
#define INVOKED_VECTOR 2
#define INVOKED_ARRAY 4
#define INVOKED_PERATOM 8

#define BIG 1.0e20

/* ---------------------------------------------------------------------- */

Variable::Variable(LAMMPS *lmp) : Pointers(lmp)
{
  MPI_Comm_rank(world,&me);

  nvar = maxvar = 0;
  names = NULL;
  style = NULL;
  num = NULL;
  which = NULL;
  pad = NULL;
  data = NULL;

  randomequal = NULL;
  randomatom = NULL;

  precedence[DONE] = 0;
  precedence[OR] = 1;
  precedence[AND] = 2;
  precedence[EQ] = precedence[NE] = 3;
  precedence[LT] = precedence[LE] = precedence[GT] = precedence[GE] = 4;
  precedence[ADD] = precedence[SUBTRACT] = 5;
  precedence[MULTIPLY] = precedence[DIVIDE] = 6;
  precedence[CARAT] = 7;
  precedence[UNARY] = precedence[NOT] = 8;

  PI = 4.0*atan(1.0);
}

/* ---------------------------------------------------------------------- */

Variable::~Variable()
{
  for (int i = 0; i < nvar; i++) {
    delete [] names[i];
    if (style[i] == LOOP || style[i] == ULOOP) delete [] data[i][0];
    else for (int j = 0; j < num[i]; j++) delete [] data[i][j];
    delete [] data[i];
  }
  memory->sfree(names);
  memory->destroy(style);
  memory->destroy(num);
  memory->destroy(which);
  memory->destroy(pad);
  memory->sfree(data);

  delete randomequal;
  delete randomatom;
}

/* ----------------------------------------------------------------------
   called by variable command in input script
------------------------------------------------------------------------- */

void Variable::set(int narg, char **arg)
{
  if (narg < 2) error->all("Illegal variable command");

  // DELETE
  // doesn't matter if variable no longer exists

  if (strcmp(arg[1],"delete") == 0) {
    if (narg != 2) error->all("Illegal variable command");
    if (find(arg[0]) >= 0) remove(find(arg[0]));
    return;

  // INDEX
  // num = listed args, which = 1st value, data = copied args

  } else if (strcmp(arg[1],"index") == 0) {
    if (narg < 3) error->all("Illegal variable command");
    if (find(arg[0]) >= 0) return;
    if (nvar == maxvar) extend();
    style[nvar] = INDEX;
    num[nvar] = narg - 2;
    which[nvar] = 0;
    pad[nvar] = 0;
    data[nvar] = new char*[num[nvar]];
    copy(num[nvar],&arg[2],data[nvar]);

  // LOOP
  // 1 arg + pad: num = N, which = 1st value, data = single string
  // 2 args + pad: num = N2, which = N1, data = single string

  } else if (strcmp(arg[1],"loop") == 0) {
    if (find(arg[0]) >= 0) return;
    if (nvar == maxvar) extend();
    style[nvar] = LOOP;
    int nfirst,nlast;
    if (narg == 3 || (narg == 4 && strcmp(arg[3],"pad") == 0)) {
      nfirst = 1;
      nlast = atoi(arg[2]);
      if (nlast <= 0) error->all("Illegal variable command");
      if (narg == 4 && strcmp(arg[3],"pad") == 0) {
	char digits[12];
	sprintf(digits,"%d",nlast);
	pad[nvar] = strlen(digits);
      } else pad[nvar] = 0;
    } else if (narg == 4 || (narg == 5 && strcmp(arg[4],"pad") == 0)) {
      nfirst = atoi(arg[2]);
      nlast = atoi(arg[3]);
      if (nfirst > nlast || nlast <= 0) error->all("Illegal variable command");
      if (narg == 5 && strcmp(arg[4],"pad") == 0) {
	char digits[12];
	sprintf(digits,"%d",nlast);
	pad[nvar] = strlen(digits);
      } else pad[nvar] = 0;
    } else error->all("Illegal variable command");
    num[nvar] = nlast;
    which[nvar] = nfirst-1;
    data[nvar] = new char*[1];
    data[nvar][0] = NULL;

  // WORLD
  // num = listed args, which = partition this proc is in, data = copied args
  // error check that num = # of worlds in universe

  } else if (strcmp(arg[1],"world") == 0) {
    if (narg < 3) error->all("Illegal variable command");
    if (find(arg[0]) >= 0) return;
    if (nvar == maxvar) extend();
    style[nvar] = WORLD;
    num[nvar] = narg - 2;
    if (num[nvar] != universe->nworlds)
      error->all("World variable count doesn't match # of partitions");
    which[nvar] = universe->iworld;
    pad[nvar] = 0;
    data[nvar] = new char*[num[nvar]];
    copy(num[nvar],&arg[2],data[nvar]);

  // UNIVERSE and ULOOP
  // for UNIVERSE: num = listed args, data = copied args
  // for ULOOP: num = N, data = single string
  // which = partition this proc is in
  // universe proc 0 creates lock file
  // error check that all other universe/uloop variables are same length

  } else if (strcmp(arg[1],"universe") == 0 || strcmp(arg[1],"uloop") == 0) {
    if (strcmp(arg[1],"universe") == 0) {
      if (narg < 3) error->all("Illegal variable command");
      if (find(arg[0]) >= 0) return;
      if (nvar == maxvar) extend();
      style[nvar] = UNIVERSE;
      num[nvar] = narg - 2;
      data[nvar] = new char*[num[nvar]];
      copy(num[nvar],&arg[2],data[nvar]);
    } else if (strcmp(arg[1],"uloop") == 0) {
      if (narg < 3 || narg > 4 || (narg == 4 && strcmp(arg[3],"pad") != 0))
	error->all("Illegal variable command");
      if (find(arg[0]) >= 0) return;
      if (nvar == maxvar) extend();
      style[nvar] = ULOOP;
      num[nvar] = atoi(arg[2]);
      data[nvar] = new char*[1];
      data[nvar][0] = NULL;
      if (narg == 4) {
	char digits[12];
	sprintf(digits,"%d",num[nvar]);
	pad[nvar] = strlen(digits);
      } else pad[nvar] = 0;
    }

    if (num[nvar] < universe->nworlds)
      error->all("Universe/uloop variable count < # of partitions");
    which[nvar] = universe->iworld;

    if (universe->me == 0) {
      FILE *fp = fopen("tmp.lammps.variable","w");
      fprintf(fp,"%d\n",universe->nworlds);
      fclose(fp);
    }

    for (int jvar = 0; jvar < nvar; jvar++)
      if (num[jvar] && (style[jvar] == UNIVERSE || style[jvar] == ULOOP) && 
	  num[nvar] != num[jvar])
	error->all("All universe/uloop variables must have same # of values");

  // STRING
  // remove pre-existing var if also style STRING (allows it to be reset)
  // num = 1, which = 1st value
  // data = 1 value, string to eval

  } else if (strcmp(arg[1],"string") == 0) {
    if (narg != 3) error->all("Illegal variable command");
    if (find(arg[0]) >= 0) {
      if (style[find(arg[0])] != STRING)
	error->all("Cannot redefine variable as a different style");
      remove(find(arg[0]));
    }
    if (nvar == maxvar) extend();
    style[nvar] = STRING;
    num[nvar] = 1;
    which[nvar] = 0;
    pad[nvar] = 0;
    data[nvar] = new char*[num[nvar]];
    copy(1,&arg[2],data[nvar]);
    
  // EQUAL
  // remove pre-existing var if also style EQUAL (allows it to be reset)
  // num = 2, which = 1st value
  // data = 2 values, 1st is string to eval, 2nd is filled on retrieval

  } else if (strcmp(arg[1],"equal") == 0) {
    if (narg != 3) error->all("Illegal variable command");
    if (find(arg[0]) >= 0) {
      if (style[find(arg[0])] != EQUAL)
	error->all("Cannot redefine variable as a different style");
      remove(find(arg[0]));
    }
    if (nvar == maxvar) extend();
    style[nvar] = EQUAL;
    num[nvar] = 2;
    which[nvar] = 0;
    pad[nvar] = 0;
    data[nvar] = new char*[num[nvar]];
    copy(1,&arg[2],data[nvar]);
    data[nvar][1] = NULL;
    
  // ATOM
  // remove pre-existing var if also style ATOM (allows it to be reset)
  // num = 1, which = 1st value
  // data = 1 value, string to eval

  } else if (strcmp(arg[1],"atom") == 0) {
    if (narg != 3) error->all("Illegal variable command");
    if (find(arg[0]) >= 0) {
      if (style[find(arg[0])] != ATOM)
	error->all("Cannot redefine variable as a different style");
      remove(find(arg[0]));
    }
    if (nvar == maxvar) extend();
    style[nvar] = ATOM;
    num[nvar] = 1;
    which[nvar] = 0;
    pad[nvar] = 0;
    data[nvar] = new char*[num[nvar]];
    copy(1,&arg[2],data[nvar]);
    
  } else error->all("Illegal variable command");

  // set name of variable
  // must come at end, since STRING/EQUAL/ATOM reset may have removed name
  // name must be all alphanumeric chars or underscores

  int n = strlen(arg[0]) + 1;
  names[nvar] = new char[n];
  strcpy(names[nvar],arg[0]);

  for (int i = 0; i < n-1; i++)
    if (!isalnum(names[nvar][i]) && names[nvar][i] != '_')
      error->all("Variable name must be alphanumeric or "
		 "underscore characters");
  nvar++;
}

/* ----------------------------------------------------------------------
   INDEX variable created by command-line argument
   make it INDEX rather than STRING so cannot be re-defined in input script
------------------------------------------------------------------------- */

void Variable::set(char *name, int narg, char **arg)
{
  char **newarg = new char*[2+narg];
  newarg[0] = name;
  newarg[1] = (char *) "index";
  for (int i = 0; i < narg; i++) newarg[2+i] = arg[i];
  set(2+narg,newarg);
  delete [] newarg;
}

/* ----------------------------------------------------------------------
   increment variable(s)
   return 0 if OK if successfully incremented
   return 1 if any variable is exhausted, free the variable to allow re-use
------------------------------------------------------------------------- */

int Variable::next(int narg, char **arg)
{
  int ivar;

  if (narg == 0) error->all("Illegal next command");

  // check that variables exist and are all the same style
  // exception: UNIVERSE and ULOOP variables can be mixed in same next command

  for (int iarg = 0; iarg < narg; iarg++) {
    ivar = find(arg[iarg]);
    if (ivar == -1) error->all("Invalid variable in next command");
    if (style[ivar] == ULOOP && style[find(arg[0])] == UNIVERSE) continue;
    else if (style[ivar] == UNIVERSE && style[find(arg[0])] == ULOOP) continue;
    else if (style[ivar] != style[find(arg[0])])
      error->all("All variables in next command must be same style");
  }

  // invalid styles STRING or EQUAL or WORLD or ATOM

  int istyle = style[find(arg[0])];
  if (istyle == STRING || istyle == EQUAL || istyle == WORLD || istyle == ATOM)
    error->all("Invalid variable style with next command");

  // increment all variables in list
  // if any variable is exhausted, set flag = 1 and remove var to allow re-use

  int flag = 0;

  if (istyle == INDEX || istyle == LOOP) {
    for (int iarg = 0; iarg < narg; iarg++) {
      ivar = find(arg[iarg]);
      which[ivar]++;
      if (which[ivar] >= num[ivar]) {
	flag = 1;
	remove(ivar);
      }
    }

  } else if (istyle == UNIVERSE || istyle == ULOOP) {

    // wait until lock file can be created and owned by proc 0 of this world
    // read next available index and Bcast it within my world
    // set all variables in list to nextindex

    int nextindex;
    if (me == 0) {
      while (1) {
	if (!rename("tmp.lammps.variable","tmp.lammps.variable.lock")) break;
	usleep(100000);
      }
      FILE *fp = fopen("tmp.lammps.variable.lock","r");
      fscanf(fp,"%d",&nextindex);
      fclose(fp);
      fp = fopen("tmp.lammps.variable.lock","w");
      fprintf(fp,"%d\n",nextindex+1);
      fclose(fp);
      rename("tmp.lammps.variable.lock","tmp.lammps.variable");
      if (universe->uscreen)
	fprintf(universe->uscreen,
		"Increment via next: value %d on partition %d\n",
		nextindex+1,universe->iworld);
      if (universe->ulogfile)
	fprintf(universe->ulogfile,
		"Increment via next: value %d on partition %d\n",
		nextindex+1,universe->iworld);
    }
    MPI_Bcast(&nextindex,1,MPI_INT,0,world);

    for (int iarg = 0; iarg < narg; iarg++) {
      ivar = find(arg[iarg]);
      which[ivar] = nextindex;
      if (which[ivar] >= num[ivar]) {
	flag = 1;
	remove(ivar);
      }
    }
  }

  return flag;
}

/* ----------------------------------------------------------------------
   return ptr to the data text associated with a variable
   if INDEX or WORLD or UNIVERSE or STRING var, return ptr to stored string
   if LOOP or ULOOP var, write int to data[0] and return ptr to string
   if EQUAL var, evaluate variable and put result in str
   if ATOM var, return NULL
   return NULL if no variable or which is bad, caller must respond
------------------------------------------------------------------------- */

char *Variable::retrieve(char *name)
{
  int ivar = find(name);
  if (ivar == -1) return NULL;
  if (which[ivar] >= num[ivar]) return NULL;

  char *str;
  if (style[ivar] == INDEX || style[ivar] == WORLD || 
      style[ivar] == UNIVERSE || style[ivar] == STRING) {
    str = data[ivar][which[ivar]];
  } else if (style[ivar] == LOOP || style[ivar] == ULOOP) {
    char result[16];
    if (pad[ivar] == 0) sprintf(result,"%d",which[ivar]+1);
    else {
      char padstr[16];
      sprintf(padstr,"%%0%dd",pad[ivar]);
      sprintf(result,padstr,which[ivar]+1);
    }
    int n = strlen(result) + 1;
    delete [] data[ivar][0];
    data[ivar][0] = new char[n];
    strcpy(data[ivar][0],result);
    str = data[ivar][0];
  } else if (style[ivar] == EQUAL) {
    char result[32];
    double answer = evaluate(data[ivar][0],NULL);
    sprintf(result,"%.10g",answer);
    int n = strlen(result) + 1;
    if (data[ivar][1]) delete [] data[ivar][1];
    data[ivar][1] = new char[n];
    strcpy(data[ivar][1],result);
    str = data[ivar][1];
  } else if (style[ivar] == ATOM) return NULL;

  return str;
}

/* ----------------------------------------------------------------------
   return result of equal-style variable evaluation
------------------------------------------------------------------------- */

double Variable::compute_equal(int ivar)
{
  return evaluate(data[ivar][0],NULL);
}

/* ----------------------------------------------------------------------
   compute result of atom-style variable evaluation
   only computed for atoms in igroup, else result is 0.0
   answers are placed every stride locations into result
   if sumflag, add variable values to existing result
------------------------------------------------------------------------- */

void Variable::compute_atom(int ivar, int igroup,
			    double *result, int stride, int sumflag)
{
  Tree *tree;
  double tmp = evaluate(data[ivar][0],&tree);
  tmp = collapse_tree(tree);

  int groupbit = group->bitmask[igroup];
  int *mask = atom->mask;
  int nlocal = atom->nlocal;

  if (sumflag == 0) {
    int m = 0;
    for (int i = 0; i < nlocal; i++) {
      if (mask[i] && groupbit) result[m] = eval_tree(tree,i);
      else result[m] = 0.0;
      m += stride;
    }

  } else {
    int m = 0;
    for (int i = 0; i < nlocal; i++) {
      if (mask[i] && groupbit) result[m] += eval_tree(tree,i);
      m += stride;
    }
  }

  free_tree(tree);
}

/* ----------------------------------------------------------------------
   search for name in list of variables names
   return index or -1 if not found
------------------------------------------------------------------------- */
  
int Variable::find(char *name)
{
  for (int i = 0; i < nvar; i++)
    if (strcmp(name,names[i]) == 0) return i;
  return -1;
}

/* ----------------------------------------------------------------------
   return 1 if variable is EQUAL style, 0 if not
------------------------------------------------------------------------- */
  
int Variable::equalstyle(int ivar)
{
  if (style[ivar] == EQUAL) return 1;
  return 0;
}

/* ----------------------------------------------------------------------
   return 1 if variable is ATOM style, 0 if not
------------------------------------------------------------------------- */
  
int Variable::atomstyle(int ivar)
{
  if (style[ivar] == ATOM) return 1;
  return 0;
}

/* ----------------------------------------------------------------------
   remove Nth variable from list and compact list
------------------------------------------------------------------------- */
  
void Variable::remove(int n)
{
  delete [] names[n];
  if (style[n] == LOOP || style[n] == ULOOP) delete [] data[n][0];
  else for (int i = 0; i < num[n]; i++) delete [] data[n][i];
  delete [] data[n];

  for (int i = n+1; i < nvar; i++) {
    names[i-1] = names[i];
    style[i-1] = style[i];
    num[i-1] = num[i];
    which[i-1] = which[i];
    pad[i-1] = pad[i];
    data[i-1] = data[i];
  }
  nvar--;
}

/* ----------------------------------------------------------------------
  make space in arrays for new variable
------------------------------------------------------------------------- */

void Variable::extend()
{
  maxvar += VARDELTA;
  names = (char **)
    memory->srealloc(names,maxvar*sizeof(char *),"var:names");
  memory->grow(style,maxvar,"var:style");
  memory->grow(num,maxvar,"var:num");
  memory->grow(which,maxvar,"var:which");
  memory->grow(pad,maxvar,"var:pad");
  data = (char ***) 
    memory->srealloc(data,maxvar*sizeof(char **),"var:data");
}

/* ----------------------------------------------------------------------
   copy narg strings from **from to **to, and allocate space for them
------------------------------------------------------------------------- */
  
void Variable::copy(int narg, char **from, char **to)
{
  int n;
  for (int i = 0; i < narg; i++) {
    n = strlen(from[i]) + 1;
    to[i] = new char[n];
    strcpy(to[i],from[i]);
  }
}

/* ----------------------------------------------------------------------
   recursive evaluation of a string str
   str is an equal-style or atom-style formula containing one or more items:
     number = 0.0, -5.45, 2.8e-4, ...
     constant = PI
     thermo keyword = ke, vol, atoms, ...
     math operation = (),-x,x+y,x-y,x*y,x/y,x^y,
                      x==y,x!=y,x<y,x<=y,x>y,x>=y,x&&y,x||y,
                      sqrt(x),exp(x),ln(x),log(x),
		      sin(x),cos(x),tan(x),asin(x),atan2(y,x),...
     group function = count(group), mass(group), xcm(group,x), ...
     special function = sum(x),min(x), ...
     atom value = x[i], y[i], vx[i], ...
     atom vector = x, y, vx, ...
     compute = c_ID, c_ID[i], c_ID[i][j]
     fix = f_ID, f_ID[i], f_ID[i][j]
     variable = v_name, v_name[i]
   equal-style variables passes in tree = NULL:
     evaluate the formula, return result as a double
   atom-style variable passes in tree = non-NULL:
     parse the formula but do not evaluate it
     create a parse tree and return it
------------------------------------------------------------------------- */

double Variable::evaluate(char *str, Tree **tree)
{
  int op,opprevious;
  double value1,value2;
  char onechar;
  char *ptr;

  double argstack[MAXLEVEL];
  Tree *treestack[MAXLEVEL];
  int opstack[MAXLEVEL];
  int nargstack = 0;
  int ntreestack = 0;
  int nopstack = 0;

  int i = 0;
  int expect = ARG;

  while (1) {
    onechar = str[i];

    // whitespace: just skip

    if (isspace(onechar)) i++;

    // ----------------
    // parentheses: recursively evaluate contents of parens
    // ----------------

    else if (onechar == '(') {
      if (expect == OP) error->all("Invalid syntax in variable formula");
      expect = OP;

      char *contents;
      i = find_matching_paren(str,i,contents);
      i++;

      // evaluate contents and push on stack

      if (tree) {
	Tree *newtree;
	double tmp = evaluate(contents,&newtree);
	treestack[ntreestack++] = newtree;
      } else argstack[nargstack++] = evaluate(contents,NULL);

      delete [] contents;

    // ----------------
    // number: push value onto stack
    // ----------------

    } else if (isdigit(onechar) || onechar == '.') {
      if (expect == OP) error->all("Invalid syntax in variable formula");
      expect = OP;

      // istop = end of number, including scientific notation

      int istart = i;
      while (isdigit(str[i]) || str[i] == '.') i++;
      if (str[i] == 'e' || str[i] == 'E') {
	i++;
	if (str[i] == '+' || str[i] == '-') i++;
	while (isdigit(str[i])) i++;
      }
      int istop = i - 1;

      int n = istop - istart + 1;
      char *number = new char[n+1];
      strncpy(number,&str[istart],n);
      number[n] = '\0';

      if (tree) {
        Tree *newtree = new Tree();
	newtree->type = VALUE;
	newtree->value = atof(number);
	newtree->left = newtree->middle = newtree->right = NULL;
	treestack[ntreestack++] = newtree;
      } else argstack[nargstack++] = atof(number);

      delete [] number;

    // ----------------
    // letter: c_ID, c_ID[], c_ID[][], f_ID, f_ID[], f_ID[][],
    //         v_name, v_name[], exp(), xcm(,), x, x[], PI, vol
    // ----------------

    } else if (isalpha(onechar)) {
      if (expect == OP) error->all("Invalid syntax in variable formula");
      expect = OP;

      // istop = end of word
      // word = all alphanumeric or underscore

      int istart = i;
      while (isalnum(str[i]) || str[i] == '_') i++;
      int istop = i-1;

      int n = istop - istart + 1;
      char *word = new char[n+1];
      strncpy(word,&str[istart],n);
      word[n] = '\0';

      // ----------------
      // compute
      // ----------------

      if (strncmp(word,"c_",2) == 0) {
	if (domain->box_exist == 0)
	  error->all("Variable evaluation before simulation box is defined");
 
	n = strlen(word) - 2 + 1;
	char *id = new char[n];
	strcpy(id,&word[2]);

	int icompute = modify->find_compute(id);
	if (icompute < 0) error->all("Invalid compute ID in variable formula");
	Compute *compute = modify->compute[icompute];
	delete [] id;

	// parse zero or one or two trailing brackets
	// point i beyond last bracket
	// nbracket = # of bracket pairs
	// index1,index2 = int inside each bracket pair

	int nbracket,index1,index2;
	if (str[i] != '[') nbracket = 0;
	else {
	  nbracket = 1;
	  ptr = &str[i];
	  index1 = int_between_brackets(ptr);
	  i = ptr-str+1;
	  if (str[i] == '[') {
	    nbracket = 2;
	    ptr = &str[i];
	    index2 = int_between_brackets(ptr);
	    i = ptr-str+1;
	  }
	}

        // c_ID = scalar from global scalar

	if (nbracket == 0 && compute->scalar_flag) {

	  if (update->whichflag == 0) {
	    if (compute->invoked_scalar != update->ntimestep)
	      error->all("Compute used in variable between runs "
			 "is not current");
	  } else if (!(compute->invoked_flag & INVOKED_SCALAR)) {
	    compute->compute_scalar();
	    compute->invoked_flag |= INVOKED_SCALAR;
	  }

	  value1 = compute->scalar;
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = value1;
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = value1;

        // c_ID[i] = scalar from global vector

	} else if (nbracket == 1 && compute->vector_flag) {

	  if (index1 > compute->size_vector)
	    error->all("Variable formula compute vector "
		       "is accessed out-of-range");
	  if (update->whichflag == 0) {
	    if (compute->invoked_vector != update->ntimestep)
	      error->all("Compute used in variable between runs "
			 "is not current");
	  } else if (!(compute->invoked_flag & INVOKED_VECTOR)) {
	    compute->compute_vector();
	    compute->invoked_flag |= INVOKED_VECTOR;
	  }

	  value1 = compute->vector[index1-1];
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = value1;
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = value1;

        // c_ID[i][j] = scalar from global array

	} else if (nbracket == 2 && compute->array_flag) {

	  if (index1 > compute->size_array_rows)
	    error->all("Variable formula compute array "
		       "is accessed out-of-range");
	  if (index2 > compute->size_array_cols)
	    error->all("Variable formula compute array "
		       "is accessed out-of-range");
	  if (update->whichflag == 0) {
	    if (compute->invoked_array != update->ntimestep)
	      error->all("Compute used in variable between runs "
			 "is not current");
	  } else if (!(compute->invoked_flag & INVOKED_ARRAY)) {
	    compute->compute_array();
	    compute->invoked_flag |= INVOKED_ARRAY;
	  }

	  value1 = compute->array[index1-1][index2-1];
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = value1;
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = value1;

        // c_ID[i] = scalar from per-atom vector

	} else if (nbracket == 1 && compute->peratom_flag && 
		   compute->size_peratom_cols == 0) {

	  if (update->whichflag == 0) {
	    if (compute->invoked_peratom != update->ntimestep)
	      error->all("Compute used in variable between runs "
			 "is not current");
	  } else if (!(compute->invoked_flag & INVOKED_PERATOM)) {
	    compute->compute_peratom();
	    compute->invoked_flag |= INVOKED_PERATOM;
	  }

	  peratom2global(1,NULL,compute->vector_atom,1,index1,
			 tree,treestack,ntreestack,argstack,nargstack);

        // c_ID[i][j] = scalar from per-atom array

	} else if (nbracket == 2 && compute->peratom_flag &&
		   compute->size_peratom_cols > 0) {

	  if (index2 > compute->size_peratom_cols)
	    error->all("Variable formula compute array "
		       "is accessed out-of-range");
	  if (update->whichflag == 0) {
	    if (compute->invoked_peratom != update->ntimestep)
	      error->all("Compute used in variable between runs "
			 "is not current");
	  } else if (!(compute->invoked_flag & INVOKED_PERATOM)) {
	    compute->compute_peratom();
	    compute->invoked_flag |= INVOKED_PERATOM;
	  }

	  peratom2global(1,NULL,&compute->array_atom[0][index2-1],
			 compute->size_peratom_cols,index1,
			 tree,treestack,ntreestack,argstack,nargstack);

        // c_ID = vector from per-atom vector

	} else if (nbracket == 0 && compute->peratom_flag && 
		   compute->size_peratom_cols == 0) {

	  if (tree == NULL)
	    error->all("Per-atom compute in equal-style variable formula");
	  if (update->whichflag == 0) {
	    if (compute->invoked_peratom != update->ntimestep)
	      error->all("Compute used in variable between runs "
			 "is not current");
	  } else if (!(compute->invoked_flag & INVOKED_PERATOM)) {
	    compute->compute_peratom();
	    compute->invoked_flag |= INVOKED_PERATOM;
	  }

	  Tree *newtree = new Tree();
	  newtree->type = ATOMARRAY;
	  newtree->array = compute->vector_atom;
	  newtree->nstride = 1;
	  newtree->left = newtree->middle = newtree->right = NULL;
	  treestack[ntreestack++] = newtree;

        // c_ID[i] = vector from per-atom array

	} else if (nbracket == 1 && compute->peratom_flag &&
		   compute->size_peratom_cols > 0) {

	  if (tree == NULL)
	    error->all("Per-atom compute in equal-style variable formula");
	  if (index1 > compute->size_peratom_cols)
	    error->all("Variable formula compute array "
		       "is accessed out-of-range");
	  if (update->whichflag == 0) {
	    if (compute->invoked_peratom != update->ntimestep)
	      error->all("Compute used in variable between runs "
			 "is not current");
	  } else if (!(compute->invoked_flag & INVOKED_PERATOM)) {
	    compute->compute_peratom();
	    compute->invoked_flag |= INVOKED_PERATOM;
	  }

	  Tree *newtree = new Tree();
	  newtree->type = ATOMARRAY;
	  newtree->array = &compute->array_atom[0][index1-1];
	  newtree->nstride = compute->size_peratom_cols;
	  newtree->left = newtree->middle = newtree->right = NULL;
	  treestack[ntreestack++] = newtree;

	} else error->all("Mismatched compute in variable formula");

      // ----------------
      // fix
      // ----------------

      } else if (strncmp(word,"f_",2) == 0) {
	if (domain->box_exist == 0)
	  error->all("Variable evaluation before simulation box is defined");
 
	n = strlen(word) - 2 + 1;
	char *id = new char[n];
	strcpy(id,&word[2]);

	int ifix = modify->find_fix(id);
	if (ifix < 0) error->all("Invalid fix ID in variable formula");
	Fix *fix = modify->fix[ifix];
	delete [] id;

	// parse zero or one or two trailing brackets
	// point i beyond last bracket
	// nbracket = # of bracket pairs
	// index1,index2 = int inside each bracket pair

	int nbracket,index1,index2;
	if (str[i] != '[') nbracket = 0;
	else {
	  nbracket = 1;
	  ptr = &str[i];
	  index1 = int_between_brackets(ptr);
	  i = ptr-str+1;
	  if (str[i] == '[') {
	    nbracket = 2;
	    ptr = &str[i];
	    index2 = int_between_brackets(ptr);
	    i = ptr-str+1;
	  }
	}

        // f_ID = scalar from global scalar

	if (nbracket == 0 && fix->scalar_flag) {

	  if (update->whichflag > 0 && update->ntimestep % fix->global_freq)
	    error->all("Fix in variable not computed at compatible time");

	  value1 = fix->compute_scalar();
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = value1;
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = value1;

        // f_ID[i] = scalar from global vector

	} else if (nbracket == 1 && fix->vector_flag) {

	  if (index1 > fix->size_vector)
	    error->all("Variable formula fix vector is accessed out-of-range");
	  if (update->whichflag > 0 && update->ntimestep % fix->global_freq)
	    error->all("Fix in variable not computed at compatible time");

	  value1 = fix->compute_vector(index1-1);
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = value1;
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = value1;

        // f_ID[i][j] = scalar from global array

	} else if (nbracket == 2 && fix->array_flag) {

	  if (index1 > fix->size_array_rows)
	    error->all("Variable formula fix array is accessed out-of-range");
	  if (index2 > fix->size_array_cols)
	    error->all("Variable formula fix array is accessed out-of-range");
	  if (update->whichflag > 0 && update->ntimestep % fix->global_freq)
	    error->all("Fix in variable not computed at compatible time");

	  value1 = fix->compute_array(index1-1,index2-1);
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = value1;
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = value1;

        // f_ID[i] = scalar from per-atom vector

	} else if (nbracket == 1 && fix->peratom_flag && 
		   fix->size_peratom_cols == 0) {

	  if (update->whichflag > 0 && 
	      update->ntimestep % fix->peratom_freq)
	    error->all("Fix in variable not computed at compatible time");

	  peratom2global(1,NULL,fix->vector_atom,1,index1,
			 tree,treestack,ntreestack,argstack,nargstack);

        // f_ID[i][j] = scalar from per-atom array

	} else if (nbracket == 2 && fix->peratom_flag &&
		   fix->size_peratom_cols > 0) {

	  if (index2 > fix->size_peratom_cols)
	    error->all("Variable formula fix array is accessed out-of-range");
	  if (update->whichflag > 0 && 
	      update->ntimestep % fix->peratom_freq)
	    error->all("Fix in variable not computed at compatible time");

	  peratom2global(1,NULL,&fix->array_atom[0][index2-1],
			 fix->size_peratom_cols,index1,
			 tree,treestack,ntreestack,argstack,nargstack);

        // f_ID = vector from per-atom vector

	} else if (nbracket == 0 && fix->peratom_flag && 
		   fix->size_peratom_cols == 0) {

	  if (tree == NULL)
	    error->all("Per-atom fix in equal-style variable formula");
	  if (update->whichflag > 0 && 
	      update->ntimestep % fix->peratom_freq)
	    error->all("Fix in variable not computed at compatible time");

	  Tree *newtree = new Tree();
	  newtree->type = ATOMARRAY;
	  newtree->array = fix->vector_atom;
	  newtree->nstride = 1;
	  newtree->left = newtree->middle = newtree->right = NULL;
	  treestack[ntreestack++] = newtree;

        // f_ID[i] = vector from per-atom array

	} else if (nbracket == 1 && fix->peratom_flag &&
		   fix->size_peratom_cols > 0) {

	  if (tree == NULL)
	    error->all("Per-atom fix in equal-style variable formula");
	  if (index1 > fix->size_peratom_cols)
	    error->all("Variable formula fix array is accessed out-of-range");
	  if (update->whichflag > 0 && 
	      update->ntimestep % fix->peratom_freq)
	    error->all("Fix in variable not computed at compatible time");

	  Tree *newtree = new Tree();
	  newtree->type = ATOMARRAY;
	  newtree->array = &fix->array_atom[0][index1-1];
	  newtree->nstride = fix->size_peratom_cols;
	  newtree->left = newtree->middle = newtree->right = NULL;
	  treestack[ntreestack++] = newtree;


	} else error->all("Mismatched fix in variable formula");

      // ----------------
      // variable
      // ----------------

      } else if (strncmp(word,"v_",2) == 0) {
	n = strlen(word) - 2 + 1;
	char *id = new char[n];
	strcpy(id,&word[2]);

	int ivar = find(id);
	if (ivar < 0) error->all("Invalid variable name in variable formula");

	// parse zero or one trailing brackets
	// point i beyond last bracket
	// nbracket = # of bracket pairs
	// index = int inside bracket

	int nbracket,index;
	if (str[i] != '[') nbracket = 0;
	else {
	  nbracket = 1;
	  ptr = &str[i];
	  index = int_between_brackets(ptr);
	  i = ptr-str+1;
	}

        // v_name = scalar from non atom-style global scalar

	if (nbracket == 0 && style[ivar] != ATOM) {

	  char *var = retrieve(id);
	  if (var == NULL)
	    error->all("Invalid variable evaluation in variable formula");
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = atof(var);
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = atof(var);

        // v_name = vector from atom-style per-atom vector

	} else if (nbracket == 0 && style[ivar] == ATOM) {

	  if (tree == NULL)
	    error->all("Atom-style variable in equal-style variable formula");
	  Tree *newtree;
	  double tmp = evaluate(data[ivar][0],&newtree);
	  treestack[ntreestack++] = newtree;

        // v_name[N] = scalar from atom-style per-atom vector
	// compute the per-atom variable in result
	// use peratom2global to extract single value from result

	} else if (nbracket && style[ivar] == ATOM) {

	  double *result;
	  memory->create(result,atom->nlocal,"variable:result");
	  compute_atom(ivar,0,result,1,0);
	  peratom2global(1,NULL,result,1,index,
			 tree,treestack,ntreestack,argstack,nargstack);
	  memory->destroy(result);

	} else error->all("Mismatched variable in variable formula");

	delete [] id;

      // ----------------
      // math/group/special function or atom value/vector or
      // constant or thermo keyword
      // ----------------

      } else {

	// ----------------
	// math or group or special function
	// ----------------

	if (str[i] == '(') {
	  char *contents;
	  i = find_matching_paren(str,i,contents);
	  i++;

	  if (math_function(word,contents,tree,
			    treestack,ntreestack,argstack,nargstack));
	  else if (group_function(word,contents,tree,
				  treestack,ntreestack,argstack,nargstack));
	  else if (special_function(word,contents,tree,
				    treestack,ntreestack,argstack,nargstack));
	  else error->all("Invalid math/group/special function "
			  "in variable formula");
	  delete [] contents;

	// ----------------
	// atom value
	// ----------------

	} else if (str[i] == '[') {
	  if (domain->box_exist == 0)
	    error->all("Variable evaluation before simulation box is defined");

	  ptr = &str[i];
	  int id = int_between_brackets(ptr);
	  i = ptr-str+1;
 
	  peratom2global(0,word,NULL,0,id,
			 tree,treestack,ntreestack,argstack,nargstack);

	// ----------------
	// atom vector
	// ----------------

	} else if (is_atom_vector(word)) {
	  if (domain->box_exist == 0)
	    error->all("Variable evaluation before simulation box is defined");

	  atom_vector(word,tree,treestack,ntreestack);

	// ----------------
	// constant
	// ----------------

	} else if (is_constant(word)) {
	  value1 = constant(word);
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = value1;
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = value1;

	// ----------------
	// thermo keyword
	// ----------------

	} else {
	  if (domain->box_exist == 0)
	    error->all("Variable evaluation before simulation box is defined");
 
	  int flag = output->thermo->evaluate_keyword(word,&value1);
	  if (flag) error->all("Invalid thermo keyword in variable formula");
	  if (tree) {
	    Tree *newtree = new Tree();
	    newtree->type = VALUE;
	    newtree->value = value1;
	    newtree->left = newtree->middle = newtree->right = NULL;
	    treestack[ntreestack++] = newtree;
	  } else argstack[nargstack++] = value1;
	}
      }

      delete [] word;

    // ----------------
    // math operator, including end-of-string
    // ----------------

    } else if (strchr("+-*/^<>=!&|\0",onechar)) {
      if (onechar == '+') op = ADD;
      else if (onechar == '-') op = SUBTRACT;
      else if (onechar == '*') op = MULTIPLY;
      else if (onechar == '/') op = DIVIDE;
      else if (onechar == '^') op = CARAT;
      else if (onechar == '=') {
	if (str[i+1] != '=') error->all("Invalid syntax in variable formula");
	op = EQ;
	i++;
      } else if (onechar == '!') {
	if (str[i+1] == '=') {
	  op = NE;
	  i++;
	} else op = NOT;
      } else if (onechar == '<') {
	if (str[i+1] != '=') op = LT;
	else {
	  op = LE;
	  i++;
	}
      } else if (onechar == '>') {
	if (str[i+1] != '=') op = GT;
	else {
	  op = GE;
	  i++;
	}
      } else if (onechar == '&') {
	if (str[i+1] != '&') error->all("Invalid syntax in variable formula");
	op = AND;
	i++;
      } else if (onechar == '|') {
	if (str[i+1] != '|') error->all("Invalid syntax in variable formula");
	op = OR;
	i++;
      } else op = DONE;

      i++;

      if (op == SUBTRACT && expect == ARG) {
	opstack[nopstack++] = UNARY;
	continue;
      }
      if (op == NOT && expect == ARG) {
	opstack[nopstack++] = op;
	continue;
      }

      if (expect == ARG) error->all("Invalid syntax in variable formula");
      expect = ARG;

      // evaluate stack as deep as possible while respecting precedence
      // before pushing current op onto stack

      while (nopstack && precedence[opstack[nopstack-1]] >= precedence[op]) {
	opprevious = opstack[--nopstack];

	if (tree) {
	  Tree *newtree = new Tree();
	  newtree->type = opprevious;
	  if (opprevious == UNARY) {
	    newtree->left = treestack[--ntreestack];
	    newtree->middle = newtree->right = NULL;
	  } else {
	    newtree->right = treestack[--ntreestack];
	    newtree->middle = NULL;
	    newtree->left = treestack[--ntreestack];
	  }
	  treestack[ntreestack++] = newtree;

	} else {
	  value2 = argstack[--nargstack];
	  if (opprevious != UNARY && opprevious != NOT)
	    value1 = argstack[--nargstack];

	  if (opprevious == ADD)
	    argstack[nargstack++] = value1 + value2;
	  else if (opprevious == SUBTRACT)
	    argstack[nargstack++] = value1 - value2;
	  else if (opprevious == MULTIPLY)
	    argstack[nargstack++] = value1 * value2;
	  else if (opprevious == DIVIDE) {
	    if (value2 == 0.0) error->all("Divide by 0 in variable formula");
	    argstack[nargstack++] = value1 / value2;
	  } else if (opprevious == CARAT) {
	    if (value2 == 0.0) error->all("Power by 0 in variable formula");
	    argstack[nargstack++] = pow(value1,value2);
	  } else if (opprevious == UNARY) {
	    argstack[nargstack++] = -value2;
	  } else if (opprevious == NOT) {
	    if (value2 == 0.0) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  } else if (opprevious == EQ) {
	    if (value1 == value2) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  } else if (opprevious == NE) {
	    if (value1 != value2) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  } else if (opprevious == LT) {
	    if (value1 < value2) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  } else if (opprevious == LE) {
	    if (value1 <= value2) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  } else if (opprevious == GT) {
	    if (value1 > value2) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  } else if (opprevious == GE) {
	    if (value1 >= value2) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  } else if (opprevious == AND) {
	    if (value1 != 0.0 && value2 != 0.0) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  } else if (opprevious == OR) {
	    if (value1 != 0.0 || value2 != 0.0) argstack[nargstack++] = 1.0;
	    else argstack[nargstack++] = 0.0;
	  }
	}
      }

      // if end-of-string, break out of entire formula evaluation loop

      if (op == DONE) break;

      // push current operation onto stack

      opstack[nopstack++] = op;

    } else error->all("Invalid syntax in variable formula");
  }

  if (nopstack) error->all("Invalid syntax in variable formula");

  // for atom-style variable, return remaining tree
  // for equal-style variable, return remaining arg

  if (tree) {
    if (ntreestack != 1) error->all("Invalid syntax in variable formula");
    *tree = treestack[0];
    return 0.0;
  } else {
    if (nargstack != 1) error->all("Invalid syntax in variable formula");
    return argstack[0];
  }
}

/* ----------------------------------------------------------------------
   one-time collapse of an atom-style variable parse tree
   tree was created by one-time parsing of formula string via evaulate()
   only keep tree nodes that depend on ATOMARRAY, TYPEARRAY, INTARRAY
   remainder is converted to single VALUE
   this enables optimal eval_tree loop over atoms
   customize by adding a function:
     sqrt(),exp(),ln(),log(),sin(),cos(),tan(),asin(),acos(),atan(),
     atan2(y,x),random(x,y,z),normal(x,y,z),ceil(),floor(),round(),
     ramp(x,y),stagger(x,y),logfreq(x,y,z),
     vdisplace(x,y),swiggle(x,y,z),cwiggle(x,y,z),
     gmask(x),rmask(x),grmask(x,y)
---------------------------------------------------------------------- */

double Variable::collapse_tree(Tree *tree)
{
  double arg1,arg2,arg3;

  if (tree->type == VALUE) return tree->value;
  if (tree->type == ATOMARRAY) return 0.0;
  if (tree->type == TYPEARRAY) return 0.0;
  if (tree->type == INTARRAY) return 0.0;

  if (tree->type == ADD) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = arg1 + arg2;
    return tree->value;
  }

  if (tree->type == SUBTRACT) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = arg1 - arg2;
    return tree->value;
  }

  if (tree->type == MULTIPLY) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = arg1 * arg2;
    return tree->value;
  }

  if (tree->type == DIVIDE) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg2 == 0.0) error->one("Divide by 0 in variable formula");
    tree->value = arg1 / arg2;
    return tree->value;
  }

  if (tree->type == CARAT) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg2 == 0.0) error->one("Power by 0 in variable formula");
    tree->value = pow(arg1,arg2);
    return tree->value;
  }

  if (tree->type == UNARY) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = -arg1;
    return tree->value;
  }

  if (tree->type == NOT) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 == 0.0) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == EQ) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 == arg2) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == NE) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 != arg2) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == LT) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 < arg2) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == LE) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 <= arg2) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == GT) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 > arg2) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == GE) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 >= arg2) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == AND) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 != 0.0 && arg2 != 0.0) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == OR) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 != 0.0 || arg2 != 0.0) tree->value = 1.0;
    else tree->value = 0.0;
    return tree->value;
  }

  if (tree->type == SQRT) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 < 0.0) error->one("Sqrt of negative value in variable formula");
    tree->value = sqrt(arg1);
    return tree->value;
  }

  if (tree->type == EXP) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = exp(arg1);
    return tree->value;
  }

  if (tree->type == LN) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 <= 0.0) 
      error->one("Log of zero/negative value in variable formula");
    tree->value = log(arg1);
    return tree->value;
  }

  if (tree->type == LOG) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 <= 0.0) 
      error->one("Log of zero/negative value in variable formula");
    tree->value = log10(arg1);
    return tree->value;
  }

  if (tree->type == SIN) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = sin(arg1);
    return tree->value;
  }

  if (tree->type == COS) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = cos(arg1);
    return tree->value;
  }

  if (tree->type == TAN) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = tan(arg1);
    return tree->value;
  }

  if (tree->type == ASIN) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 < -1.0 || arg1 > 1.0)
      error->one("Arcsin of invalid value in variable formula");
    tree->value = asin(arg1);
    return tree->value;
  }

  if (tree->type == ACOS) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg1 < -1.0 || arg1 > 1.0)
      error->one("Arccos of invalid value in variable formula");
    tree->value = acos(arg1);
    return tree->value;
  }

  if (tree->type == ATAN) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = atan(arg1);
    return tree->value;
  }

  if (tree->type == ATAN2) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = atan2(arg1,arg2);
    return tree->value;
  }

  // random() or normal() do not become a single collapsed value

  if (tree->type == RANDOM) {
    double lower = collapse_tree(tree->left);
    double upper = collapse_tree(tree->middle);
    if (randomatom == NULL) {
      int seed = static_cast<int> (collapse_tree(tree->right));
      if (seed <= 0) error->one("Invalid math function in variable formula");
      randomatom = new RanMars(lmp,seed+me);
    }
    return 0.0;
  }

  if (tree->type == NORMAL) {
    double mu = collapse_tree(tree->left);
    double sigma = collapse_tree(tree->middle);
    if (sigma < 0.0) error->one("Invalid math function in variable formula");
    if (randomatom == NULL) {
      int seed = static_cast<int> (collapse_tree(tree->right));
      if (seed <= 0) error->one("Invalid math function in variable formula");
      randomatom = new RanMars(lmp,seed+me);
    }
    return 0.0;
  }

  if (tree->type == CEIL) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = ceil(arg1);
    return tree->value;
  }

  if (tree->type == FLOOR) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = floor(arg1);
    return tree->value;
  }

  if (tree->type == ROUND) {
    arg1 = collapse_tree(tree->left);
    if (tree->left->type != VALUE) return 0.0;
    tree->type = VALUE;
    tree->value = MYROUND(arg1);
    return tree->value;
  }

  if (tree->type == RAMP) {
    arg1 = collapse_tree(tree->left);
    arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    double delta = update->ntimestep - update->beginstep;
    delta /= update->endstep - update->beginstep;
    tree->value = arg1 + delta*(arg2-arg1);
    return tree->value;
  }

  if (tree->type == STAGGER) {
    int ivalue1 = static_cast<int> (collapse_tree(tree->left));
    int ivalue2 = static_cast<int> (collapse_tree(tree->right));
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (ivalue1 <= 0 || ivalue2 <= 0 || ivalue1 <= ivalue2)
      error->one("Invalid math function in variable formula");
    int lower = update->ntimestep/ivalue1 * ivalue1;
    int delta = update->ntimestep - lower;
    if (delta < ivalue2) tree->value = lower+ivalue2;
    else tree->value = lower+ivalue1;
    return tree->value;
  }

  if (tree->type == LOGFREQ) {
    int ivalue1 = static_cast<int> (collapse_tree(tree->left));
    int ivalue2 = static_cast<int> (collapse_tree(tree->middle));
    int ivalue3 = static_cast<int> (collapse_tree(tree->right));
    if (tree->left->type != VALUE || tree->middle->type != VALUE ||
	tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (ivalue1 <= 0 || ivalue2 <= 0 || ivalue3 <= 0 || ivalue2 >= ivalue3)
      error->one("Invalid math function in variable formula");
    if (update->ntimestep < ivalue1) tree->value = ivalue1;
    else {
      int lower = ivalue1;
      while (update->ntimestep >= ivalue3*lower) lower *= ivalue3;
      int multiple = update->ntimestep/lower;
      if (multiple < ivalue2) tree->value = (multiple+1)*lower;
      else tree->value = lower*ivalue3;
    }
    return tree->value;
  }

  if (tree->type == VDISPLACE) {
    double arg1 = collapse_tree(tree->left);
    double arg2 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    double delta = update->ntimestep - update->beginstep;
    tree->value = arg1 + arg2*delta*update->dt;
    return tree->value;
  }

  if (tree->type == SWIGGLE) {
    double arg1 = collapse_tree(tree->left);
    double arg2 = collapse_tree(tree->middle);
    double arg3 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->middle->type != VALUE ||
	tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg3 == 0.0) error->one("Invalid math function in variable formula");
    double delta = update->ntimestep - update->beginstep;
    double omega = 2.0*PI/arg3;
    tree->value = arg1 + arg2*sin(omega*delta*update->dt);
    return tree->value;
  }

  if (tree->type == CWIGGLE) {
    double arg1 = collapse_tree(tree->left);
    double arg2 = collapse_tree(tree->middle);
    double arg3 = collapse_tree(tree->right);
    if (tree->left->type != VALUE || tree->middle->type != VALUE ||
	tree->right->type != VALUE) return 0.0;
    tree->type = VALUE;
    if (arg3 == 0.0) error->one("Invalid math function in variable formula");
    double delta = update->ntimestep - update->beginstep;
    double omega = 2.0*PI/arg3;
    tree->value = arg1 + arg2*(1.0-cos(omega*delta*update->dt));
    return tree->value;
  }

  // mask functions do not become a single collapsed value

  if (tree->type == GMASK) return 0.0;
  if (tree->type == RMASK) return 0.0;
  if (tree->type == GRMASK) return 0.0;

  return 0.0;
}

/* ----------------------------------------------------------------------
   evaluate an atom-style variable parse tree for atom I
   tree was created by one-time parsing of formula string via evaulate()
   customize by adding a function:
     sqrt(),exp(),ln(),log(),sin(),cos(),tan(),asin(),acos(),atan(),
     atan2(y,x),random(x,y,z),normal(x,y,z),ceil(),floor(),round(),
     ramp(x,y),stagger(x,y),logfreq(x,y,z),
     vdisplace(x,y),swiggle(x,y,z),cwiggle(x,y,z),
     gmask(x),rmask(x),grmask(x,y)
---------------------------------------------------------------------- */

double Variable::eval_tree(Tree *tree, int i)
{
  double arg,arg1,arg2,arg3;

  if (tree->type == VALUE) return tree->value;
  if (tree->type == ATOMARRAY) return tree->array[i*tree->nstride];
  if (tree->type == TYPEARRAY) return tree->array[atom->type[i]];
  if (tree->type == INTARRAY) return (double) tree->iarray[i*tree->nstride];

  if (tree->type == ADD)
    return eval_tree(tree->left,i) + eval_tree(tree->right,i);
  if (tree->type == SUBTRACT)
    return eval_tree(tree->left,i) - eval_tree(tree->right,i);
  if (tree->type == MULTIPLY)
    return eval_tree(tree->left,i) * eval_tree(tree->right,i);
  if (tree->type == DIVIDE) {
    double denom = eval_tree(tree->right,i);
    if (denom == 0.0) error->one("Divide by 0 in variable formula");
    return eval_tree(tree->left,i) / denom;
  }
  if (tree->type == CARAT) {
    double exponent = eval_tree(tree->right,i);
    if (exponent == 0.0) error->one("Power by 0 in variable formula");
    return pow(eval_tree(tree->left,i),exponent);
  }
  if (tree->type == UNARY) return -eval_tree(tree->left,i);

  if (tree->type == NOT) {
    if (eval_tree(tree->left,i) == 0.0) return 1.0;
    else return 0.0;
  }
  if (tree->type == EQ) {
    if (eval_tree(tree->left,i) == eval_tree(tree->right,i)) return 1.0;
    else return 0.0;
  }
  if (tree->type == NE) {
    if (eval_tree(tree->left,i) != eval_tree(tree->right,i)) return 1.0;
    else return 0.0;
  }
  if (tree->type == LT) {
    if (eval_tree(tree->left,i) < eval_tree(tree->right,i)) return 1.0;
    else return 0.0;
  }
  if (tree->type == LE) {
    if (eval_tree(tree->left,i) <= eval_tree(tree->right,i)) return 1.0;
    else return 0.0;
  }
  if (tree->type == GT) {
    if (eval_tree(tree->left,i) > eval_tree(tree->right,i)) return 1.0;
    else return 0.0;
  }
  if (tree->type == GE) {
    if (eval_tree(tree->left,i) >= eval_tree(tree->right,i)) return 1.0;
    else return 0.0;
  }
  if (tree->type == AND) {
    if (eval_tree(tree->left,i) != 0.0 && eval_tree(tree->right,i) != 0.0)
      return 1.0;
    else return 0.0;
  }
  if (tree->type == OR) {
    if (eval_tree(tree->left,i) != 0.0 || eval_tree(tree->right,i) != 0.0)
      return 1.0;
    else return 0.0;
  }

  if (tree->type == SQRT) {
    arg1 = eval_tree(tree->left,i);
    if (arg1 < 0.0) error->one("Sqrt of negative value in variable formula");
    return sqrt(arg1);
  }
  if (tree->type == EXP)
    return exp(eval_tree(tree->left,i));
  if (tree->type == LN) {
    arg1 = eval_tree(tree->left,i);
    if (arg1 <= 0.0) 
      error->one("Log of zero/negative value in variable formula");
    return log(arg1);
  }
  if (tree->type == LOG) {
    arg1 = eval_tree(tree->left,i);
    if (arg1 <= 0.0) 
      error->one("Log of zero/negative value in variable formula");
    return log10(arg1);
  }

  if (tree->type == SIN)
    return sin(eval_tree(tree->left,i));
  if (tree->type == COS)
    return cos(eval_tree(tree->left,i));
  if (tree->type == TAN)
    return tan(eval_tree(tree->left,i));

  if (tree->type == ASIN) {
    arg1 = eval_tree(tree->left,i);
    if (arg1 < -1.0 || arg1 > 1.0)
      error->one("Arcsin of invalid value in variable formula");
    return asin(arg1);
  }
  if (tree->type == ACOS) {
    arg1 = eval_tree(tree->left,i);
    if (arg1 < -1.0 || arg1 > 1.0)
      error->one("Arccos of invalid value in variable formula");
    return acos(arg1);
  }
  if (tree->type == ATAN)
    return atan(eval_tree(tree->left,i));
  if (tree->type == ATAN2)
    return atan2(eval_tree(tree->left,i),eval_tree(tree->right,i));

  if (tree->type == RANDOM) {
    double lower = eval_tree(tree->left,i);
    double upper = eval_tree(tree->middle,i);
    if (randomatom == NULL) {
      int seed = static_cast<int> (eval_tree(tree->right,i));
      if (seed <= 0) error->one("Invalid math function in variable formula");
      randomatom = new RanMars(lmp,seed+me);
    }
    return randomatom->uniform()*(upper-lower)+lower;
  }
  if (tree->type == NORMAL) {
    double mu = eval_tree(tree->left,i);
    double sigma = eval_tree(tree->middle,i);
    if (sigma < 0.0) error->one("Invalid math function in variable formula");
    if (randomatom == NULL) {
      int seed = static_cast<int> (eval_tree(tree->right,i));
      if (seed <= 0) error->one("Invalid math function in variable formula");
      randomatom = new RanMars(lmp,seed+me);
    }
    return mu + sigma*randomatom->gaussian();
  }

  if (tree->type == CEIL)
    return ceil(eval_tree(tree->left,i));
  if (tree->type == FLOOR)
    return floor(eval_tree(tree->left,i));
  if (tree->type == ROUND)
    return MYROUND(eval_tree(tree->left,i));

  if (tree->type == RAMP) {
    arg1 = eval_tree(tree->left,i);
    arg2 = eval_tree(tree->right,i);
    double delta = update->ntimestep - update->beginstep;
    delta /= update->endstep - update->beginstep;
    arg = arg1 + delta*(arg2-arg1);
    return arg;
  }

  if (tree->type == STAGGER) {
    int ivalue1 = static_cast<int> (eval_tree(tree->left,i));
    int ivalue2 = static_cast<int> (eval_tree(tree->right,i));
    if (ivalue1 <= 0 || ivalue2 <= 0 || ivalue1 <= ivalue2)
      error->one("Invalid math function in variable formula");
    int lower = update->ntimestep/ivalue1 * ivalue1;
    int delta = update->ntimestep - lower;
    if (delta < ivalue2) arg = lower+ivalue2;
    else arg = lower+ivalue1;
    return arg;
  }

  if (tree->type == LOGFREQ) {
    int ivalue1 = static_cast<int> (eval_tree(tree->left,i));
    int ivalue2 = static_cast<int> (eval_tree(tree->middle,i));
    int ivalue3 = static_cast<int> (eval_tree(tree->right,i));
    if (ivalue1 <= 0 || ivalue2 <= 0 || ivalue3 <= 0 || ivalue2 >= ivalue3)
      error->one("Invalid math function in variable formula");
    if (update->ntimestep < ivalue1) arg = ivalue1;
    else {
      int lower = ivalue1;
      while (update->ntimestep >= ivalue3*lower) lower *= ivalue3;
      int multiple = update->ntimestep/lower;
      if (multiple < ivalue2) arg = (multiple+1)*lower;
      else arg = lower*ivalue3;
    }
    return arg;
  }

  if (tree->type == VDISPLACE) {
    arg1 = eval_tree(tree->left,i);
    arg2 = eval_tree(tree->right,i);
    double delta = update->ntimestep - update->beginstep;
    arg = arg1 + arg2*delta*update->dt;
    return arg;
  }

  if (tree->type == SWIGGLE) {
    arg1 = eval_tree(tree->left,i);
    arg2 = eval_tree(tree->middle,i);
    arg3 = eval_tree(tree->right,i);
    if (arg3 == 0.0) error->one("Invalid math function in variable formula");
    double delta = update->ntimestep - update->beginstep;
    double omega = 2.0*PI/arg3;
    arg = arg1 + arg2*sin(omega*delta*update->dt);
    return arg;
  }

  if (tree->type == CWIGGLE) {
    arg1 = eval_tree(tree->left,i);
    arg2 = eval_tree(tree->middle,i);
    arg3 = eval_tree(tree->right,i);
    if (arg3 == 0.0) error->one("Invalid math function in variable formula");
    double delta = update->ntimestep - update->beginstep;
    double omega = 2.0*PI/arg3;
    arg = arg1 + arg2*(1.0-cos(omega*delta*update->dt));
    return arg;
  }

  if (tree->type == GMASK) {
    if (atom->mask[i] & tree->ivalue1) return 1.0;
    else return 0.0;
  }

  if (tree->type == RMASK) {
    if (domain->regions[tree->ivalue1]->inside(atom->x[i][0],
					       atom->x[i][1],
					       atom->x[i][2])) return 1.0;
    else return 0.0;
  }

  if (tree->type == GRMASK) {
    if ((atom->mask[i] & tree->ivalue1) &&
	(domain->regions[tree->ivalue2]->inside(atom->x[i][0],
						atom->x[i][1],
						atom->x[i][2]))) return 1.0;
    else return 0.0;
  }

  return 0.0;
}

/* ---------------------------------------------------------------------- */

void Variable::free_tree(Tree *tree)
{
  if (tree->left) free_tree(tree->left);
  if (tree->middle) free_tree(tree->middle);
  if (tree->right) free_tree(tree->right);
  delete tree;
}

/* ----------------------------------------------------------------------
   find matching parenthesis in str, allocate contents = str between parens
   i = left paren
   return loc or right paren
------------------------------------------------------------------------- */

int Variable::find_matching_paren(char *str, int i,char *&contents)
{
  // istop = matching ')' at same level, allowing for nested parens

  int istart = i;
  int ilevel = 0;
  while (1) {
    i++;
    if (!str[i]) break;
    if (str[i] == '(') ilevel++;
    else if (str[i] == ')' && ilevel) ilevel--;
    else if (str[i] == ')') break;
  }
  if (!str[i]) error->all("Invalid syntax in variable formula");
  int istop = i;

  int n = istop - istart - 1;
  contents = new char[n+1];
  strncpy(contents,&str[istart+1],n);
  contents[n] = '\0';

  return istop;
}

/* ----------------------------------------------------------------------
   find int between brackets and return it
   ptr initially points to left bracket
   return it pointing to right bracket
   error if no right bracket or brackets are empty
   error if any between-bracket chars are non-digits or value == 0
------------------------------------------------------------------------- */

int Variable::int_between_brackets(char *&ptr)
{
  char *start = ++ptr;

  while (*ptr && *ptr != ']') {
    if (!isdigit(*ptr)) 
      error->all("Non digit character between brackets in variable");
    ptr++;
  }

  if (*ptr != ']') error->all("Mismatched brackets in variable");
  if (ptr == start) error->all("Empty brackets in variable");

  *ptr = '\0';
  int index = atoi(start);
  *ptr = ']';

  if (index == 0) 
    error->all("Index between variable brackets must be positive");
  return index;
}

/* ----------------------------------------------------------------------
   process a math function in formula
   push result onto tree or arg stack
   word = math function
   contents = str between parentheses with one,two,three args
   return 0 if not a match, 1 if successfully processed
   customize by adding a math function:
     sqrt(),exp(),ln(),log(),sin(),cos(),tan(),asin(),acos(),atan(),
     atan2(y,x),random(x,y,z),normal(x,y,z),ceil(),floor(),round(),
     ramp(x,y),stagger(x,y),logfreq(x,y,z),
     vdisplace(x,y),swiggle(x,y,z),cwiggle(x,y,z)
------------------------------------------------------------------------- */

int Variable::math_function(char *word, char *contents, Tree **tree,
			    Tree **treestack, int &ntreestack,
			    double *argstack, int &nargstack)
{
  // word not a match to any math function

  if (strcmp(word,"sqrt") && strcmp(word,"exp") && 
      strcmp(word,"ln") && strcmp(word,"log") &&
      strcmp(word,"sin") && strcmp(word,"cos") &&
      strcmp(word,"tan") && strcmp(word,"asin") &&
      strcmp(word,"acos") && strcmp(word,"atan") && 
      strcmp(word,"atan2") && strcmp(word,"random") && 
      strcmp(word,"normal") && strcmp(word,"ceil") && 
      strcmp(word,"floor") && strcmp(word,"round") &&
      strcmp(word,"ramp") && strcmp(word,"stagger") &&
      strcmp(word,"logfreq") && strcmp(word,"vdisplace") &&
      strcmp(word,"swiggle") && strcmp(word,"cwiggle"))
    return 0;

  // parse contents for arg1,arg2,arg3 separated by commas
  // ptr1,ptr2 = location of 1st and 2nd comma, NULL if none

  char *arg1,*arg2,*arg3;
  char *ptr1,*ptr2;

  ptr1 = strchr(contents,',');
  if (ptr1) {
    *ptr1 = '\0';
    ptr2 = strchr(ptr1+1,',');
    if (ptr2) *ptr2 = '\0';
  } else ptr2 = NULL;

  int n = strlen(contents) + 1;
  arg1 = new char[n];
  strcpy(arg1,contents);
  int narg = 1;
  if (ptr1) {
    n = strlen(ptr1+1) + 1;
    arg2 = new char[n];
    strcpy(arg2,ptr1+1);
    narg = 2;
  } else arg2 = NULL;
  if (ptr2) {
    n = strlen(ptr2+1) + 1;
    arg3 = new char[n];
    strcpy(arg3,ptr2+1);
    narg = 3;
  } else arg3 = NULL;

  // evaluate args
    
  Tree *newtree;
  double tmp,value1,value2,value3;

  if (tree) {
    newtree = new Tree();
    Tree *argtree;
    if (narg == 1) {
      tmp = evaluate(arg1,&argtree);
      newtree->left = argtree;
      newtree->middle = newtree->right = NULL;
    } else if (narg == 2) {
      tmp = evaluate(arg1,&argtree);
      newtree->left = argtree;
      newtree->middle = NULL;
      tmp = evaluate(arg2,&argtree);
      newtree->right = argtree;
    } else if (narg == 3) {
      tmp = evaluate(arg1,&argtree);
      newtree->left = argtree;
      tmp = evaluate(arg2,&argtree);
      newtree->middle = argtree;
      tmp = evaluate(arg3,&argtree);
      newtree->right = argtree;
    }
    treestack[ntreestack++] = newtree;
  } else {
    if (narg == 1) {
      value1 = evaluate(arg1,NULL);
    } else if (narg == 2) {
      value1 = evaluate(arg1,NULL);
      value2 = evaluate(arg2,NULL);
    } else if (narg == 3) {
      value1 = evaluate(arg1,NULL);
      value2 = evaluate(arg2,NULL);
      value3 = evaluate(arg3,NULL);
    }
  }
    
  if (strcmp(word,"sqrt") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = SQRT;
    else {
      if (value1 < 0.0) 
	error->all("Sqrt of negative value in variable formula");
      argstack[nargstack++] = sqrt(value1);
    }

  } else if (strcmp(word,"exp") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = EXP;
    else argstack[nargstack++] = exp(value1);
  } else if (strcmp(word,"ln") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = LN;
    else {
      if (value1 <= 0.0) 
	error->all("Log of zero/negative value in variable formula");
      argstack[nargstack++] = log(value1);
    }
  } else if (strcmp(word,"log") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = LOG;
    else {
      if (value1 <= 0.0) 
	error->all("Log of zero/negative value in variable formula");
      argstack[nargstack++] = log10(value1);
    }

  } else if (strcmp(word,"sin") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = SIN;
    else argstack[nargstack++] = sin(value1);
  } else if (strcmp(word,"cos") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = COS;
    else argstack[nargstack++] = cos(value1);
  } else if (strcmp(word,"tan") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = TAN;
    else argstack[nargstack++] = tan(value1);

  } else if (strcmp(word,"asin") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = ASIN;
    else {
      if (value1 < -1.0 || value1 > 1.0) 
	error->all("Arcsin of invalid value in variable formula");
      argstack[nargstack++] = asin(value1);
    }
  } else if (strcmp(word,"acos") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = ACOS;
    else {
      if (value1 < -1.0 || value1 > 1.0) 
	error->all("Arccos of invalid value in variable formula");
      argstack[nargstack++] = acos(value1);
    }
  } else if (strcmp(word,"atan") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = ATAN;
    else argstack[nargstack++] = atan(value1);
  } else if (strcmp(word,"atan2") == 0) {
    if (narg != 2) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = ATAN2;
    else argstack[nargstack++] = atan2(value1,value2);

  } else if (strcmp(word,"random") == 0) {
    if (narg != 3) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = RANDOM;
    else {
      if (randomequal == NULL) {
	int seed = static_cast<int> (value3);
	if (seed <= 0) error->all("Invalid math function in variable formula");
	randomequal = new RanMars(lmp,seed);
      }
      argstack[nargstack++] = randomequal->uniform()*(value2-value1) + value1;
    }
  } else if (strcmp(word,"normal") == 0) {
    if (narg != 3) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = NORMAL;
    else {
      if (value2 < 0.0) 
	error->all("Invalid math function in variable formula");
      if (randomequal == NULL) {
	int seed = static_cast<int> (value3);
	if (seed <= 0) error->all("Invalid math function in variable formula");
	randomequal = new RanMars(lmp,seed);
      }
      argstack[nargstack++] = value1 + value2*randomequal->gaussian();
    }

  } else if (strcmp(word,"ceil") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = CEIL;
    else argstack[nargstack++] = ceil(value1);

  } else if (strcmp(word,"floor") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = FLOOR;
    else argstack[nargstack++] = floor(value1);

  } else if (strcmp(word,"round") == 0) {
    if (narg != 1) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = ROUND;
    else argstack[nargstack++] = MYROUND(value1);

  } else if (strcmp(word,"ramp") == 0) {
    if (narg != 2) error->all("Invalid math function in variable formula");
    if (update->whichflag == 0)
      error->all("Cannot use ramp in variable formula between runs");
    if (tree) newtree->type = RAMP;
    else {
      double delta = update->ntimestep - update->beginstep;
      delta /= update->endstep - update->beginstep;
      double value = value1 + delta*(value2-value1);
      argstack[nargstack++] = value;
    }

  } else if (strcmp(word,"stagger") == 0) {
    if (narg != 2) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = STAGGER;
    else {
      int ivalue1 = static_cast<int> (value1);
      int ivalue2 = static_cast<int> (value2);
      if (ivalue1 <= 0 || ivalue2 <= 0 || ivalue1 <= ivalue2)
	error->all("Invalid math function in variable formula");
      int lower = update->ntimestep/ivalue1 * ivalue1;
      int delta = update->ntimestep - lower;
      double value;
      if (delta < ivalue2) value = lower+ivalue2;
      else value = lower+ivalue1;
      argstack[nargstack++] = value;
    }

  } else if (strcmp(word,"logfreq") == 0) {
    if (narg != 3) error->all("Invalid math function in variable formula");
    if (tree) newtree->type = LOGFREQ;
    else {
      int ivalue1 = static_cast<int> (value1);
      int ivalue2 = static_cast<int> (value2);
      int ivalue3 = static_cast<int> (value3);
      if (ivalue1 <= 0 || ivalue2 <= 0 || ivalue3 <= 0 || ivalue2 >= ivalue3)
	error->all("Invalid math function in variable formula");
      double value;
      if (update->ntimestep < ivalue1) value = ivalue1;
      else {
	int lower = ivalue1;
	while (update->ntimestep >= ivalue3*lower) lower *= ivalue3;
	int multiple = update->ntimestep/lower;
	if (multiple < ivalue2) value = (multiple+1)*lower;
	else value = lower*ivalue3;
      }
      argstack[nargstack++] = value;
    }

  } else if (strcmp(word,"vdisplace") == 0) {
    if (narg != 2) error->all("Invalid math function in variable formula");
    if (update->whichflag == 0)
      error->all("Cannot use vdisplace in variable formula between runs");
    if (tree) newtree->type = VDISPLACE;
    else {
      double delta = update->ntimestep - update->beginstep;
      double value = value1 + value2*delta*update->dt;
      argstack[nargstack++] = value;
    }

  } else if (strcmp(word,"swiggle") == 0) {
    if (narg != 3) error->all("Invalid math function in variable formula");
    if (update->whichflag == 0)
      error->all("Cannot use swiggle in variable formula between runs");
    if (tree) newtree->type = CWIGGLE;
    else {
      if (value3 == 0.0)
	error->all("Invalid math function in variable formula");
      double delta = update->ntimestep - update->beginstep;
      double omega = 2.0*PI/value3;
      double value = value1 + value2*sin(omega*delta*update->dt);
      argstack[nargstack++] = value;
    }

  } else if (strcmp(word,"cwiggle") == 0) {
    if (narg != 3) error->all("Invalid math function in variable formula");
    if (update->whichflag == 0)
      error->all("Cannot use cwiggle in variable formula between runs");
    if (tree) newtree->type = CWIGGLE;
    else {
      if (value3 == 0.0)
	error->all("Invalid math function in variable formula");
      double delta = update->ntimestep - update->beginstep;
      double omega = 2.0*PI/value3;
      double value = value1 + value2*(1.0-cos(omega*delta*update->dt));
      argstack[nargstack++] = value;
    }
  }

  delete [] arg1;
  delete [] arg2;
  delete [] arg3;

  return 1;
}

/* ----------------------------------------------------------------------
   process a group function in formula with optional region arg
   push result onto tree or arg stack
   word = group function
   contents = str between parentheses with one,two,three args
   return 0 if not a match, 1 if successfully processed
   customize by adding a group function with optional region arg:
     count(group),mass(group),charge(group),
     xcm(group,dim),vcm(group,dim),fcm(group,dim),
     bound(group,xmin),gyration(group),ke(group),angmom(group,dim),
     torque(group,dim),inertia(group,dim),omega(group,dim)
------------------------------------------------------------------------- */

int Variable::group_function(char *word, char *contents, Tree **tree,
			     Tree **treestack, int &ntreestack,
			     double *argstack, int &nargstack)
{
  // word not a match to any group function

  if (strcmp(word,"count") && strcmp(word,"mass") && 
      strcmp(word,"charge") && strcmp(word,"xcm") &&
      strcmp(word,"vcm") && strcmp(word,"fcm") &&
      strcmp(word,"bound") && strcmp(word,"gyration") &&
      strcmp(word,"ke") && strcmp(word,"angmom") &&
      strcmp(word,"torque") && strcmp(word,"inertia") && 
      strcmp(word,"omega"))
    return 0;

  // parse contents for arg1,arg2,arg3 separated by commas
  // ptr1,ptr2 = location of 1st and 2nd comma, NULL if none

  char *arg1,*arg2,*arg3;
  char *ptr1,*ptr2;

  ptr1 = strchr(contents,',');
  if (ptr1) {
    *ptr1 = '\0';
    ptr2 = strchr(ptr1+1,',');
    if (ptr2) *ptr2 = '\0';
  } else ptr2 = NULL;

  int n = strlen(contents) + 1;
  arg1 = new char[n];
  strcpy(arg1,contents);
  int narg = 1;
  if (ptr1) {
    n = strlen(ptr1+1) + 1;
    arg2 = new char[n];
    strcpy(arg2,ptr1+1);
    narg = 2;
  } else arg2 = NULL;
  if (ptr2) {
    n = strlen(ptr2+1) + 1;
    arg3 = new char[n];
    strcpy(arg3,ptr2+1);
    narg = 3;
  } else arg3 = NULL;

  // group to operate on

  int igroup = group->find(arg1);
  if (igroup == -1)
    error->all("Group ID in variable formula does not exist");

  // match word to group function

  double value;

  if (strcmp(word,"count") == 0) {
    if (narg == 1) value = group->count(igroup);
    else if (narg == 2) value = group->count(igroup,region_function(arg2));
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"mass") == 0) {
    if (narg == 1) value = group->mass(igroup);
    else if (narg == 2) value = group->mass(igroup,region_function(arg2));
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"charge") == 0) {
    if (narg == 1) value = group->charge(igroup);
    else if (narg == 2) value = group->charge(igroup,region_function(arg2));
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"xcm") == 0) {
    atom->check_mass();
    double xcm[3];
    if (narg == 2) {
      double masstotal = group->mass(igroup);
      group->xcm(igroup,masstotal,xcm);
    } else if (narg == 3) {
      int iregion = region_function(arg3);
      double masstotal = group->mass(igroup,iregion);
      group->xcm(igroup,masstotal,xcm,iregion);
    } else error->all("Invalid group function in variable formula");
    if (strcmp(arg2,"x") == 0) value = xcm[0];
    else if (strcmp(arg2,"y") == 0) value = xcm[1];
    else if (strcmp(arg2,"z") == 0) value = xcm[2];
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"vcm") == 0) {
    atom->check_mass();
    double vcm[3];
    if (narg == 2) {
      double masstotal = group->mass(igroup);
      group->vcm(igroup,masstotal,vcm);
    } else if (narg == 3) {
      int iregion = region_function(arg3);
      double masstotal = group->mass(igroup,iregion);
      group->vcm(igroup,masstotal,vcm,iregion);
    } else error->all("Invalid group function in variable formula");
    if (strcmp(arg2,"x") == 0) value = vcm[0];
    else if (strcmp(arg2,"y") == 0) value = vcm[1];
    else if (strcmp(arg2,"z") == 0) value = vcm[2];
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"fcm") == 0) {
    double fcm[3];
    if (narg == 2) group->fcm(igroup,fcm);
    else if (narg == 3) group->fcm(igroup,fcm,region_function(arg3));
    else error->all("Invalid group function in variable formula");
    if (strcmp(arg2,"x") == 0) value = fcm[0];
    else if (strcmp(arg2,"y") == 0) value = fcm[1];
    else if (strcmp(arg2,"z") == 0) value = fcm[2];
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"bound") == 0) {
    double minmax[6];
    if (narg == 2) group->bounds(igroup,minmax);
    else if (narg == 3) group->bounds(igroup,minmax,region_function(arg3));
    else error->all("Invalid group function in variable formula");
    if (strcmp(arg2,"xmin") == 0) value = minmax[0];
    else if (strcmp(arg2,"xmax") == 0) value = minmax[1];
    else if (strcmp(arg2,"ymin") == 0) value = minmax[2];
    else if (strcmp(arg2,"ymax") == 0) value = minmax[3];
    else if (strcmp(arg2,"zmin") == 0) value = minmax[4];
    else if (strcmp(arg2,"zmax") == 0) value = minmax[5];
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"gyration") == 0) {
    atom->check_mass();
    double xcm[3];
    if (narg == 1) {
      double masstotal = group->mass(igroup);
      group->xcm(igroup,masstotal,xcm);
      value = group->gyration(igroup,masstotal,xcm);
    } else if (narg == 2) {
      int iregion = region_function(arg2);
      double masstotal = group->mass(igroup,iregion);
      group->xcm(igroup,masstotal,xcm,iregion);
      value = group->gyration(igroup,masstotal,xcm,iregion);
    } else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"ke") == 0) {
    if (narg == 1) value = group->ke(igroup);
    else if (narg == 2) value = group->ke(igroup,region_function(arg2));
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"angmom") == 0) {
    atom->check_mass();
    double xcm[3],lmom[3];
    if (narg == 2) {
      double masstotal = group->mass(igroup);
      group->xcm(igroup,masstotal,xcm);
      group->angmom(igroup,xcm,lmom);
    } else if (narg == 3) {
      int iregion = region_function(arg3);
      double masstotal = group->mass(igroup,iregion);
      group->xcm(igroup,masstotal,xcm,iregion);
      group->angmom(igroup,xcm,lmom,iregion);
    } else error->all("Invalid group function in variable formula");
    if (strcmp(arg2,"x") == 0) value = lmom[0];
    else if (strcmp(arg2,"y") == 0) value = lmom[1];
    else if (strcmp(arg2,"z") == 0) value = lmom[2];
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"torque") == 0) {
    atom->check_mass();
    double xcm[3],tq[3];
    if (narg == 2) {
      double masstotal = group->mass(igroup);
      group->xcm(igroup,masstotal,xcm);
      group->torque(igroup,xcm,tq);
    } else if (narg == 3) {
      int iregion = region_function(arg3);
      double masstotal = group->mass(igroup,iregion);
      group->xcm(igroup,masstotal,xcm,iregion);
      group->torque(igroup,xcm,tq,iregion);
    } else error->all("Invalid group function in variable formula");
    if (strcmp(arg2,"x") == 0) value = tq[0];
    else if (strcmp(arg2,"y") == 0) value = tq[1];
    else if (strcmp(arg2,"z") == 0) value = tq[2];
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"inertia") == 0) {
    atom->check_mass();
    double xcm[3],inertia[3][3];
    if (narg == 2) {
      double masstotal = group->mass(igroup);
      group->xcm(igroup,masstotal,xcm);
      group->inertia(igroup,xcm,inertia);
    } else if (narg == 3) {
      int iregion = region_function(arg3);
      double masstotal = group->mass(igroup,iregion);
      group->xcm(igroup,masstotal,xcm,iregion);
      group->inertia(igroup,xcm,inertia,iregion);
    } else error->all("Invalid group function in variable formula");
    if (strcmp(arg2,"xx") == 0) value = inertia[0][0];
    else if (strcmp(arg2,"yy") == 0) value = inertia[1][1];
    else if (strcmp(arg2,"zz") == 0) value = inertia[2][2];
    else if (strcmp(arg2,"xy") == 0) value = inertia[0][1];
    else if (strcmp(arg2,"yz") == 0) value = inertia[1][2];
    else if (strcmp(arg2,"xz") == 0) value = inertia[0][2];
    else error->all("Invalid group function in variable formula");

  } else if (strcmp(word,"omega") == 0) {
    atom->check_mass();
    double xcm[3],angmom[3],inertia[3][3],omega[3];
    if (narg == 2) {
      double masstotal = group->mass(igroup);
      group->xcm(igroup,masstotal,xcm);
      group->angmom(igroup,xcm,angmom);
      group->inertia(igroup,xcm,inertia);
      group->omega(angmom,inertia,omega);
    } else if (narg == 3) {
      int iregion = region_function(arg3);
      double masstotal = group->mass(igroup,iregion);
      group->xcm(igroup,masstotal,xcm,iregion);
      group->angmom(igroup,xcm,angmom,iregion);
      group->inertia(igroup,xcm,inertia,iregion);
      group->omega(angmom,inertia,omega);
    } else error->all("Invalid group function in variable formula");
    if (strcmp(arg2,"x") == 0) value = omega[0];
    else if (strcmp(arg2,"y") == 0) value = omega[1];
    else if (strcmp(arg2,"z") == 0) value = omega[2];
    else error->all("Invalid group function in variable formula");
  }
    
  delete [] arg1;
  delete [] arg2;
  delete [] arg3;

  // save value in tree or on argstack

  if (tree) {
    Tree *newtree = new Tree();
    newtree->type = VALUE;
    newtree->value = value;
    newtree->left = newtree->middle = newtree->right = NULL;
    treestack[ntreestack++] = newtree;
  } else argstack[nargstack++] = value;

  return 1;
}

/* ---------------------------------------------------------------------- */

int Variable::region_function(char *id)
{
  int iregion = domain->find_region(id);
  if (iregion == -1)
    error->all("Region ID in variable formula does not exist");
  return iregion;
}

/* ----------------------------------------------------------------------
   process a special function in formula
   push result onto tree or arg stack
   word = special function
   contents = str between parentheses with one,two,three args
   return 0 if not a match, 1 if successfully processed
   customize by adding a special function:
     sum(x),min(x),max(x),ave(x),trap(x),gmask(x),rmask(x),grmask(x,y)
------------------------------------------------------------------------- */

int Variable::special_function(char *word, char *contents, Tree **tree,
			       Tree **treestack, int &ntreestack,
			       double *argstack, int &nargstack)
{
  // word not a match to any special function

  if (strcmp(word,"sum") && strcmp(word,"min") && strcmp(word,"max") &&
      strcmp(word,"ave") && strcmp(word,"trap") && strcmp(word,"gmask") &&
      strcmp(word,"rmask") && strcmp(word,"grmask"))
    return 0;

  // parse contents for arg1,arg2,arg3 separated by commas
  // ptr1,ptr2 = location of 1st and 2nd comma, NULL if none

  char *arg1,*arg2,*arg3;
  char *ptr1,*ptr2;

  ptr1 = strchr(contents,',');
  if (ptr1) {
    *ptr1 = '\0';
    ptr2 = strchr(ptr1+1,',');
    if (ptr2) *ptr2 = '\0';
  } else ptr2 = NULL;

  int n = strlen(contents) + 1;
  arg1 = new char[n];
  strcpy(arg1,contents);
  int narg = 1;
  if (ptr1) {
    n = strlen(ptr1+1) + 1;
    arg2 = new char[n];
    strcpy(arg2,ptr1+1);
    narg = 2;
  } else arg2 = NULL;
  if (ptr2) {
    n = strlen(ptr2+1) + 1;
    arg3 = new char[n];
    strcpy(arg3,ptr2+1);
    narg = 3;
  } else arg3 = NULL;

  // special functions that operate on global vectors

  if (strcmp(word,"sum") == 0 || strcmp(word,"min") == 0 ||
      strcmp(word,"max") == 0 || strcmp(word,"ave") == 0 ||
      strcmp(word,"trap") == 0) {

    int method;
    if (strcmp(word,"sum") == 0) method = SUM;
    else if (strcmp(word,"min") == 0) method = XMIN;
    else if (strcmp(word,"max") == 0) method = XMAX;
    else if (strcmp(word,"ave") == 0) method = AVE;
    else if (strcmp(word,"trap") == 0) method = TRAP;
    
    if (narg != 1) error->all("Invalid special function in variable formula");
  
    Compute *compute = NULL;
    Fix *fix = NULL;
    int index,nvec,nstride;

    if (strstr(arg1,"c_") == arg1) {
      ptr1 = strchr(arg1,'[');
      if (ptr1) {
	ptr2 = ptr1;
	index = int_between_brackets(ptr2);
	*ptr1 = '\0';
      } else index = 0;

      int icompute = modify->find_compute(&arg1[2]);
      if (icompute < 0) error->all("Invalid compute ID in variable formula");
      compute = modify->compute[icompute];
      if (index == 0 && compute->vector_flag) {
	if (update->whichflag == 0) {
	  if (compute->invoked_vector != update->ntimestep)
	    error->all("Compute used in variable between runs is not current");
	} else if (!(compute->invoked_flag & INVOKED_VECTOR)) {
	  compute->compute_vector();
	  compute->invoked_flag |= INVOKED_VECTOR;
	}
	nvec = compute->size_vector;
	nstride = 1;
      } else if (index && compute->array_flag) {
	if (index > compute->size_array_cols)
	  error->all("Variable formula compute array "
		     "is accessed out-of-range");
	if (update->whichflag == 0) {
	  if (compute->invoked_array != update->ntimestep)
	    error->all("Compute used in variable between runs is not current");
	} else if (!(compute->invoked_flag & INVOKED_ARRAY)) {
	  compute->compute_array();
	  compute->invoked_flag |= INVOKED_ARRAY;
	}
	nvec = compute->size_array_rows;
	nstride = compute->size_array_cols;
      } else error->all("Mismatched compute in variable formula");
      
    } else if (strstr(arg1,"f_") == arg1) {
      ptr1 = strchr(arg1,'[');
      if (ptr1) {
	ptr2 = ptr1;
	index = int_between_brackets(ptr2);
	*ptr1 = '\0';
      } else index = 0;
      
      int ifix = modify->find_fix(&arg1[2]);
      if (ifix < 0) error->all("Invalid fix ID in variable formula");
      fix = modify->fix[ifix];
      if (index == 0 && fix->vector_flag) {
	if (update->whichflag > 0 && update->ntimestep % fix->global_freq)
	  error->all("Fix in variable not computed at compatible time");
	nvec = fix->size_vector;
	nstride = 1;
      } else if (index && fix->array_flag) {
	if (index > fix->size_array_cols)
	  error->all("Variable formula fix array is accessed out-of-range");
	if (update->whichflag > 0 && update->ntimestep % fix->global_freq)
	  error->all("Fix in variable not computed at compatible time");
	nvec = fix->size_array_rows;
	nstride = fix->size_array_cols;
      } else error->all("Mismatched fix in variable formula");
      
    } else error->all("Invalid special function in variable formula");
    
    double value = 0.0;
    if (method == XMIN) value = BIG;
    if (method == XMAX) value = -BIG;
    
    if (compute) {
      double *vec;
      if (index) vec = &compute->array[0][index-1];
      else vec = compute->vector; 
      
      int j = 0;
      for (int i = 0; i < nvec; i++) {
	if (method == SUM) value += vec[j];
	else if (method == XMIN) value = MIN(value,vec[j]);
	else if (method == XMAX) value = MAX(value,vec[j]);
	else if (method == AVE) value += vec[j];
	else if (method == TRAP) {
	  if (i > 0 && i < nvec-1) value += vec[j];
	  else value += 0.5*vec[j];
	}
	j += nstride;
      }
    }
    
    if (fix) {
      double one;
      for (int i = 0; i < nvec; i++) {
	if (index) one = fix->compute_array(i,index-1);
	else one = fix->compute_vector(i);
	if (method == SUM) value += one;
	else if (method == XMIN) value = MIN(value,one);
	else if (method == XMAX) value = MAX(value,one);
	else if (method == AVE) value += one;
	else if (method == TRAP) {
	  if (i > 1 && i < nvec) value += one;
	  else value += 0.5*one;
	}
      }
    }
    
    if (method == AVE) value /= nvec;
    
    delete [] arg1;
    delete [] arg2;
    delete [] arg3;
    
    // save value in tree or on argstack
    
    if (tree) {
      Tree *newtree = new Tree();
      newtree->type = VALUE;
      newtree->value = value;
      newtree->left = newtree->middle = newtree->right = NULL;
      treestack[ntreestack++] = newtree;
    } else argstack[nargstack++] = value;

  // mask special functions

  } else if (strcmp(word,"gmask") == 0) {
    if (tree == NULL)
      error->all("Gmask function in equal-style variable formula");
    if (narg != 1) error->all("Invalid special function in variable formula");

    int igroup = group->find(arg1);
    if (igroup == -1)
      error->all("Group ID in variable formula does not exist");
    
    Tree *newtree = new Tree();
    newtree->type = GMASK;
    newtree->ivalue1 = group->bitmask[igroup];
    newtree->left = newtree->middle = newtree->right = NULL;
    treestack[ntreestack++] = newtree;

  } else if (strcmp(word,"rmask") == 0) {
    if (tree == NULL)
      error->all("Rmask function in equal-style variable formula");
    if (narg != 1) error->all("Invalid special function in variable formula");

    int iregion = region_function(arg1);
    
    Tree *newtree = new Tree();
    newtree->type = RMASK;
    newtree->ivalue1 = iregion;
    newtree->left = newtree->middle = newtree->right = NULL;
    treestack[ntreestack++] = newtree;

  } else if (strcmp(word,"grmask") == 0) {
    if (tree == NULL)
      error->all("Grmask function in equal-style variable formula");
    if (narg != 2) error->all("Invalid special function in variable formula");

    int igroup = group->find(arg1);
    if (igroup == -1)
      error->all("Group ID in variable formula does not exist");
    int iregion = region_function(arg2);
    
    Tree *newtree = new Tree();
    newtree->type = RMASK;
    newtree->ivalue1 = group->bitmask[igroup];
    newtree->ivalue2 = iregion;
    newtree->left = newtree->middle = newtree->right = NULL;
    treestack[ntreestack++] = newtree;
  }

  return 1;
}

/* ----------------------------------------------------------------------
   extract a global value from a per-atom quantity in a formula
   flag = 0 -> word is an atom vector
   flag = 1 -> vector is a per-atom compute or fix quantity with nstride
   id = positive global ID of atom, converted to local index
   push result onto tree or arg stack
   customize by adding an atom vector:
     mass,type,x,y,z,vx,vy,vz,fx,fy,fz
------------------------------------------------------------------------- */

void Variable::peratom2global(int flag, char *word,
			      double *vector, int nstride, int id,
			      Tree **tree, Tree **treestack, int &ntreestack,
			      double *argstack, int &nargstack)
{
  if (atom->map_style == 0)
    error->all("Indexed per-atom vector in variable formula without atom map");

  int index = atom->map(id);

  double mine;
  if (index >= 0 && index < atom->nlocal) {

    if (flag == 0) {
      if (strcmp(word,"mass") == 0) {
	if (atom->rmass) mine = atom->rmass[index];
	else mine = atom->mass[atom->type[index]];
      }
      else if (strcmp(word,"type") == 0) mine = atom->type[index];
      else if (strcmp(word,"x") == 0) mine = atom->x[index][0];
      else if (strcmp(word,"y") == 0) mine = atom->x[index][1];
      else if (strcmp(word,"z") == 0) mine = atom->x[index][2];
      else if (strcmp(word,"vx") == 0) mine = atom->v[index][0];
      else if (strcmp(word,"vy") == 0) mine = atom->v[index][1];
      else if (strcmp(word,"vz") == 0) mine = atom->v[index][2];
      else if (strcmp(word,"fx") == 0) mine = atom->f[index][0];
      else if (strcmp(word,"fy") == 0) mine = atom->f[index][1];
      else if (strcmp(word,"fz") == 0) mine = atom->f[index][2];
      
      else error->one("Invalid atom vector in variable formula");

    } else mine = vector[index*nstride];
    
  } else mine = 0.0;

  double value;
  MPI_Allreduce(&mine,&value,1,MPI_DOUBLE,MPI_SUM,world);
  
  if (tree) {
    Tree *newtree = new Tree();
    newtree->type = VALUE;
    newtree->value = value;
    newtree->left = newtree->middle = newtree->right = NULL;
    treestack[ntreestack++] = newtree;
  } else argstack[nargstack++] = value;
}

/* ----------------------------------------------------------------------
   check if word matches an atom vector
   return 1 if yes, else 0
   customize by adding an atom vector:
     mass,type,x,y,z,vx,vy,vz,fx,fy,fz
------------------------------------------------------------------------- */

int Variable::is_atom_vector(char *word)
{
  if (strcmp(word,"mass") == 0) return 1;
  if (strcmp(word,"type") == 0) return 1;
  if (strcmp(word,"x") == 0) return 1;
  if (strcmp(word,"y") == 0) return 1;
  if (strcmp(word,"z") == 0) return 1;
  if (strcmp(word,"vx") == 0) return 1;
  if (strcmp(word,"vy") == 0) return 1;
  if (strcmp(word,"vz") == 0) return 1;
  if (strcmp(word,"fx") == 0) return 1;
  if (strcmp(word,"fy") == 0) return 1;
  if (strcmp(word,"fz") == 0) return 1;
  return 0;
}

/* ----------------------------------------------------------------------
   process an atom vector in formula
   push result onto tree
   word = atom vector
   customize by adding an atom vector:
     mass,type,x,y,z,vx,vy,vz,fx,fy,fz
------------------------------------------------------------------------- */

void Variable::atom_vector(char *word, Tree **tree,
			   Tree **treestack, int &ntreestack)
{
  if (tree == NULL)
    error->all("Atom vector in equal-style variable formula");

  Tree *newtree = new Tree();
  newtree->type = ATOMARRAY;
  newtree->nstride = 3;
  newtree->left = newtree->middle = newtree->right = NULL;
  treestack[ntreestack++] = newtree;
	    
  if (strcmp(word,"mass") == 0) {
    if (atom->rmass) {
      newtree->nstride = 1;
      newtree->array = atom->rmass;
    } else {
      newtree->type = TYPEARRAY;
      newtree->array = atom->mass;
    }
  } else if (strcmp(word,"type") == 0) {
    newtree->type = INTARRAY;
    newtree->nstride = 1;
    newtree->iarray = atom->type;
  }
  else if (strcmp(word,"x") == 0) newtree->array = &atom->x[0][0];
  else if (strcmp(word,"y") == 0) newtree->array = &atom->x[0][1];
  else if (strcmp(word,"z") == 0) newtree->array = &atom->x[0][2];
  else if (strcmp(word,"vx") == 0) newtree->array = &atom->v[0][0];
  else if (strcmp(word,"vy") == 0) newtree->array = &atom->v[0][1];
  else if (strcmp(word,"vz") == 0) newtree->array = &atom->v[0][2];
  else if (strcmp(word,"fx") == 0) newtree->array = &atom->f[0][0];
  else if (strcmp(word,"fy") == 0) newtree->array = &atom->f[0][1];
  else if (strcmp(word,"fz") == 0) newtree->array = &atom->f[0][2];
}

/* ----------------------------------------------------------------------
   check if word matches a constant
   return 1 if yes, else 0
   customize by adding a constant: PI
------------------------------------------------------------------------- */

int Variable::is_constant(char *word)
{
  if (strcmp(word,"PI") == 0) return 1;
  return 0;
}

/* ----------------------------------------------------------------------
   process a constant in formula
   customize by adding a constant: PI
------------------------------------------------------------------------- */

double Variable::constant(char *word)
{
  if (strcmp(word,"PI") == 0) return PI;
  return 0.0;
}

/* ----------------------------------------------------------------------
   read a floating point value from a string
   generate an error if not a legitimate floating point value
------------------------------------------------------------------------- */

double Variable::numeric(char *str)
{
  int n = strlen(str);
  for (int i = 0; i < n; i++) {
    if (isdigit(str[i])) continue;
    if (str[i] == '-' || str[i] == '+' || str[i] == '.') continue;
    if (str[i] == 'e' || str[i] == 'E') continue;
    error->all("Expected floating point parameter in variable definition");
  }

  return atof(str);
}

/* ----------------------------------------------------------------------
   read an integer value from a string
   generate an error if not a legitimate integer value
------------------------------------------------------------------------- */

int Variable::inumeric(char *str)
{
  int n = strlen(str);
  for (int i = 0; i < n; i++) {
    if (isdigit(str[i]) || str[i] == '-' || str[i] == '+') continue;
    error->all("Expected integer parameter in variable definition");
  }

  return atoi(str);
}

/* ----------------------------------------------------------------------
   debug routine for printing formula tree recursively
------------------------------------------------------------------------- */

void Variable::print_tree(Tree *tree, int level)
{
  printf("TREE %d: %d %g\n",level,tree->type,tree->value);
  if (tree->left) print_tree(tree->left,level+1);
  if (tree->middle) print_tree(tree->middle,level+1);
  if (tree->right) print_tree(tree->right,level+1);
  return;
}

/* ----------------------------------------------------------------------
   recursive evaluation of string str
   called from "if" command in input script
   str is a boolean expression containing one or more items:
     number = 0.0, -5.45, 2.8e-4, ...
     math operation = (),x==y,x!=y,x<y,x<=y,x>y,x>=y,x&&y,x||y
------------------------------------------------------------------------- */

double Variable::evaluate_boolean(char *str)
{
  int op,opprevious;
  double value1,value2;
  char onechar;
  char *ptr;

  double argstack[MAXLEVEL];
  int opstack[MAXLEVEL];
  int nargstack = 0;
  int nopstack = 0;

  int i = 0;
  int expect = ARG;

  while (1) {
    onechar = str[i];
    
    // whitespace: just skip
    
    if (isspace(onechar)) i++;
    
    // ----------------
    // parentheses: recursively evaluate contents of parens
    // ----------------
    
    else if (onechar == '(') {
      if (expect == OP) error->all("Invalid Boolean syntax in if command");
      expect = OP;
      
      char *contents;
      i = find_matching_paren(str,i,contents);
      i++;
      
      // evaluate contents and push on stack
      
      argstack[nargstack++] = evaluate_boolean(contents);
      
      delete [] contents;
      
    // ----------------
    // number: push value onto stack
    // ----------------
      
    } else if (isdigit(onechar) || onechar == '.' || onechar == '-') {
      if (expect == OP) error->all("Invalid Boolean syntax in if command");
      expect = OP;
      
      // istop = end of number, including scientific notation
      
      int istart = i++;
      while (isdigit(str[i]) || str[i] == '.') i++;
      if (str[i] == 'e' || str[i] == 'E') {
	i++;
	if (str[i] == '+' || str[i] == '-') i++;
	while (isdigit(str[i])) i++;
      }
      int istop = i - 1;
      
      int n = istop - istart + 1;
      char *number = new char[n+1];
      strncpy(number,&str[istart],n);
      number[n] = '\0';
      
      argstack[nargstack++] = atof(number);
      
      delete [] number;
      
    // ----------------
    // Boolean operator, including end-of-string
    // ----------------
      
    } else if (strchr("<>=!&|\0",onechar)) {
      if (onechar == '=') {
	if (str[i+1] != '=') 
	  error->all("Invalid Boolean syntax in if command");
	op = EQ;
	i++;
      } else if (onechar == '!') {
	if (str[i+1] == '=') {
	  op = NE;
	  i++;
	} else op = NOT;
      } else if (onechar == '<') {
	if (str[i+1] != '=') op = LT;
	else {
	  op = LE;
	  i++;
	}
      } else if (onechar == '>') {
	if (str[i+1] != '=') op = GT;
	else {
	  op = GE;
	  i++;
	}
      } else if (onechar == '&') {
	if (str[i+1] != '&') 
	  error->all("Invalid Boolean syntax in if command");
	op = AND;
	i++;
      } else if (onechar == '|') {
	if (str[i+1] != '|') 
	  error->all("Invalid Boolean syntax in if command");
	op = OR;
	i++;
      } else op = DONE;
      
      i++;
      
      if (op == NOT && expect == ARG) {
	opstack[nopstack++] = op;
	continue;
      }

      if (expect == ARG) error->all("Invalid Boolean syntax in if command");
      expect = ARG;
      
      // evaluate stack as deep as possible while respecting precedence
      // before pushing current op onto stack

      while (nopstack && precedence[opstack[nopstack-1]] >= precedence[op]) {
	opprevious = opstack[--nopstack];
	
	value2 = argstack[--nargstack];
	if (opprevious != NOT) value1 = argstack[--nargstack];
	
	if (opprevious == NOT) {
	  if (value2 == 0.0) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	} else if (opprevious == EQ) {
	  if (value1 == value2) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	} else if (opprevious == NE) {
	  if (value1 != value2) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	} else if (opprevious == LT) {
	  if (value1 < value2) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	} else if (opprevious == LE) {
	  if (value1 <= value2) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	} else if (opprevious == GT) {
	  if (value1 > value2) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	} else if (opprevious == GE) {
	  if (value1 >= value2) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	} else if (opprevious == AND) {
	  if (value1 != 0.0 && value2 != 0.0) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	} else if (opprevious == OR) {
	  if (value1 != 0.0 || value2 != 0.0) argstack[nargstack++] = 1.0;
	  else argstack[nargstack++] = 0.0;
	}
      }
      
      // if end-of-string, break out of entire formula evaluation loop
      
      if (op == DONE) break;
      
      // push current operation onto stack
      
      opstack[nopstack++] = op;

    } else error->all("Invalid Boolean syntax in if command");
  }

  if (nopstack) error->all("Invalid Boolean syntax in if command");
  if (nargstack != 1) error->all("Invalid Boolean syntax in if command");
  return argstack[0];
}