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/*
* Mathomatic differentiation routines and commands.
*
* Copyright (C) 1987-2006 George Gesslein II.
*/
#include "includes.h"
static int d_recurse();
/*
* Compute the derivative of an equation side, with respect to variable "v",
* using the fast, rule based transform method.
* The result must be simplified by the caller.
*
* Return true if successful.
*/
int
differentiate(equation, np, v)
token_type *equation; /* pointer to source and destination equation side */
int *np; /* pointer to the length of the equation side */
long v; /* differentiation variable */
{
int i;
organize(equation, np);
/* First put every times and divide on a level by itself. */
for (i = 1; i < *np; i += 2) {
switch (equation[i].token.operatr) {
case TIMES:
case DIVIDE:
binary_parenthesize(equation, *np, i);
}
}
return d_recurse(equation, np, 0, 1, v);
}
/*
* Recursive differentiation routine.
*
* Symbolically differentiate expression in "equation"
* (which is a standard equation side) starting at "loc".
* The current level of parentheses is "level" and
* do the differentation with respect to variable "v".
*
* Return "true" if successful, "false" if it is beyond this program's
* capabilities or an error was encountered.
*/
static int
d_recurse(equation, np, loc, level, v)
token_type *equation;
int *np, loc, level;
long v;
{
int i, j;
int n;
int op;
int oploc, endloc;
complexs c;
if (equation[loc].level < level) {
/* First differentiate if it is a single variable or constant. */
/* If it is the specified variable, change it to the constant "1.0", */
/* otherwise change it to "0.0". */
if (equation[loc].kind == VARIABLE
&& ((v == MATCH_ANY && (equation[loc].token.variable & VAR_MASK) > SIGN)
|| equation[loc].token.variable == v)) {
equation[loc].kind = CONSTANT;
equation[loc].token.constant = 1.0;
} else {
equation[loc].kind = CONSTANT;
equation[loc].token.constant = 0.0;
}
return true;
}
op = 0;
for (oploc = endloc = loc + 1; endloc < *np && equation[endloc].level >= level; endloc += 2) {
if (equation[endloc].level == level) {
switch (op) {
case 0:
case PLUS:
case MINUS:
break;
default:
/* Oops. More than one operator on the same level in this expression. */
error("Error in d_recurse().");
return false;
}
op = equation[endloc].token.operatr;
oploc = endloc;
}
}
switch (op) {
case 0:
case PLUS:
case MINUS:
break;
case TIMES:
goto d_times;
case DIVIDE:
goto d_divide;
case POWER:
goto d_power;
default:
/* Differentiate an unsupported operator. */
/* This is possible if the expression doesn't contain the specified variable. */
/* In that case, the expression is replaced with "0", otherwise return false. */
for (i = loc; i < endloc; i += 2) {
if (equation[i].kind == VARIABLE
&& ((v == MATCH_ANY && (equation[i].token.variable & VAR_MASK) > SIGN)
|| equation[i].token.variable == v)) {
return false;
}
}
blt(&equation[loc+1], &equation[endloc], (*np - endloc) * sizeof(token_type));
*np -= (endloc - (loc + 1));
equation[loc].level = level;
equation[loc].kind = CONSTANT;
equation[loc].token.constant = 0.0;
return true;
}
/* Differentiate PLUS and MINUS operators. */
/* Use rule: d(u+v) = d(u) + d(v), */
/* where "d()" is the derivative function, */
/* "u" and "v" are expressions. */
for (i = loc; i < *np && equation[i].level >= level;) {
if (equation[i].kind != OPERATOR) {
if (!d_recurse(equation, np, i, level + 1, v))
return false;
i++;
for (; i < *np && equation[i].level > level; i += 2)
;
continue;
}
i++;
}
return true;
d_times:
/* Differentiate TIMES operator. */
/* Use rule: d(u*v) = u*d(v) + v*d(u). */
if (*np + 1 + (endloc - loc) > n_tokens) {
error_huge();
}
for (i = loc; i < endloc; i++)
equation[i].level++;
blt(&equation[endloc+1], &equation[loc], (*np - loc) * sizeof(token_type));
*np += 1 + (endloc - loc);
equation[endloc].level = level;
equation[endloc].kind = OPERATOR;
equation[endloc].token.operatr = PLUS;
if (!d_recurse(equation, np, endloc + (oploc - loc) + 2, level + 2, v))
return false;
return(d_recurse(equation, np, loc, level + 2, v));
d_divide:
/* Differentiate DIVIDE operator. */
/* Use rule: d(u/v) = (v*d(u) - u*d(v))/v^2. */
if (*np + 3 + (endloc - loc) + (endloc - oploc) > n_tokens) {
error_huge();
}
for (i = loc; i < endloc; i++)
equation[i].level += 2;
equation[oploc].token.operatr = TIMES;
j = 1 + (endloc - loc);
blt(&equation[endloc+1], &equation[loc], (*np - loc) * sizeof(token_type));
*np += j;
equation[endloc].level = level + 1;
equation[endloc].kind = OPERATOR;
equation[endloc].token.operatr = MINUS;
j += endloc;
blt(&equation[j+2+(endloc-oploc)], &equation[j], (*np - j) * sizeof(token_type));
*np += 2 + (endloc - oploc);
equation[j].level = level;
equation[j].kind = OPERATOR;
equation[j].token.operatr = DIVIDE;
blt(&equation[j+1], &equation[oploc+1], (endloc - (oploc + 1)) * sizeof(token_type));
j += endloc - oploc;
equation[j].level = level + 1;
equation[j].kind = OPERATOR;
equation[j].token.operatr = POWER;
j++;
equation[j].level = level + 1;
equation[j].kind = CONSTANT;
equation[j].token.constant = 2.0;
if (!d_recurse(equation, np, endloc + (oploc - loc) + 2, level + 3, v))
return false;
return(d_recurse(equation, np, loc, level + 3, v));
d_power:
/* Differentiate POWER operator. */
/* Since we don't have logarithms, just do what we can without them. */
for (i = oploc; i < endloc; i++) {
if (equation[i].kind == VARIABLE
&& ((v == MATCH_ANY && (equation[i].token.variable & VAR_MASK) > SIGN)
|| equation[i].token.variable == v)) {
if (parse_complex(&equation[loc], oploc - loc, &c)) {
c = complex_log(c);
n = (endloc - oploc) + 6;
if (*np + n > n_tokens) {
error_huge();
}
blt(&equation[endloc+n], &equation[endloc], (*np - endloc) * sizeof(token_type));
*np += n;
n = endloc;
equation[n].level = level;
equation[n].kind = OPERATOR;
equation[n].token.operatr = TIMES;
n++;
equation[n].level = level + 1;
equation[n].kind = CONSTANT;
equation[n].token.constant = c.re;
n++;
equation[n].level = level + 1;
equation[n].kind = OPERATOR;
equation[n].token.operatr = PLUS;
n++;
equation[n].level = level + 2;
equation[n].kind = CONSTANT;
equation[n].token.constant = c.im;
n++;
equation[n].level = level + 2;
equation[n].kind = OPERATOR;
equation[n].token.operatr = TIMES;
n++;
equation[n].level = level + 2;
equation[n].kind = VARIABLE;
equation[n].token.variable = IMAGINARY;
n++;
equation[n].level = level;
equation[n].kind = OPERATOR;
equation[n].token.operatr = TIMES;
n++;
blt(&equation[n], &equation[oploc+1], (endloc - (oploc + 1)) * sizeof(token_type));
for (i = loc; i < endloc; i++) {
equation[i].level++;
}
return(d_recurse(equation, np, n, level + 1, v));
}
return false;
}
}
blt(scratch, &equation[oploc+1], (endloc - (oploc + 1)) * sizeof(token_type));
n = endloc - (oploc + 1);
scratch[n].level = level;
scratch[n].kind = OPERATOR;
scratch[n].token.operatr = TIMES;
n++;
if (n + (endloc - loc) + 2 > n_tokens) {
error_huge();
}
blt(&scratch[n], &equation[loc], (endloc - loc) * sizeof(token_type));
i = n;
n += oploc + 1 - loc;
for (; i < n; i++)
scratch[i].level++;
n += endloc - (oploc + 1);
for (; i < n; i++)
scratch[i].level += 2;
scratch[n].level = level + 2;
scratch[n].kind = OPERATOR;
scratch[n].token.operatr = MINUS;
n++;
scratch[n].level = level + 2;
scratch[n].kind = CONSTANT;
scratch[n].token.constant = 1.0;
n++;
if (n + (oploc - loc) + 1 > n_tokens) {
error_huge();
}
scratch[n].level = level;
scratch[n].kind = OPERATOR;
scratch[n].token.operatr = TIMES;
n++;
j = n;
blt(&scratch[n], &equation[loc], (oploc - loc) * sizeof(token_type));
n += oploc - loc;
if (*np - (endloc - loc) + n > n_tokens) {
error_huge();
}
blt(&equation[loc+n], &equation[endloc], (*np - endloc) * sizeof(token_type));
*np += loc + n - endloc;
blt(&equation[loc], scratch, n * sizeof(token_type));
return(d_recurse(equation, np, loc + j, level + 1, v));
}
/*
* The derivative command.
*/
int
derivative_cmd(cp)
char *cp;
{
long v = 0;
int i;
long l1, order = 1;
token_type *source, *dest;
int n1, *nps, *np;
if (current_not_defined()) {
return false;
}
i = next_espace();
if (n_rhs[cur_equation]) {
source = rhs[cur_equation];
nps = &n_rhs[cur_equation];
dest = rhs[i];
np = &n_rhs[i];
} else {
source = lhs[cur_equation];
nps = &n_lhs[cur_equation];
dest = lhs[i];
np = &n_lhs[i];
}
if (*cp) {
if (is_all(cp)) {
cp = skip_param(cp);
v = MATCH_ANY;
} else {
if (!isdigit(*cp)) {
cp = parse_var2(&v, cp);
if (cp == NULL) {
return false;
}
}
}
if (*cp) {
order = decstrtol(cp, &cp);
}
if (*cp || order <= 0) {
error(_("The order must be a positive integer."));
return false;
}
}
if (no_vars(source, *nps, &v)) {
error(_("Current expression contains no variables, result would be zero."));
return false;
}
if (v == 0) {
if (!prompt_var(&v)) {
return false;
}
}
if (v != MATCH_ANY && !found_var(source, *nps, v)) {
error(_("Variable not found, result would be zero."));
return false;
}
#if !SILENT
list_var(v, 0);
printf(_("Differentiating with respect to (%s) and simplifying"), var_str);
if (n_rhs[cur_equation]) {
printf(_(" the RHS...\n"));
} else {
printf("...\n");
}
#endif
blt(dest, source, *nps * sizeof(token_type));
n1 = *nps;
for (l1 = 0; l1 < order; l1++) {
if (!differentiate(dest, &n1, v)) {
error(_("Differentiation failed."));
return false;
}
simpa_side(dest, &n1, true);
}
if (n_rhs[cur_equation]) {
blt(lhs[i], lhs[cur_equation], n_lhs[cur_equation] * sizeof(token_type));
n_lhs[i] = n_lhs[cur_equation];
}
*np = n1;
cur_equation = i;
return return_result(cur_equation);
}
/*
* The extrema command.
*/
int
extrema_cmd(cp)
char *cp;
{
int i;
long v = 0;
token_type want;
token_type *source;
int n;
if (current_not_defined()) {
return false;
}
if (n_rhs[cur_equation]) {
if (!solved_equation(cur_equation)) {
error(_("The current equation is not solved for a variable."));
return false;
}
source = rhs[cur_equation];
n = n_rhs[cur_equation];
} else {
source = lhs[cur_equation];
n = n_lhs[cur_equation];
}
if (*cp) {
cp = parse_var2(&v, cp);
if (cp == NULL) {
return false;
}
if (extra_garbage(cp)) {
return false;
}
}
if (no_vars(source, n, &v)) {
error(_("Current expression contains no variables, there are no extrema."));
return false;
}
if (v == 0) {
if (!prompt_var(&v)) {
return false;
}
}
if (!found_var(source, n, v)) {
error(_("Variable not found."));
return false;
}
i = next_espace();
blt(rhs[i], source, n * sizeof(token_type));
/* take the derivative with respect to the specified variable and simplify: */
if (!differentiate(rhs[i], &n, v)) {
error(_("Differentiation failed."));
return false;
}
simpa_side(rhs[i], &n, true);
if (!found_var(rhs[i], n, v)) {
error(_("There are no extrema for this equation."));
return false;
}
n_rhs[i] = n;
/* set equal to zero: */
n_lhs[i] = 1;
lhs[i][0] = zero_token;
cur_equation = i;
/* lastly, solve for the specified variable and simplify: */
want.level = 1;
want.kind = VARIABLE;
want.token.variable = v;
if (solve_sub(&want, 1, lhs[i], &n_lhs[i], rhs[i], &n_rhs[i]) <= 0) {
error(_("Solve failed."));
return false;
}
simpa_side(rhs[i], &n_rhs[i], false);
return return_result(cur_equation);
}
/*
* The taylor command.
*/
int
taylor_cmd(cp)
char *cp;
{
long v = 0;
int i, j, k, i1;
int level;
long l1, n, order = -1L;
double d;
char *cp_start, *cp1, buf[MAX_CMD_LEN];
int our;
int our_nlhs, our_nrhs;
token_type *ep, *source, *dest;
int n1, *nps, *np;
cp_start = cp;
if (current_not_defined()) {
return false;
}
i = next_espace();
blt(lhs[i], lhs[cur_equation], n_lhs[cur_equation] * sizeof(token_type));
n_lhs[i] = n_lhs[cur_equation];
n_rhs[i] = 0;
our = alloc_next_espace();
n_lhs[i] = 0;
if (our < 0) {
error(_("Out of free equation spaces."));
return false;
}
if (n_rhs[cur_equation]) {
source = rhs[cur_equation];
nps = &n_rhs[cur_equation];
dest = rhs[i];
np = &n_rhs[i];
} else {
source = lhs[cur_equation];
nps = &n_lhs[cur_equation];
dest = lhs[i];
np = &n_lhs[i];
}
if (*cp && !isdigit(*cp)) {
cp = parse_var2(&v, cp);
if (cp == NULL) {
return false;
}
}
if (*cp) {
order = decstrtol(cp, &cp1);
if (cp1 != skip_param(cp) || order < 0) {
error(_("Positive integer required for order."));
return false;
}
cp = cp1;
}
if (no_vars(source, *nps, &v)) {
error(_("Current expression contains no variables."));
return false;
}
if (v == 0) {
if (!prompt_var(&v)) {
return false;
}
}
if (!found_var(source, *nps, v)) {
error(_("Variable not found."));
return false;
}
blt(rhs[our], source, *nps * sizeof(token_type));
our_nrhs = *nps;
if (!differentiate(rhs[our], &our_nrhs, v)) {
error(_("Differentiation failed."));
return false;
}
if (*cp) {
input_column += (cp - cp_start);
if (!case_sensitive_flag) {
str_tolower(cp);
}
if ((cp = parse_section(lhs[our], &our_nlhs, cp)) == NULL || our_nlhs <= 0) {
return false;
}
if (extra_garbage(cp))
return false;
} else {
#if !SILENT
list_var(v, 0);
printf(_("Taylor series expansion about (%s) = point.\n"), var_str);
#endif
for (;;) {
my_strlcpy(prompt_str, _("Enter point: "), sizeof(prompt_str));
if (!get_expr(lhs[our], &our_nlhs)) {
error(_("Enter \"0\" if you don't know."));
continue;
}
break;
}
}
if (order < 0) {
my_strlcpy(prompt_str, _("Enter order (number of derivatives to take): "), sizeof(prompt_str));
if ((cp1 = get_string(buf, sizeof(buf))) == NULL)
return false;
if (*cp1) {
order = decstrtol(cp1, &cp);
if (*cp || order < 0) {
error(_("Positive integer required for order."));
return false;
}
} else {
printf(_("Derivatives will be taken until they reach zero...\n"));
order = LONG_MAX - 1L;
}
}
n = 0;
i1 = 0;
blt(dest, source, *nps * sizeof(token_type));
n1 = *nps;
loop_again:
for (k = i1; k < n1; k += 2) {
if (dest[k].kind == VARIABLE && dest[k].token.variable == v) {
level = dest[k].level;
if ((n1 + our_nlhs - 1) > n_tokens)
error_huge();
blt(&dest[k+our_nlhs], &dest[k+1], (n1 - (k + 1)) * sizeof(token_type));
n1 += our_nlhs - 1;
j = k;
blt(&dest[k], lhs[our], our_nlhs * sizeof(token_type));
k += our_nlhs;
for (; j < k; j++)
dest[j].level += level;
k--;
}
}
if ((n1 + our_nlhs + 7) > n_tokens)
error_huge();
for (k = i1; k < n1; k++)
dest[k].level++;
ep = &dest[n1];
ep->level = 1;
ep->kind = OPERATOR;
ep->token.operatr = TIMES;
ep++;
ep->level = 3;
ep->kind = VARIABLE;
ep->token.variable = v;
ep++;
ep->level = 3;
ep->kind = OPERATOR;
ep->token.operatr = MINUS;
n1 += 3;
j = n1;
blt(&dest[n1], lhs[our], our_nlhs * sizeof(token_type));
n1 += our_nlhs;
for (; j < n1; j++)
dest[j].level += 3;
ep = &dest[n1];
ep->level = 2;
ep->kind = OPERATOR;
ep->token.operatr = POWER;
ep++;
ep->level = 2;
ep->kind = CONSTANT;
ep->token.constant = n;
ep++;
ep->level = 1;
ep->kind = OPERATOR;
ep->token.operatr = DIVIDE;
ep++;
for (d = 1.0, l1 = 2; l1 <= n; l1++)
d *= l1;
ep->level = 1;
ep->kind = CONSTANT;
ep->token.constant = d;
n1 += 4;
for (; i1 < n1; i1++)
dest[i1].level++;
simp_loop(dest, &n1);
/* side_debug(1, dest, n1); */
if (n < order) {
if (n > 0) {
if (!differentiate(rhs[our], &our_nrhs, v)) {
error(_("Differentiation failed."));
return false;
}
}
simpa_side(rhs[our], &our_nrhs, true);
if (our_nrhs != 1 || rhs[our][0].kind != CONSTANT || rhs[our][0].token.constant != 0.0) {
i1 = n1;
if ((i1 + 1 + our_nrhs) > n_tokens)
error_huge();
for (j = 0; j < i1; j++)
dest[j].level++;
dest[i1].level = 1;
dest[i1].kind = OPERATOR;
dest[i1].token.operatr = PLUS;
i1++;
blt(&dest[i1], rhs[our], our_nrhs * sizeof(token_type));
n1 = i1 + our_nrhs;
n++;
goto loop_again;
}
}
if (n_rhs[cur_equation]) {
n_lhs[i] = n_lhs[cur_equation];
}
*np = n1;
cur_equation = i;
return return_result(cur_equation);
}
/*
* The limit command.
*/
int
limit_cmd(cp)
char *cp;
{
int i, j, k;
long v = 0;
token_type solved_v, want;
char *cp_start;
int nl;
int infinity_flag, found_den, complex_den, num, den;
int level;
cp_start = cp;
if (current_not_defined()) {
return false;
}
if (n_rhs[cur_equation] == 0) {
/* make expression into equation: */
blt(rhs[cur_equation], lhs[cur_equation], n_lhs[cur_equation] * sizeof(token_type));
n_rhs[cur_equation] = n_lhs[cur_equation];
n_lhs[cur_equation] = 1;
lhs[cur_equation][0].level = 1;
lhs[cur_equation][0].kind = VARIABLE;
lhs[cur_equation][0].token.variable = V_ANSWER;
}
if (!solved_equation(cur_equation)) {
error(_("The current equation is not solved for a variable."));
return false;
}
solved_v = lhs[cur_equation][0];
i = next_espace();
if (*cp) {
cp = parse_var2(&v, cp);
if (cp == NULL) {
return false;
}
}
if (no_vars(rhs[cur_equation], n_rhs[cur_equation], &v)) {
error(_("Current expression contains no variables."));
return false;
}
if (v == 0) {
if (!prompt_var(&v)) {
return false;
}
}
if (!found_var(rhs[cur_equation], n_rhs[cur_equation], v)) {
error(_("Variable not found."));
return false;
}
if (*cp) {
input_column += (cp - cp_start);
if (!case_sensitive_flag) {
str_tolower(cp);
}
if ((cp = parse_section(lhs[i], &nl, cp)) == NULL || nl <= 0) {
return false;
}
} else {
list_var(v, 0);
snprintf(prompt_str, sizeof(prompt_str), _("as (%s) goes to: "), var_str);
if (!get_expr(lhs[i], &nl)) {
return false;
}
}
if (extra_garbage(cp))
return false;
simp_loop(lhs[i], &nl);
infinity_flag = exp_contains_infinity(lhs[i], nl);
simpa_side(rhs[cur_equation], &n_rhs[cur_equation], false);
group_proc(rhs[cur_equation], &n_rhs[cur_equation]);
found_den = false;
complex_den = false;
for (j = 1; j < n_rhs[cur_equation]; j += 2) {
switch (rhs[cur_equation][j].token.operatr) {
case DIVIDE:
case MODULUS:
num = false;
den = false;
level = rhs[cur_equation][j].level;
for (k = j + 2;; k += 2) {
if (rhs[cur_equation][k-1].kind == VARIABLE
&& rhs[cur_equation][k-1].token.variable == v) {
den = true;
found_den = true;
}
if (k >= n_rhs[cur_equation] || rhs[cur_equation][k].level <= level) {
break;
}
switch (rhs[cur_equation][k].token.operatr) {
case DIVIDE:
case MODULUS:
complex_den = true;
break;
}
}
for (k = j - 1; k >= 0 && rhs[cur_equation][k].level >= level; k--) {
if (rhs[cur_equation][k].kind == VARIABLE
&& rhs[cur_equation][k].token.variable == v) {
num = true;
}
}
if (num && den) {
complex_den = true;
}
break;
}
}
if (!found_den || (infinity_flag && !complex_den)) {
/* in this case, just substitute the limit variable with the limit expression and simplify: */
subst_var_with_exp(rhs[cur_equation], &n_rhs[cur_equation], lhs[i], nl, v);
simpa_side(rhs[cur_equation], &n_rhs[cur_equation], false);
return return_result(cur_equation);
}
if (infinity_flag) {
/* To take the limit to infinity, */
/* replace infinity with zero and replace the limit variable with its reciprocal: */
nl = 1;
lhs[i][0] = zero_token;
tlhs[0] = one_token;
tlhs[1].level = 1;
tlhs[1].kind = OPERATOR;
tlhs[1].token.operatr = DIVIDE;
tlhs[2].level = 1;
tlhs[2].kind = VARIABLE;
tlhs[2].token.variable = v;
n_tlhs = 3;
subst_var_with_exp(rhs[cur_equation], &n_rhs[cur_equation], tlhs, n_tlhs, v);
}
/* General limit taking, solve for the limit variable: */
want.level = 1;
want.kind = VARIABLE;
want.token.variable = v;
if (solve_sub(&want, 1, lhs[cur_equation], &n_lhs[cur_equation], rhs[cur_equation], &n_rhs[cur_equation]) <= 0) {
error(_("Can't take the limit because solve failed."));
return false;
}
/* replace the limit variable (LHS) with the limit expression: */
blt(lhs[cur_equation], lhs[i], nl * sizeof(token_type));
n_lhs[cur_equation] = nl;
/* symbolically simplify the RHS: */
symb_flag = true;
simpa_side(rhs[cur_equation], &n_rhs[cur_equation], false);
symb_flag = false;
/* solve back for the original variable: */
if (solve_sub(&solved_v, 1, lhs[cur_equation], &n_lhs[cur_equation], rhs[cur_equation], &n_rhs[cur_equation]) <= 0) {
error(_("Can't take the limit because solve failed."));
return false;
}
/* symbolically simplify before returning the result: */
symb_flag = true;
simpa_side(rhs[cur_equation], &n_rhs[cur_equation], false);
symb_flag = false;
return return_result(cur_equation);
}
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