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use crate::{
coefficient::{Coefficient, CoefficientView},
domains::integer::Integer,
representations::{
Add, AsAtomView, Atom, AtomBuilder, AtomSet, AtomView, Fun, Identifier, Mul, Num, OwnedAdd,
OwnedFun, OwnedMul, OwnedNum, OwnedPow, Pow, Var,
},
state::{State, Workspace},
};
impl<'a, P: AtomSet> AtomView<'a, P> {
/// Take a derivative of the expression with respect to `x` and
/// write the result in `out`.
/// Returns `true` if the derivative is non-zero.
pub fn derivative(
&self,
x: Identifier,
workspace: &Workspace<P>,
state: &State,
out: &mut Atom<P>,
) -> bool {
match self {
AtomView::Num(_) => {
let n = out.to_num();
n.set_from_coeff(Coefficient::zero());
false
}
AtomView::Var(v) => {
if v.get_name() == x {
let n = out.to_num();
n.set_from_coeff(1.into());
true
} else {
let n = out.to_num();
n.set_from_coeff(Coefficient::zero());
false
}
}
AtomView::Fun(f_orig) => {
// detect if the function to derive is the derivative function itself
// if so, derive the last argument of the derivative function and set
// a flag to later accumulate previous derivatives
let (to_derive, f, is_der) = if f_orig.get_name() == State::DERIVATIVE {
let to_derive = f_orig.iter().last().unwrap();
(
to_derive,
match to_derive {
AtomView::Fun(f) => f,
_ => panic!("Last argument of der function must be a function"),
},
true,
)
} else {
(*self, *f_orig, false)
};
// take derivative of all the arguments and store it in a list
let mut args_der = Vec::with_capacity(f.get_nargs());
for (i, arg) in f.iter().enumerate() {
let mut arg_der = workspace.new_atom();
if arg.derivative(x, workspace, state, &mut arg_der) {
args_der.push((i, arg_der));
}
}
if args_der.is_empty() {
let n = out.to_num();
n.set_from_coeff(Coefficient::zero());
return false;
}
// derive special functions
if f.get_nargs() == 1
&& [State::EXP, State::LOG, State::SIN, State::COS].contains(&f.get_name())
{
let mut fn_der = workspace.new_atom();
match f.get_name() {
State::EXP => {
fn_der.set_from_view(self);
}
State::LOG => {
let mut n = workspace.new_atom();
n.to_num().set_from_coeff((-1).into());
let p = fn_der.to_pow();
p.set_from_base_and_exp(f.iter().next().unwrap(), n.as_view());
p.set_dirty(true);
}
State::SIN => {
let p = fn_der.to_fun();
p.set_from_name(State::COS);
p.add_arg(f.iter().next().unwrap());
p.set_dirty(true);
}
State::COS => {
let mut n = workspace.new_atom();
n.to_num().set_from_coeff((-1).into());
let mut sin = workspace.new_atom();
let sin_fun = sin.to_fun();
sin_fun.set_from_name(State::SIN);
sin_fun.add_arg(f.iter().next().unwrap());
let m = fn_der.to_mul();
m.extend(sin.as_view());
m.extend(n.as_view());
m.set_dirty(true);
}
_ => unreachable!(),
}
let (_, mut arg_der) = args_der.pop().unwrap();
if let Atom::Mul(m) = arg_der.get_mut() {
m.extend(fn_der.as_view());
m.set_dirty(true);
arg_der.as_view().normalize(workspace, state, out);
} else {
let mut mul = workspace.new_atom();
let m = mul.to_mul();
m.extend(fn_der.as_view());
m.extend(arg_der.as_view());
m.set_dirty(true);
mul.as_view().normalize(workspace, state, out);
}
return true;
}
// create a derivative function that tags which index was derived
let mut add = workspace.new_atom();
let a = add.to_add();
let mut fn_der = workspace.new_atom();
let mut n = workspace.new_atom();
let mut mul = workspace.new_atom();
for (index, arg_der) in args_der {
let p = fn_der.to_fun();
p.set_from_name(State::DERIVATIVE);
if is_der {
for (i, x_orig) in f_orig.iter().take(f.get_nargs()).enumerate() {
if let AtomView::Num(nn) = x_orig {
let num = nn.get_coeff_view().add(
&CoefficientView::Natural(if i == index { 1 } else { 0 }, 1),
state,
);
n.to_num().set_from_coeff(num);
p.add_arg(n.as_view());
} else {
panic!("Derivative function must contain numbers for all but the last position");
}
}
} else {
for i in 0..f.get_nargs() {
n.to_num()
.set_from_coeff((if i == index { 1 } else { 0 }, 1).into());
p.add_arg(n.as_view());
}
}
p.add_arg(to_derive);
p.set_dirty(true);
let m = mul.to_mul();
m.extend(fn_der.as_view());
m.extend(arg_der.as_view());
m.set_dirty(true);
mul.as_view().normalize(workspace, state, out);
a.extend(mul.as_view());
a.set_dirty(true);
}
add.as_view().normalize(workspace, state, out);
true
}
AtomView::Pow(p) => {
let (base, exp) = p.get_base_exp();
let mut exp_der = workspace.new_atom();
let exp_der_non_zero = exp.derivative(x, workspace, state, &mut exp_der);
let mut base_der = workspace.new_atom();
let base_der_non_zero = base.derivative(x, workspace, state, &mut base_der);
if !exp_der_non_zero && !base_der_non_zero {
let n = out.to_num();
n.set_from_coeff(Coefficient::zero());
return false;
}
let mut exp_der_contrib = workspace.new_atom();
if exp_der_non_zero {
// create log(base)
let mut log_base = workspace.new_atom();
let lb = log_base.to_fun();
lb.set_from_name(State::LOG);
lb.add_arg(base);
if let Atom::Mul(m) = exp_der.get_mut() {
m.extend(*self);
m.extend(log_base.as_view());
m.set_dirty(true);
exp_der
.as_view()
.normalize(workspace, state, &mut exp_der_contrib);
} else {
let mut mul = workspace.new_atom();
let m = mul.to_mul();
m.extend(*self);
m.extend(exp_der.as_view());
m.extend(log_base.as_view());
m.set_dirty(true);
mul.as_view()
.normalize(workspace, state, &mut exp_der_contrib);
}
if !base_der_non_zero {
out.set_from_view(&exp_der_contrib.as_view());
return true;
}
}
let mut mul_h = workspace.new_atom();
let mul = mul_h.to_mul();
mul.extend(base_der.as_view());
let mut new_exp = workspace.new_atom();
if let AtomView::Num(n) = exp {
mul.extend(exp);
let pow_min_one = new_exp.to_num();
let res = n
.get_coeff_view()
.add(&CoefficientView::Natural(-1, 1), state);
pow_min_one.set_from_coeff(res);
} else {
mul.extend(exp);
let ao = new_exp.to_add();
ao.extend(exp);
let mut min_one = workspace.new_atom();
min_one.to_num().set_from_coeff((-1).into());
ao.extend(min_one.as_view());
ao.set_dirty(true);
}
let mut pow_h = workspace.new_atom();
let pow = pow_h.to_pow();
pow.set_from_base_and_exp(base, new_exp.as_view());
pow.set_dirty(true);
mul.extend(pow_h.as_view());
mul.set_dirty(true);
if exp_der_non_zero {
let mut add = workspace.new_atom();
let a = add.to_add();
a.extend(mul_h.as_view());
a.extend(exp_der_contrib.as_view());
a.set_dirty(true);
add.as_view().normalize(workspace, state, out);
} else {
mul_h.as_view().normalize(workspace, state, out);
}
true
}
AtomView::Mul(args) => {
let mut add_h = workspace.new_atom();
let add = add_h.to_add();
let mut mul_h = workspace.new_atom();
let mut non_zero = false;
for arg in args.iter() {
let mut arg_der = workspace.new_atom();
if arg.derivative(x, workspace, state, &mut arg_der) {
if let Atom::Mul(mm) = arg_der.get_mut() {
for other_arg in args.iter() {
if other_arg != arg {
mm.extend(other_arg);
mm.set_dirty(true);
}
}
add.extend(arg_der.as_view());
add.set_dirty(true);
} else {
let mm = mul_h.to_mul();
mm.extend(arg_der.as_view());
for other_arg in args.iter() {
if other_arg != arg {
mm.extend(other_arg);
mm.set_dirty(true);
}
}
add.extend(mul_h.as_view());
add.set_dirty(true);
}
non_zero = true;
}
}
if non_zero {
add_h.as_view().normalize(workspace, state, out);
true
} else {
let n = out.to_num();
n.set_from_coeff(Coefficient::zero());
false
}
}
AtomView::Add(args) => {
let mut add_h = workspace.new_atom();
let add = add_h.to_add();
let mut arg_der = workspace.new_atom();
let mut non_zero = false;
for arg in args.iter() {
if arg.derivative(x, workspace, state, &mut arg_der) {
add.extend(arg_der.as_view());
non_zero = true;
add.set_dirty(true);
}
}
if non_zero {
add_h.as_view().normalize(workspace, state, out);
true
} else {
let n = out.to_num();
n.set_from_coeff(Coefficient::zero());
false
}
}
}
}
/// Taylor expand in `x` around `expansion_point` to depth `depth`.
pub fn taylor_series(
&self,
x: Identifier,
expansion_point: AtomView<P>,
depth: u32,
workspace: &Workspace<P>,
state: &State,
out: &mut Atom<P>,
) -> bool {
let mut current_order = workspace.new_atom();
current_order.set_from_view(self);
let mut next_order = workspace.new_atom();
let var = workspace.new_var(x);
let var_pat = var.into_pattern(state);
let expansion_point_pat = expansion_point.into_pattern(state);
// construct x - expansion_point
// TODO: check that expansion_point does not involve `x`
let mut dist = AtomBuilder::new(var.as_view(), state, workspace, workspace.new_atom());
dist = dist - expansion_point;
let mut series = workspace.new_atom();
let series_sum = series.to_add();
let mut series_contrib = workspace.new_atom();
for d in 0..=depth {
// replace x by expansion_point
var_pat.replace_all(
current_order.as_view(),
&expansion_point_pat,
state,
workspace,
None,
None,
&mut next_order,
);
if d > 0 {
let m = series_contrib.to_mul();
m.extend(next_order.as_view());
if d > 1 {
let mut exp = workspace.new_atom();
let e = exp.to_pow();
e.set_from_base_and_exp(
dist.as_atom_view(),
workspace.new_num(d as i64).as_view(),
);
m.extend(exp.as_atom_view());
} else if d == 1 {
m.extend(dist.as_atom_view());
}
let mut fact = workspace.new_atom();
fact.to_num()
.set_from_coeff((Integer::one(), Integer::factorial(d)).into());
m.extend(fact.as_atom_view());
m.set_dirty(true);
series_sum.extend(series_contrib.as_view());
} else {
series_sum.extend(next_order.as_view());
}
if d < depth
&& current_order
.as_view()
.derivative(x, workspace, state, &mut next_order)
{
std::mem::swap(&mut current_order, &mut next_order);
} else {
if d == 0 {
out.set_from_view(&workspace.new_num(0).as_view());
return false;
}
break;
}
}
series_sum.set_dirty(true);
series.as_view().normalize(workspace, state, out);
true
}
}