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//! Explicit Runge-Kutta method with Dormand-Prince coefficients of order 5(4) and dense output of order 4.
use crate::butcher_tableau::dopri54;
use crate::controller::Controller;
use crate::dop_shared::*;
use nalgebra::{allocator::Allocator, DefaultAllocator, Dim, OVector};
trait DefaultController<T: FloatNumber> {
fn default(x: T, x_end: T) -> Self;
}
impl<T: FloatNumber> DefaultController<T> for Controller<T> {
fn default(x: T, x_end: T) -> Self {
let alpha = T::from(0.2 - 0.04 * 0.75).unwrap();
Controller::new(
alpha,
T::from(0.04).unwrap(),
T::from(10.0).unwrap(),
T::from(0.2).unwrap(),
x_end - x,
T::from(0.9).unwrap(),
sign(T::one(), x_end - x),
)
}
}
/// Structure containing the parameters for the numerical integration.
pub struct Dopri5<T, V, F>
where
T: FloatNumber,
F: System<T, V>,
{
f: F,
x: T,
x_old: T,
x_end: T,
xd: T,
dx: T,
y: V,
rtol: T,
atol: T,
results: SolverResult<T, V>,
uround: T,
h: T,
h_old: T,
n_max: u32,
n_stiff: u32,
controller: Controller<T>,
out_type: OutputType,
rcont: [V; 5],
stats: Stats,
}
impl<T, D: Dim, F> Dopri5<T, OVector<T, D>, F>
where
f64: From<T>,
T: FloatNumber,
F: System<T, OVector<T, D>>,
OVector<T, D>: std::ops::Mul<T, Output = OVector<T, D>>,
DefaultAllocator: Allocator<T, D>,
{
/// Default initializer for the structure
///
/// # Arguments
///
/// * `f` - Structure implementing the System<V> trait
/// * `x` - Initial value of the independent variable (usually time)
/// * `x_end` - Final value of the independent variable
/// * `dx` - Increment in the dense output. This argument has no effect if the output type is Sparse
/// * `y` - Initial value of the dependent variable(s)
/// * `rtol` - Relative tolerance used in the computation of the adaptive step size
/// * `atol` - Absolute tolerance used in the computation of the adaptive step size
///
pub fn new(f: F, x: T, x_end: T, dx: T, y: OVector<T, D>, rtol: T, atol: T) -> Self {
let (rows, cols) = y.shape_generic();
Self {
f,
x,
xd: x,
dx,
x_old: x,
x_end,
y,
rtol,
atol,
results: SolverResult::default(),
uround: T::epsilon(),
h: T::zero(),
h_old: T::zero(),
n_max: 100000,
n_stiff: 1000,
controller: Controller::default(x, x_end),
out_type: OutputType::Dense,
rcont: [
OVector::zeros_generic(rows, cols),
OVector::zeros_generic(rows, cols),
OVector::zeros_generic(rows, cols),
OVector::zeros_generic(rows, cols),
OVector::zeros_generic(rows, cols),
],
stats: Stats::new(),
}
}
/// Advanced initializer for the structure.
///
/// # Arguments
///
/// * `f` - Structure implementing the System<V> trait
/// * `x` - Initial value of the independent variable (usually time)
/// * `x_end` - Final value of the independent variable
/// * `dx` - Increment in the dense output. This argument has no effect if the output type is Sparse
/// * `y` - Initial value of the dependent variable(s)
/// * `rtol` - Relative tolerance used in the computation of the adaptive step size
/// * `atol` - Absolute tolerance used in the computation of the adaptive step size
/// * `safety_factor` - Safety factor used in the computation of the adaptive step size
/// * `beta` - Value of the beta coefficient of the PI controller. Default is 0.04
/// * `fac_min` - Minimum factor between two successive steps. Default is 0.2
/// * `fac_max` - Maximum factor between two successive steps. Default is 10.0
/// * `h_max` - Maximum step size. Default is `x_end-x`
/// * `h` - Initial value of the step size. If h = 0.0, the intial value of h is computed automatically
/// * `n_max` - Maximum number of iterations. Default is 100000
/// * `n_stiff` - Stifness is tested when the number of iterations is a multiple of n_stiff. Default is 1000
/// * `out_type` - Type of the output. Must be a variant of the OutputType enum. Default is Dense
///
#[allow(clippy::too_many_arguments)]
pub fn from_param(
f: F,
x: T,
x_end: T,
dx: T,
y: OVector<T, D>,
rtol: T,
atol: T,
safety_factor: T,
beta: T,
fac_min: T,
fac_max: T,
h_max: T,
h: T,
n_max: u32,
n_stiff: u32,
out_type: OutputType,
) -> Self {
let alpha = T::from(0.2).unwrap() - beta * T::from(0.75).unwrap();
let (rows, cols) = y.shape_generic();
Self {
f,
x,
xd: x,
x_old: T::zero(),
x_end,
dx,
y,
rtol,
atol,
results: SolverResult::default(),
uround: T::from(f64::EPSILON).unwrap(),
h,
h_old: T::zero(),
n_max,
n_stiff,
controller: Controller::new(
alpha,
beta,
fac_max,
fac_min,
h_max,
safety_factor,
sign(T::one(), x_end - x),
),
out_type,
rcont: [
OVector::zeros_generic(rows, cols),
OVector::zeros_generic(rows, cols),
OVector::zeros_generic(rows, cols),
OVector::zeros_generic(rows, cols),
OVector::zeros_generic(rows, cols),
],
stats: Stats::new(),
}
}
/// Compute the initial stepsize
fn hinit(&self) -> T {
let (rows, cols) = self.y.shape_generic();
let mut f0 = OVector::zeros_generic(rows, cols);
self.f.system(self.x, &self.y, &mut f0);
let posneg = sign(T::from(1.0).unwrap(), self.x_end - self.x);
// Compute the norm of y0 and f0
let dim = rows.value();
let mut d0 = T::zero();
let mut d1 = T::zero();
for i in 0..dim {
let y_i = T::from(self.y[i]).unwrap();
let sci = self.atol + y_i.abs() * self.rtol;
d0 += (y_i / sci) * (y_i / sci);
let f0_i = T::from(f0[i]).unwrap();
d1 += (f0_i / sci) * (f0_i / sci);
}
// Compute h0
let tol = T::from(1.0E-10).unwrap();
let mut h0 = if d0 < tol || d1 < tol {
T::from(1.0E-6).unwrap()
} else {
T::from(0.01).unwrap() * (d0 / d1).sqrt()
};
h0 = h0.min(self.controller.h_max());
h0 = sign(h0, posneg);
let y1 = &self.y + &f0 * h0;
let mut f1 = OVector::zeros_generic(rows, cols);
self.f.system(self.x + h0, &y1, &mut f1);
// Compute the norm of f1-f0 divided by h0
let mut d2 = T::zero();
for i in 0..dim {
let f0_i = f0[i];
let f1_i = f1[i];
let y_i = self.y[i];
let sci = self.atol + y_i.abs() * self.rtol;
d2 += ((f1_i - f0_i) / sci) * ((f1_i - f0_i) / sci);
}
d2 = d2.sqrt() / h0;
let h1 = if d1.sqrt().max(d2.abs()) <= T::from(1.0E-15).unwrap() {
T::from(1.0E-6_f64)
.unwrap()
.max(h0.abs() * T::from(1.0E-3).unwrap())
} else {
(T::from(0.01).unwrap() / (d1.sqrt().max(d2))).powf(T::one() / T::from(5.0).unwrap())
};
sign(
(T::from(100.0).unwrap() * h0.abs()).min(h1.min(self.controller.h_max())),
posneg,
)
}
/// Core integration method.
pub fn integrate(&mut self) -> Result<Stats, IntegrationError> {
// Initilization
let (rows, cols) = self.y.shape_generic();
self.x_old = self.x;
let mut n_step = 0;
let mut last = false;
let mut h_new = T::zero();
let dim = rows.value();
let mut non_stiff = 0;
let mut iasti = 0;
let posneg = sign(T::one(), self.x_end - self.x);
if self.h == T::zero() {
self.h = self.hinit();
self.stats.num_eval += 2;
}
self.h_old = self.h;
// Save initial values
if self.out_type == OutputType::Sparse {
self.results.push(self.x, self.y.clone());
}
let mut k = vec![OVector::zeros_generic(rows, cols); 7];
self.f.system(self.x, &self.y, &mut k[0]);
self.stats.num_eval += 1;
// Main loop
while !last {
// Check if step number is within allowed range
if n_step > self.n_max {
self.h_old = self.h;
return Err(IntegrationError::MaxNumStepReached {
x: f64::from(self.x),
n_step,
});
}
// Check for step size underflow
if T::from(0.1).unwrap() * self.h.abs() <= self.uround * self.x.abs() {
self.h_old = self.h;
return Err(IntegrationError::StepSizeUnderflow {
x: f64::from(self.x),
});
}
// Check if it's the last iteration
if (self.x + T::from(1.01).unwrap() * self.h - self.x_end) * posneg > T::zero() {
self.h = self.x_end - self.x;
last = true;
}
n_step += 1;
let h = self.h;
// 6 Stages
let mut y_next = OVector::zeros_generic(rows, cols);
let mut y_stiff = OVector::zeros_generic(rows, cols);
for s in 1..7 {
y_next = self.y.clone();
for (j, k_value) in k.iter().enumerate().take(s) {
y_next += k_value * h * dopri54::a(s + 1, j + 1);
}
self.f
.system(self.x + self.h * dopri54::c::<T>(s + 1), &y_next, &mut k[s]);
if s == 5 {
y_stiff = y_next.clone();
}
}
k[1] = k[6].clone();
self.stats.num_eval += 6;
// Prepare dense output
if self.out_type == OutputType::Dense {
self.rcont[4] = (&k[0] * dopri54::d::<T>(1)
+ &k[2] * dopri54::d::<T>(3)
+ &k[3] * dopri54::d::<T>(4)
+ &k[4] * dopri54::d::<T>(5)
+ &k[5] * dopri54::d::<T>(6)
+ &k[1] * dopri54::d::<T>(7))
* h;
}
// Compute error estimate
k[3] = (&k[0] * dopri54::e::<T>(1)
+ &k[2] * dopri54::e::<T>(3)
+ &k[3] * dopri54::e::<T>(4)
+ &k[4] * dopri54::e::<T>(5)
+ &k[5] * dopri54::e::<T>(6)
+ &k[1] * dopri54::e::<T>(7))
* h;
// Compute error
let mut err = T::zero();
for i in 0..dim {
let y_i = T::from(self.y[i]).unwrap();
let y_next_i = T::from(y_next[i]).unwrap();
let sc_i: T = self.atol + y_i.abs().max(y_next_i.abs()) * self.rtol;
let err_est_i = T::from(k[3][i]).unwrap();
err += (err_est_i / sc_i) * (err_est_i / sc_i);
}
err = (err / T::from(dim).unwrap()).sqrt();
// Step size control
if self.controller.accept(err, self.h, &mut h_new) {
self.stats.accepted_steps += 1;
// Stifness detection
if self.stats.accepted_steps % self.n_stiff == 0 || iasti > 0 {
let num = T::from((&k[1] - &k[5]).dot(&(&k[1] - &k[5]))).unwrap();
let den = T::from((&y_next - &y_stiff).dot(&(&y_next - &y_stiff))).unwrap();
let h_lamb = if den > T::zero() {
self.h * (num / den).sqrt()
} else {
T::zero()
};
if h_lamb > T::from(3.25).unwrap() {
iasti += 1;
non_stiff = 0;
if iasti == 15 {
self.h_old = self.h;
return Err(IntegrationError::StiffnessDetected {
x: f64::from(self.x),
});
}
} else {
non_stiff += 1;
if non_stiff == 6 {
iasti = 0;
}
}
}
// Prepare dense output
if self.out_type == OutputType::Dense {
let h = self.h;
let ydiff = &y_next - &self.y;
let bspl = &k[0] * h - &ydiff;
self.rcont[0] = self.y.clone();
self.rcont[1] = ydiff.clone();
self.rcont[2] = bspl.clone();
self.rcont[3] = -&k[1] * h + ydiff - bspl;
}
k[0] = k[1].clone();
self.y = y_next.clone();
self.x_old = self.x;
self.x += self.h;
self.h_old = self.h;
self.solution_output(y_next, &k);
if self
.f
.solout(self.x, self.results.get().1.last().unwrap(), &k[0])
{
last = true;
}
// Normal exit
if last {
self.h_old = posneg * h_new;
return Ok(self.stats);
}
} else {
last = false;
if self.stats.accepted_steps >= 1 {
self.stats.rejected_steps += 1;
}
}
self.h = h_new;
}
Ok(self.stats)
}
fn solution_output(&mut self, y_next: OVector<T, D>, _k: &[OVector<T, D>]) {
if self.out_type == OutputType::Dense {
while self.xd.abs() <= self.x.abs() {
if self.x_old.abs() <= self.xd.abs() && self.x.abs() >= self.xd.abs() {
let theta = (self.xd - self.x_old) / self.h_old;
let theta1 = T::one() - theta;
let y_out = &self.rcont[0]
+ (&self.rcont[1]
+ (&self.rcont[2]
+ (&self.rcont[3] + &self.rcont[4] * theta1) * theta)
* theta1)
* theta;
self.results.push(self.xd, y_out);
self.xd += self.dx;
}
}
} else {
self.results.push(self.x, y_next)
}
}
/// Getter for the independent variable's output.
pub fn x_out(&self) -> &Vec<T> {
&self.results.get().0
}
/// Getter for the dependent variables' output.
pub fn y_out(&self) -> &Vec<OVector<T, D>> {
&self.results.get().1
}
/// Getter for the results type, a pair of independent and dependent variables
pub fn results(&self) -> &SolverResult<T, OVector<T, D>> {
&self.results
}
}
impl<T, D: Dim, F> Into<SolverResult<T, OVector<T, D>>> for Dopri5<T, OVector<T, D>, F>
where
T: FloatNumber,
F: System<T, OVector<T, D>>,
DefaultAllocator: Allocator<T, D>,
{
fn into(self) -> SolverResult<T, OVector<T, D>> {
self.results
}
}
fn sign<T: FloatNumber>(a: T, b: T) -> T {
if b > T::zero() {
a.abs()
} else {
-a.abs()
}
}
#[cfg(test)]
mod tests {
use super::*;
use crate::{OVector, System, Vector1};
use nalgebra::{allocator::Allocator, DefaultAllocator, Dim};
// Same as Test3 from rk4.rs, but aborts after x is greater/equal than 0.5
struct Test1 {}
impl<D: Dim> System<f64, OVector<f64, D>> for Test1
where
DefaultAllocator: Allocator<f64, D>,
{
fn system(&self, x: f64, y: &OVector<f64, D>, dy: &mut OVector<f64, D>) {
dy[0] = (5. * x * x - y[0]) / (x + y[0]).exp();
}
fn solout(&mut self, x: f64, _y: &OVector<f64, D>, _dy: &OVector<f64, D>) -> bool {
return x >= 0.5;
}
}
#[test]
fn test_integrate_test1_svector() {
let system = Test1 {};
let mut stepper = Dopri5::new(system, 0., 1., 0.1, Vector1::new(1.), 1e-12, 1e-6);
let _ = stepper.integrate();
let x = stepper.x_out();
assert!((*x.last().unwrap() - 0.5).abs() < 1.0E-9); //
let out = stepper.y_out();
assert!((&out[5][0] - 0.913059243).abs() < 1.0E-9);
}
}