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Genericization and Autodiff #4

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connor merged 4 commits from feature/generic-scalar into main 2026-09-29 18:24:02 -04:00
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Cargo.toml
+6 -2
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@@ -9,16 +9,20 @@ repository = "https://git.rcjohnstone.com/connor/ordinary-diffeq"
license = "MIT"
exclude = ["roadmap/"]
# See more keys and their definitions at https://doc.rust-lang.org/cargo/reference/manifest.html
[features]
default = ["autodiff"]
autodiff = ["dep:num-dual"]
[dependencies]
serde = { version = "1.0", features = ["derive"] }
nalgebra = { version = "0.34", features = ["serde-serialize"] }
nalgebra = { version = "0.35", features = ["serde-serialize"] }
num-traits = "0.2.19"
num-dual = { version = "0.15", optional = true }
roots = "0.0.8"
[dev-dependencies]
approx = "0.5"
num-dual = "0.15"
criterion = "0.7.0"
[[bench]]
VERN7_BENCHMARK_REPORT.md
+51 -44
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@@ -1,17 +1,17 @@
# Vern7 Performance Benchmark Report
**Date**: 2025-10-24
**Test System**: Linux 6.17.4-arch2-1
**Date**: 2026-09-29
**Test System**: Linux 7.2.6-arch2-1
**Optimization Level**: Release build with full optimizations
## Executive Summary
Vern7 demonstrates **substantial performance advantages** over lower-order methods (BS3 and DP5) at tight tolerances (1e-8 to 1e-12), achieving:
- **2.7x faster** than DP5 at 1e-10 tolerance (exponential problem)
- **3.8x faster** than DP5 in harmonic oscillator
- **8.8x faster** than DP5 for orbital mechanics
- **51x faster** than BS3 in harmonic oscillator
- **1.65x faster** than DP5 for interpolation workloads
- **2.9x faster** than DP5 at 1e-10 tolerance (exponential problem)
- **4.4x faster** than DP5 in harmonic oscillator
- **1.1x faster** than DP5 for orbital mechanics
- **83x faster** than BS3 in harmonic oscillator
- **2.3x faster** than DP5 for interpolation workloads
These results confirm Vern7's design goal: **maximum efficiency for high-accuracy requirements**.
@@ -23,14 +23,14 @@ These results confirm Vern7's design goal: **maximum efficiency for high-accurac
| Method | Time (μs) | Relative Speed | Speedup vs BS3 |
|--------|-----------|----------------|----------------|
| **Vern7** | **3.81** | **1.00x** (baseline) | **51.8x** |
| DP5 | 10.43 | 2.74x slower | 18.9x |
| BS3 | 197.37 | 51.8x slower | 1.0x |
| **Vern7** | **2.96** | **1.00x** (baseline) | **53.2x** |
| DP5 | 8.46 | 2.86x slower | 18.6x |
| BS3 | 157.35 | 53.2x slower | 1.0x |
**Analysis**:
- Vern7 is **2.7x faster** than DP5 and **51x faster** than BS3
- Vern7 is **2.9x faster** than DP5 and **53x faster** than BS3
- BS3's 3rd-order method requires many tiny steps to maintain 1e-10 accuracy
- DP5's 5th-order is better but still requires ~2.7x more work than Vern7
- DP5's 5th-order is better but still requires ~2.9x more work than Vern7
- Vern7's 7th-order allows much larger step sizes while maintaining accuracy
---
@@ -41,15 +41,15 @@ These results confirm Vern7's design goal: **maximum efficiency for high-accurac
| Method | Time (μs) | Relative Speed | Speedup vs BS3 |
|--------|-----------|----------------|----------------|
| **Vern7** | **26.89** | **1.00x** (baseline) | **55.1x** |
| DP5 | 102.74 | 3.82x slower | 14.4x |
| BS3 | 1,481.4 | 55.1x slower | 1.0x |
| **Vern7** | **19.33** | **1.00x** (baseline) | **83.2x** |
| DP5 | 84.19 | 4.36x slower | 19.1x |
| BS3 | 1,608.7 | 83.2x slower | 1.0x |
**Analysis**:
- Vern7 is **3.8x faster** than DP5 and **55x faster** than BS3
- Vern7 is **4.4x faster** than DP5 and **83x faster** than BS3
- Smooth periodic problems like harmonic oscillators are ideal for high-order methods
- BS3 requires ~1.5ms due to tiny steps needed for tight tolerance
- DP5 needs ~103μs, still significantly more than Vern7's 27μs
- BS3 requires ~1.6ms due to tiny steps needed for tight tolerance
- DP5 needs ~84μs, still significantly more than Vern7's 19μs
- Higher dimensionality (2D vs 1D) amplifies the advantage of larger steps
---
@@ -60,15 +60,18 @@ These results confirm Vern7's design goal: **maximum efficiency for high-accurac
| Method | Time (μs) | Relative Speed | Speedup |
|--------|-----------|----------------|---------|
| **Vern7** | **98.75** | **1.00x** (baseline) | **8.77x** |
| DP5 | 865.79 | 8.77x slower | 1.0x |
| **Vern7** | **83.88** | **1.00x** (baseline) | **1.13x** |
| DP5 | 94.68 | 1.13x slower | 1.0x |
**Analysis**:
- Vern7 is **8.8x faster** than DP5 for this challenging 6D problem
- Vern7 is **1.1x faster** than DP5 on this problem
- Step counts: Vern7 184, DP5 440. Vern7 takes 2.4x fewer steps, but each step
costs 16 stages against DP5's 7, so the advantage very nearly cancels
- Orbital mechanics requires tight tolerances to maintain energy conservation
- BS3 was too slow to include in the benchmark at this tolerance
- 6D problem with long integration time shows Vern7's scalability
- This represents realistic astrodynamics/orbital mechanics workloads
- Vern7's advantage here is in step count, which is what matters once the
right-hand side is expensive - a real gravity model rather than a two-body
point mass
---
@@ -78,17 +81,20 @@ These results confirm Vern7's design goal: **maximum efficiency for high-accurac
| Method | Time (μs) | Relative Speed | Notes |
|--------|-----------|----------------|-------|
| **Vern7** | **11.05** | **1.00x** (baseline) | Lazy extra stages |
| DP5 | 18.27 | 1.65x slower | Standard dense output |
| **Vern7** | **7.28** | **1.00x** (baseline) | Lazy extra stages |
| DP5 | 16.73 | 2.30x slower | Standard dense output |
**Analysis**:
- Vern7 with lazy computation is **1.65x faster** than DP5
- Vern7 with lazy computation is **2.3x faster** than DP5
- First interpolation triggers lazy computation of 6 extra stages (k11-k16)
- Subsequent interpolations reuse cached extra stages (~10ns RefCell overhead)
- Despite computing extra stages, Vern7 is still faster overall due to:
1. Fewer total integration steps (larger step sizes)
2. Higher accuracy interpolation (7th order vs 5th order)
- Lazy computation adds minimal overhead (~6μs for 6 stages, amortized over 100 interpolations)
- Lazy computation adds minimal overhead, amortized over 100 interpolations
- Each benchmark iteration builds a fresh `Solution`, so the lazy stages are
computed once per iteration rather than served from the `RefCell` cache - the
cost of interpolating into a solution you have just produced
---
@@ -100,11 +106,11 @@ These results confirm Vern7's design goal: **maximum efficiency for high-accurac
| Tolerance | DP5 (μs) | Vern7 (μs) | Speedup | Winner |
|-----------|----------|------------|---------|--------|
| 1e-6 | 2.63 | 2.05 | 1.28x | Vern7 |
| 1e-7 | 3.71 | 2.74 | 1.35x | Vern7 |
| 1e-8 | 5.43 | 3.12 | 1.74x | Vern7 |
| 1e-9 | 7.97 | 3.86 | 2.06x | **Vern7** |
| 1e-10 | 11.33 | 5.33 | 2.13x | **Vern7** |
| 1e-6 | 1.94 | 1.51 | 1.29x | Vern7 |
| 1e-7 | 2.82 | 1.93 | 1.46x | Vern7 |
| 1e-8 | 4.20 | 2.29 | 1.83x | Vern7 |
| 1e-9 | 6.25 | 2.97 | 2.11x | **Vern7** |
| 1e-10 | 9.52 | 3.89 | 2.45x | **Vern7** |
### Performance Scaling Chart (Conceptual)
@@ -128,7 +134,7 @@ Time (μs)
**Analysis**:
- At **moderate tolerances (1e-6)**: Vern7 is 1.3x faster
- At **tight tolerances (1e-10)**: Vern7 is 2.1x faster
- At **tight tolerances (1e-10)**: Vern7 is 2.5x faster
- **Crossover point**: Vern7 becomes increasingly advantageous as tolerance tightens
- DP5's time scales roughly quadratically with tolerance
- Vern7's time scales more slowly (higher order = larger steps)
@@ -157,22 +163,23 @@ Time (μs)
### Lazy Computation Impact
The lazy computation of extra stages (k11-k16) provides:
- **Minimal overhead**: ~6μs to compute 6 extra stages
- **Minimal overhead**: the 6 extra stages are computed once per step that is interpolated into
- **Cache efficiency**: Extra stages computed once per interval, reused for multiple interpolations
- **Memory efficiency**: Only computed when interpolation is requested
- **Performance**: Despite extra computation, still 1.65x faster than DP5 for interpolation workloads
- **Performance**: Despite extra computation, still 2.3x faster than DP5 for interpolation workloads
### Step Size Comparison
Estimated step sizes at 1e-10 tolerance for exponential problem:
Measured at 1e-10 tolerance on the exponential problem (`y' = y`, t = 0 to 4):
| Method | Avg Step Size | Steps Required | Function Evals |
|--------|---------------|----------------|----------------|
| BS3 | ~0.002 | ~2000 | ~8000 |
| DP5 | ~0.01 | ~400 | ~2400 |
| **Vern7** | ~0.05 | **~80** | **~800** |
| BS3 | 0.0014 | 2,916 | 11,664 |
| DP5 | 0.0354 | 113 | 791 |
| **Vern7** | 0.1379 | **29** | **290** |
**Vern7 requires ~3x fewer function evaluations than DP5.**
**Vern7 requires 2.7x fewer function evaluations than DP5**, taking 3.9x fewer
steps at 10 stages each against DP5's 7.
---
@@ -185,7 +192,7 @@ Our Rust implementation achieves performance comparable to Julia's highly-optimi
| Step computation | Highly optimized, FSAL | Optimized, no FSAL |
| Lazy interpolation | ✓ | ✓ |
| Stage caching | RefCell-based | RefCell-based (~10ns) |
| Memory allocation | Minimal | Minimal |
| Memory allocation | Cached in the integrator | Two heap allocations per step |
| Relative speed | Baseline | ~Comparable |
**Note**: Direct comparison difficult due to different hardware and problems, but algorithmic approach is identical.
@@ -213,8 +220,8 @@ Our Rust implementation achieves performance comparable to Julia's highly-optimi
Vern7 successfully achieves its design goal of being the **most efficient method for high-accuracy non-stiff problems**. The implementation with lazy computation of extra stages provides:
- ✅ **2-9x speedup** over DP5 at tight tolerances
- ✅ **50x+ speedup** over BS3 at tight tolerances
- ✅ **1.1-4.4x speedup** over DP5 at tight tolerances
- ✅ **53-83x speedup** over BS3 at tight tolerances
- ✅ **Efficient lazy interpolation** with minimal overhead
- ✅ **Full 7th-order accuracy** for both steps and interpolation
- ✅ **Memory-efficient caching** with RefCell
@@ -225,7 +232,7 @@ The results validate the effort invested in implementing the complex 16-stage in
## Appendix: Benchmark Configuration
**Hardware**: Not specified (Linux system)
**Hardware**: Linux 7.2.6-arch2-1
**Compiler**: rustc (release mode, full optimizations)
**Measurement Tool**: Criterion.rs v0.7.0
**Sample Size**: 100 samples per benchmark
benches/BS3_VS_DP5_RESULTS.md
+23 -23
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@@ -1,6 +1,6 @@
# BS3 vs DP5 Benchmark Results
Generated: 2025-10-23
Generated: 2026-09-29
## Summary
@@ -10,7 +10,7 @@ Comprehensive performance comparison between **BS3** (Bogacki-Shampine 3rd order
### Overall Performance Comparison
**DP5 is consistently faster than BS3 across all tested scenarios**, typically by a factor of **1.5x to 4.3x**.
**DP5 is consistently faster than BS3 across all tested scenarios**, typically by a factor of **1.2x to 5.1x**.
This might seem counterintuitive since BS3 uses fewer stages (4 vs 7), but several factors explain DP5's superior performance:
@@ -32,8 +32,8 @@ Despite being slower in these benchmarks, BS3 still has value:
| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 3.28 µs | 1.92x slower |
| **DP5** | 1.70 µs | baseline |
| **BS3** | 2.99 µs | 2.21x slower |
| **DP5** | 1.35 µs | baseline |
Simple 1D problem with smooth exponential solution.
@@ -41,8 +41,8 @@ Simple 1D problem with smooth exponential solution.
| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 30.70 µs | 2.25x slower |
| **DP5** | 13.67 µs | baseline |
| **BS3** | 27.37 µs | 2.76x slower |
| **DP5** | 9.92 µs | baseline |
2D conservative system with periodic solution.
@@ -50,8 +50,8 @@ Simple 1D problem with smooth exponential solution.
| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 132.35 µs | 3.57x slower |
| **DP5** | 37.11 µs | baseline |
| **BS3** | 113.44 µs | 3.81x slower |
| **DP5** | 29.74 µs | baseline |
Nonlinear 2D system with trigonometric terms.
@@ -59,8 +59,8 @@ Nonlinear 2D system with trigonometric terms.
| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 124.72 µs | 1.45x slower |
| **DP5** | 86.10 µs | baseline |
| **BS3** | 98.02 µs | 5.07x slower |
| **DP5** | 19.32 µs | baseline |
Higher-dimensional problem with gravitational dynamics.
@@ -68,8 +68,8 @@ Higher-dimensional problem with gravitational dynamics.
| Method | Time (solve + 100 interpolations) | Ratio |
|--------|-----------------------------------|-------|
| **BS3** | 19.68 µs | 4.81x slower |
| **DP5** | 4.09 µs | baseline |
| **BS3** | 16.34 µs | 4.67x slower |
| **DP5** | 3.50 µs | baseline |
BS3 uses cubic Hermite interpolation, DP5 uses optimized 5th order interpolation.
@@ -79,14 +79,11 @@ Performance across different tolerance levels (`y' = -y` problem):
| Tolerance | BS3 Time | DP5 Time | Ratio (BS3/DP5) |
|-----------|----------|----------|-----------------|
| 1e-3 | 1.63 µs | 1.26 µs | 1.30x |
| 1e-4 | 2.61 µs | 1.54 µs | 1.70x |
| 1e-5 | 4.64 µs | 2.03 µs | 2.28x |
| 1e-6 | 8.76 µs | ~2.6 µs* | ~3.4x* |
| 1e-7 | -** | -** | - |
\* Estimated from trend (benchmark timed out)
\** Not completed
| 1e-3 | 1.10 µs | 0.91 µs | 1.21x |
| 1e-4 | 1.87 µs | 1.19 µs | 1.57x |
| 1e-5 | 3.46 µs | 1.44 µs | 2.40x |
| 1e-6 | 6.77 µs | 1.95 µs | 3.46x |
| 1e-7 | 13.73 µs | 2.79 µs | 4.91x |
**Observation**: The performance gap widens as tolerance tightens, because DP5's higher order allows it to take larger steps while maintaining accuracy.
@@ -96,15 +93,18 @@ Performance across different tolerance levels (`y' = -y` problem):
1. **DP5 is the better default choice** for most problems requiring moderate to high accuracy
2. **Performance gap increases** with tighter tolerances (favoring DP5)
3. **Higher dimensions** slightly favor BS3 relative to DP5 (1.45x vs 3.57x slowdown)
4. **Interpolation** strongly favors DP5 (4.8x faster)
3. **Higher dimensions favor DP5 most strongly** (5.07x on the 6D orbit against
2.21x on 1D exponential decay)
4. **Interpolation** strongly favors DP5 (4.7x faster)
### Implementation Quality
Both integrators pass all accuracy and convergence tests:
- ✅ BS3: 3rd order convergence rate verified
- ✅ DP5: 5th order convergence rate verified (existing implementation)
- ✅ Both: FSAL property correctly implemented
- ⚠️ Both: FSAL is a property of the *methods*, but neither implementation
exploits it - each step recomputes `k[0] = f(t, y)` rather than reusing the
previous step's last stage. See roadmap feature 36.
- ✅ Both: Dense output accurate to specified order
### Future Optimizations
benches/bs3_vs_dp5.rs
+28 -42
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@@ -27,9 +27,8 @@ fn bench_exponential_decay(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, params);
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
let mut problem = Problem::new(ode, bs3, controller);
black_box(problem.solve());
});
});
@@ -37,9 +36,8 @@ fn bench_exponential_decay(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, params);
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
@@ -66,9 +64,8 @@ fn bench_harmonic_oscillator(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 20.0, y0, ());
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
let mut problem = Problem::new(ode, bs3, controller);
black_box(problem.solve());
});
});
@@ -76,9 +73,8 @@ fn bench_harmonic_oscillator(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 20.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
@@ -109,9 +105,8 @@ fn bench_pendulum(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, params);
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
let mut problem = Problem::new(ode, bs3, controller);
black_box(problem.solve());
});
});
@@ -119,9 +114,8 @@ fn bench_pendulum(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, params);
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
@@ -160,9 +154,8 @@ fn bench_orbit_6d(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10000.0, y0, params);
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
let mut problem = Problem::new(ode, bs3, controller);
black_box(problem.solve());
});
});
@@ -170,9 +163,8 @@ fn bench_orbit_6d(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10000.0, y0, params);
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
@@ -199,12 +191,10 @@ fn bench_interpolation(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
let mut problem = Problem::new(ode, bs3, controller);
let solution = problem.solve();
// Interpolate at 100 points
let _: Vec<_> = (0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect();
});
let mut problem = Problem::new(ode, bs3, controller);
let solution = problem.solve();
// Interpolate at 100 points
black_box((0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect::<Vec<_>>());
});
});
@@ -213,12 +203,10 @@ fn bench_interpolation(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
let mut problem = Problem::new(ode, dp45, controller);
let solution = problem.solve();
// Interpolate at 100 points
let _: Vec<_> = (0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect();
});
let mut problem = Problem::new(ode, dp45, controller);
let solution = problem.solve();
// Interpolate at 100 points
black_box((0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect::<Vec<_>>());
});
});
@@ -245,9 +233,8 @@ fn bench_tolerance_scaling(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, ());
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
let mut problem = Problem::new(ode, bs3, controller);
black_box(problem.solve());
});
});
@@ -255,9 +242,8 @@ fn bench_tolerance_scaling(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
}
benches/orbit.rs
+2 -3
View File
@@ -29,9 +29,8 @@ fn bench_orbit(c: &mut Criterion) {
c.bench_function("bench_orbit", |b| {
b.iter(|| {
std::hint::black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
std::hint::black_box(problem.solve());
});
});
}
benches/simple_1d.rs
+9 -8
View File
@@ -20,9 +20,8 @@ fn bench_simple_1d(c: &mut Criterion) {
c.bench_function("bench_simple_1d", |b| {
b.iter(|| {
std::hint::black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
std::hint::black_box(problem.solve());
});
});
}
@@ -44,11 +43,13 @@ fn bench_interpolation_1d(c: &mut Criterion) {
c.bench_function("bench_interpolation_1d", |b| {
b.iter(|| {
std::hint::black_box({
let mut problem = Problem::new(ode, dp45, controller);
let solution = problem.solve();
let _ = (0..100).map(|t| solution.interpolate(t as f64 * 0.1)[0]);
});
let mut problem = Problem::new(ode, dp45, controller);
let solution = problem.solve();
std::hint::black_box(
(0..100)
.map(|t| solution.interpolate(t as f64 * 0.1)[0])
.sum::<f64>(),
);
});
});
}
benches/vern7_comparison.rs
+31 -45
View File
@@ -28,9 +28,8 @@ fn bench_exponential_tight_tol(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 4.0, y0, ());
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
let mut problem = Problem::new(ode, bs3, controller);
black_box(problem.solve());
});
});
@@ -38,9 +37,8 @@ fn bench_exponential_tight_tol(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 4.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
@@ -48,9 +46,8 @@ fn bench_exponential_tight_tol(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 4.0, y0, ());
let vern7 = Vern7::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, vern7, controller).solve();
});
let mut problem = Problem::new(ode, vern7, controller);
black_box(problem.solve());
});
});
@@ -77,9 +74,8 @@ fn bench_harmonic_oscillator_tight_tol(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 20.0, y0, ());
let bs3 = BS3::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, bs3, controller).solve();
});
let mut problem = Problem::new(ode, bs3, controller);
black_box(problem.solve());
});
});
@@ -87,9 +83,8 @@ fn bench_harmonic_oscillator_tight_tol(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 20.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
@@ -97,9 +92,8 @@ fn bench_harmonic_oscillator_tight_tol(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 20.0, y0, ());
let vern7 = Vern7::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, vern7, controller).solve();
});
let mut problem = Problem::new(ode, vern7, controller);
black_box(problem.solve());
});
});
@@ -138,9 +132,8 @@ fn bench_orbit_tight_tol(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10000.0, y0, params);
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
@@ -148,9 +141,8 @@ fn bench_orbit_tight_tol(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10000.0, y0, params);
let vern7 = Vern7::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, vern7, controller).solve();
});
let mut problem = Problem::new(ode, vern7, controller);
black_box(problem.solve());
});
});
@@ -175,27 +167,23 @@ fn bench_vern7_interpolation(c: &mut Criterion) {
// Vern7 with interpolation (should compute extra stages lazily)
group.bench_function("vern7_with_interpolation", |b| {
b.iter(|| {
black_box({
let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
let vern7 = Vern7::new().a_tol(tol).r_tol(tol);
let mut problem = Problem::new(ode, vern7, controller);
let solution = problem.solve();
// Interpolate at 100 points - first one computes extra stages
let _: Vec<_> = (0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect();
});
let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
let vern7 = Vern7::new().a_tol(tol).r_tol(tol);
let mut problem = Problem::new(ode, vern7, controller);
let solution = problem.solve();
// Interpolate at 100 points - first one computes extra stages
black_box((0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect::<Vec<_>>());
});
});
// DP5 with interpolation for comparison
group.bench_function("dp5_with_interpolation", |b| {
b.iter(|| {
black_box({
let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
let mut problem = Problem::new(ode, dp45, controller);
let solution = problem.solve();
let _: Vec<_> = (0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect();
});
let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
let mut problem = Problem::new(ode, dp45, controller);
let solution = problem.solve();
black_box((0..100).map(|i| solution.interpolate(i as f64 * 0.05)).collect::<Vec<_>>());
});
});
@@ -223,9 +211,8 @@ fn bench_tolerance_scaling_vern7(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, ());
let dp45 = DormandPrince45::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, dp45, controller).solve();
});
let mut problem = Problem::new(ode, dp45, controller);
black_box(problem.solve());
});
});
@@ -233,9 +220,8 @@ fn bench_tolerance_scaling_vern7(c: &mut Criterion) {
let ode = ODE::new(&derivative, 0.0, 10.0, y0, ());
let vern7 = Vern7::new().a_tol(tol).r_tol(tol);
b.iter(|| {
black_box({
Problem::new(ode, vern7, controller).solve();
});
let mut problem = Problem::new(ode, vern7, controller);
black_box(problem.solve());
});
});
}
readme.md
+69 -5
View File
@@ -15,8 +15,8 @@ and field line tracing:
| **Vern7** (Verner) | 7(6) | 10+6 | 7th order | High accuracy (rtol ~ 1e-8 to 1e-12) |
**Performance at 1e-10 tolerance:**
- Vern7: **2.7-8.8x faster** than DP5
- Vern7: **50x+ faster** than BS3
- Vern7: **1.1-4.4x faster** than DP5
- Vern7: **53-83x faster** than BS3
See [benchmark report](VERN7_BENCHMARK_REPORT.md) for detailed performance analysis.
@@ -27,6 +27,8 @@ See [benchmark report](VERN7_BENCHMARK_REPORT.md) for detailed performance analy
- **Dense output interpolation** at any time point
- **Parameters** in derivative and callback functions
- **Lazy computation** of extra interpolation stages (Vern7)
- **PID step size control** as an alternative to the PI controller
- **Automatic Differentiation**, the same problem can be solved with floats or also differentiated with dual numbers
### Future Improvements
@@ -68,15 +70,77 @@ let ode = ODE::new(&derivative, 0.0, 6.3, y0, params);
let dp45 = DormandPrince45::new().a_tol(1e-12).r_tol(1e-6);
let controller = PIController::default();
let value_too_high = Callback {
event: &|t: f64, _y: Vector2<f64>, _p: &Params| { 5.0 - t },
let value_too_high = FnCallback {
event: &|t: f64, _y: &Vector2<f64>, _p: &Params| { 5.0 - t },
effect: &stop,
};
// Solve the problem
let mut problem = Problem::new(ode, dp45, controller).with_callback(value_too_high);
let mut problem = Problem::new(ode, dp45, controller).with_callback(&value_too_high);
let solution = problem.solve();
// Can interpolate solutions to whatever you want
let _interpolated_answer = solution.interpolate(4.4);
```
## Automatic differentiation
State, time and step size are generic over the `Scalar` trait. `f64` implements it by default,
and so do the forward-mode dual numbers of [`num-dual`](https://crates.io/crates/num-dual) (the
`autodiff` feature, on by default). Solving at a dual scalar propagates exact derivatives alongside
the state.
The only change needed is to the signature of the right-hand side. Here is the pendulum from above,
before and after:
```rust,ignore
fn derivative(_t: f64, y: Vector2<f64>, p: &Params) -> Vector2<f64> {
let &(g, l) = p;
Vector2::new( y[1], -(g/l) * y[0].sin() )
}
fn derivative<T: Scalar>(_t: T, y: Vector2<T>, p: &Params) -> Vector2<T> {
let &(g, l) = p;
Vector2::new( y[1], -y[0].sin() * (g/l) )
}
```
`Params` stays `(f64, f64)` — parameters only need to become `T` themselves if you want to
differentiate *with respect to* them. Real constants mix into `T` arithmetic directly; `Scalar::lift`
is only needed to scale a vector by a real, or to build a `T` out of nothing.
With that one change, the same function can be solved or differentiated:
```rust
use nalgebra::{Matrix2, Vector2};
use num_dual::DualSVec64;
use ordinary_diffeq::prelude::*;
use std::f64::consts::PI;
type Params = (f64, f64);
fn derivative<T: Scalar>(_t: T, y: Vector2<T>, p: &Params) -> Vector2<T> {
let &(g, l) = p;
Vector2::new(y[1], -y[0].sin() * (g / l))
}
// One propagation in `DualSVec64<2>` yields the whole 2x2 Jacobian of the
// final state with respect to the initial one.
let (y_end, jacobian): (Vector2<f64>, Matrix2<f64>) = num_dual::jacobian(
|y0: Vector2<DualSVec64<2>>| {
type T = DualSVec64<2>;
let ode = ODE::new(&derivative, T::lift(0.0), T::lift(1.0), y0, (9.81, 1.0));
let vern7 = Vern7::new().a_tol(1e-12).r_tol(1e-12);
let mut problem = Problem::new(ode, vern7, PIController::default());
let solution = problem.solve();
*solution.states.last().unwrap()
},
&Vector2::new(0.0, PI / 2.0),
);
// The pendulum is Hamiltonian, so its flow is area-preserving
assert!((jacobian.determinant() - 1.0).abs() < 1e-9);
```
These are exact derivatives of the scheme as it ran, at machine precision, with no finite
differencing and no step size to tune.
roadmap/README.md
+6
View File
@@ -298,6 +298,12 @@ Each feature below links to a detailed implementation plan in the `features/` di
- **Dependencies**: None
- **Effort**: Small
- [ ] **[Reduce Per-Step Allocation](features/39-per-step-allocation.md)**
- Two heap allocations per accepted step (stage array, dense coefficients)
- Against ~150 ns of arithmetic; matters only for cheap right-hand sides
- **Dependencies**: None
- **Effort**: Medium
---
## Implementation Notes
roadmap/features/19-jacobian-computation.md
+25
View File
@@ -4,6 +4,31 @@
Finite difference and auto-diff Jacobians
## Note - 2026-09-29
Forward-mode AD arrived by a different route than this entry assumes. On the
`feature/generic-scalar` branch, state, time and step size became generic over a
`Scalar` trait, implemented by `f64` and by `num-dual`'s dual numbers. That gives
differentiation *through the whole solve* - d y(t_f)/d y0, d/d t0, d/d t_end and
d/d params, so a full 6x6 state transition matrix comes out of one propagation.
This entry is framed around the narrower case, df/dy at a point for implicit
methods, which now falls out of the same machinery.
What that leaves:
- `rosenbrock::compute_jacobian` is still forward finite differences
(eps = sqrt(machine_eps) * max(|y[j]|, 1), D+1 evaluations, ~8 digits).
Pointing it at a dual evaluation is now a small change, not a large one.
- Sparse Jacobian support: untouched.
- The **Effort: Large** estimate below predates the genericisation and wants
revisiting.
Deferred deliberately, as scope creep against the genericisation: how a Jacobian
should be obtained for whichever algorithms need one. Whether the integrator
requests it through a trait, whether AD or finite differences is a per-problem
choice, and how either interacts with the `JacobianUpdateStrategy` reuse policy
already in `rosenbrock.rs`. To be thought through properly on its own.
## Why This Feature Matters
(To be detailed)
roadmap/features/39-per-step-allocation.md
+81
View File
@@ -0,0 +1,81 @@
# Feature: Reduce Per-Step Allocation
## Overview
Remove the heap allocations every accepted step performs.
## Why This Feature Matters
Each call to `step()` allocates a `Vec` for the stage values
(`vec![SVector::zeros(); STAGES]`) and a second `Vec` for the dense output
coefficients. `Problem::solve` then wraps the latter in a `RefCell` and pushes
it onto a growing vector. That is two allocations per step against roughly
150 ns of arithmetic.
Measured 2026-09-29 on the harmonic oscillator at 1e-10, where the right-hand
side is two flops so the timing is almost entirely integrator overhead:
| Method | Steps | Total | Per step |
|--------|-------|-------|----------|
| BS3 | 13,927 | 1.58 ms | 113 ns |
| DP45 | 722 | 112 us | 155 ns |
| Vern7 | 124 | 26.6 us | 215 ns |
`BS3::step` in isolation is 44.6 ns for four stages.
The gain is largest exactly where it is least important: problems with cheap
right-hand sides. For the orbit propagation this crate is aimed at, a
spherical-harmonic gravity evaluation dominates and per-step overhead is
rounding error. Worth doing, not worth doing first.
## Dependencies
None. `STAGES` is already an associated const, so the stage array can be a
fixed-size array rather than a `Vec`.
## Implementation Approach
(To be detailed.) Sketch: stage storage becomes `[SVector<T, D>; STAGES]` if
const generics allow it at the trait level, or a caller-supplied scratch buffer
threaded through `step()`. Dense coefficients want a pre-reserved buffer sized
from an estimate of the step count, or a flat `Vec<SVector<T, D>>` indexed by
stride rather than a `Vec` of `Vec`s.
Note the interaction with the `RefCell` in `Solution::dense`: it exists so
Vern7 can append its lazily computed extra stages. A flat buffer has to keep
that possible.
## Implementation Tasks
- [ ] Core implementation
- [ ] Integration with existing code
- [ ] Testing
- [ ] Benchmarking
## Testing Requirements
Numerics must not change. The existing benches plus a step-count and
final-state comparison against the previous release are sufficient: results
should be bit-identical.
## References
1. Julia implementation: OrdinaryDiffEq.jl, which caches stage storage in the
integrator rather than allocating per step
## Complexity Estimate
**Effort**: Medium
**Risk**: Low - no change to the numerics, so the test suite is a strong guard
## Success Criteria
- [ ] No allocation in the steady-state step loop
- [ ] Bit-identical results against the previous release
- [ ] Measurable improvement on the cheap-RHS benchmarks
## Future Enhancements
Supersedes the "Optimize memory allocation in step method" bullet under Future
Enhancements in `01-bs3-method.md`.
src/callback.rs
+42 -9
View File
@@ -1,21 +1,54 @@
use nalgebra::SVector;
use super::ode::ODE;
use crate::ode::ODE;
use crate::scalar::Scalar;
/// A function that takes in a time and a state and outputs a single float value
/// A callback function you can utilize to change your model mid-run without stopping.
///
/// The integration solver will check this function for zero crossings
/// The solver evaluates [`event`](Callback::event) at each end of every accepted
/// step and looks for a sign change. When it finds one it locates the crossing
/// and applies [`effect`](Callback::effect) to the ODE there.
/// ```
/// # use ordinary_diffeq::prelude::*;
/// # use nalgebra::Vector2;
/// # type Params = ();
/// let too_high = FnCallback {
/// event: &|_t: f64, y: &Vector2<f64>, _p: &Params| 10.0 - y[0],
/// effect: &stop,
/// };
/// ```
pub trait Callback<T: Scalar, const D: usize, P> {
/// The function whose zero crossings mark the event.
fn event(&self, t: T, y: &SVector<T, D>, params: &P) -> T;
/// What to do to the ODE once the crossing has been located.
fn effect(&self, ode: &mut ODE<'_, T, D, P>);
}
/// A [`Callback`] built from a pair of closures.
#[derive(Clone, Copy)]
pub struct Callback<'a, const D: usize, P> {
pub struct FnCallback<'a, T: Scalar, const D: usize, P> {
/// The function to check for zero crossings
pub event: &'a dyn Fn(f64, SVector<f64, D>, &P) -> f64,
pub event: &'a dyn Fn(T, &SVector<T, D>, &P) -> T,
/// The function to change the ODE
pub effect: &'a dyn Fn(&mut ODE<D, P>),
pub effect: &'a dyn Fn(&mut ODE<'_, T, D, P>),
}
impl<T: Scalar, const D: usize, P> Callback<T, D, P> for FnCallback<'_, T, D, P> {
#[inline]
fn event(&self, t: T, y: &SVector<T, D>, params: &P) -> T {
(self.event)(t, y, params)
}
#[inline]
fn effect(&self, ode: &mut ODE<'_, T, D, P>) {
(self.effect)(ode)
}
}
/// A convenience function for stopping the integration
pub fn stop<const D: usize, P>(ode: &mut ODE<D, P>) {
pub fn stop<T: Scalar, const D: usize, P>(ode: &mut ODE<'_, T, D, P>) {
ode.t_end = ode.t;
}
@@ -26,8 +59,8 @@ mod tests {
#[test]
fn test_basic_callbacks() {
type Params = ();
let _value_too_high = Callback {
event: &|_: f64, y: SVector<f64, 3>, _p: &Params| 10.0 - y[0],
let _value_too_high = FnCallback {
event: &|_: f64, y: &SVector<f64, 3>, _p: &Params| 10.0 - y[0],
effect: &stop,
};
}
src/controller.rs
+132 -100
View File
@@ -1,11 +1,13 @@
use crate::scalar::Scalar;
#[derive(Debug, Clone, Copy, PartialEq)]
pub enum TryStep {
Accepted(f64, f64),
NotYetAccepted(f64),
pub enum TryStep<T> {
Accepted(T, T),
NotYetAccepted(T),
}
impl TryStep {
pub fn extract(&self) -> f64 {
impl<T: Copy> TryStep<T> {
pub fn extract(&self) -> T {
match self {
TryStep::Accepted(h, _) => *h,
TryStep::NotYetAccepted(h) => *h,
@@ -16,7 +18,7 @@ impl TryStep {
matches!(self, TryStep::Accepted(_, _))
}
pub fn reset(&mut self) -> Result<TryStep, &str> {
pub fn reset(&mut self) -> Result<TryStep<T>, &str> {
match self {
TryStep::Accepted(_, h) => Ok(TryStep::NotYetAccepted(*h)),
TryStep::NotYetAccepted(_) => Err("Cannot reset a NotYetAccepted TryStep"),
@@ -24,12 +26,18 @@ impl TryStep {
}
}
pub trait Controller<const D: usize> {
fn determine_step(&mut self, h: f64, err: f64) -> TryStep;
/// Chooses the next step size from the last one and its normalized error.
pub trait Controller<T: Scalar> {
fn determine_step(&mut self, h: T, err: f64) -> TryStep<T>;
/// The step size to attempt next
fn next_step_guess(&self) -> TryStep<T>;
fn set_next_step_guess(&mut self, guess: TryStep<T>);
}
#[derive(Debug, Clone, Copy)]
pub struct PIController {
pub struct PIController<T: Scalar = f64> {
pub alpha: f64,
pub beta: f64,
pub factor_c1: f64,
@@ -37,13 +45,13 @@ pub struct PIController {
pub factor_old: f64,
pub h_max: f64,
pub safety_factor: f64,
pub next_step_guess: TryStep,
pub next_step_guess: TryStep<T>,
}
impl<const D: usize> Controller<D> for PIController {
impl<T: Scalar> Controller<T> for PIController<T> {
/// Determines if the previously run step size and error were valid or not. Either way, it also
/// returns what the next step size should be
fn determine_step(&mut self, prev_step: f64, err: f64) -> TryStep {
fn determine_step(&mut self, prev_step: T, err: f64) -> TryStep<T> {
let factor_11 = err.powf(self.alpha);
let factor = self.factor_c2.max(
self.factor_c1
@@ -53,19 +61,30 @@ impl<const D: usize> Controller<D> for PIController {
let mut h = prev_step / factor;
// Accept the stepsize and provide what the next step size should be
self.factor_old = err.max(1.0e-4);
if h.abs() > self.h_max {
if h.re().abs() > self.h_max {
// If the step goes past the maximum allowed, though, we shrink it
h = self.h_max.copysign(h);
h = T::lift(self.h_max.copysign(h.re()));
}
TryStep::Accepted(prev_step, h)
} else {
// Reject the stepsize and propose a smaller one for the current step
TryStep::NotYetAccepted(prev_step / (self.factor_c1.min(factor_11 / self.safety_factor)))
TryStep::NotYetAccepted(
prev_step / self.factor_c1.min(factor_11 / self.safety_factor),
)
}
}
fn next_step_guess(&self) -> TryStep<T> {
self.next_step_guess
}
fn set_next_step_guess(&mut self, guess: TryStep<T>) {
self.next_step_guess = guess;
}
}
impl PIController {
impl<T: Scalar> PIController<T> {
pub fn new(
alpha: f64,
beta: f64,
@@ -83,17 +102,18 @@ impl PIController {
factor_old: 1.0e-4,
h_max: h_max.abs(),
safety_factor,
next_step_guess: TryStep::NotYetAccepted(initial_h),
next_step_guess: TryStep::NotYetAccepted(T::lift(initial_h)),
}
}
}
impl Default for PIController {
impl<T: Scalar> Default for PIController<T> {
fn default() -> Self {
Self::new(0.17, 0.04, 10.0, 0.2, 100000.0, 0.9, 1e-4)
}
}
/// PID (Proportional-Integral-Derivative) step size controller
///
/// The PID controller is an advanced adaptive time-stepping controller that provides
@@ -111,9 +131,9 @@ impl Default for PIController {
/// Where:
/// - ε_n = normalized error estimate at current step
/// - ε_{n-1} = normalized error estimate at previous step
/// - β₁ = proportional coefficient (controls reaction to current error)
/// - β₂ = integral coefficient (controls reaction to error history)
/// - β₃ = derivative coefficient (controls reaction to error rate of change)
/// - β₁ (`p_term`) = proportional coefficient (controls reaction to current error)
/// - β₂ (`i_term`) = integral coefficient (controls reaction to error history)
/// - β₃ (`d_term`) = derivative coefficient (controls reaction to error rate of change)
///
/// # When to Use
///
@@ -138,11 +158,11 @@ impl Default for PIController {
/// let controller = PIDController::for_order(5);
/// ```
#[derive(Debug, Clone, Copy)]
pub struct PIDController {
pub struct PIDController<T: Scalar = f64> {
// PID Coefficients
pub beta1: f64, // Proportional: reaction to current error
pub beta2: f64, // Integral: reaction to error history
pub beta3: f64, // Derivative: reaction to error rate of change
pub p_term: f64, // Proportional: reaction to current error
pub i_term: f64, // Integral: reaction to error history
pub d_term: f64, // Derivative: reaction to error rate of change
// Constraints
pub factor_c1: f64, // 1/min_factor (maximum step decrease)
@@ -156,35 +176,35 @@ pub struct PIDController {
pub h_old: f64, // h_{n-1}: previous step size
// Next step guess
pub next_step_guess: TryStep,
pub next_step_guess: TryStep<T>,
}
impl<const D: usize> Controller<D> for PIDController {
impl<T: Scalar> Controller<T> for PIDController<T> {
/// Determines if the previously run step was acceptable and computes the next step size
/// using PID control theory
fn determine_step(&mut self, h: f64, err: f64) -> TryStep {
fn determine_step(&mut self, h: T, err: f64) -> TryStep<T> {
// Compute PID control factor
// For first step or when history isn't available, fall back to simpler control
let factor = if self.err_old <= 0.0 {
// First step: use only proportional control
let factor_11 = err.powf(self.beta1);
let factor_11 = err.powf(self.p_term);
self.factor_c2.max(
self.factor_c1.min(factor_11 / self.safety_factor)
)
} else if self.err_older <= 0.0 {
// Second step: use PI control (proportional + integral)
let factor_11 = err.powf(self.beta1);
let factor_12 = self.err_old.powf(-self.beta2);
let factor_11 = err.powf(self.p_term);
let factor_12 = self.err_old.powf(-self.i_term);
self.factor_c2.max(
self.factor_c1.min(factor_11 * factor_12 / self.safety_factor)
)
} else {
// Full PID control (proportional + integral + derivative)
let factor_11 = err.powf(self.beta1);
let factor_12 = self.err_old.powf(-self.beta2);
let factor_11 = err.powf(self.p_term);
let factor_12 = self.err_old.powf(-self.i_term);
// Derivative term uses ratio of consecutive step sizes
let factor_13 = if self.h_old > 0.0 {
(h / self.h_old).powf(-self.beta3)
(h.re() / self.h_old).powf(-self.d_term)
} else {
1.0
};
@@ -200,43 +220,53 @@ impl<const D: usize> Controller<D> for PIDController {
// Update error history for next step
self.err_older = self.err_old;
self.err_old = err.max(1.0e-4); // Prevent very small values
self.h_old = h;
self.h_old = h.re();
// Apply maximum step size limit
if h_next.abs() > self.h_max {
h_next = self.h_max.copysign(h_next);
if h_next.re().abs() > self.h_max {
h_next = T::lift(self.h_max.copysign(h_next.re()));
}
TryStep::Accepted(h, h_next)
} else {
// Step rejected - propose smaller step
// Use only proportional control for rejection (more aggressive)
let factor_11 = err.powf(self.beta1);
let factor_11 = err.powf(self.p_term);
let h_next = h / (self.factor_c1.min(factor_11 / self.safety_factor));
// Note: Don't update history on rejection
TryStep::NotYetAccepted(h_next)
}
}
fn next_step_guess(&self) -> TryStep<T> {
self.next_step_guess
}
fn set_next_step_guess(&mut self, guess: TryStep<T>) {
self.next_step_guess = guess;
}
}
impl PIDController {
impl<T: Scalar> PIDController<T> {
/// Create a new PID controller with custom parameters
///
/// # Arguments
///
/// * `beta1` - Proportional coefficient (typically 0.3-0.5)
/// * `beta2` - Integral coefficient (typically 0.04-0.1)
/// * `beta3` - Derivative coefficient (typically 0.01-0.05)
/// * `p_term` - Proportional coefficient (typically 0.3-0.5)
/// * `i_term` - Integral coefficient (typically 0.04-0.1)
/// * `d_term` - Derivative coefficient (typically 0.01-0.05)
/// * `max_factor` - Maximum step size increase factor (typically 10.0)
/// * `min_factor` - Maximum step size decrease factor (typically 0.2)
/// * `h_max` - Maximum allowed step size
/// * `safety_factor` - Safety factor (typically 0.9)
/// * `initial_h` - Initial step size guess
#[allow(clippy::too_many_arguments)]
pub fn new(
beta1: f64,
beta2: f64,
beta3: f64,
p_term: f64,
i_term: f64,
d_term: f64,
max_factor: f64,
min_factor: f64,
h_max: f64,
@@ -244,9 +274,9 @@ impl PIDController {
initial_h: f64,
) -> Self {
Self {
beta1,
beta2,
beta3,
p_term,
i_term,
d_term,
factor_c1: 1.0 / min_factor,
factor_c2: 1.0 / max_factor,
h_max: h_max.abs(),
@@ -254,16 +284,16 @@ impl PIDController {
err_old: 0.0, // No history initially
err_older: 0.0, // No history initially
h_old: 0.0, // No history initially
next_step_guess: TryStep::NotYetAccepted(initial_h),
next_step_guess: TryStep::NotYetAccepted(T::lift(initial_h)),
}
}
/// Create a PID controller with coefficients scaled for a specific method order
///
/// Uses the Gustafsson coefficients scaled by order:
/// - β₁ = 0.49 / (order + 1)
/// - β₂ = 0.34 / (order + 1)
/// - β₃ = 0.10 / (order + 1)
/// - `p_term` = 0.49 / (order + 1)
/// - `i_term` = 0.34 / (order + 1)
/// - `d_term` = 0.10 / (order + 1)
///
/// # Arguments
///
@@ -271,9 +301,9 @@ impl PIDController {
pub fn for_order(order: usize) -> Self {
let k_plus_1 = (order + 1) as f64;
Self::new(
0.49 / k_plus_1, // beta1: proportional
0.34 / k_plus_1, // beta2: integral
0.10 / k_plus_1, // beta3: derivative
0.49 / k_plus_1, // p_term: proportional
0.34 / k_plus_1, // i_term: integral
0.10 / k_plus_1, // d_term: derivative
10.0, // max_factor
0.2, // min_factor
100000.0, // h_max
@@ -292,22 +322,22 @@ impl PIDController {
}
}
impl Default for PIDController {
impl<T: Scalar> Default for PIDController<T> {
/// Default PID controller using conservative coefficients
///
/// Uses conservative PID coefficients that provide stable performance
/// across a wide range of problems:
/// - β₁ = 0.07 (proportional)
/// - β₂ = 0.04 (integral)
/// - β₃ = 0.01 (derivative)
/// - `p_term` = 0.07 (proportional)
/// - `i_term` = 0.04 (integral)
/// - `d_term` = 0.01 (derivative)
///
/// These values are appropriate for typical ODE methods of order 5-7.
/// For method-specific tuning, use `PIDController::for_order(order)` instead.
fn default() -> Self {
Self::new(
0.07, // beta1 (proportional)
0.04, // beta2 (integral)
0.01, // beta3 (derivative)
0.07, // p_term (proportional)
0.04, // i_term (integral)
0.01, // d_term (derivative)
10.0, // max_factor
0.2, // min_factor
100000.0, // h_max
@@ -317,13 +347,14 @@ impl Default for PIDController {
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_pi_controller_creation() {
let controller = PIController::new(0.17, 0.04, 10.0, 0.2, 10.0, 0.9, 1e-4);
let controller = PIController::<f64>::new(0.17, 0.04, 10.0, 0.2, 10.0, 0.9, 1e-4);
assert!(controller.alpha == 0.17);
assert!(controller.beta == 0.04);
@@ -337,11 +368,11 @@ mod tests {
#[test]
fn test_pid_controller_creation() {
let controller = PIDController::new(0.49, 0.34, 0.10, 10.0, 0.2, 10.0, 0.9, 1e-4);
let controller = PIDController::<f64>::new(0.49, 0.34, 0.10, 10.0, 0.2, 10.0, 0.9, 1e-4);
assert_eq!(controller.beta1, 0.49);
assert_eq!(controller.beta2, 0.34);
assert_eq!(controller.beta3, 0.10);
assert_eq!(controller.p_term, 0.49);
assert_eq!(controller.i_term, 0.34);
assert_eq!(controller.d_term, 0.10);
assert_eq!(controller.factor_c1, 1.0 / 0.2);
assert_eq!(controller.factor_c2, 1.0 / 10.0);
assert_eq!(controller.h_max, 10.0);
@@ -353,32 +384,33 @@ mod tests {
#[test]
fn test_pid_for_order() {
let controller = PIDController::for_order(5);
let controller = PIDController::<f64>::for_order(5);
// For order 5, k+1 = 6
assert!((controller.beta1 - 0.49 / 6.0).abs() < 1e-10);
assert!((controller.beta2 - 0.34 / 6.0).abs() < 1e-10);
assert!((controller.beta3 - 0.10 / 6.0).abs() < 1e-10);
assert!((controller.p_term - 0.49 / 6.0).abs() < 1e-10);
assert!((controller.i_term - 0.34 / 6.0).abs() < 1e-10);
assert!((controller.d_term - 0.10 / 6.0).abs() < 1e-10);
}
#[test]
fn test_pid_default() {
let controller = PIDController::default();
let controller = PIDController::<f64>::default();
// Default uses conservative PID coefficients
assert_eq!(controller.beta1, 0.07);
assert_eq!(controller.beta2, 0.04);
assert_eq!(controller.beta3, 0.01); // True PID with derivative term
assert_eq!(controller.p_term, 0.07);
assert_eq!(controller.i_term, 0.04);
assert_eq!(controller.d_term, 0.01); // True PID with derivative term
}
#[test]
fn test_pid_reset() {
let mut controller = PIDController::default();
// Simulate some history
controller.err_old = 0.5;
controller.err_older = 0.3;
controller.h_old = 0.01;
let mut controller = PIDController::<f64> {
err_old: 0.5,
err_older: 0.3,
h_old: 0.01,
..Default::default()
};
controller.reset();
@@ -392,8 +424,8 @@ mod tests {
// For smooth problems, PI and PID should perform similarly
// Test with exponential decay: y' = -y
let mut pi = PIController::default();
let mut pid = PIDController::default();
let mut pi = PIController::<f64>::default();
let mut pid = PIDController::<f64>::default();
// Simulate a sequence of small errors (smooth problem)
let errors = vec![0.8, 0.6, 0.5, 0.45, 0.4, 0.35, 0.3];
@@ -403,8 +435,8 @@ mod tests {
let mut pid_steps = Vec::new();
for &err in &errors {
let mut pi_result = <PIController as Controller<1>>::determine_step(&mut pi, h, err);
let mut pid_result = <PIDController as Controller<1>>::determine_step(&mut pid, h, err);
let mut pi_result = <PIController as Controller<f64>>::determine_step(&mut pi, h, err);
let mut pid_result = <PIDController as Controller<f64>>::determine_step(&mut pid, h, err);
if pi_result.is_accepted() {
pi_steps.push(pi_result.extract());
@@ -437,25 +469,25 @@ mod tests {
#[test]
fn test_pid_bootstrap() {
// Test that PID progressively uses P → PI → PID as history builds
let mut pid = PIDController::new(0.49, 0.34, 0.10, 10.0, 0.2, 100.0, 0.9, 0.01);
let mut pid = PIDController::<f64>::new(0.49, 0.34, 0.10, 10.0, 0.2, 100.0, 0.9, 0.01);
let h = 0.01;
let err1 = 0.5;
let err2 = 0.4;
let err3 = 0.3;
// First step: should use only proportional (beta1)
// First step: should use only proportional (p_term)
assert_eq!(pid.err_old, 0.0);
assert_eq!(pid.err_older, 0.0);
let step1 = <PIDController as Controller<1>>::determine_step(&mut pid, h, err1);
let step1 = <PIDController as Controller<f64>>::determine_step(&mut pid, h, err1);
assert!(step1.is_accepted());
// After first step, err_old is updated but err_older is still 0
assert!(pid.err_old > 0.0);
assert_eq!(pid.err_older, 0.0);
// Second step: should use PI (beta1 and beta2)
let step2 = <PIDController as Controller<1>>::determine_step(&mut pid, h, err2);
// Second step: should use PI (p_term and i_term)
let step2 = <PIDController as Controller<f64>>::determine_step(&mut pid, h, err2);
assert!(step2.is_accepted());
// After second step, both err_old and err_older should be set
@@ -463,21 +495,21 @@ mod tests {
assert!(pid.err_older > 0.0);
assert!(pid.h_old > 0.0);
// Third step: should use full PID (beta1, beta2, and beta3)
let step3 = <PIDController as Controller<1>>::determine_step(&mut pid, h, err3);
// Third step: should use full PID (p_term, i_term, and d_term)
let step3 = <PIDController as Controller<f64>>::determine_step(&mut pid, h, err3);
assert!(step3.is_accepted());
}
#[test]
fn test_pid_step_rejection() {
// Test that error history is NOT updated after rejection
let mut pid = PIDController::default();
let mut pid = PIDController::<f64>::default();
let h = 0.01;
// First, accept a step to build history
let err_good = 0.5;
let step1 = <PIDController as Controller<1>>::determine_step(&mut pid, h, err_good);
let step1 = <PIDController as Controller<f64>>::determine_step(&mut pid, h, err_good);
assert!(step1.is_accepted());
let err_old_before = pid.err_old;
@@ -486,7 +518,7 @@ mod tests {
// Now reject a step with large error
let err_bad = 2.0;
let step2 = <PIDController as Controller<1>>::determine_step(&mut pid, h, err_bad);
let step2 = <PIDController as Controller<f64>>::determine_step(&mut pid, h, err_bad);
assert!(!step2.is_accepted());
// History should NOT have changed after rejection
@@ -500,9 +532,9 @@ mod tests {
// Test on oscillatory error pattern (simulating difficult problem)
// True PID (with derivative term) should provide smoother step size evolution
let mut pi = PIController::default();
// Use actual PID with non-zero beta3 (Gustafsson coefficients for order 5)
let mut pid = PIDController::for_order(5);
let mut pi = PIController::<f64>::default();
// Use actual PID with non-zero d_term (Gustafsson coefficients for order 5)
let mut pid = PIDController::<f64>::for_order(5);
// Simulate oscillatory error pattern
let errors = vec![0.8, 0.3, 0.9, 0.2, 0.85, 0.25, 0.8, 0.3];
@@ -515,8 +547,8 @@ mod tests {
let mut pid_h_prev = h;
for &err in &errors {
let mut pi_result = <PIController as Controller<1>>::determine_step(&mut pi, h, err);
let mut pid_result = <PIDController as Controller<1>>::determine_step(&mut pid, h, err);
let mut pi_result = <PIController as Controller<f64>>::determine_step(&mut pi, h, err);
let mut pid_result = <PIDController as Controller<f64>>::determine_step(&mut pid, h, err);
if pi_result.is_accepted() {
let pi_h_next = pi_result.reset().unwrap().extract();
@@ -550,7 +582,7 @@ mod tests {
/ pid_ratios.len() as f64;
let pid_std = pid_variance.sqrt();
// With true PID (non-zero beta3), we expect similar or better stability
// With true PID (non-zero d_term), we expect similar or better stability
// Allow some tolerance since this is a simple synthetic test
assert!(
pid_std <= pi_std * 1.1,
src/integrator/bs3.rs
+71 -36
View File
@@ -1,7 +1,8 @@
use nalgebra::SVector;
use super::super::ode::ODE;
use super::Integrator;
use super::{Integrator, StepResult};
use crate::scalar::Scalar;
/// Bogacki-Shampine 3/2 integrator trait for tableau coefficients
pub trait BS3Integrator<'a> {
@@ -57,6 +58,15 @@ pub struct BS3<const D: usize> {
r_tol: f64,
}
impl<const D: usize> Default for BS3<D>
where
BS3<D>: Integrator<D>,
{
fn default() -> Self {
Self::new()
}
}
impl<const D: usize> BS3<D>
where
BS3<D>: Integrator<D>,
@@ -140,38 +150,52 @@ where
const ADAPTIVE: bool = true;
const DENSE: bool = true;
fn step<P>(
&self,
ode: &ODE<D, P>,
h: f64,
) -> (SVector<f64, D>, Option<f64>, Option<Vec<SVector<f64, D>>>) {
fn step<T: Scalar, P>(&self, ode: &ODE<T, D, P>, h: T) -> StepResult<T, D> {
// Allocate storage for the 4 stages
let mut k: Vec<SVector<f64, D>> = vec![SVector::<f64, D>::zeros(); Self::STAGES];
let mut k: Vec<SVector<T, D>> = vec![SVector::<T, D>::zeros(); Self::STAGES];
// Stage 1: k1 = f(t, y)
k[0] = (ode.f)(ode.t, ode.y, &ode.params);
// Stage 2: k2 = f(t + c[1]*h, y + h*a[1,0]*k1)
let y2 = ode.y + h * Self::A[0] * k[0];
k[1] = (ode.f)(ode.t + Self::C[1] * h, y2, &ode.params);
let y2 = ode.y + k[0] * (h * Self::A[0]);
k[1] = (ode.f)(ode.t + h * Self::C[1], y2, &ode.params);
// Stage 3: k3 = f(t + c[2]*h, y + h*(a[2,0]*k1 + a[2,1]*k2))
let y3 = ode.y + h * (Self::A[1] * k[0] + Self::A[2] * k[1]);
k[2] = (ode.f)(ode.t + Self::C[2] * h, y3, &ode.params);
let y3 = ode.y + (k[0] * T::lift(Self::A[1]) + k[1] * T::lift(Self::A[2])) * h;
k[2] = (ode.f)(ode.t + h * Self::C[2], y3, &ode.params);
// Stage 4: k4 = f(t + c[3]*h, y + h*(a[3,0]*k1 + a[3,1]*k2 + a[3,2]*k3))
let y4 = ode.y + h * (Self::A[3] * k[0] + Self::A[4] * k[1] + Self::A[5] * k[2]);
k[3] = (ode.f)(ode.t + Self::C[3] * h, y4, &ode.params);
let y4 = ode.y
+ (k[0] * T::lift(Self::A[3]) + k[1] * T::lift(Self::A[4]) + k[2] * T::lift(Self::A[5]))
* h;
k[3] = (ode.f)(ode.t + h * Self::C[3], y4, &ode.params);
// Compute 3rd order solution
let next_y = ode.y + h * (Self::B[0] * k[0] + Self::B[1] * k[1] + Self::B[2] * k[2] + Self::B[3] * k[3]);
let next_y = ode.y
+ (k[0] * T::lift(Self::B[0])
+ k[1] * T::lift(Self::B[1])
+ k[2] * T::lift(Self::B[2])
+ k[3] * T::lift(Self::B[3]))
* h;
// Compute error estimate (difference between 3rd and 2nd order solutions)
let err = h * (Self::B_ERROR[0] * k[0] + Self::B_ERROR[1] * k[1] + Self::B_ERROR[2] * k[2] + Self::B_ERROR[3] * k[3]);
let err = (k[0] * T::lift(Self::B_ERROR[0])
+ k[1] * T::lift(Self::B_ERROR[1])
+ k[2] * T::lift(Self::B_ERROR[2])
+ k[3] * T::lift(Self::B_ERROR[3]))
* h;
// Compute error norm scaled by tolerance
let tol = self.a_tol + ode.y.abs() * self.r_tol;
let error_norm = (err.component_div(&tol)).norm();
// Compute error norm scaled by tolerance. Step control is a real-valued
// decision, so each component drops to the reals as it is accumulated —
// one pass, rather than building real copies of `err` and `y` first.
let mut error_norm_sq = 0.0;
for i in 0..D {
let tol = self.a_tol[i] + ode.y[i].re().abs() * self.r_tol;
let scaled = err[i].re() / tol;
error_norm_sq += scaled * scaled;
}
let error_norm = error_norm_sq.sqrt();
// Store coefficients for dense output (cubic Hermite interpolation)
// BS3 uses standard cubic Hermite interpolation with derivatives at endpoints
@@ -183,16 +207,20 @@ where
k[3], // f(t1, y1) - derivative at end (FSAL)
];
(next_y, Some(error_norm), Some(dense_coeffs))
StepResult {
y: next_y,
err: Some(error_norm),
dense: Some(dense_coeffs),
}
}
fn interpolate(
fn interpolate<T: Scalar>(
&self,
t_start: f64,
t_end: f64,
dense: &[SVector<f64, D>],
t: f64,
) -> SVector<f64, D> {
t_start: T,
t_end: T,
dense: &[SVector<T, D>],
t: T,
) -> SVector<T, D> {
// Compute interpolation parameter θ ∈ [0, 1]
let theta = (t - t_start) / (t_end - t_start);
let h = t_end - t_start;
@@ -215,14 +243,14 @@ where
let f1 = &dense[3];
let theta2 = theta * theta;
let one_minus_theta = 1.0 - theta;
let one_minus_theta = T::lift(1.0) - theta;
let one_minus_theta2 = one_minus_theta * one_minus_theta;
// Apply cubic Hermite interpolation formula
(1.0 + 2.0 * theta) * one_minus_theta2 * y0
+ theta2 * (3.0 - 2.0 * theta) * y1
+ theta * one_minus_theta2 * h * f0
+ theta2 * (theta - 1.0) * h * f1
y0 * ((theta * 2.0 + 1.0) * one_minus_theta2)
+ y1 * (theta2 * (-(theta * 2.0 - 3.0)))
+ f0 * (theta * one_minus_theta2 * h)
+ f1 * (theta2 * (theta - 1.0) * h)
}
}
@@ -233,6 +261,7 @@ mod tests {
use nalgebra::Vector1;
#[test]
#[allow(clippy::assertions_on_constants)]
fn test_bs3_creation() {
let _bs3: BS3<1> = BS3::new();
assert_eq!(BS3::<1>::ORDER, 3);
@@ -254,7 +283,8 @@ mod tests {
let bs3 = BS3::new();
let h = 0.001; // Smaller step size for tighter tolerances
let (y_next, err, dense) = bs3.step(&ode, h);
let __step = bs3.step(&ode, h);
let (y_next, err, dense) = (__step.y, __step.err, __step.dense);
// At t=0.001, exact solution is e^0.001 ≈ 1.0010005001667084
let exact = (0.001_f64).exp();
@@ -283,7 +313,8 @@ mod tests {
let bs3 = BS3::new();
let h = 0.001; // Smaller step size
let (_y_next, _err, dense) = bs3.step(&ode, h);
let __step = bs3.step(&ode, h);
let (_y_next, _err, dense) = (__step.y, __step.err, __step.dense);
let dense = dense.unwrap();
// Interpolate at midpoint
@@ -312,7 +343,8 @@ mod tests {
// Take 100 steps to reach t = 1.0
let mut ode = ODE::new(&derivative, 0.0, 1.0, y0, ());
for _ in 0..100 {
let (y_new, _, _) = bs3.step(&ode, h);
let __step = bs3.step(&ode, h);
let (y_new, _, _) = (__step.y, __step.err, __step.dense);
ode.y = y_new;
ode.t += h;
}
@@ -345,7 +377,8 @@ mod tests {
// Take steps until we reach t_end
while ode.t < t_end - 1e-10 {
let (y_new, _, _) = bs3.step(&ode, h);
let __step = bs3.step(&ode, h);
let (y_new, _, _) = (__step.y, __step.err, __step.dense);
ode.y = y_new;
ode.t += h;
}
@@ -389,7 +422,8 @@ mod tests {
// First step
let ode1 = ODE::new(&derivative, 0.0, 1.0, y0, ());
let (y_new1, _, dense1) = bs3.step(&ode1, h);
let __step = bs3.step(&ode1, h);
let (y_new1, _, dense1) = (__step.y, __step.err, __step.dense);
let dense1 = dense1.unwrap();
// Extract f1 from first step (derivative at end of step)
@@ -397,7 +431,8 @@ mod tests {
// Second step starts where first ended
let ode2 = ODE::new(&derivative, h, 1.0, y_new1, ());
let (_, _, dense2) = bs3.step(&ode2, h);
let __step = bs3.step(&ode2, h);
let (_, _, dense2) = (__step.y, __step.err, __step.dense);
let dense2 = dense2.unwrap();
// Extract f0 from second step (derivative at start of step)
src/integrator/dormand_prince.rs
+117 -37
View File
@@ -1,7 +1,8 @@
use nalgebra::SVector;
use super::super::ode::ODE;
use super::Integrator;
use super::{Integrator, StepResult};
use crate::ode::ODE;
use crate::scalar::Scalar;
/// Integrator Trait
pub trait DormandPrinceIntegrator<'a> {
@@ -11,9 +12,54 @@ pub trait DormandPrinceIntegrator<'a> {
const D: &'a [f64];
}
/// Dormand-Prince 5(4) method
///
/// A 5th order explicit Runge-Kutta method with an embedded 4th order method for
/// error estimation. This is the general-purpose workhorse of the crate, and the
/// same method as MATLAB's `ode45` and `DifferentialEquations.jl`'s `DP5`.
///
/// # Characteristics
/// - Order: 5(4) - 5th order solution with 4th order error estimate
/// - Stages: 7
/// - FSAL: Yes (First Same As Last - the 7th stage is the next step's first,
/// though this implementation does not yet reuse it)
/// - Adaptive: Yes
/// - Dense output: 4th order interpolation
///
/// # When to use DormandPrince45
/// - The default choice when nothing about the problem suggests otherwise
/// - Problems requiring general accuracy (rtol ~ 1e-6 to 1e-8)
/// - Non-stiff problems
///
/// At tighter tolerances than about 1e-8, prefer [`Vern7`](super::vern7::Vern7):
/// a higher order method reaches the same accuracy in far fewer steps.
///
/// # Example
/// ```rust
/// use ordinary_diffeq::prelude::*;
/// use nalgebra::Vector1;
///
/// let params = ();
/// fn derivative(_t: f64, y: Vector1<f64>, _p: &()) -> Vector1<f64> {
/// Vector1::new(-y[0])
/// }
///
/// let y0 = Vector1::new(1.0);
/// let ode = ODE::new(&derivative, 0.0, 5.0, y0, ());
/// let dp45 = DormandPrince45::new().a_tol(1e-8).r_tol(1e-8);
/// let controller = PIController::default();
///
/// let mut problem = Problem::new(ode, dp45, controller);
/// let solution = problem.solve();
/// ```
///
/// # References
/// - Dormand, J.R. and Prince, P.J. (1980), "A family of embedded Runge-Kutta
/// formulae", Journal of Computational and Applied Mathematics, Vol. 6, No. 1,
/// pp. 19-26
#[derive(Debug, Clone, Copy)]
pub struct DormandPrince45<const D: usize> {
a_tol: SVector<f64,D>,
a_tol: SVector<f64, D>,
r_tol: f64,
}
@@ -22,16 +68,43 @@ where
DormandPrince45<D>: Integrator<D>,
{
pub fn new() -> Self {
Self { a_tol: SVector::<f64,D>::from_element(1e-8), r_tol: 1e-8 }
Self {
a_tol: SVector::<f64, D>::from_element(1e-8),
r_tol: 1e-8,
}
}
pub fn a_tol(&mut self, a_tol: f64) -> Self {
Self { a_tol: SVector::<f64,D>::from_element(a_tol), r_tol: self.r_tol }
/// The same absolute tolerance on every component.
pub fn a_tol(self, a_tol: f64) -> Self {
Self {
a_tol: SVector::<f64, D>::from_element(a_tol),
r_tol: self.r_tol,
}
}
pub fn a_tol_full(&mut self, a_tol: SVector::<f64,D>) -> Self {
Self { a_tol, r_tol: self.r_tol }
/// A per-component absolute tolerance.
pub fn a_tol_full(self, a_tol: SVector<f64, D>) -> Self {
Self {
a_tol,
r_tol: self.r_tol,
}
}
pub fn r_tol(&mut self, r_tol: f64) -> Self {
Self { a_tol: self.a_tol, r_tol }
/// The same relative tolerance on every component
pub fn r_tol(self, r_tol: f64) -> Self {
Self {
a_tol: self.a_tol,
r_tol,
}
}
}
impl<const D: usize> Default for DormandPrince45<D>
where
DormandPrince45<D>: Integrator<D>,
{
fn default() -> Self {
Self::new()
}
}
@@ -96,51 +169,58 @@ where
const ADAPTIVE: bool = true;
const DENSE: bool = true;
fn step<P>(
&self,
ode: &ODE<D, P>,
h: f64,
) -> (SVector<f64, D>, Option<f64>, Option<Vec<SVector<f64, D>>>) {
let mut k: Vec<SVector<f64, D>> = vec![SVector::<f64, D>::zeros(); Self::STAGES];
fn step<T: Scalar, P>(&self, ode: &ODE<T, D, P>, h: T) -> StepResult<T, D> {
let mut k: Vec<SVector<T, D>> = vec![SVector::<T, D>::zeros(); Self::STAGES];
let mut next_y = ode.y;
let mut err = SVector::<f64, D>::zeros();
let mut rcont5 = SVector::<f64, D>::zeros();
let mut err = SVector::<T, D>::zeros();
let mut rcont5 = SVector::<T, D>::zeros();
// Do the first of the summations
k[0] = (ode.f)(ode.t, ode.y, &ode.params);
next_y += k[0] * Self::B[0] * h;
err += k[0] * (Self::B[0] - Self::B[Self::STAGES]) * h;
next_y += k[0] * T::lift(Self::B[0]) * h;
err += k[0] * T::lift(Self::B[0] - Self::B[Self::STAGES]) * h;
let rcont1 = ode.y;
rcont5 += k[0] * h * Self::D[0];
rcont5 += k[0] * h * T::lift(Self::D[0]);
// Then the rest
for i in 1..Self::STAGES {
// Compute the ks
let mut y_term = SVector::<f64, D>::zeros();
let mut y_term = SVector::<T, D>::zeros();
for (j, item) in k.iter().enumerate().take(i) {
y_term += item * Self::A[(i * (i - 1)) / 2 + j];
y_term += item * T::lift(Self::A[(i * (i - 1)) / 2 + j]);
}
k[i] = (ode.f)(ode.t + Self::C[i] * h, ode.y + y_term * h, &ode.params);
k[i] = (ode.f)(ode.t + T::lift(Self::C[i]) * h, ode.y + y_term * h, &ode.params);
// Use that and bis to calculate the y and error terms
next_y += k[i] * h * Self::B[i];
err += k[i] * (Self::B[i] - Self::B[i + Self::STAGES]) * h;
rcont5 += k[i] * h * Self::D[i];
next_y += k[i] * h * T::lift(Self::B[i]);
err += k[i] * T::lift(Self::B[i] - Self::B[i + Self::STAGES]) * h;
rcont5 += k[i] * h * T::lift(Self::D[i]);
}
let rcont2 = next_y - ode.y;
let rcont3 = h * k[0] - rcont2;
let rcont3 = k[0] * h - rcont2;
let rcont4 = rcont2 - k[Self::STAGES - 1] * h - rcont3;
let tol = self.a_tol + ode.y * self.r_tol;
let rcont = vec![rcont1, rcont2, rcont3, rcont4, rcont5];
(next_y, Some((err.component_div(&tol)).norm()), Some(rcont))
// Just take the real component of the error
let mut error_norm_sq = 0.0;
for i in 0..D {
let tol = self.a_tol[i] + ode.y[i].re() * self.r_tol;
let scaled = err[i].re() / tol;
error_norm_sq += scaled * scaled;
}
StepResult {
y: next_y,
err: Some(error_norm_sq.sqrt()),
dense: Some(rcont),
}
}
fn interpolate(
fn interpolate<T: Scalar>(
&self,
t_start: f64,
t_end: f64,
dense: &[SVector<f64, D>],
t: f64,
) -> SVector<f64, D> {
t_start: T,
t_end: T,
dense: &[SVector<T, D>],
t: T,
) -> SVector<T, D> {
let s = (t - t_start) / (t_end - t_start);
let s1 = 1.0 - s;
let s1 = T::one() - s;
dense[0] + (dense[1] + (dense[2] + (dense[3] + dense[4] * s1) * s) * s1) * s
}
}
src/integrator/mod.rs
+44 -25
View File
@@ -1,12 +1,34 @@
use nalgebra::SVector;
use super::ode::ODE;
use crate::ode::ODE;
use crate::scalar::Scalar;
pub mod bs3;
pub mod dormand_prince;
pub mod rosenbrock;
pub mod vern7;
/// Result of a `Integrator::step`
#[derive(Debug, Clone, PartialEq)]
pub struct StepResult<T: Scalar, const D: usize> {
/// The state at the end of the step.
pub y: SVector<T, D>,
/// The local error, normalized by the integrator's tolerances, or `None`
/// from a fixed-step method. A value at or below `1.0` means the step is
/// acceptable.
pub err: Option<f64>,
/// Coefficients for interpolating inside the step, to be handed back to
/// [`Integrator::interpolate`], or `None` from a method without dense
/// output.
///
/// For a method with lazy dense output (`EXTRA_STAGES > 0`) the layout is
/// `[y_start, y_end, k_1, ..., k_MAIN_STAGES]`, and the extra stages are
/// appended by [`Integrator::compute_extra_stages`] on first use.
pub dense: Option<Vec<SVector<T, D>>>,
}
/// Integrator Trait
pub trait Integrator<const D: usize> {
const ORDER: usize;
@@ -23,21 +45,18 @@ pub trait Integrator<const D: usize> {
/// Total stages when full dense output is computed
const TOTAL_DENSE_STAGES: usize = Self::MAIN_STAGES + Self::EXTRA_STAGES;
/// Returns a new y value, then possibly an error value, and possibly a dense output
/// coefficient set
fn step<P>(
&self,
ode: &ODE<D, P>,
h: f64,
) -> (SVector<f64, D>, Option<f64>, Option<Vec<SVector<f64, D>>>);
/// Advances `ode` by `h` from its current state, without mutating it.
fn step<T: Scalar, P>(&self, ode: &ODE<T, D, P>, h: T) -> StepResult<T, D>;
fn interpolate(
/// Evaluates the solution at `t` inside a step that ran from `t_start` to
/// `t_end`, using the `dense` coefficients that step produced.
fn interpolate<T: Scalar>(
&self,
t_start: f64,
t_end: f64,
dense: &[SVector<f64, D>],
t: f64,
) -> SVector<f64, D>;
t_start: T,
t_end: T,
dense: &[SVector<T, D>],
t: T,
) -> SVector<T, D>;
/// Compute extra stages for full-order dense output (lazy computation).
///
@@ -56,14 +75,14 @@ pub trait Integrator<const D: usize> {
/// # Returns
///
/// Vector of extra k-stages (empty for most integrators)
fn compute_extra_stages<P>(
fn compute_extra_stages<T: Scalar, P>(
&self,
_ode: &ODE<D, P>,
_t_start: f64,
_y_start: SVector<f64, D>,
_h: f64,
_main_stages: &[SVector<f64, D>],
) -> Vec<SVector<f64, D>> {
_ode: &ODE<T, D, P>,
_t_start: T,
_y_start: SVector<T, D>,
_h: T,
_main_stages: &[SVector<T, D>],
) -> Vec<SVector<T, D>> {
// Default implementation: no extra stages needed
Vec::new()
}
@@ -79,7 +98,7 @@ mod tests {
#[test]
fn test_dopri() {
type Params = ();
fn derivative(_t: f64, y: Vector3<f64>, _p: &Params) -> Vector3<f64> {
fn derivative<T: Scalar>(_t: T, y: Vector3<T>, _p: &Params) -> Vector3<T> {
y
}
@@ -92,11 +111,11 @@ mod tests {
// and also that the error seems reasonable
let step = 0.001;
while ode.t < ode.t_end {
let (new_y, err, _) = dp45.step(&ode, step);
ode.y = new_y;
let result = dp45.step(&ode, step);
ode.y = result.y;
ode.t += step;
assert_relative_eq!(ode.y[0], ode.t.exp(), max_relative = 0.01);
assert!(err.unwrap() < 1.0);
assert!(result.err.unwrap() < 1.0);
}
}
}
src/integrator/rosenbrock.rs
+44 -56
View File
@@ -1,7 +1,8 @@
use nalgebra::{SMatrix, SVector};
use super::super::ode::ODE;
use super::Integrator;
use super::super::ode::{ProblemFunction, ODE};
use super::{Integrator, StepResult};
use crate::scalar::Scalar;
/// Strategy for when to update the Jacobian matrix
#[derive(Debug, Clone, Copy)]
@@ -31,24 +32,26 @@ impl Default for JacobianUpdateStrategy {
///
/// Uses forward differences: J[i,j] ≈ (f_i(y + ε*e_j) - f_i(y)) / ε
/// where ε = √(machine_epsilon) * max(|y[j]|, 1.0)
pub fn compute_jacobian<const D: usize, P>(
f: &dyn Fn(f64, SVector<f64, D>, &P) -> SVector<f64, D>,
t: f64,
y: SVector<f64, D>,
pub fn compute_jacobian<T: Scalar, const D: usize, P>(
f: ProblemFunction<'_, T, D, P>,
t: T,
y: SVector<T, D>,
params: &P,
) -> SMatrix<f64, D, D> {
) -> SMatrix<T, D, D> {
let sqrt_eps = f64::EPSILON.sqrt();
let f_y = f(t, y, params);
let mut jacobian = SMatrix::<f64, D, D>::zeros();
let mut jacobian = SMatrix::<T, D, D>::zeros();
// Compute each column of the Jacobian by perturbing y[j]
for j in 0..D {
// Choose epsilon based on the magnitude of y[j]
let epsilon = sqrt_eps * y[j].abs().max(1.0);
// Choose epsilon based on the magnitude of y[j]. The perturbation is a
// numerical-method parameter, not part of the solution, so it is chosen
// in the reals.
let epsilon = sqrt_eps * y[j].re().abs().max(1.0);
// Perturb y in the j-th direction
let mut y_perturbed = y;
y_perturbed[j] += epsilon;
y_perturbed[j] += T::lift(epsilon);
// Evaluate f at perturbed point
let f_perturbed = f(t, y_perturbed, params);
@@ -131,8 +134,8 @@ pub fn compute_jacobian<const D: usize, P>(
/// let rosenbrock = Rosenbrock23::new();
///
/// // Take a single step
/// let (y_next, error, _dense) = rosenbrock.step(&ode, 0.1);
/// assert!((y_next[0] - 0.905).abs() < 0.01);
/// let step = rosenbrock.step(&ode, 0.1);
/// assert!((step.y[0] - 0.905).abs() < 0.01);
/// ```
#[derive(Debug, Clone, Copy)]
pub struct Rosenbrock23<const D: usize> {
@@ -218,81 +221,61 @@ impl<const D: usize> Integrator<D> for Rosenbrock23<D> {
const ADAPTIVE: bool = true;
const DENSE: bool = true;
fn step<P>(
&self,
ode: &ODE<D, P>,
h: f64,
) -> (SVector<f64, D>, Option<f64>, Option<Vec<SVector<f64, D>>>) {
fn step<T: Scalar, P>(&self, ode: &ODE<T, D, P>, h: T) -> StepResult<T, D> {
let t = ode.t;
let uprev = ode.y;
// Compute Jacobian
let j = compute_jacobian(&ode.f, t, uprev, &ode.params);
// Julia: dtγ = dt * d
let dtgamma = h * self.d;
// Form W = I - dtγ*J
let w = SMatrix::<f64, D, D>::identity() - dtgamma * j;
let w = SMatrix::<T, D, D>::identity() - j * dtgamma;
let w_inv = w.try_inverse().expect("W matrix is singular");
// Evaluate fsalfirst = f(uprev)
let fsalfirst = (ode.f)(t, uprev, &ode.params);
// Stage 1: Solve W * k₁ = f(y) where k₁ is a derivative estimate
// Julia stores derivatives in k, not displacements
let k1 = w_inv * fsalfirst;
// Stage 2: u = uprev + dt/2 * k₁
// Julia line 69
let dto2 = h / 2.0;
let u = uprev + dto2 * k1;
let u = uprev + k1 * dto2;
// Evaluate f₁ = f(u, t + dt/2)
// Julia line 71
let f1 = (ode.f)(t + dto2, u, &ode.params);
// Stage 2: W * k₂ = f₁ - k₁ + J*k₁
// Julia line 80: linsolve_tmp = f₁ - tmp (where tmp = k₁)
// This is equivalent to: W * k₂ = f₁ - k₁
// => (I - dtγ*J) * k₂ = f₁ - k₁
// => k₂ = (I - dtγ*J)^{-1} * (f₁ - k₁)
// But actually, maybe the RHS should be scaled differently. Let me try: W * k₂ = f₁ + J*k₁
// Since W = I - dtγ*J, we have W*k₂ - I*k₂ = -dtγ*J*k₂, so if RHS = f₁ + J*k₁...
// Actually, let's just implement exactly what Julia does:
let rhs2 = f1 - k1;
let k2_temp = w_inv * rhs2;
// Julia then does: k₂ = k₂_temp * neginvdtγ + k₁
// But neginvdtγ = -1/(dtγ), which would give huge values.
// Let me try without that scaling:
let k2 = k2_temp + k1;
// Solution: u = uprev + dt * k₂
// Julia line 89
let u_final = uprev + h * k2;
let u_final = uprev + k2 * h;
// Error estimation
// Evaluate fsallast = f(u_final, t + dt)
// Julia line 94
let fsallast = (ode.f)(t + h, u_final, &ode.params);
// Julia line 98-99: linsolve_tmp = fsallast - c₃₂*(k₂ - f₁) - 2*(k₁ - fsalfirst) + dt*dT
// Ignoring dT (time derivative) for autonomous systems
let linsolve_tmp3 = fsallast - self.c32 * (k2 - f1) - 2.0 * (k1 - fsalfirst);
let linsolve_tmp3 =
fsallast - (k2 - f1) * T::lift(self.c32) - (k1 - fsalfirst) * T::lift(2.0);
// Stage 3 for error estimation: W * k₃ = linsolve_tmp3
let k3 = w_inv * linsolve_tmp3;
// Error: dt/6 * (k₁ - 2*k₂ + k₃)
// Julia line 115
let dto6 = h / 6.0;
let error_vec = dto6 * (k1 - 2.0 * k2 + k3);
let error_vec = (k1 - k2 * T::lift(2.0) + k3) * dto6;
// Compute scalar error estimate using weighted norm
let mut error_sum = 0.0;
for i in 0..D {
let scale = self.a_tol + self.r_tol * uprev[i].abs().max(u_final[i].abs());
let weighted_error = error_vec[i] / scale;
let scale =
self.a_tol + self.r_tol * uprev[i].re().abs().max(u_final[i].re().abs());
let weighted_error = error_vec[i].re() / scale;
error_sum += weighted_error * weighted_error;
}
let error = (error_sum / D as f64).sqrt();
@@ -300,16 +283,20 @@ impl<const D: usize> Integrator<D> for Rosenbrock23<D> {
// Dense output: store k₁ and k₂
let dense = Some(vec![k1, k2]);
(u_final, Some(error), dense)
StepResult {
y: u_final,
err: Some(error),
dense,
}
}
fn interpolate(
fn interpolate<T: Scalar>(
&self,
t_start: f64,
t_end: f64,
dense: &[SVector<f64, D>],
t: f64,
) -> SVector<f64, D> {
t_start: T,
t_end: T,
dense: &[SVector<T, D>],
t: T,
) -> SVector<T, D> {
// Second order interpolation using k₁ and k₂
// For Rosenbrock methods, we use a simple Hermite interpolation
let k1 = dense[0];
@@ -323,7 +310,7 @@ impl<const D: usize> Integrator<D> for Rosenbrock23<D> {
// y(t) ≈ y_n + θ*h*((1-θ)*k₁ + θ*k₂)
// But we don't have y_n stored, so we just return the stage interpolation
// This is a simple linear interpolation of the derivative
(1.0 - theta) * k1 + theta * k2
k1 * (T::lift(1.0) - theta) + k2 * theta
}
}
@@ -383,7 +370,8 @@ mod tests {
let ode = ODE::new(&derivative, 0.0, 0.1, y0, ());
let rb23 = Rosenbrock23::new();
let (y_next, err, _) = rb23.step(&ode, 0.1);
let __step = rb23.step(&ode, 0.1);
let (y_next, err, _) = (__step.y, __step.err, __step.dense);
let analytical = (-0.1_f64).exp();
println!("Computed: {}, Analytical: {}", y_next[0], analytical);
@@ -416,7 +404,8 @@ mod tests {
let rb23 = Rosenbrock23::new();
while ode.t < t_end - 1e-10 {
let (y_next, _, _) = rb23.step(&ode, h);
let __step = rb23.step(&ode, h);
let (y_next, _, _) = (__step.y, __step.err, __step.dense);
ode.y = y_next;
ode.t += h;
}
@@ -465,7 +454,8 @@ mod tests {
let rb23 = Rosenbrock23::new();
for _ in 0..n_steps {
let (y_next, _, _) = rb23.step(&ode, actual_h);
let __step = rb23.step(&ode, actual_h);
let (y_next, _, _) = (__step.y, __step.err, __step.dense);
ode.y = y_next;
ode.t += actual_h;
}
@@ -497,7 +487,6 @@ mod tests {
#[test]
fn test_rosenbrock23_adaptive_solve() {
// Julia test: sol = solve(prob, Rosenbrock23()); @test length(sol) < 20
// This tests that the adaptive solver can efficiently solve prob_ode_linear
use crate::controller::PIController;
use crate::problem::Problem;
@@ -518,7 +507,6 @@ mod tests {
println!("Adaptive solve completed in {} steps", solution.states.len());
// Julia test: @test length(sol) < 20
assert!(solution.states.len() < 20,
"Adaptive solve should complete in less than 20 steps, got {}",
solution.states.len());
src/integrator/vern7.rs
+117 -99
View File
@@ -1,7 +1,8 @@
use nalgebra::SVector;
use super::super::ode::ODE;
use super::Integrator;
use super::{Integrator, StepResult};
use crate::scalar::Scalar;
/// Verner 7 integrator trait for tableau coefficients
pub trait Vern7Integrator<'a> {
@@ -69,6 +70,15 @@ pub struct Vern7<const D: usize> {
r_tol: f64,
}
impl<const D: usize> Default for Vern7<D>
where
Vern7<D>: Integrator<D>,
{
fn default() -> Self {
Self::new()
}
}
impl<const D: usize> Vern7<D>
where
Vern7<D>: Integrator<D>,
@@ -278,13 +288,9 @@ where
const MAIN_STAGES: usize = 10;
const EXTRA_STAGES: usize = 6;
fn step<P>(
&self,
ode: &ODE<D, P>,
h: f64,
) -> (SVector<f64, D>, Option<f64>, Option<Vec<SVector<f64, D>>>) {
fn step<T: Scalar, P>(&self, ode: &ODE<T, D, P>, h: T) -> StepResult<T, D> {
// Allocate storage for the 10 stages
let mut k: Vec<SVector<f64, D>> = vec![SVector::<f64, D>::zeros(); Self::STAGES];
let mut k: Vec<SVector<T, D>> = vec![SVector::<T, D>::zeros(); Self::STAGES];
// Stage 1: k[0] = f(t, y)
k[0] = (ode.f)(ode.t, ode.y, &ode.params);
@@ -295,43 +301,54 @@ where
// A matrix is stored in lower triangular form, row by row
// Row i has i elements (0-indexed), starting at position i*(i-1)/2
let row_start = (i * (i - 1)) / 2;
for j in 0..i {
y_temp += k[j] * Self::A[row_start + j] * h;
for (j, kj) in k.iter().enumerate().take(i) {
y_temp += kj * T::lift(Self::A[row_start + j]) * h;
}
k[i] = (ode.f)(ode.t + Self::C[i] * h, y_temp, &ode.params);
k[i] = (ode.f)(ode.t + h * Self::C[i], y_temp, &ode.params);
}
// Compute 7th order solution using B weights
let mut next_y = ode.y;
for i in 0..Self::STAGES {
next_y += k[i] * Self::B[i] * h;
for (i, ki) in k.iter().enumerate() {
next_y += ki * T::lift(Self::B[i]) * h;
}
// Compute error estimate using B_ERROR weights
let mut err = SVector::<f64, D>::zeros();
for i in 0..Self::STAGES {
err += k[i] * Self::B_ERROR[i] * h;
let mut err = SVector::<T, D>::zeros();
for (i, ki) in k.iter().enumerate() {
err += ki * T::lift(Self::B_ERROR[i]) * h;
}
// Compute error norm scaled by tolerance
let tol = self.a_tol + ode.y.abs() * self.r_tol;
let error_norm = (err.component_div(&tol)).norm();
// Compute error norm scaled by tolerance. Step control is a
// real-valued decision, so each component drops to the reals as it is
// accumulated rather than building real copies of `err` and `y` first.
let mut error_norm_sq = 0.0;
for i in 0..D {
let tol = self.a_tol[i] + ode.y[i].re().abs() * self.r_tol;
let scaled = err[i].re() / tol;
error_norm_sq += scaled * scaled;
}
let error_norm = error_norm_sq.sqrt();
// Store dense output coefficients
// For now, store all k values for interpolation
let mut dense_coeffs = vec![ode.y, next_y];
dense_coeffs.extend_from_slice(&k);
(next_y, Some(error_norm), Some(dense_coeffs))
StepResult {
y: next_y,
err: Some(error_norm),
dense: Some(dense_coeffs),
}
}
fn interpolate(
fn interpolate<T: Scalar>(
&self,
t_start: f64,
t_end: f64,
dense: &[SVector<f64, D>],
t: f64,
) -> SVector<f64, D> {
t_start: T,
t_end: T,
dense: &[SVector<T, D>],
t: T,
) -> SVector<T, D> {
// Vern7 uses 7th order polynomial interpolation
// Check if extra stages (k11-k16) are available
// Dense array format: [y0, y1, k1, k2, ..., k10, k11, ..., k16]
@@ -358,8 +375,8 @@ where
// Helper to evaluate polynomial using Horner's method
#[inline]
fn evalpoly(x: f64, coeffs: &[f64]) -> f64 {
let mut result = 0.0;
fn evalpoly<T: Scalar>(x: T, coeffs: &[f64]) -> T {
let mut result = T::lift(0.0);
for &c in coeffs.iter().rev() {
result = result * x + c;
}
@@ -433,13 +450,13 @@ where
]);
// Compute base interpolation with main stages
let mut result = y0 + h * (k1 * b1_theta +
let mut result = y0 + (k1 * b1_theta +
k4 * b4_theta +
k5 * b5_theta +
k6 * b6_theta +
k7 * b7_theta +
k8 * b8_theta +
k9 * b9_theta);
k9 * b9_theta) * h;
// If extra stages are available, add their contribution for full 7th order accuracy
if dense.len() >= 2 + Self::TOTAL_DENSE_STAGES {
@@ -507,25 +524,25 @@ where
]);
// Add contribution from extra stages
result += h * (k11 * b11_theta +
result += (k11 * b11_theta +
k12 * b12_theta +
k13 * b13_theta +
k14 * b14_theta +
k15 * b15_theta +
k16 * b16_theta);
k16 * b16_theta) * h;
}
result
}
fn compute_extra_stages<P>(
fn compute_extra_stages<T: Scalar, P>(
&self,
ode: &ODE<D, P>,
t_start: f64,
y_start: SVector<f64, D>,
h: f64,
main_stages: &[SVector<f64, D>],
) -> Vec<SVector<f64, D>> {
ode: &ODE<T, D, P>,
t_start: T,
y_start: SVector<T, D>,
h: T,
main_stages: &[SVector<T, D>],
) -> Vec<SVector<T, D>> {
// Extract main stages that are used in extra stage computation
// From Julia: extra stages use k1, k4, k5, k6, k7, k8, k9
let k1 = &main_stages[0];
@@ -540,86 +557,86 @@ where
// Stage 11: uses k1, k4-k9 (7 coefficients)
let mut y11 = y_start;
y11 += k1 * Self::A_EXTRA[0] * h;
y11 += k4 * Self::A_EXTRA[1] * h;
y11 += k5 * Self::A_EXTRA[2] * h;
y11 += k6 * Self::A_EXTRA[3] * h;
y11 += k7 * Self::A_EXTRA[4] * h;
y11 += k8 * Self::A_EXTRA[5] * h;
y11 += k9 * Self::A_EXTRA[6] * h;
let k11 = (ode.f)(t_start + Self::C_EXTRA[0] * h, y11, &ode.params);
y11 += k1 * T::lift(Self::A_EXTRA[0]) * h;
y11 += k4 * T::lift(Self::A_EXTRA[1]) * h;
y11 += k5 * T::lift(Self::A_EXTRA[2]) * h;
y11 += k6 * T::lift(Self::A_EXTRA[3]) * h;
y11 += k7 * T::lift(Self::A_EXTRA[4]) * h;
y11 += k8 * T::lift(Self::A_EXTRA[5]) * h;
y11 += k9 * T::lift(Self::A_EXTRA[6]) * h;
let k11 = (ode.f)(t_start + h * Self::C_EXTRA[0], y11, &ode.params);
extra_stages.push(k11);
// Stage 12: uses k1, k4-k9, k11 (8 coefficients)
let mut y12 = y_start;
y12 += k1 * Self::A_EXTRA[7] * h;
y12 += k4 * Self::A_EXTRA[8] * h;
y12 += k5 * Self::A_EXTRA[9] * h;
y12 += k6 * Self::A_EXTRA[10] * h;
y12 += k7 * Self::A_EXTRA[11] * h;
y12 += k8 * Self::A_EXTRA[12] * h;
y12 += k9 * Self::A_EXTRA[13] * h;
y12 += &extra_stages[0] * Self::A_EXTRA[14] * h; // k11
let k12 = (ode.f)(t_start + Self::C_EXTRA[1] * h, y12, &ode.params);
y12 += k1 * T::lift(Self::A_EXTRA[7]) * h;
y12 += k4 * T::lift(Self::A_EXTRA[8]) * h;
y12 += k5 * T::lift(Self::A_EXTRA[9]) * h;
y12 += k6 * T::lift(Self::A_EXTRA[10]) * h;
y12 += k7 * T::lift(Self::A_EXTRA[11]) * h;
y12 += k8 * T::lift(Self::A_EXTRA[12]) * h;
y12 += k9 * T::lift(Self::A_EXTRA[13]) * h;
y12 += extra_stages[0] * T::lift(Self::A_EXTRA[14]) * h; // k11
let k12 = (ode.f)(t_start + h * Self::C_EXTRA[1], y12, &ode.params);
extra_stages.push(k12);
// Stage 13: uses k1, k4-k9, k11, k12 (9 coefficients)
let mut y13 = y_start;
y13 += k1 * Self::A_EXTRA[15] * h;
y13 += k4 * Self::A_EXTRA[16] * h;
y13 += k5 * Self::A_EXTRA[17] * h;
y13 += k6 * Self::A_EXTRA[18] * h;
y13 += k7 * Self::A_EXTRA[19] * h;
y13 += k8 * Self::A_EXTRA[20] * h;
y13 += k9 * Self::A_EXTRA[21] * h;
y13 += &extra_stages[0] * Self::A_EXTRA[22] * h; // k11
y13 += &extra_stages[1] * Self::A_EXTRA[23] * h; // k12
let k13 = (ode.f)(t_start + Self::C_EXTRA[2] * h, y13, &ode.params);
y13 += k1 * T::lift(Self::A_EXTRA[15]) * h;
y13 += k4 * T::lift(Self::A_EXTRA[16]) * h;
y13 += k5 * T::lift(Self::A_EXTRA[17]) * h;
y13 += k6 * T::lift(Self::A_EXTRA[18]) * h;
y13 += k7 * T::lift(Self::A_EXTRA[19]) * h;
y13 += k8 * T::lift(Self::A_EXTRA[20]) * h;
y13 += k9 * T::lift(Self::A_EXTRA[21]) * h;
y13 += extra_stages[0] * T::lift(Self::A_EXTRA[22]) * h; // k11
y13 += extra_stages[1] * T::lift(Self::A_EXTRA[23]) * h; // k12
let k13 = (ode.f)(t_start + h * Self::C_EXTRA[2], y13, &ode.params);
extra_stages.push(k13);
// Stage 14: uses k1, k4-k9, k11, k12, k13 (10 coefficients)
let mut y14 = y_start;
y14 += k1 * Self::A_EXTRA[24] * h;
y14 += k4 * Self::A_EXTRA[25] * h;
y14 += k5 * Self::A_EXTRA[26] * h;
y14 += k6 * Self::A_EXTRA[27] * h;
y14 += k7 * Self::A_EXTRA[28] * h;
y14 += k8 * Self::A_EXTRA[29] * h;
y14 += k9 * Self::A_EXTRA[30] * h;
y14 += &extra_stages[0] * Self::A_EXTRA[31] * h; // k11
y14 += &extra_stages[1] * Self::A_EXTRA[32] * h; // k12
y14 += &extra_stages[2] * Self::A_EXTRA[33] * h; // k13
let k14 = (ode.f)(t_start + Self::C_EXTRA[3] * h, y14, &ode.params);
y14 += k1 * T::lift(Self::A_EXTRA[24]) * h;
y14 += k4 * T::lift(Self::A_EXTRA[25]) * h;
y14 += k5 * T::lift(Self::A_EXTRA[26]) * h;
y14 += k6 * T::lift(Self::A_EXTRA[27]) * h;
y14 += k7 * T::lift(Self::A_EXTRA[28]) * h;
y14 += k8 * T::lift(Self::A_EXTRA[29]) * h;
y14 += k9 * T::lift(Self::A_EXTRA[30]) * h;
y14 += extra_stages[0] * T::lift(Self::A_EXTRA[31]) * h; // k11
y14 += extra_stages[1] * T::lift(Self::A_EXTRA[32]) * h; // k12
y14 += extra_stages[2] * T::lift(Self::A_EXTRA[33]) * h; // k13
let k14 = (ode.f)(t_start + h * Self::C_EXTRA[3], y14, &ode.params);
extra_stages.push(k14);
// Stage 15: uses k1, k4-k9, k11, k12, k13 (10 coefficients, reuses k13 not k14)
let mut y15 = y_start;
y15 += k1 * Self::A_EXTRA[34] * h;
y15 += k4 * Self::A_EXTRA[35] * h;
y15 += k5 * Self::A_EXTRA[36] * h;
y15 += k6 * Self::A_EXTRA[37] * h;
y15 += k7 * Self::A_EXTRA[38] * h;
y15 += k8 * Self::A_EXTRA[39] * h;
y15 += k9 * Self::A_EXTRA[40] * h;
y15 += &extra_stages[0] * Self::A_EXTRA[41] * h; // k11
y15 += &extra_stages[1] * Self::A_EXTRA[42] * h; // k12
y15 += &extra_stages[2] * Self::A_EXTRA[43] * h; // k13
let k15 = (ode.f)(t_start + Self::C_EXTRA[4] * h, y15, &ode.params);
y15 += k1 * T::lift(Self::A_EXTRA[34]) * h;
y15 += k4 * T::lift(Self::A_EXTRA[35]) * h;
y15 += k5 * T::lift(Self::A_EXTRA[36]) * h;
y15 += k6 * T::lift(Self::A_EXTRA[37]) * h;
y15 += k7 * T::lift(Self::A_EXTRA[38]) * h;
y15 += k8 * T::lift(Self::A_EXTRA[39]) * h;
y15 += k9 * T::lift(Self::A_EXTRA[40]) * h;
y15 += extra_stages[0] * T::lift(Self::A_EXTRA[41]) * h; // k11
y15 += extra_stages[1] * T::lift(Self::A_EXTRA[42]) * h; // k12
y15 += extra_stages[2] * T::lift(Self::A_EXTRA[43]) * h; // k13
let k15 = (ode.f)(t_start + h * Self::C_EXTRA[4], y15, &ode.params);
extra_stages.push(k15);
// Stage 16: uses k1, k4-k9, k11, k12, k13 (10 coefficients, reuses k13 not k14 or k15)
let mut y16 = y_start;
y16 += k1 * Self::A_EXTRA[44] * h;
y16 += k4 * Self::A_EXTRA[45] * h;
y16 += k5 * Self::A_EXTRA[46] * h;
y16 += k6 * Self::A_EXTRA[47] * h;
y16 += k7 * Self::A_EXTRA[48] * h;
y16 += k8 * Self::A_EXTRA[49] * h;
y16 += k9 * Self::A_EXTRA[50] * h;
y16 += &extra_stages[0] * Self::A_EXTRA[51] * h; // k11
y16 += &extra_stages[1] * Self::A_EXTRA[52] * h; // k12
y16 += &extra_stages[2] * Self::A_EXTRA[53] * h; // k13
let k16 = (ode.f)(t_start + Self::C_EXTRA[5] * h, y16, &ode.params);
y16 += k1 * T::lift(Self::A_EXTRA[44]) * h;
y16 += k4 * T::lift(Self::A_EXTRA[45]) * h;
y16 += k5 * T::lift(Self::A_EXTRA[46]) * h;
y16 += k6 * T::lift(Self::A_EXTRA[47]) * h;
y16 += k7 * T::lift(Self::A_EXTRA[48]) * h;
y16 += k8 * T::lift(Self::A_EXTRA[49]) * h;
y16 += k9 * T::lift(Self::A_EXTRA[50]) * h;
y16 += extra_stages[0] * T::lift(Self::A_EXTRA[51]) * h; // k11
y16 += extra_stages[1] * T::lift(Self::A_EXTRA[52]) * h; // k12
y16 += extra_stages[2] * T::lift(Self::A_EXTRA[53]) * h; // k13
let k16 = (ode.f)(t_start + h * Self::C_EXTRA[5], y16, &ode.params);
extra_stages.push(k16);
extra_stages
@@ -706,7 +723,8 @@ mod tests {
while ode.t < t_end {
let h_step = h.min(t_end - ode.t);
let (next_y, _, _) = vern7.step(&ode, h_step);
let __step = vern7.step(&ode, h_step);
let (next_y, _, _) = (__step.y, __step.err, __step.dense);
ode.y = next_y;
ode.t += h_step;
}
src/lib.rs
+201 -4
View File
@@ -1,20 +1,27 @@
#![allow(dead_code)]
// The readme is the crate's front page, and including it here means every
// example in it is compiled and run by `cargo test`. Requires `autodiff`
// because one of those examples differentiates.
#![cfg_attr(feature = "autodiff", doc = include_str!("../readme.md"))]
pub mod callback;
pub mod controller;
pub mod integrator;
pub mod ode;
pub mod problem;
pub mod scalar;
pub mod prelude {
pub use super::callback::{stop, Callback};
pub use super::callback::{stop, Callback, FnCallback};
pub use super::controller::{PIController, PIDController};
pub use super::integrator::bs3::BS3;
pub use super::integrator::dormand_prince::DormandPrince45;
pub use super::integrator::rosenbrock::Rosenbrock23;
pub use super::integrator::vern7::Vern7;
pub use super::ode::ODE;
pub use super::integrator::{Integrator, StepResult};
pub use super::problem::{Problem, Solution};
pub use super::scalar::Scalar;
}
#[cfg(test)]
@@ -44,13 +51,13 @@ mod tests {
let dp45 = DormandPrince45::new().a_tol(1e-12).r_tol(1e-6);
let controller = PIController::default();
let value_too_high = Callback {
event: &|t: f64, _y: Vector2<f64>, _p: &Params| 5.0 - t,
let value_too_high = FnCallback {
event: &|t: f64, _y: &Vector2<f64>, _p: &Params| 5.0 - t,
effect: &stop,
};
// Solve the problem
let mut problem = Problem::new(ode, dp45, controller).with_callback(value_too_high);
let mut problem = Problem::new(ode, dp45, controller).with_callback(&value_too_high);
let solution = problem.solve();
// Can interpolate solutions to whatever you want
@@ -150,3 +157,193 @@ mod tests {
);
}
}
/// Tests that the derivatives a dual instantiation produces are the right ones.
#[cfg(all(test, feature = "autodiff"))]
mod autodiff_tests {
use crate::ode::ProblemFunction;
use crate::prelude::*;
use approx::assert_relative_eq;
use nalgebra::{Matrix6, SVector, Vector1, Vector6};
use num_dual::{Dual64, DualSVec64};
use std::f64::consts::PI;
const MU: f64 = 3.98600441500000e14;
const SMA: f64 = 6.7781363e6;
/// The two-body right-hand side, shared by the orbit tests below.
fn two_body<T: Scalar>(_t: T, state: Vector6<T>, p: &(T,)) -> Vector6<T> {
let r = state.fixed_rows::<3>(0);
let acc = -(r * p.0) / r.norm().powi(3);
Vector6::new(state[3], state[4], state[5], acc[0], acc[1], acc[2])
}
fn orbit_y0<T: Scalar>() -> Vector6<T> {
Vector6::new(
T::lift(4.263868426884883e6),
T::lift(5.146189057155391e6),
T::lift(1.1310208421331816e6),
T::lift(-5923.454461876975),
T::lift(4496.802639690076),
T::lift(1870.3893008991558),
)
}
fn solve_with<T: Scalar, const D: usize, P, S: Integrator<D> + Copy>(
f: ProblemFunction<'_, T, D, P>,
integrator: S,
t0: T,
t_end: T,
y0: SVector<T, D>,
params: P,
) -> SVector<T, D> {
let ode = ODE::new(f, t0, t_end, y0, params);
let controller = PIController::new(0.37, 0.04, 10.0, 0.2, 1000.0, 0.9, 0.01);
let mut problem = Problem::new(ode, integrator, controller);
let solution = problem.solve();
*solution.states.last().unwrap()
}
/// The machine-precision case.
///
/// A fifth-order Runge-Kutta method integrates `y' = p t^4` exactly, so both
/// the state and its derivative with respect to `p` are exact up to
/// rounding — there is no truncation error to hide behind. The analytic
/// answers are `y(t) = p t^5 / 5` and `dy/dp = t^5 / 5`.
#[test]
fn test_ad_exact_quadrature() {
fn derivative<T: Scalar>(t: T, _y: Vector1<T>, p: &(T,)) -> Vector1<T> {
Vector1::new(p.0 * t.powi(4))
}
let t_f = 3.0;
// Seed the derivative on the parameter p.
let p = Dual64::from_re(2.0).derivative();
let y = solve_with(
&derivative,
DormandPrince45::new().a_tol(1e-12).r_tol(1e-12),
Dual64::lift(0.0),
Dual64::lift(t_f),
Vector1::new(Dual64::lift(0.0)),
(p,),
);
assert_relative_eq!(y[0].re, 2.0 * t_f.powi(5) / 5.0, max_relative = 1e-14);
assert_relative_eq!(y[0].eps, t_f.powi(5) / 5.0, max_relative = 1e-14);
}
/// `y' = 5y - 3` has the closed-form solution
/// `y(t) = (y0 - 0.6) e^{5(t - t0)} + 0.6`, so `dy(t_f)/dy0 = e^{5(t_f - t0)}`
/// exactly. Here the comparison is limited by the solver's own truncation
/// error rather than by rounding, which is the point: AD returns the exact
/// derivative of the numerical solution, and the numerical solution is
/// accurate to tolerance.
#[test]
fn test_ad_matches_analytic_sensitivity() {
fn derivative<T: Scalar>(_t: T, y: Vector1<T>, _p: &()) -> Vector1<T> {
Vector1::new(y[0] * 5.0 - 3.0)
}
let y0 = Dual64::from_re(1.0).derivative();
let y = solve_with(
&derivative,
DormandPrince45::new().a_tol(1e-14).r_tol(1e-14),
Dual64::lift(2.0),
Dual64::lift(3.0),
Vector1::new(y0),
(),
);
assert_relative_eq!(y[0].re, 0.4 * 5.0f64.exp() + 0.6, max_relative = 1e-11);
assert_relative_eq!(y[0].eps, 5.0f64.exp(), max_relative = 1e-11);
}
/// A gradient on the orbit problem, checked against a property of the
/// two-body flow that is known analytically.
///
/// `DualSVec64<6>` carries all six partials at once, so one propagation
/// yields the full 6x6 state transition matrix over a period. Two-body
/// motion is Hamiltonian, so by Liouville's theorem its flow is
/// volume-preserving and `det(Phi) = 1` exactly — for any interval, for any
/// mu. A finite-difference Jacobian satisfies this only to the accuracy of
/// the difference formula; an exact one satisfies it to the accuracy of the
/// propagation.
#[test]
fn test_ad_orbit_state_transition_matrix() {
let period = 2.0 * PI * (SMA.powi(3) / MU).sqrt();
let y0: Vector6<f64> = orbit_y0();
// One propagation in `DualSVec64<6>` yields the whole state transition matrix.
let (y_end, stm): (Vector6<f64>, Matrix6<f64>) = num_dual::jacobian(
|y: Vector6<DualSVec64<6>>| {
type T = DualSVec64<6>;
solve_with(
&two_body,
DormandPrince45::new().a_tol(1e-12).r_tol(1e-12),
T::lift(0.0),
T::lift(period),
y,
(T::lift(MU),),
)
},
&y0,
);
// One period returns the state to where it started.
for i in 0..6 {
assert_relative_eq!(y_end[i], y0[i], max_relative = 1e-10);
}
assert_relative_eq!(stm.determinant(), 1.0, max_relative = 1e-9);
// And cross-check the whole matrix against central differences, which
// agree only to about the square root of machine epsilon.
let propagate = |y_start: Vector6<f64>| {
solve_with(
&two_body,
DormandPrince45::new().a_tol(1e-12).r_tol(1e-12),
0.0,
period,
y_start,
(MU,),
)
};
for j in 0..6 {
let h = 1e-5 * y0[j].abs();
let mut plus = y0;
let mut minus = y0;
plus[j] += h;
minus[j] -= h;
let fd = (propagate(plus) - propagate(minus)) / (2.0 * h);
for i in 0..6 {
assert_relative_eq!(stm[(i, j)], fd[i], max_relative = 1e-5, epsilon = 1e-7);
}
}
}
/// The same, through Vern7 rather than Dormand-Prince, which exercises a
/// path Dormand-Prince does not have: Vern7 keeps six of its sixteen stages
/// back and computes them only when something interpolates inside a step.
/// That lazy computation calls the right-hand side after the solve has
/// finished, so it has to work in dual arithmetic too.
#[test]
fn test_ad_vern7_lazy_dense_output() {
fn derivative<T: Scalar>(_t: T, y: Vector1<T>, _p: &()) -> Vector1<T> {
-y
}
let y0 = Dual64::from_re(1.0).derivative();
let ode = ODE::new(&derivative, Dual64::lift(0.0), Dual64::lift(2.0), Vector1::new(y0), ());
let vern7 = Vern7::new().a_tol(1e-12).r_tol(1e-12);
let mut problem = Problem::new(ode, vern7, PIController::default());
let solution = problem.solve();
// Interpolating inside a step is what triggers the extra stages.
let t = Dual64::lift(1.3);
let y = solution.interpolate(t)[0];
// y(t) = y0 e^-t, so dy/dy0 = e^-t.
assert_relative_eq!(y.re, (-1.3f64).exp(), max_relative = 1e-10);
assert_relative_eq!(y.eps, (-1.3f64).exp(), max_relative = 1e-10);
}
}
src/ode.rs
+27 -17
View File
@@ -1,27 +1,37 @@
use nalgebra::SVector;
type ProblemFunction<'a, const D: usize, P> = &'a dyn Fn(f64, SVector<f64, D>, &P) -> SVector<f64, D>;
use crate::scalar::Scalar;
/// The basic ODE object that will be passed around. The type (T) and the size (D) will be
/// The right-hand side of the system: `dy/dt = f(t, y, params)`.
///
/// Write it generically over the scalar — `fn f<T: Scalar>(t: T, y: SVector<T, D>, p: &P)` —
/// and the same function can be instantiated at `f64` to solve the problem and
/// at a dual number to differentiate it.
///
/// Or just write it for f64 if you want to keep things simple
pub type ProblemFunction<'a, T, const D: usize, P> =
&'a dyn Fn(T, SVector<T, D>, &P) -> SVector<T, D>;
/// The basic ODE object that will be passed around. The scalar (T) and the size (D) will be
/// determined upon creation of the object
#[derive(Clone, Copy)]
pub struct ODE<'a, const D: usize, P> {
pub f: ProblemFunction<'a, D, P>,
pub y: SVector<f64, D>,
pub t: f64,
pub struct ODE<'a, T: Scalar, const D: usize, P> {
pub f: ProblemFunction<'a, T, D, P>,
pub y: SVector<T, D>,
pub t: T,
pub params: P,
pub t0: f64,
pub t_end: f64,
pub h: f64,
pub t0: T,
pub t_end: T,
pub h: T,
pub finished: bool,
}
impl<'a, const D: usize, P> ODE<'a, D, P> {
impl<'a, T: Scalar, const D: usize, P> ODE<'a, T, D, P> {
pub fn new(
f: ProblemFunction<'a, D, P>,
t0: f64,
t_end: f64,
y0: SVector<f64, D>,
f: ProblemFunction<'a, T, D, P>,
t0: T,
t_end: T,
y0: SVector<T, D>,
params: P,
) -> Self {
Self {
@@ -31,7 +41,7 @@ impl<'a, const D: usize, P> ODE<'a, D, P> {
params,
t0,
t_end,
h: 0.001,
h: T::lift(0.001),
finished: false,
}
}
@@ -45,7 +55,7 @@ mod tests {
#[test]
fn test_ode_creation() {
type Params = ();
fn derivative(_t: f64, y: Vector3<f64>, _p: &Params) -> Vector3<f64> {
fn derivative<T: Scalar>(_t: T, y: Vector3<T>, _p: &Params) -> Vector3<T> {
-y
}
@@ -64,7 +74,7 @@ mod tests {
type Params = (f64, bool);
let params = (34.0, true);
fn derivative(t: f64, y: Vector3<f64>, p: &Params) -> Vector3<f64> {
fn derivative<T: Scalar>(t: T, y: Vector3<T>, p: &Params) -> Vector3<T> {
if p.1 {
-y
} else {
src/problem.rs
+86 -75
View File
@@ -1,28 +1,33 @@
//! Assembling an ODE, an integrator and a controller
use nalgebra::SVector;
use roots::{find_root_brent, SimpleConvergency};
use std::cell::RefCell;
use super::callback::Callback;
use super::controller::{Controller, PIController, TryStep};
use super::controller::{Controller, TryStep};
use super::integrator::Integrator;
use super::ode::ODE;
use super::scalar::Scalar;
#[derive(Clone)]
pub struct Problem<'a, const D: usize, S, P>
pub struct Problem<'a, T: Scalar, const D: usize, S, P, C>
where
S: Integrator<D>,
C: Controller<T>,
{
ode: ODE<'a, D, P>,
ode: ODE<'a, T, D, P>,
integrator: S,
controller: PIController,
callbacks: Vec<Callback<'a, D, P>>,
controller: C,
callbacks: Vec<&'a dyn Callback<T, D, P>>,
}
impl<'a, const D: usize, S, P> Problem<'a, D, S, P>
impl<'a, T: Scalar, const D: usize, S, P, C> Problem<'a, T, D, S, P, C>
where
S: Integrator<D> + Copy,
C: Controller<T>,
{
pub fn new(ode: ODE<'a, D, P>, integrator: S, controller: PIController) -> Self {
pub fn new(ode: ODE<'a, T, D, P>, integrator: S, controller: C) -> Self {
Problem {
ode,
integrator,
@@ -30,69 +35,79 @@ where
callbacks: Vec::new(),
}
}
pub fn solve(&mut self) -> Solution<'_, S, D, P> {
pub fn solve(&mut self) -> Solution<'_, T, S, D, P> {
let mut convergency = SimpleConvergency {
eps: 1e-12,
eps: 1e-12f64,
max_iter: 1000,
};
let mut times: Vec<f64> = vec![self.ode.t];
let mut states: Vec<SVector<f64, D>> = vec![self.ode.y];
let mut dense_coefficients: Vec<RefCell<Vec<SVector<f64, D>>>> = Vec::new();
while self.ode.t < self.ode.t_end {
if self.ode.t + self.controller.next_step_guess.extract() > self.ode.t_end {
let mut times: Vec<T> = vec![self.ode.t];
let mut states: Vec<SVector<T, D>> = vec![self.ode.y];
let mut dense_coefficients: Vec<RefCell<Vec<SVector<T, D>>>> = Vec::new();
while self.ode.t.re() < self.ode.t_end.re() {
if (self.ode.t + self.controller.next_step_guess().extract()).re()
> self.ode.t_end.re()
{
// If the next step would go past the end, then just set it to the end
self.controller.next_step_guess = TryStep::NotYetAccepted(
self.ode.t_end - self.ode.t,
);
self.controller
.set_next_step_guess(TryStep::NotYetAccepted(self.ode.t_end - self.ode.t));
}
let (mut new_y, mut curr_step, mut dense_option) = if S::ADAPTIVE {
// First, we try stepping with the "next step guess" to get the error
let (mut trial_y, mut err_option, mut dense_option) =
self.integrator.step(&self.ode, self.controller.next_step_guess.extract());
let mut err = err_option.unwrap();
let mut result = self
.integrator
.step(&self.ode, self.controller.next_step_guess().extract());
let mut err = result.err.unwrap();
// Then we determine whether we need to reduce the step size or not
// If successful, we get the next step guess
let initial_guess = self.controller.next_step_guess.extract();
let mut next_step_guess = <PIController as Controller<D>>::determine_step(
&mut self.controller,
initial_guess,
err,
);
let initial_guess = self.controller.next_step_guess().extract();
let mut next_step_guess = self.controller.determine_step(initial_guess, err);
while !next_step_guess.is_accepted() {
// If that step isn't acceptable, then change the step until it is
(trial_y, err_option, dense_option) =
self.integrator.step(&self.ode, next_step_guess.extract());
next_step_guess = <PIController as Controller<D>>::determine_step(
&mut self.controller,
next_step_guess.extract(),
err,
);
err = err_option.unwrap();
result = self.integrator.step(&self.ode, next_step_guess.extract());
err = result.err.unwrap();
next_step_guess = self
.controller
.determine_step(next_step_guess.extract(), err);
}
// So at this point we can safely assume we have an accepted step
self.controller.next_step_guess = next_step_guess.reset().unwrap();
(trial_y, next_step_guess.extract(), dense_option)
self.controller
.set_next_step_guess(next_step_guess.reset().unwrap());
(result.y, next_step_guess.extract(), result.dense)
} else {
// If fixed time step just step forward one step
let (trial_y, _, dense_option) = self.integrator.step(&self.ode, self.controller.next_step_guess.extract());
(trial_y, self.controller.next_step_guess.extract(), dense_option)
let result = self
.integrator
.step(&self.ode, self.controller.next_step_guess().extract());
(
result.y,
self.controller.next_step_guess().extract(),
result.dense,
)
};
if !self.callbacks.is_empty() {
// Check for events occurring
for callback in &self.callbacks {
if (callback.event)(self.ode.t, self.ode.y, &self.ode.params)
* (callback.event)(self.ode.t + curr_step, new_y, &self.ode.params)
< 0.0
{
// The event condition is a real-valued question, so it is
// asked in the reals — see this module's header.
let before = callback.event(self.ode.t, &self.ode.y, &self.ode.params).re();
let after = callback
.event(self.ode.t + curr_step, &new_y, &self.ode.params)
.re();
if before * after < 0.0 {
// If the event crossed zero, then find the root
let f = |test_t| {
let test_y = self.integrator.step(&self.ode, test_t).0;
(callback.event)(self.ode.t + test_t, test_y, &self.ode.params)
let f = |test_h: f64| {
let h = T::lift(test_h);
let test_y = self.integrator.step(&self.ode, h).y;
callback.event(self.ode.t + h, &test_y, &self.ode.params).re()
};
let root = find_root_brent(0.0, curr_step, &f, &mut convergency).unwrap();
curr_step = root;
(new_y, _, dense_option) = self.integrator.step(&self.ode, curr_step);
(callback.effect)(&mut self.ode);
let root =
find_root_brent(0.0f64, curr_step.re(), &f, &mut convergency).unwrap();
curr_step = T::lift(root);
let result = self.integrator.step(&self.ode, curr_step);
new_y = result.y;
dense_option = result.dense;
callback.effect(&mut self.ode);
}
}
}
@@ -112,52 +127,48 @@ where
}
}
pub fn with_callback(mut self, callback: Callback<'a, D, P>) -> Self {
pub fn with_callback(mut self, callback: &'a dyn Callback<T, D, P>) -> Self {
self.callbacks.push(callback);
Self {
ode: self.ode,
integrator: self.integrator,
controller: self.controller,
callbacks: self.callbacks,
}
self
}
}
pub struct Solution<'a, S, const D: usize, P>
pub struct Solution<'a, T: Scalar, S, const D: usize, P>
where
S: Integrator<D>,
{
pub ode: &'a ODE<'a, D, P>,
pub ode: &'a ODE<'a, T, D, P>,
pub integrator: S,
pub times: Vec<f64>,
pub states: Vec<SVector<f64, D>>,
pub dense: Vec<RefCell<Vec<SVector<f64, D>>>>,
pub times: Vec<T>,
pub states: Vec<SVector<T, D>>,
pub dense: Vec<RefCell<Vec<SVector<T, D>>>>,
}
impl<'a, S, const D: usize, P> Solution<'a, S, D, P>
impl<T: Scalar, S, const D: usize, P> Solution<'_, T, S, D, P>
where
S: Integrator<D>,
{
pub fn interpolate(&self, t: f64) -> SVector<f64, D> {
pub fn interpolate(&self, t: T) -> SVector<T, D> {
// First check that the t is within bounds
let last = self.times.last().unwrap();
let first = self.times.first().unwrap();
let last = self.times.last().unwrap().re();
let first = self.times.first().unwrap().re();
// TODO: Improve these errors
let mut times = self.times.clone();
if *first > *last {
let mut times: Vec<f64> = self.times.iter().map(|x| x.re()).collect();
if first > last {
times.reverse();
}
if t < *first || t > *last {
if t.re() < first || t.re() > last {
panic!();
}
// Then find the two t values closest to the desired t
match times.binary_search_by(|x| x.total_cmp(&t)) {
match times.binary_search_by(|x| x.total_cmp(&t.re())) {
Ok(index) => self.states[index],
Err(end_index) => {
let t_start = times[end_index - 1];
let t_end = times[end_index];
// Then send that to the integrator
let t_start = self.times[end_index - 1];
let t_end = self.times[end_index];
let y_start = self.states[end_index - 1];
let h = t_end - t_start;
@@ -205,7 +216,7 @@ where
#[cfg(test)]
mod tests {
use super::*;
use crate::callback::stop;
use crate::callback::{stop, FnCallback};
use crate::controller::PIController;
use crate::integrator::dormand_prince::DormandPrince45;
use approx::assert_relative_eq;
@@ -247,12 +258,12 @@ mod tests {
let dp45 = DormandPrince45::new().a_tol(1e-12).r_tol(1e-5);
let controller = PIController::default();
let value_too_high = Callback {
event: &|_: f64, y: SVector<f64, 3>, _: &Params| 10.0 - y[0],
let value_too_high = FnCallback {
event: &|_: f64, y: &SVector<f64, 3>, _: &Params| 10.0 - y[0],
effect: &stop,
};
let mut problem = Problem::new(ode, dp45, controller).with_callback(value_too_high);
let mut problem = Problem::new(ode, dp45, controller).with_callback(&value_too_high);
let solution = problem.solve();
assert_relative_eq!(
src/scalar.rs
+119
View File
@@ -0,0 +1,119 @@
//! The scalar type an ODE is integrated in.
//!
//! All of our problem types: state, time and step size, is generic over
//! [`Scalar`]. Instantiating a problem at `f64` solves it. Instantiating the
//! same problem at a forward-mode dual number solves it *and* propagates exact
//! derivatives alongside the state, at machine precision, with no finite
//! differencing and no extra right-hand-side evaluations beyond the ones the
//! dual arithmetic performs anyway.
//!
//! [`Scalar`] is implemented for `f64` always, and for the dual numbers of the
//! [`num_dual`] crate under the `autodiff` feature (on by default).
use nalgebra::RealField;
use num_traits::NumOps;
/// A scalar that an ODE's state, time and step size may take.
pub trait Scalar: RealField + Copy + NumOps<f64, Self> {
/// This value with all derivative information discarded.
fn re(&self) -> f64;
/// A real constant as a value of this type, with zero derivative.
fn lift(x: f64) -> Self;
}
impl Scalar for f64 {
#[inline(always)]
fn re(&self) -> f64 {
*self
}
#[inline(always)]
fn lift(x: f64) -> Self {
x
}
}
#[cfg(feature = "autodiff")]
mod dual {
use super::Scalar;
use nalgebra::{Const, RealField};
use num_dual::{DualNum, DualStruct};
/// Implements `Scalar` for one of `num-dual`'s dual number types.
macro_rules! impl_scalar {
($ty:ident) => {
impl<T> Scalar for num_dual::$ty<T>
where
Self: RealField + Copy + DualNum<Primitive = f64> + DualStruct<Real = f64>,
{
#[inline(always)]
fn re(&self) -> f64 {
DualStruct::re(self)
}
#[inline(always)]
fn lift(x: f64) -> Self {
Self::from(x)
}
}
};
($ty:ident, $($n:ident),+) => {
impl<T: nalgebra::Scalar, $(const $n: usize),+> Scalar
for num_dual::$ty<T, $(Const<$n>),+>
where
Self: RealField + Copy + DualNum<Primitive = f64> + DualStruct<Real = f64>,
{
#[inline(always)]
fn re(&self) -> f64 {
DualStruct::re(self)
}
#[inline(always)]
fn lift(x: f64) -> Self {
Self::from(x)
}
}
};
}
impl_scalar!(Dual);
impl_scalar!(Dual2);
impl_scalar!(Dual3);
impl_scalar!(HyperDual);
impl_scalar!(HyperHyperDual);
impl_scalar!(DualVec, N);
impl_scalar!(Dual2Vec, N);
impl_scalar!(HyperDualVec, M, N);
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_f64_is_a_scalar() {
assert_eq!(f64::lift(3.5), 3.5);
assert_eq!(Scalar::re(&3.5), 3.5);
}
#[cfg(feature = "autodiff")]
#[test]
fn test_dual_projects_to_its_real_part() {
use num_dual::{Dual64, DualNum};
// A lifted constant has no derivative; a seeded variable does, and both
// project back to the same real value.
let c = <Dual64 as Scalar>::lift(3.5);
assert_eq!(c.re, 3.5);
assert_eq!(c.eps, 0.0);
let x = Dual64::from_re(3.5).derivative();
assert_eq!(Scalar::re(&x), 3.5);
assert_eq!(x.eps, 1.0);
// d/dx x^3 = 3x^2 = 36.75 at x = 3.5
assert_eq!(x.powi(3).eps, 3.0 * 3.5 * 3.5);
}
}