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‎doc/solvers/RungeKuttaIntegrators.md‎

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## Complexity Analysis
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| Integrator | RHS evaluations per accepted step | State memory |
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|----------------|----------------------------------|----------------|
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| RK4 (fixed) | 4 (always) | $O(n_s)$ |
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| Dormand-Prince | 6 (with FSAL), 7 on first step | $O(n_s)$ |
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| Integrator | RHS evaluations per accepted step | State memory |
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|----------------|-----------------------------------|--------------|
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| RK4 (fixed) | 4 (always) | $O(n_s)$ |
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| Dormand-Prince | 6 (with FSAL), 7 on first step | $O(n_s)$ |
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All intermediate stage vectors are stack-allocated. No heap is used. The cost of one step is $O(s \cdot n_s)$ where $s$ is the stage count plus the cost of evaluating $f$.
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## Step-by-Step Walkthrough
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**Scalar decay** $\dot{x} = -x$, $x(0) = 1$, exact solution $x(t) = e^{-t}$, $h = 0.1$:
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| Stage | Formula | Value |
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|-------|---------------------------------------------------|-------------|
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| $k_1$ | $f(1, 0) = -1$ | $-1$ |
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| $k_2$ | $f(1 - 0.05, 0.05) = -0.95$ | $-0.95$ |
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| $k_3$ | $f(1 - 0.0475, 0.05) = -0.9525$ | $-0.9525$ |
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| $k_4$ | $f(1 - 0.09525, 0.1) = -0.90475$ | $-0.90475$ |
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| $x_1$ | $1 + (0.1/6)(-1 - 1.9 - 1.905 - 0.90475)$ | $\approx 0.90484$ |
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| Stage | Formula | Value |
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|-------|-------------------------------------------|-------------------|
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| $k_1$ | $f(1, 0) = -1$ | $-1$ |
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| $k_2$ | $f(1 - 0.05, 0.05) = -0.95$ | $-0.95$ |
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| $k_3$ | $f(1 - 0.0475, 0.05) = -0.9525$ | $-0.9525$ |
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| $k_4$ | $f(1 - 0.09525, 0.1) = -0.90475$ | $-0.90475$ |
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| $x_1$ | $1 + (0.1/6)(-1 - 1.9 - 1.905 - 0.90475)$ | $\approx 0.90484$ |
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Exact: $e^{-0.1} \approx 0.90484$. Agreement to six significant figures — consistent with $O(h^5)$ local error.
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## Variants & Generalizations
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| Variant | Key Difference |
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|------------------------------|---------------------------------------------------------------------------------------|
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| **Euler (1st order)** | One stage; $O(h)$ global error; useful only for rough prototyping |
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| **RK4 (this)** | Four stages; $O(h^4)$ global error; standard fixed-step workhorse |
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| **Dormand-Prince (this)** | Seven stages; $O(h^5)$ propagator with built-in $O(h^4)$ error estimate |
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| **Bogacki-Shampine RK23** | Three-stage embedded pair; lower overhead for mildly stiff or smooth problems |
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| **Adams-Bashforth** | Multi-step; reuses past evaluations; efficient but requires startup phase |
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| **Implicit RK / SDIRK** | Solves a nonlinear system at each stage; suitable for stiff problems at the cost of a linear solve per step |
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| Variant | Key Difference |
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|---------------------------|-------------------------------------------------------------------------------------------------------------|
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| **Euler (1st order)** | One stage; $O(h)$ global error; useful only for rough prototyping |
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| **RK4 (this)** | Four stages; $O(h^4)$ global error; standard fixed-step workhorse |
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| **Dormand-Prince (this)** | Seven stages; $O(h^5)$ propagator with built-in $O(h^4)$ error estimate |
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| **Bogacki-Shampine RK23** | Three-stage embedded pair; lower overhead for mildly stiff or smooth problems |
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| **Adams-Bashforth** | Multi-step; reuses past evaluations; efficient but requires startup phase |
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| **Implicit RK / SDIRK** | Solves a nonlinear system at each stage; suitable for stiff problems at the cost of a linear solve per step |
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## Applications
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C2D -.->|"exact linear alternative"| RK
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```
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| Algorithm | Relationship |
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|----------------------------|-----------------------------------------------------------------------------------------------|
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| `dynamics/` models | Provide the right-hand side $f(x, u, t)$ that RK integrates |
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| Extended Kalman Filter | Uses RK to propagate the state prediction step between measurements |
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| MPC Controller | Uses RK to simulate the plant over a prediction horizon |
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| ContinuousToDiscrete | Exact matrix-exponential discretization — an alternative for linear, time-invariant systems |
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| Algorithm | Relationship |
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|------------------------|---------------------------------------------------------------------------------------------|
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| `dynamics/` models | Provide the right-hand side $f(x, u, t)$ that RK integrates |
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| Extended Kalman Filter | Uses RK to propagate the state prediction step between measurements |
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| MPC Controller | Uses RK to simulate the plant over a prediction horizon |
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| ContinuousToDiscrete | Exact matrix-exponential discretization — an alternative for linear, time-invariant systems |
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## References & Further Reading
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