HMath.ODE¶

Description

Checked real ODE/DAE integration. Fixed-step methods use the requested grid. Adaptive methods have explicit tolerances, step bounds and work budgets. Callbacks must be pure, total and retain the state dimension. Dense output uses a cubic Hermite interpolant, independently of the integration method.

On this page¶

API	Purpose
ODE	Time and state callback, returning a derivative of exactly the same length.
Method	Explicit, multistep, implicit, and linearly implicit methods.
Euler	Declaration and type contract below.
Heun	Declaration and type contract below.
Midpoint	Declaration and type contract below.
RK2	Declaration and type contract below.
RK4	Declaration and type contract below.
RK23	Declaration and type contract below.
RK45	Declaration and type contract below.
DormandPrince	Declaration and type contract below.
DOP853	Declaration and type contract below.
AdamsBashforth	Declaration and type contract below.
AdamsMoulton	Declaration and type contract below.
PredictorCorrector	Declaration and type contract below.
BackwardEuler	Declaration and type contract below.
BDF	Declaration and type contract below.
Radau	Declaration and type contract below.
Rosenbrock	Declaration and type contract below.
ODEOptions	Positive step bounds and tolerances; maximumSteps bounds accepted and rejected attempts, and newtonIterations bounds each implicit correction.
ODEOptions	Declaration and type contract below.
odeAbsoluteTolerance	Finite positive absolute local error tolerance.
odeRelativeTolerance	Finite nonnegative local error tolerance relative to state scale.
initialStep	Requested positive initial step magnitude; integration direction follows time bounds.
minimumStep	Positive minimum permitted step magnitude before reporting failure.
maximumStep	Positive maximum permitted step magnitude.
maximumSteps	Maximum integration-step work budget.
newtonIterations	Iteration budget for implicit-method nonlinear solves.
defaultODEOptions	Conservative default adaptive settings for Double callbacks.
Trajectory	Strictly monotonic time samples, states and endpoint derivatives.
Trajectory	Declaration and type contract below.
trajectoryTimes	Accepted sample times in integration order.
trajectoryStates	State vector at each trajectory time.
trajectoryDerivatives	ODE derivative vector at each trajectory sample, used by dense output.
solveIVP	Integrate on n equal panels.
solveAdaptive	Adaptive explicit RK integration.
interpolateTrajectory	Cubic Hermite interpolation inside the trajectory's closed time range.
Event	A located event time and its dense-interpolated state.
Event	Declaration and type contract below.
eventTime	Located event crossing time.
eventState	Interpolated state vector at eventTime.
EventDirection	Event direction in increasing physical time, even for backward integration.
AnyDirection	Declaration and type contract below.
Rising	Declaration and type contract below.
Falling	Declaration and type contract below.
locateEvents	Locate bracketed sign-changing events by dense-output bisection.
solveBoundary	Shooting boundary-value solver: all initial-state components are unknown.
solveDAE	Index-one implicit DAE driver F(t,y,y')=0 with backward differences.
solveHamiltonian	Fixed-step symplectic integration; output states concatenate q then p.
HamiltonianMethod	Separable Hamiltonian methods for q'=velocity(p), p'=force(q).
SymplecticEuler	Declaration and type contract below.
Verlet	Declaration and type contract below.
Leapfrog	Declaration and type contract below.
solveNystrom	Fourth-order explicit Runge-Kutta-Nystrom method for q''=a(t,q), with acceleration independent of velocity.
sensitivities	Terminal-state parameter sensitivities by centered perturbation of the complete IVP solve.
equilibrium	Find an autonomous equilibrium by a locally convergent Newton solve.
phaseTrajectories	Compute autonomous phase-space trajectories for independently supplied starts.
lyapunovExponents	Full Lyapunov spectrum by the Benettin tangent-space method: integrate variational equations with RK4, then modified Gram-Schmidt at each sample.

Documentation

ODE

type ODE = Double -> [Double] -> [Double] Source #

Time and state callback, returning a derivative of exactly the same length.

Method

data Method Source #

Explicit, multistep, implicit, and linearly implicit methods. RK45 is the Cash-Karp 5(4) pair; DormandPrince is the Dormand-Prince 5(4) pair. Radau is two-stage Radau IIA (order 3), BDF is order 2 with backward-Euler startup, and Rosenbrock is linearly implicit Rosenbrock-Euler (order 1).

Constructors

Euler
Heun
Midpoint
RK2
RK4
RK23
RK45
DormandPrince
DOP853
AdamsBashforth
AdamsMoulton
PredictorCorrector
BackwardEuler
BDF
Radau
Rosenbrock

Instances

Instances details

Show Method Source #
Instance details Defined in HMath.ODE Methods showsPrec :: Int -> Method -> ShowS # show :: Method -> String # showList :: [Method] -> ShowS #
Eq Method Source #
Instance details Defined in HMath.ODE Methods (==) :: Method -> Method -> Bool # (/=) :: Method -> Method -> Bool #

ODEOptions

data ODEOptions Source #

Positive step bounds and tolerances; maximumSteps bounds accepted and rejected attempts, and newtonIterations bounds each implicit correction.

Constructors

ODEOptions

Fields

odeAbsoluteTolerance :: Double
Finite positive absolute local error tolerance.
odeRelativeTolerance :: Double
Finite nonnegative local error tolerance relative to state scale.
initialStep :: Double
Requested positive initial step magnitude; integration direction follows time bounds.
minimumStep :: Double
Positive minimum permitted step magnitude before reporting failure.
maximumStep :: Double
Positive maximum permitted step magnitude.
maximumSteps :: Int
Maximum integration-step work budget.
newtonIterations :: Int
Iteration budget for implicit-method nonlinear solves.

Instances

Instances details

Show ODEOptions Source #
Instance details Defined in HMath.ODE Methods showsPrec :: Int -> ODEOptions -> ShowS # show :: ODEOptions -> String # showList :: [ODEOptions] -> ShowS #
Eq ODEOptions Source #
Instance details Defined in HMath.ODE Methods (==) :: ODEOptions -> ODEOptions -> Bool # (/=) :: ODEOptions -> ODEOptions -> Bool #

defaultODEOptions

defaultODEOptions :: ODEOptions Source #

Conservative default adaptive settings for Double callbacks.

Trajectory

data Trajectory Source #

Strictly monotonic time samples, states and endpoint derivatives. The constructor is public for inspection; interpolation validates all dimensions.

Constructors

Trajectory
Fields trajectoryTimes :: [Double] Accepted sample times in integration order. trajectoryStates :: [[Double]] State vector at each trajectory time. trajectoryDerivatives :: [[Double]] ODE derivative vector at each trajectory sample, used by dense output.

Instances

Instances details

Show Trajectory Source #
Instance details Defined in HMath.ODE Methods showsPrec :: Int -> Trajectory -> ShowS # show :: Trajectory -> String # showList :: [Trajectory] -> ShowS #
Eq Trajectory Source #
Instance details Defined in HMath.ODE Methods (==) :: Trajectory -> Trajectory -> Bool # (/=) :: Trajectory -> Trajectory -> Bool #

solveIVP

solveIVP :: Method -> ODEOptions -> Int -> ODE -> Double -> Double -> [Double] -> Either MathError Trajectory Source #

Integrate on n equal panels. Multistep methods use genuine derivative history; implicit methods use checked finite-difference Newton iterations.

solveAdaptive

solveAdaptive :: Method -> ODEOptions -> ODE -> Double -> Double -> [Double] -> Either MathError Trajectory Source #

Adaptive explicit RK integration. EulerHeunMidpoint/RK4 use step-doubling error estimates. RK23RK45DormandPrince/DOP853 use their embedded estimates. Multistep and implicit methods require solveIVP to preserve their grid/history.

interpolateTrajectory

interpolateTrajectory :: Trajectory -> Double -> Either MathError [Double] Source #

Cubic Hermite interpolation inside the trajectory's closed time range. Extrapolation, malformed public trajectories and repeated sample times fail.

Event

data Event Source #

A located event time and its dense-interpolated state.

Constructors

Event
Fields eventTime :: Double Located event crossing time. eventState :: [Double] Interpolated state vector at eventTime.

Instances

Instances details

Show Event Source #
Instance details Defined in HMath.ODE Methods showsPrec :: Int -> Event -> ShowS # show :: Event -> String # showList :: [Event] -> ShowS #
Eq Event Source #
Instance details Defined in HMath.ODE Methods (==) :: Event -> Event -> Bool # (/=) :: Event -> Event -> Bool #

EventDirection

data EventDirection Source #

Event direction in increasing physical time, even for backward integration.

Constructors

AnyDirection
Rising
Falling

Instances

Instances details

Show EventDirection Source #
Instance details Defined in HMath.ODE Methods showsPrec :: Int -> EventDirection -> ShowS # show :: EventDirection -> String # showList :: [EventDirection] -> ShowS #
Eq EventDirection Source #
Instance details Defined in HMath.ODE Methods (==) :: EventDirection -> EventDirection -> Bool # (/=) :: EventDirection -> EventDirection -> Bool #

locateEvents

locateEvents :: EventDirection -> Double -> (Double -> [Double] -> Double) -> Trajectory -> Either MathError [Event] Source #

Locate bracketed sign-changing events by dense-output bisection. Tangential roots and multiple roots inside one sample interval are not guaranteed.

solveBoundary

solveBoundary :: Method -> ODEOptions -> Int -> ODE -> ([Double] -> [Double] -> [Double]) -> Double -> Double -> [Double] -> Either MathError Trajectory Source #

Shooting boundary-value solver: all initial-state components are unknown. The boundary callback takes initial/final states and returns n residuals. Returns the trajectory matching a locally converged shooting root; convergence is local and depends on the supplied initial guess.

solveDAE

solveDAE :: ODEOptions -> Int -> (Double -> [Double] -> [Double] -> [Double]) -> Double -> Double -> [Double] -> [Double] -> Either MathError Trajectory Source #

Index-one implicit DAE driver F(t,y,y')=0 with backward differences. Initial y and y' must be consistent to the configured tolerance. Each subsequent state is obtained by a full residual Newton solve.

solveHamiltonian

solveHamiltonian :: HamiltonianMethod -> Int -> ([Double] -> [Double]) -> ([Double] -> [Double]) -> Double -> Double -> [Double] -> [Double] -> Either MathError Trajectory Source #

Fixed-step symplectic integration; output states concatenate q then p. Verlet is kick-drift-kick; Leapfrog is drift-kick-drift with integer-time output momenta. These equivalent second-order splittings have distinct order.

HamiltonianMethod

data HamiltonianMethod Source #

Separable Hamiltonian methods for q'=velocity(p), p'=force(q).

Constructors

SymplecticEuler
Verlet
Leapfrog

Instances

Instances details

Show HamiltonianMethod Source #
Instance details Defined in HMath.ODE Methods showsPrec :: Int -> HamiltonianMethod -> ShowS # show :: HamiltonianMethod -> String # showList :: [HamiltonianMethod] -> ShowS #
Eq HamiltonianMethod Source #
Instance details Defined in HMath.ODE Methods (==) :: HamiltonianMethod -> HamiltonianMethod -> Bool # (/=) :: HamiltonianMethod -> HamiltonianMethod -> Bool #

solveNystrom

solveNystrom :: Int -> (Double -> [Double] -> [Double]) -> Double -> Double -> [Double] -> [Double] -> Either MathError Trajectory Source #

Fourth-order explicit Runge-Kutta-Nystrom method for q''=a(t,q), with acceleration independent of velocity. State output concatenates q and q'.

sensitivities

sensitivities :: Method -> ODEOptions -> Int -> Double -> ([Double] -> ODE) -> [Double] -> Double -> Double -> [Double] -> Either MathError (Matrix Double) Source #

Terminal-state parameter sensitivities by centered perturbation of the complete IVP solve. Parameters use the same explicit finite-difference step.

equilibrium

equilibrium :: ODEOptions -> ([Double] -> [Double]) -> [Double] -> Either MathError [Double] Source #

Find an autonomous equilibrium by a locally convergent Newton solve.

phaseTrajectories

phaseTrajectories :: Method -> ODEOptions -> Int -> ([Double] -> [Double]) -> Double -> Double -> [[Double]] -> Either MathError [Trajectory] Source #

Compute autonomous phase-space trajectories for independently supplied starts.

lyapunovExponents

lyapunovExponents :: ODEOptions -> Int -> Double -> ODE -> Double -> Double -> [Double] -> Either MathError [Double] Source #

Full Lyapunov spectrum by the Benettin tangent-space method: integrate variational equations with RK4, then modified Gram-Schmidt at each sample. Finite-difference Jacobians are used; results are finite-time estimates.