| Stability | experimental |
|---|---|
| Maintainer | Scott N. Walck <walck@lvc.edu> |
| Safe Haskell | Trustworthy |
Physics.Learn.StateSpace
Description
A StateSpace is an affine space where the associated vector space
has scalars that are instances of Fractional.
If p is an instance of StateSpace, then the associated vectorspace
Diff p is intended to represent the space of (time) derivatives
of paths in p.
StateSpace is very similar to Conal Elliott's AffineSpace.
- class (VectorSpace (Diff p), Fractional (Scalar (Diff p))) => StateSpace p where
- (.-^) :: StateSpace p => p -> Diff p -> p
- type Time p = Scalar (Diff p)
- type DifferentialEquation state = state -> Diff state
- type InitialValueProblem state = (DifferentialEquation state, state)
- type EvolutionMethod state = DifferentialEquation state -> Time state -> state -> state
- type SolutionMethod state = InitialValueProblem state -> [state]
- stepSolution :: EvolutionMethod state -> Time state -> SolutionMethod state
- eulerMethod :: StateSpace state => EvolutionMethod state
Documentation
class (VectorSpace (Diff p), Fractional (Scalar (Diff p))) => StateSpace p whereSource
An instance of StateSpace is a data type that can serve as the state
of some system. Alternatively, a StateSpace is a collection of dependent
variables for a differential equation.
A StateSpace has an associated vector space for the (time) derivatives
of the state. The associated vector space is a linearized version of
the StateSpace.
Methods
(.-.) :: p -> p -> Diff pSource
Subtract points
(.+^) :: p -> Diff p -> pSource
Point plus vector
Instances
| StateSpace Double | |
| StateSpace Vec | |
| StateSpace Position | Position is not a vector, but displacement (difference in position) is a vector. |
| StateSpace St | |
| StateSpace p => StateSpace [p] | |
| (StateSpace p, StateSpace q, ~ * (Time p) (Time q)) => StateSpace (p, q) | |
| (StateSpace p, StateSpace q, StateSpace r, ~ * (Time p) (Time q), ~ * (Time q) (Time r)) => StateSpace (p, q, r) |
(.-^) :: StateSpace p => p -> Diff p -> pSource
Point minus vector
type Time p = Scalar (Diff p)Source
The scalars of the associated vector space can be thought of as time intervals.
type DifferentialEquation state = state -> Diff stateSource
A differential equation expresses how the dependent variables (state) change with the independent variable (time). A differential equation is specified by giving the (time) derivative of the state as a function of the state. The (time) derivative of a state is an element of the associated vector space.
type InitialValueProblem state = (DifferentialEquation state, state)Source
An initial value problem is a differential equation along with an initial state.
type EvolutionMethod stateSource
Arguments
| = DifferentialEquation state | differential equation |
| -> Time state | time interval |
| -> state | initial state |
| -> state | evolved state |
An evolution method is a way of approximating the state after advancing a finite interval in the independent variable (time) from a given state.
type SolutionMethod state = InitialValueProblem state -> [state]Source
A (numerical) solution method is a way of converting an initial value problem into a list of states (a solution). The list of states need not be equally spaced in time.
stepSolution :: EvolutionMethod state -> Time state -> SolutionMethod stateSource
Given an evolution method and a time step, return the solution method which applies the evolution method repeatedly with with given time step. The solution method returned will produce an infinite list of states.
eulerMethod :: StateSpace state => EvolutionMethod stateSource
The Euler method is the simplest evolution method. It increments the state by the derivative times the time step.