Copyright | (c) Scott N. Walck 2014 |
---|---|

License | BSD3 (see LICENSE) |

Maintainer | Scott N. Walck <walck@lvc.edu> |

Stability | experimental |

Safe Haskell | Trustworthy |

Language | Haskell98 |

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 where Source

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`

.

(.-.) :: p -> p -> Diff p infix 6 Source

Subtract points

(.+^) :: p -> Diff p -> p infixl 6 Source

Point plus vector

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 -> p infixl 6 Source

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 state Source

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 state Source

= 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 state Source

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 state Source

The Euler method is the simplest evolution method. It increments the state by the derivative times the time step.