Module `Odepack`

Binding to ODEPACK. This is a collection of solvers for the initial value problem for ordinary differential equation systems. See the ODEPACK page and Netlib.

You can jump to the interface of the Odepack library.

Example of use

To solve the equation ∂ₜ²u = f(t,u) with initial conditions u(t₀) = u₀ and ∂ₜu(t₀) = u'₀, you must first reduce it to a first order ODE: ∂ₜ(y₁,y₂) = (y₂, f(t,y₁)) with the initial condition y(t₀) = (u₀, u'₀). Then write an OCaml function to evaluate the right hand side of this ODE:

let ode t (y: vec) (dy: vec) =
  dy.{1} <- y.{2};
  dy.{2} <- f t y.{1}

and get an approximate value of the vector y(t) with

let init = Array1.of_array float64 fortran_layout [|u₀; u'₀|] in
Odepack.vec(Odepack.lsoda ode init t₀ t)

You must explicitly project the return value of Odepack.lsoda with Odepack.vec to get the state of the system because there are several other operations that you can perform on this value (see above). The value of u(t) is the given by the first component of y, so you can define (an approximation of) u with

let u ~u0 ~u'0 t =
  let init = Array1.of_array float64 fortran_layout [|u0; u'0|] in
  Odepack.vec(Odepack.lsoda ode init t₀ t).{1}

version: 0.7.1

author: Christophe Troestler (Christophe.Troestler@umons.ac.be)

Odepack library

type vec = (float, Stdlib.Bigarray.float64_elt, Stdlib.Bigarray.fortran_layout) Stdlib.Bigarray.Array1.t: Representation of vectors.

type mat = (float, Stdlib.Bigarray.float64_elt, Stdlib.Bigarray.fortran_layout) Stdlib.Bigarray.Array2.t: Representation of matrices.

type t: A mutable value holding the current state of solving the ODE.

type jacobian = | Auto_full Internally generated (difference quotient) full Jacobian | Auto_band of int * int Internally generated (difference quotient) band Jacobian. It takes (l,u) where l (resp. u) is the number of lines below (resp. above) the diagonal (excluded). | Full of float -> vec -> mat -> unit Full df means that a function df is provided to compute the full Jacobian matrix (∂fᵢ/∂yⱼ) of the vector field f(t,y). df t y jac must store ∂fᵢ/∂yⱼ(t,y) into jac.{i,j}. | Band of int * int * float -> vec -> int -> mat -> unit Band(l, u, df) means that a function df is provided to compute the banded Jacobian matrix with l (resp. u) diagonals below (resp. above) the main one (not counted). df t y d jac must store ∂fᵢ/∂yⱼ(t,y) into jac.{i-j+d, j}. d is the row of jac corresponding to the main diagonal of the Jacobian matrix.: Types of Jacobian matrices.

val lsoda : ?⁠rtol:float -> ?⁠rtol_vec:vec -> ?⁠atol:float -> ?⁠atol_vec:vec -> ?⁠jac:jacobian -> ?⁠mxstep:int -> ?⁠copy_y0:bool -> ?⁠debug:bool -> ?⁠debug_switches:bool -> (float -> vec -> vec -> unit) -> vec -> float -> float -> t

lsoda f y0 t0 t solves the ODE dy/dt = F(t,y) with initial condition y(t0) = y0. The execution of f t y y' must compute the value of the F(t, y) and store it in y'. It uses a dense or banded Jacobian when the problem is stiff, but it automatically selects between nonstiff (Adams) and stiff (BDF) methods. It uses the nonstiff method initially, and dynamically monitors data in order to decide which method to use.

parameter rtol: relative error tolerance parameter. Default 1e-6.

parameter rtol_vec: relative error tolerance vector.

parameter atol: absolute error tolerance parameter. Default 1e-6.

parameter atol_vec

absolute error tolerance vector.

If rtol_vec (resp. atol_vec) is specified, it is used in place of rtol (resp. atol). Specifying only rtol (resp. atol) is equivalent to pass a constant rtol_vec (resp. atol_vec). The solver will control the vector E = (E(i)) of estimated local errors in y, according to an inequality of the form max-norm(E(i)/EWT(i)) <= 1, where EWT(i) = rtol_vec.{i} * abs_float(y.{i}) + atol_vec.{i}.

parameter jac: is an optional Jabobian matrix. If the problem is expected to be stiff much of the time, you are encouraged to supply jac, for the sake of efficiency. Default: Auto_full.

parameter mxstep: maximum number of (internally defined) steps allowed during one call to the solver. The default value is 500.

parameter copy_y0: if false, the vector y0 is MODIFIED to contain the value of the solution at time t. Otherwise y0 is unchanged (the current solution vector is then obtained by Odepack.vec). Default: true.

parameter debug: allows lsoda to print messages. Default true. The messages contain valuable information, it is not recommended to turn them off.

parameter debug_switches: prints a message to stdout on each (automatic) method switch (between nonstiff and stiff). Default: false.

val lsodar : ?⁠rtol:float -> ?⁠rtol_vec:vec -> ?⁠atol:float -> ?⁠atol_vec:vec -> ?⁠jac:jacobian -> ?⁠mxstep:int -> ?⁠copy_y0:bool -> ?⁠debug:bool -> ?⁠debug_switches:bool -> g:(float -> vec -> vec -> unit) -> ng:int -> (float -> vec -> vec -> unit) -> vec -> float -> float -> t: lsodar f y0 t0 t ~g ~ng is like lsoda but has root searching capabilities. The algorithm will stop before reaching time t if a root of one of the ng constraints is found. You can determine whether the lsodar stopped at a root using has_root. It only finds those roots for which some component of g, as a function of t, changes sign in the interval of integration. The function g is evaluated like f, that is: g t y gout must write to gout.{1},..., gout.{ng} the values of the ng constraints.

val vec : t -> vec: vec ode returns the current value of the solution vector.

val time : t -> float: t ode returns the current time at which the solution vector was computed.

val advance : ?⁠time:float -> t -> unit: advance ode ~time:t modifies ode so that an approximation of the value of the solution at times t is computed. Note that, if the solver has root searching capabilities and a time is provided, the solver may stop before that time if a root is found. The time is recorded for future calls to advance ode. If the solver has no root finding capabilities and no time is provided, this function does nothing.

val has_root : t -> bool: has_root ode says wheter the solver stopped (i.e. the current state of ode is) because a root was found. If the solver has no root searching capabilities, this returns false.

val root : t -> int -> bool: root t i returns true iff the ith constraint in lsodar has a root. It raises Invalid_argument if i is not between 1 and ng, the number of constraints (included). This only makes sense if has_root t holds.

val roots : t -> bool array: roots t returns an array r such that r.(i) holds if and only if the ith constraint has a root.

val sol : t -> float -> vec: sol ode t modifies ode so that it holds an approximation of the solution at t and returns this approximation. Any root that might be found is ignored.

`\|` `Auto_full`	Internally generated (difference quotient) full Jacobian
`\|` `Auto_band of int * int`	Internally generated (difference quotient) band Jacobian. It takes `(l,u)` where `l` (resp. `u`) is the number of lines below (resp. above) the diagonal (excluded).
`\|` `Full of float -> vec -> mat -> unit`	`Full df` means that a function `df` is provided to compute the full Jacobian matrix (∂fᵢ/∂yⱼ) of the vector field f(t,y). `df t y jac` must store ∂fᵢ/∂yⱼ(`t`,`y`) into `jac.{i,j}`.
`\|` `Band of int * int * float -> vec -> int -> mat -> unit`	`Band(l, u, df)` means that a function `df` is provided to compute the banded Jacobian matrix with `l` (resp. `u`) diagonals below (resp. above) the main one (not counted). `df t y d jac` must store ∂fᵢ/∂yⱼ(`t`,`y`) into `jac.{i-j+d, j}`. `d` is the row of `jac` corresponding to the main diagonal of the Jacobian matrix.