# scipy/scipy

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 # Author: Travis Oliphant __all__ = ['odeint'] import _odepack from copy import copy _msgs = {2: "Integration successful.", -1: "Excess work done on this call (perhaps wrong Dfun type).", -2: "Excess accuracy requested (tolerances too small).", -3: "Illegal input detected (internal error).", -4: "Repeated error test failures (internal error).", -5: "Repeated convergence failures (perhaps bad Jacobian or tolerances).", -6: "Error weight became zero during problem.", -7: "Internal workspace insufficient to finish (internal error)." } def odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0, ml=None, mu=None, rtol=None, atol=None, tcrit=None, h0=0.0, hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12, mxords=5, printmessg=0): """ Integrate a system of ordinary differential equations. Solve a system of ordinary differential equations using lsoda from the FORTRAN library odepack. Solves the initial value problem for stiff or non-stiff systems of first order ode-s:: dy/dt = func(y,t0,...) where y can be a vector. Parameters ---------- func : callable(y, t0, ...) Computes the derivative of y at t0. y0 : array Initial condition on y (can be a vector). t : array A sequence of time points for which to solve for y. The initial value point should be the first element of this sequence. args : tuple Extra arguments to pass to function. Dfun : callable(y, t0, ...) Gradient (Jacobian) of func. col_deriv : boolean True if Dfun defines derivatives down columns (faster), otherwise Dfun should define derivatives across rows. full_output : boolean True if to return a dictionary of optional outputs as the second output printmessg : boolean Whether to print the convergence message Returns ------- y : array, shape (len(t), len(y0)) Array containing the value of y for each desired time in t, with the initial value y0 in the first row. infodict : dict, only returned if full_output == True Dictionary containing additional output information ======= ============================================================ key meaning ======= ============================================================ 'hu' vector of step sizes successfully used for each time step. 'tcur' vector with the value of t reached for each time step. (will always be at least as large as the input times). 'tolsf' vector of tolerance scale factors, greater than 1.0, computed when a request for too much accuracy was detected. 'tsw' value of t at the time of the last method switch (given for each time step) 'nst' cumulative number of time steps 'nfe' cumulative number of function evaluations for each time step 'nje' cumulative number of jacobian evaluations for each time step 'nqu' a vector of method orders for each successful step. 'imxer' index of the component of largest magnitude in the weighted local error vector (e / ewt) on an error return, -1 otherwise. 'lenrw' the length of the double work array required. 'leniw' the length of integer work array required. 'mused' a vector of method indicators for each successful time step: 1: adams (nonstiff), 2: bdf (stiff) ======= ============================================================ Other Parameters ---------------- ml, mu : int If either of these are not None or non-negative, then the Jacobian is assumed to be banded. These give the number of lower and upper non-zero diagonals in this banded matrix. For the banded case, Dfun should return a matrix whose columns contain the non-zero bands (starting with the lowest diagonal). Thus, the return matrix from Dfun should have shape ``len(y0) * (ml + mu + 1)`` when ``ml >=0`` or ``mu >=0``. rtol, atol : float The input parameters `rtol` and `atol` determine the error control performed by the solver. The solver will control the vector, e, of estimated local errors in y, according to an inequality of the form ``max-norm of (e / ewt) <= 1``, where ewt is a vector of positive error weights computed as ``ewt = rtol * abs(y) + atol``. rtol and atol can be either vectors the same length as y or scalars. Defaults to 1.49012e-8. tcrit : ndarray Vector of critical points (e.g. singularities) where integration care should be taken. h0 : float, (0: solver-determined) The step size to be attempted on the first step. hmax : float, (0: solver-determined) The maximum absolute step size allowed. hmin : float, (0: solver-determined) The minimum absolute step size allowed. ixpr : bool Whether to generate extra printing at method switches. mxstep : int, (0: solver-determined) Maximum number of (internally defined) steps allowed for each integration point in t. mxhnil : int, (0: solver-determined) Maximum number of messages printed. mxordn : int, (0: solver-determined) Maximum order to be allowed for the non-stiff (Adams) method. mxords : int, (0: solver-determined) Maximum order to be allowed for the stiff (BDF) method. See Also -------- ode : a more object-oriented integrator based on VODE. quad : for finding the area under a curve. """ if ml is None: ml = -1 # changed to zero inside function call if mu is None: mu = -1 # changed to zero inside function call t = copy(t) y0 = copy(y0) output = _odepack.odeint(func, y0, t, args, Dfun, col_deriv, ml, mu, full_output, rtol, atol, tcrit, h0, hmax, hmin, ixpr, mxstep, mxhnil, mxordn, mxords) if output[-1] < 0: print _msgs[output[-1]] print "Run with full_output = 1 to get quantitative information." else: if printmessg: print _msgs[output[-1]] if full_output: output[1]['message'] = _msgs[output[-1]] output = output[:-1] if len(output) == 1: return output[0] else: return output
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