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Intro to ODEs
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"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Initial imports"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"%pylab inline"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.kernel.zmq.pylab.backend_inline].\n",
"For more information, type 'help(pylab)'.\n"
]
}
],
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{
"cell_type": "code",
"collapsed": false,
"input": [
"from matplotlib import pyplot as plt\n",
"import numpy as np\n",
"\n",
"from scipy.integrate import odeint\n",
"\n",
"from colorline import colorline"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 4
},
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"cell_type": "code",
"collapsed": false,
"input": [
"# Display the help for the odeint function:\n",
"# The output can be shortened or suppressed by clicking or double clicking on the region to the left of the output\n",
"\n",
"help(odeint)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Help on function odeint in module scipy.integrate.odepack:\n",
"\n",
"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)\n",
" Integrate a system of ordinary differential equations.\n",
" \n",
" Solve a system of ordinary differential equations using lsoda from the\n",
" FORTRAN library odepack.\n",
" \n",
" Solves the initial value problem for stiff or non-stiff systems\n",
" of first order ode-s::\n",
" \n",
" dy/dt = func(y,t0,...)\n",
" \n",
" where y can be a vector.\n",
" \n",
" Parameters\n",
" ----------\n",
" func : callable(y, t0, ...)\n",
" Computes the derivative of y at t0.\n",
" y0 : array\n",
" Initial condition on y (can be a vector).\n",
" t : array\n",
" A sequence of time points for which to solve for y. The initial\n",
" value point should be the first element of this sequence.\n",
" args : tuple, optional\n",
" Extra arguments to pass to function.\n",
" Dfun : callable(y, t0, ...)\n",
" Gradient (Jacobian) of `func`.\n",
" col_deriv : bool, optional\n",
" True if `Dfun` defines derivatives down columns (faster),\n",
" otherwise `Dfun` should define derivatives across rows.\n",
" full_output : bool, optional\n",
" True if to return a dictionary of optional outputs as the second output\n",
" printmessg : bool, optional\n",
" Whether to print the convergence message\n",
" \n",
" Returns\n",
" -------\n",
" y : array, shape (len(t), len(y0))\n",
" Array containing the value of y for each desired time in t,\n",
" with the initial value `y0` in the first row.\n",
" infodict : dict, only returned if full_output == True\n",
" Dictionary containing additional output information\n",
" \n",
" ======= ============================================================\n",
" key meaning\n",
" ======= ============================================================\n",
" 'hu' vector of step sizes successfully used for each time step.\n",
" 'tcur' vector with the value of t reached for each time step.\n",
" (will always be at least as large as the input times).\n",
" 'tolsf' vector of tolerance scale factors, greater than 1.0,\n",
" computed when a request for too much accuracy was detected.\n",
" 'tsw' value of t at the time of the last method switch\n",
" (given for each time step)\n",
" 'nst' cumulative number of time steps\n",
" 'nfe' cumulative number of function evaluations for each time step\n",
" 'nje' cumulative number of jacobian evaluations for each time step\n",
" 'nqu' a vector of method orders for each successful step.\n",
" 'imxer' index of the component of largest magnitude in the\n",
" weighted local error vector (e / ewt) on an error return, -1\n",
" otherwise.\n",
" 'lenrw' the length of the double work array required.\n",
" 'leniw' the length of integer work array required.\n",
" 'mused' a vector of method indicators for each successful time step:\n",
" 1: adams (nonstiff), 2: bdf (stiff)\n",
" ======= ============================================================\n",
" \n",
" Other Parameters\n",
" ----------------\n",
" ml, mu : int, optional\n",
" If either of these are not None or non-negative, then the\n",
" Jacobian is assumed to be banded. These give the number of\n",
" lower and upper non-zero diagonals in this banded matrix.\n",
" For the banded case, `Dfun` should return a matrix whose\n",
" columns contain the non-zero bands (starting with the\n",
" lowest diagonal). Thus, the return matrix from `Dfun` should\n",
" have shape ``len(y0) * (ml + mu + 1)`` when ``ml >=0`` or ``mu >=0``.\n",
" rtol, atol : float, optional\n",
" The input parameters `rtol` and `atol` determine the error\n",
" control performed by the solver. The solver will control the\n",
" vector, e, of estimated local errors in y, according to an\n",
" inequality of the form ``max-norm of (e / ewt) <= 1``,\n",
" where ewt is a vector of positive error weights computed as\n",
" ``ewt = rtol * abs(y) + atol``.\n",
" rtol and atol can be either vectors the same length as y or scalars.\n",
" Defaults to 1.49012e-8.\n",
" tcrit : ndarray, optional\n",
" Vector of critical points (e.g. singularities) where integration\n",
" care should be taken.\n",
" h0 : float, (0: solver-determined), optional\n",
" The step size to be attempted on the first step.\n",
" hmax : float, (0: solver-determined), optional\n",
" The maximum absolute step size allowed.\n",
" hmin : float, (0: solver-determined), optional\n",
" The minimum absolute step size allowed.\n",
" ixpr : bool, optional\n",
" Whether to generate extra printing at method switches.\n",
" mxstep : int, (0: solver-determined), optional\n",
" Maximum number of (internally defined) steps allowed for each\n",
" integration point in t.\n",
" mxhnil : int, (0: solver-determined), optional\n",
" Maximum number of messages printed.\n",
" mxordn : int, (0: solver-determined), optional\n",
" Maximum order to be allowed for the non-stiff (Adams) method.\n",
" mxords : int, (0: solver-determined), optional\n",
" Maximum order to be allowed for the stiff (BDF) method.\n",
" \n",
" See Also\n",
" --------\n",
" ode : a more object-oriented integrator based on VODE.\n",
" quad : for finding the area under a curve.\n",
"\n"
]
}
],
"prompt_number": 5
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"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Integrating ODEs using `odeint`"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The function `odeint`, which is found in the `integrate` submodule of the `scipy` package, integrates ordinary differential equations of the form\n",
"$$\n",
"\\newcommand{\\yy}{\\mathbf{y}}\n",
"\\newcommand{\\ff}{\\mathbf{f}}\n",
"\\frac{\\mathrm{d} \\yy}{\\mathrm{d}t} = \\ff(\\yy, t),\n",
"$$where $\\yy$ is a vector in $\\mathbb{R}^n$, or\n",
"$$\n",
"\\dot{\\yy} = \\ff(\\yy, t).\n",
"$$\n",
"That is, it provides a numerical solution of the *initial value problem* given by this differential equation, together with an initial condition\n",
"$$\n",
"\\yy(t = t_0) = \\yy_0,\n",
"$$\n",
"returning the solution curve (or trajectory) $\\yy(t)$ as a function of time $t$, within a certain (finite!) range of values of $t$.\n",
"\n",
"Writing $\\yy = (y_1, \\ldots, y_n)$, we have a system of the form\n",
"\\begin{align}\n",
"\\dot{y_1} &= f_1(y_1, \\ldots, y_n, t) \\\\\n",
"\\dot{y_2} &= f_2(y_1, \\ldots, y_n, t) \\\\\n",
"&\\ldots \\\\\n",
"\\dot{y_n} &= f_n(y_1, \\ldots, y_n, t)\n",
"\\end{align}\n",
"\n",
"Note that in fact, *any* ordinary (system of) ordinary differential equation(s) can be written in this form by defining new variables to represent higher-order derivatives, \n",
"and incorporating them into the vector $\\yy$ of variables.\n",
"\n",
"To use `odeint`, we must define at least the Python function `f` which implements the mathematical function $\\ff$. It must take a vector `y` and a time `t`,\n",
"and return the new vector `f(y,t)`."
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"1D equations"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The simplest case is when the spatial dimension is $n=1$. For example, we could wish to solve the ODE\n",
"$\n",
"\\dot{y} = ay,\n",
"$\n",
"whose exact solution is, of course, $y(t) = y_0 \\exp(a t)$."
]
},
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"cell_type": "code",
"collapsed": false,
"input": [
"a = 2\n",
"\n",
"def f(y, t):\n",
" return a*y"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 6
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We must provide an initial condition for $y$, for example at $t=0$:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"y0 = 1.0"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 7
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"and an array of values of $t$ where we wish to have output:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"t_output = np.arange(0, 6, 0.1)\n",
"print t_output"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"[ 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1. 1.1 1.2 1.3 1.4\n",
" 1.5 1.6 1.7 1.8 1.9 2. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9\n",
" 3. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4. 4.1 4.2 4.3 4.4\n",
" 4.5 4.6 4.7 4.8 4.9 5. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9]\n"
]
}
],
"prompt_number": 8
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We pass the information to the `odeint` function to perform the integration:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"y_result = odeint(f, y0, t_output)\n",
"y_result = y_result[:, 0] # convert the returned 2D array to a 1D array"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 9
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we can plot the results, together with the exact solution:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#plt.plot(t_output, y_result)\n",
"colorline(t_output, y_result)\n",
"plt.plot(t_output, y0 * np.exp(a * t_output))\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 10,
"text": [
"[<matplotlib.lines.Line2D at 0x114547310>]"
]
},
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],
"metadata": {},
"output_type": "display_data",
"text": [
"<matplotlib.figure.Figure at 0x114547910>"
]
}
],
"prompt_number": 10
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can calculate the absolute error between the numerically-calculated result and the exact solution:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"plt.plot(t_output, np.abs(y_result - y0 * np.exp(a * t_output)))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 11,
"text": [
"[<matplotlib.lines.Line2D at 0x1144ef410>]"
]
},
{
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"' width=383 height=252/>"
],
"metadata": {},
"output_type": "display_data",
"text": [
"<matplotlib.figure.Figure at 0x1135d1290>"
]
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"2D equations"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's consider a simple 2nd-order ODE, that of the harmonic oscillator:\n",
"$$\n",
"\\ddot{x} + \\omega^2 x = 0.\n",
"$$\n",
"To integrate this equation, we must rewrite it as a system of first-order ODEs by introducing a new variable $y := \\dot{x}$, so that the system becomes\n",
"\\begin{align}\n",
"\\dot{x} &= y \\\\\n",
"\\dot{y} &= - \\omega^2 x\n",
"\\end{align}\n",
"\n",
"Writing the vector $\\yy = (x,y)$, we obtain the standard form\n",
"$\n",
"\\dot{\\yy} = \\ff(\\yy),\n",
"$\n",
"where $\\ff$ is given by\n",
"$$\n",
"\\ff \\left[ \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\right]\n",
"= \\begin{pmatrix} f_1 \\left[ \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\right] \\\\ \n",
"f_2 \\left[ \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\right]\n",
"\\end{pmatrix}\n",
"= \\begin{pmatrix} y \\\\ -\\omega^2 x \\end{pmatrix}\n",
"$$\n",
"That is, the functions $f_i$ are just the right-hand sides of each of the equations in the original system.\n",
"We now implement this function $\\ff$ directly as a function `f` in Python. Remember that it must also take `t` as a second variable, even if $\\ff$ is not explicitly a function of $t$:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"omega_squared = 4\n",
"\n",
"def harmonic(x_vec, t):\n",
" x, y = x_vec # tuple unpacking\n",
" \n",
" return [y, -omega_squared*x]\n",
"\n",
"y0 = [0.7, 0.5]\n",
"\n",
"t_output = np.arange(0, 5, 0.1)\n",
"\n",
"y_result = odeint(harmonic, y0, t_output)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 12
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#plt.figure(figsize = (5,5))\n",
"\n",
"fig, (ax1, ax2, ax3) = plt.subplots(ncols = 3, figsize=(10,10))\n",
"\n",
"xx, yy = y_result.T # extract x and y cols\n",
"\n",
"#ax1.plot(xx, yy)\n",
"plt.sca(ax1)\n",
"colorline(xx, yy, cmap='jet')\n",
"ax1.axis('scaled')\n",
"\n",
"#ax2.plot(t_output, xx)\n",
"plt.sca(ax2)\n",
"colorline(t_output, xx, cmap='cool')\n",
"ax2.axis('scaled')\n",
"\n",
"ax3.plot(t_output, yy)\n",
"ax3.axis('scaled')\n",
"\n",
"plt.tight_layout()\n",
"plt.show()\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
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],
"metadata": {},
"output_type": "display_data",
"text": [
"<matplotlib.figure.Figure at 0x114550f10>"
]
}
],
"prompt_number": 13
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"x = np.linspace(-1.0, 1.0, 10)\n",
"XX, YY = np.meshgrid(x, x)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 14
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"k = omega_squared\n",
"plt.quiver(XX, YY, YY, -k*XX, pivot='middle')\n",
"plt.axis('equal')"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 15,
"text": [
"(-1.0, 1.0, -1.0, 1.0)"
]
},
{
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],
"metadata": {},
"output_type": "display_data",
"text": [
"<matplotlib.figure.Figure at 0x1142b4110>"
]
}
],
"prompt_number": 15
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"plt.streamplot(XX, YY, YY, -k*XX)\n",
"plt.quiver(XX, YY, YY, -k*XX, pivot = 'middle')\n",
"plt.axis('equal')"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 16,
"text": [
"(-1.5, 1.0, -1.0, 1.0)"
]
},
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"' width=375 height=252/>"
],
"metadata": {},
"output_type": "display_data",
"text": [
"<matplotlib.figure.Figure at 0x1142f8210>"
]
}
],
"prompt_number": 16
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's try with a matrix:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"MM = np.array([[-1.5, 1.], [-1., -1.0]])\n",
"\n",
"def matrix(x_vec, t):\n",
" \n",
" return np.dot(MM, x_vec)\n",
"\n",
"y0 = [1, 1]\n",
"\n",
"t_output = np.arange(0, 10, 0.1)\n",
"y_result = odeint(matrix, y0, t_output)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 17
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"fig, (ax1, ax2, ax3) = plt.subplots(ncols = 3)\n",
"\n",
"xx, yy = y_result.T # extract x and y cols\n",
"\n",
"ax1.plot(xx, yy)\n",
"ax1.axis('scaled')\n",
"\n",
"ax2.plot(t_output, xx)\n",
"ax2.axis('scaled')\n",
"\n",
"ax3.plot(t_output, yy)\n",
"ax3.axis('scaled')\n",
"\n",
"plt.tight_layout()\n",
"plt.show()"
],
"language": "python",
"metadata": {},
"outputs": [
{
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{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Using the `ode` interface"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"An alternative interface is `ode`."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from scipy.integrate import ode"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 19
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{
"cell_type": "code",
"collapsed": false,
"input": [
"help(ode)"
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{
"output_type": "stream",
"stream": "stdout",
"text": [
"Help on class ode in module scipy.integrate._ode:\n",
"\n",
"class ode(__builtin__.object)\n",
" | A generic interface class to numeric integrators.\n",
" | \n",
" | Solve an equation system :math:`y'(t) = f(t,y)` with (optional) ``jac = df/dy``.\n",
" | \n",
" | Parameters\n",
" | ----------\n",
" | f : callable ``f(t, y, *f_args)``\n",
" | Rhs of the equation. t is a scalar, ``y.shape == (n,)``.\n",
" | ``f_args`` is set by calling ``set_f_params(*args)``.\n",
" | `f` should return a scalar, array or list (not a tuple).\n",
" | jac : callable ``jac(t, y, *jac_args)``\n",
" | Jacobian of the rhs, ``jac[i,j] = d f[i] / d y[j]``.\n",
" | ``jac_args`` is set by calling ``set_f_params(*args)``.\n",
" | \n",
" | Attributes\n",
" | ----------\n",
" | t : float\n",
" | Current time.\n",
" | y : ndarray\n",
" | Current variable values.\n",
" | \n",
" | See also\n",
" | --------\n",
" | odeint : an integrator with a simpler interface based on lsoda from ODEPACK\n",
" | quad : for finding the area under a curve\n",
" | \n",
" | Notes\n",
" | -----\n",
" | Available integrators are listed below. They can be selected using\n",
" | the `set_integrator` method.\n",
" | \n",
" | \"vode\"\n",
" | \n",
" | Real-valued Variable-coefficient Ordinary Differential Equation\n",
" | solver, with fixed-leading-coefficient implementation. It provides\n",
" | implicit Adams method (for non-stiff problems) and a method based on\n",
" | backward differentiation formulas (BDF) (for stiff problems).\n",
" | \n",
" | Source: http://www.netlib.org/ode/vode.f\n",
" | \n",
" | .. warning::\n",
" | \n",
" | This integrator is not re-entrant. You cannot have two `ode`\n",
" | instances using the \"vode\" integrator at the same time.\n",
" | \n",
" | This integrator accepts the following parameters in `set_integrator`\n",
" | method of the `ode` class:\n",
" | \n",
" | - atol : float or sequence\n",
" | absolute tolerance for solution\n",
" | - rtol : float or sequence\n",
" | relative tolerance for solution\n",
" | - lband : None or int\n",
" | - rband : None or int\n",
" | Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+rband.\n",
" | Setting these requires your jac routine to return the jacobian\n",
" | in packed format, jac_packed[i-j+lband, j] = jac[i,j].\n",
" | - method: 'adams' or 'bdf'\n",
" | Which solver to use, Adams (non-stiff) or BDF (stiff)\n",
" | - with_jacobian : bool\n",
" | Whether to use the jacobian\n",
" | - nsteps : int\n",
" | Maximum number of (internally defined) steps allowed during one\n",
" | call to the solver.\n",
" | - first_step : float\n",
" | - min_step : float\n",
" | - max_step : float\n",
" | Limits for the step sizes used by the integrator.\n",
" | - order : int\n",
" | Maximum order used by the integrator,\n",
" | order <= 12 for Adams, <= 5 for BDF.\n",
" | \n",
" | \"zvode\"\n",
" | \n",
" | Complex-valued Variable-coefficient Ordinary Differential Equation\n",
" | solver, with fixed-leading-coefficient implementation. It provides\n",
" | implicit Adams method (for non-stiff problems) and a method based on\n",
" | backward differentiation formulas (BDF) (for stiff problems).\n",
" | \n",
" | Source: http://www.netlib.org/ode/zvode.f\n",
" | \n",
" | .. warning::\n",
" | \n",
" | This integrator is not re-entrant. You cannot have two `ode`\n",
" | instances using the \"zvode\" integrator at the same time.\n",
" | \n",
" | This integrator accepts the same parameters in `set_integrator`\n",
" | as the \"vode\" solver.\n",
" | \n",
" | .. note::\n",
" | \n",
" | When using ZVODE for a stiff system, it should only be used for\n",
" | the case in which the function f is analytic, that is, when each f(i)\n",
" | is an analytic function of each y(j). Analyticity means that the\n",
" | partial derivative df(i)/dy(j) is a unique complex number, and this\n",
" | fact is critical in the way ZVODE solves the dense or banded linear\n",
" | systems that arise in the stiff case. For a complex stiff ODE system\n",
" | in which f is not analytic, ZVODE is likely to have convergence\n",
" | failures, and for this problem one should instead use DVODE on the\n",
" | equivalent real system (in the real and imaginary parts of y).\n",
" | \n",
" | \"lsoda\"\n",
" | \n",
" | Real-valued Variable-coefficient Ordinary Differential Equation\n",
" | solver, with fixed-leading-coefficient implementation. It provides\n",
" | automatic method switching between implicit Adams method (for non-stiff\n",
" | problems) and a method based on backward differentiation formulas (BDF)\n",
" | (for stiff problems).\n",
" | \n",
" | Source: http://www.netlib.org/odepack\n",
" | \n",
" | .. warning::\n",
" | \n",
" | This integrator is not re-entrant. You cannot have two `ode`\n",
" | instances using the \"lsoda\" integrator at the same time.\n",
" | \n",
" | This integrator accepts the following parameters in `set_integrator`\n",
" | method of the `ode` class:\n",
" | \n",
" | - atol : float or sequence\n",
" | absolute tolerance for solution\n",
" | - rtol : float or sequence\n",
" | relative tolerance for solution\n",
" | - lband : None or int\n",
" | - rband : None or int\n",
" | Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+rband.\n",
" | Setting these requires your jac routine to return the jacobian\n",
" | in packed format, jac_packed[i-j+lband, j] = jac[i,j].\n",
" | - with_jacobian : bool\n",
" | Whether to use the jacobian\n",
" | - nsteps : int\n",
" | Maximum number of (internally defined) steps allowed during one\n",
" | call to the solver.\n",
" | - first_step : float\n",
" | - min_step : float\n",
" | - max_step : float\n",
" | Limits for the step sizes used by the integrator.\n",
" | - max_order_ns : int\n",
" | Maximum order used in the nonstiff case (default 12).\n",
" | - max_order_s : int\n",
" | Maximum order used in the stiff case (default 5).\n",
" | - max_hnil : int\n",
" | Maximum number of messages reporting too small step size (t + h = t)\n",
" | (default 0)\n",
" | - ixpr : int\n",
" | Whether to generate extra printing at method switches (default False).\n",
" | \n",
" | \"dopri5\"\n",
" | \n",
" | This is an explicit runge-kutta method of order (4)5 due to Dormand &\n",
" | Prince (with stepsize control and dense output).\n",
" | \n",
" | Authors:\n",
" | \n",
" | E. Hairer and G. Wanner\n",
" | Universite de Geneve, Dept. de Mathematiques\n",
" | CH-1211 Geneve 24, Switzerland\n",
" | e-mail: ernst.hairer@math.unige.ch, gerhard.wanner@math.unige.ch\n",
" | \n",
" | This code is described in [HNW93]_.\n",
" | \n",
" | This integrator accepts the following parameters in set_integrator()\n",
" | method of the ode class:\n",
" | \n",
" | - atol : float or sequence\n",
" | absolute tolerance for solution\n",
" | - rtol : float or sequence\n",
" | relative tolerance for solution\n",
" | - nsteps : int\n",
" | Maximum number of (internally defined) steps allowed during one\n",
" | call to the solver.\n",
" | - first_step : float\n",
" | - max_step : float\n",
" | - safety : float\n",
" | Safety factor on new step selection (default 0.9)\n",
" | - ifactor : float\n",
" | - dfactor : float\n",
" | Maximum factor to increase/decrease step size by in one step\n",
" | - beta : float\n",
" | Beta parameter for stabilised step size control.\n",
" | \n",
" | \"dop853\"\n",
" | \n",
" | This is an explicit runge-kutta method of order 8(5,3) due to Dormand\n",
" | & Prince (with stepsize control and dense output).\n",
" | \n",
" | Options and references the same as \"dopri5\".\n",
" | \n",
" | Examples\n",
" | --------\n",
" | \n",
" | A problem to integrate and the corresponding jacobian:\n",
" | \n",
" | >>> from scipy.integrate import ode\n",
" | >>>\n",
" | >>> y0, t0 = [1.0j, 2.0], 0\n",
" | >>>\n",
" | >>> def f(t, y, arg1):\n",
" | >>> return [1j*arg1*y[0] + y[1], -arg1*y[1]**2]\n",
" | >>> def jac(t, y, arg1):\n",
" | >>> return [[1j*arg1, 1], [0, -arg1*2*y[1]]]\n",
" | \n",
" | The integration:\n",
" | \n",
" | >>> r = ode(f, jac).set_integrator('zvode', method='bdf', with_jacobian=True)\n",
" | >>> r.set_initial_value(y0, t0).set_f_params(2.0).set_jac_params(2.0)\n",
" | >>> t1 = 10\n",
" | >>> dt = 1\n",
" | >>> while r.successful() and r.t < t1:\n",
" | >>> r.integrate(r.t+dt)\n",
" | >>> print r.t, r.y\n",
" | \n",
" | References\n",
" | ----------\n",
" | .. [HNW93] E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary\n",
" | Differential Equations i. Nonstiff Problems. 2nd edition.\n",
" | Springer Series in Computational Mathematics,\n",
" | Springer-Verlag (1993)\n",
" | \n",
" | Methods defined here:\n",
" | \n",
" | __init__(self, f, jac=None)\n",
" | \n",
" | integrate(self, t, step=0, relax=0)\n",
" | Find y=y(t), set y as an initial condition, and return y.\n",
" | \n",
" | set_f_params(self, *args)\n",
" | Set extra parameters for user-supplied function f.\n",
" | \n",
" | set_initial_value(self, y, t=0.0)\n",
" | Set initial conditions y(t) = y.\n",
" | \n",
" | set_integrator(self, name, **integrator_params)\n",
" | Set integrator by name.\n",
" | \n",
" | Parameters\n",
" | ----------\n",
" | name : str\n",
" | Name of the integrator.\n",
" | integrator_params :\n",
" | Additional parameters for the integrator.\n",
" | \n",
" | set_jac_params(self, *args)\n",
" | Set extra parameters for user-supplied function jac.\n",
" | \n",
" | successful(self)\n",
" | Check if integration was successful.\n",
" | \n",
" | ----------------------------------------------------------------------\n",
" | Data descriptors defined here:\n",
" | \n",
" | __dict__\n",
" | dictionary for instance variables (if defined)\n",
" | \n",
" | __weakref__\n",
" | list of weak references to the object (if defined)\n",
" | \n",
" | y\n",
"\n"
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"An annoying 'feature' is that the function must now be written as `f(t, y)` rather than `f(y, t)` as before!\n",
"This interface includes two Runge-Kutta methods, `dopri5` and `dop853`."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's use the `harmonic` function, suitably modified to invert its arguments, to investigate `ode`:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"omega_squared = 4\n",
"\n",
"def harmonic_new(t, x_vec):\n",
" x, y = x_vec # tuple unpacking\n",
" \n",
" return [y, -omega_squared*x]"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 21
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We first create an object of type `ode` and set various parameters to indicate the type of method to use:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"t_final = 10.0\n",
"dt = 0.1\n",
"\n",
"y0 = [0.7, 0.5]\n",
"t0 = 0.0\n",
"\n",
"y_result = []\n",
"t_output = []\n",
"\n",
"# Initialisation:\n",
"\n",
"backend = \"dopri5\"\n",
"\n",
"solver = ode(harmonic_new)\n",
"solver.set_integrator(backend) # nsteps=1\n",
"solver.set_initial_value(y0, t0)\n",
"\n",
"y_result.append(y0)\n",
"t_output.append(t0)\n",
"\n",
"while solver.successful() and solver.t < t_final:\n",
" solver.integrate(solver.t + dt, step=1)\n",
" \n",
" y_result.append(solver.y)\n",
" t_output.append(solver.t)\n",
" \n",
"y_result = array(y_result)\n",
"t_output = array(t_output)\n",
"\n",
"#print y_result"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 22
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from scipy.integrate import ode\n",
"\n",
"def my_odeint(f, y0, t):\n",
" '''\n",
" ODE integrator compatible with odeint, that uses ode underneath\n",
" '''\n",
" \n",
" y0 = np.asarray(y0)\n",
" \n",
" backend = \"dopri5\"\n",
" \n",
" solver = ode(f)\n",
" solver.set_integrator(backend) # nsteps=1\n",
" \n",
" t0 = t[0]\n",
" t_final = t[-1]\n",
" \n",
" solver.set_initial_value(y0, t0)\n",
" \n",
" y_result = [y0]\n",
" \n",
" i = 1\n",
" current_t = t[i]\n",
" \n",
" while solver.successful() and solver.t < t_final:\n",
" solver.integrate(current_t, step=1)\n",
" i += 1\n",
" if i < len(t):\n",
" current_t = t[i]\n",
" \n",
" y_result.append(solver.y)\n",
" \n",
" return np.array(y_result)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 23
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"t_output = np.arange(0, 10, 0.01)\n",
"y0 = [1, 1]\n",
"\n",
"y_result = my_odeint(harmonic_new, y0, t_output)\n"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 24
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"fig, (ax1, ax2, ax3) = plt.subplots(ncols = 3, figsize=(10,10))\n",
"\n",
"xx, yy = y_result.T # extract x and y cols\n",
"\n",
"#ax1.plot(xx, yy)\n",
"plt.sca(ax1)\n",
"colorline(xx, yy, cmap='jet')\n",
"ax1.axis('scaled')\n",
"\n",
"#ax2.plot(t_output, xx)\n",
"plt.sca(ax2)\n",
"colorline(t_output, xx, cmap='cool')\n",
"ax2.axis('scaled')\n",
"\n",
"plt.sca(ax3)\n",
"colorline(t_output, yy)\n",
"ax3.axis('scaled')\n",
"\n",
"plt.tight_layout()\n",
"plt.show()"
],
"language": "python",
"metadata": {},
"outputs": [
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],
"metadata": {},
"output_type": "display_data",
"text": [
"<matplotlib.figure.Figure at 0x11433c390>"
]
}
],
"prompt_number": 25
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from numpy import zeros, linspace, array\n",
"from scipy.integrate import ode\n",
"from pylab import figure, show, xlabel, ylabel\n",
"from mpl_toolkits.mplot3d import Axes3D\n",
"\n",
"def lorenz_sys(t, q):\n",
" x = q[0]\n",
" y = q[1]\n",
" z = q[2]\n",
" # sigma, rho and beta are global.\n",
" f = [sigma * (y - x),\n",
" rho*x - y - x*z,\n",
" x*y - beta*z]\n",
" return f\n",
"\n",
"\n",
"ic = [1.0, 2.0, 1.0]\n",
"t0 = 0.0\n",
"t1 = 100.0\n",
"dt = 0.01\n",
"\n",
"sigma = 10.0\n",
"rho = 28.0\n",
"beta = 10.0/3\n",
"\n",
"solver = ode(lorenz_sys)\n",
"\n",
"t = []\n",
"sol = []\n",
"solver.set_initial_value(ic, t0)\n",
"#solver.set_integrator('dop853')\n",
"solver.set_integrator('dopri5')\n",
"\n",
"while solver.successful() and solver.t < t1:\n",
" solver.integrate(solver.t + dt)\n",
" t.append(solver.t)\n",
" sol.append(solver.y)\n",
"\n",
"t = array(t)\n",
"sol = array(sol)\n",
"\n",
"fig = figure()\n",
"ax = Axes3D(fig)\n",
"ax.plot(sol[:,0], sol[:,1], sol[:,2])\n",
"xlabel('x')\n",
"ylabel('y')\n",
"show()\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
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