package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Cumulatively integrate y(x) using the composite trapezoidal rule.

Parameters ---------- y : array_like Values to integrate. x : array_like, optional The coordinate to integrate along. If None (default), use spacing `dx` between consecutive elements in `y`. dx : float, optional Spacing between elements of `y`. Only used if `x` is None. axis : int, optional Specifies the axis to cumulate. Default is -1 (last axis). initial : scalar, optional If given, insert this value at the beginning of the returned result. Typically this value should be 0. Default is None, which means no value at ``x0`` is returned and `res` has one element less than `y` along the axis of integration.

Returns ------- res : ndarray The result of cumulative integration of `y` along `axis`. If `initial` is None, the shape is such that the axis of integration has one less value than `y`. If `initial` is given, the shape is equal to that of `y`.

See Also -------- numpy.cumsum, numpy.cumprod quad: adaptive quadrature using QUADPACK romberg: adaptive Romberg quadrature quadrature: adaptive Gaussian quadrature fixed_quad: fixed-order Gaussian quadrature dblquad: double integrals tplquad: triple integrals romb: integrators for sampled data ode: ODE integrators odeint: ODE integrators

Examples -------- >>> from scipy import integrate >>> import matplotlib.pyplot as plt

>>> x = np.linspace(-2, 2, num=20) >>> y = x >>> y_int = integrate.cumtrapz(y, x, initial=0) >>> plt.plot(x, y_int, 'ro', x, y0 + 0.5 * x**2, 'b-') >>> plt.show()

Compute a double integral.

Return the double (definite) integral of ``func(y, x)`` from ``x = a..b`` and ``y = gfun(x)..hfun(x)``.

Parameters ---------- func : callable A Python function or method of at least two variables: y must be the first argument and x the second argument. a, b : float The limits of integration in x: `a` < `b` gfun : callable or float The lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result or a float indicating a constant boundary curve. hfun : callable or float The upper boundary curve in y (same requirements as `gfun`). args : sequence, optional Extra arguments to pass to `func`. epsabs : float, optional Absolute tolerance passed directly to the inner 1-D quadrature integration. Default is 1.49e-8. epsrel : float, optional Relative tolerance of the inner 1-D integrals. Default is 1.49e-8.

Returns ------- y : float The resultant integral. abserr : float An estimate of the error.

See also -------- quad : single integral tplquad : triple integral nquad : N-dimensional integrals fixed_quad : fixed-order Gaussian quadrature quadrature : adaptive Gaussian quadrature odeint : ODE integrator ode : ODE integrator simps : integrator for sampled data romb : integrator for sampled data scipy.special : for coefficients and roots of orthogonal polynomials

Examples --------

Compute the double integral of ``x * y**2`` over the box ``x`` ranging from 0 to 2 and ``y`` ranging from 0 to 1.

>>> from scipy import integrate >>> f = lambda y, x: x*y**2 >>> integrate.dblquad(f, 0, 2, lambda x: 0, lambda x: 1) (0.6666666666666667, 7.401486830834377e-15)

Compute a definite integral using fixed-order Gaussian quadrature.

Integrate `func` from `a` to `b` using Gaussian quadrature of order `n`.

Parameters ---------- func : callable A Python function or method to integrate (must accept vector inputs). If integrating a vector-valued function, the returned array must have shape ``(..., len(x))``. a : float Lower limit of integration. b : float Upper limit of integration. args : tuple, optional Extra arguments to pass to function, if any. n : int, optional Order of quadrature integration. Default is 5.

Returns ------- val : float Gaussian quadrature approximation to the integral none : None Statically returned value of None

See Also -------- quad : adaptive quadrature using QUADPACK dblquad : double integrals tplquad : triple integrals romberg : adaptive Romberg quadrature quadrature : adaptive Gaussian quadrature romb : integrators for sampled data simps : integrators for sampled data cumtrapz : cumulative integration for sampled data ode : ODE integrator odeint : ODE integrator

Examples -------- >>> from scipy import integrate >>> f = lambda x: x**8 >>> integrate.fixed_quad(f, 0.0, 1.0, n=4) (0.1110884353741496, None) >>> integrate.fixed_quad(f, 0.0, 1.0, n=5) (0.11111111111111102, None) >>> print(1/9.0) # analytical result 0.1111111111111111

>>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=4) (0.9999999771971152, None) >>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=5) (1.000000000039565, None) >>> np.sin(np.pi/2)-np.sin(0) # analytical result 1.0

Return weights and error coefficient for Newton-Cotes integration.

Suppose we have (N+1) samples of f at the positions x_0, x_1, ..., x_N. Then an N-point Newton-Cotes formula for the integral between x_0 and x_N is:

:math:`\int_x_0^x_N f(x)dx = \Delta x \sum_=0^N a_i f(x_i)

1. B_N (\Delta x)^N+2 f^N+1 (\xi)`

where :math:`\xi \in x_0,x_N` and :math:`\Delta x = \fracx_N-x_0N` is the average samples spacing.

If the samples are equally-spaced and N is even, then the error term is :math:`B_N (\Delta x)^N+3 f^N+2(\xi)`.

Parameters ---------- rn : int The integer order for equally-spaced data or the relative positions of the samples with the first sample at 0 and the last at N, where N+1 is the length of `rn`. N is the order of the Newton-Cotes integration. equal : int, optional Set to 1 to enforce equally spaced data.

Returns ------- an : ndarray 1-D array of weights to apply to the function at the provided sample positions. B : float Error coefficient.

Examples -------- Compute the integral of sin(x) in 0, :math:`\pi`:

>>> from scipy.integrate import newton_cotes >>> def f(x): ... return np.sin(x) >>> a = 0 >>> b = np.pi >>> exact = 2 >>> for N in 2, 4, 6, 8, 10: ... x = np.linspace(a, b, N + 1) ... an, B = newton_cotes(N, 1) ... dx = (b - a) / N ... quad = dx * np.sum(an * f(x)) ... error = abs(quad - exact) ... print('2d 10.9f .5e'.format(N, quad, error)) ... 2 2.094395102 9.43951e-02 4 1.998570732 1.42927e-03 6 2.000017814 1.78136e-05 8 1.999999835 1.64725e-07 10 2.000000001 1.14677e-09

Notes ----- Normally, the Newton-Cotes rules are used on smaller integration regions and a composite rule is used to return the total integral.

Integration over multiple variables.

Wraps `quad` to enable integration over multiple variables. Various options allow improved integration of discontinuous functions, as well as the use of weighted integration, and generally finer control of the integration process.

Parameters ---------- func : callable, scipy.LowLevelCallable The function to be integrated. Has arguments of ``x0, ... xn``, ``t0, tm``, where integration is carried out over ``x0, ... xn``, which must be floats. Function signature should be ``func(x0, x1, ..., xn, t0, t1, ..., tm)``. Integration is carried out in order. That is, integration over ``x0`` is the innermost integral, and ``xn`` is the outermost.

If the user desires improved integration performance, then `f` may be a `scipy.LowLevelCallable` with one of the signatures::

double func(int n, double *xx) double func(int n, double *xx, void *user_data)

where ``n`` is the number of extra parameters and args is an array of doubles of the additional parameters, the ``xx`` array contains the coordinates. The ``user_data`` is the data contained in the `scipy.LowLevelCallable`. ranges : iterable object Each element of ranges may be either a sequence of 2 numbers, or else a callable that returns such a sequence. ``ranges0`` corresponds to integration over x0, and so on. If an element of ranges is a callable, then it will be called with all of the integration arguments available, as well as any parametric arguments. e.g. if ``func = f(x0, x1, x2, t0, t1)``, then ``ranges0`` may be defined as either ``(a, b)`` or else as ``(a, b) = range0(x1, x2, t0, t1)``. args : iterable object, optional Additional arguments ``t0, ..., tn``, required by `func`, `ranges`, and ``opts``. opts : iterable object or dict, optional Options to be passed to `quad`. May be empty, a dict, or a sequence of dicts or functions that return a dict. If empty, the default options from scipy.integrate.quad are used. If a dict, the same options are used for all levels of integraion. If a sequence, then each element of the sequence corresponds to a particular integration. e.g. opts0 corresponds to integration over x0, and so on. If a callable, the signature must be the same as for ``ranges``. The available options together with their default values are:

• epsabs = 1.49e-08
• epsrel = 1.49e-08
• limit = 50
• points = None
• weight = None
• wvar = None
• wopts = None

For more information on these options, see `quad` and `quad_explain`.

full_output : bool, optional Partial implementation of ``full_output`` from scipy.integrate.quad. The number of integrand function evaluations ``neval`` can be obtained by setting ``full_output=True`` when calling nquad.

Returns ------- result : float The result of the integration. abserr : float The maximum of the estimates of the absolute error in the various integration results. out_dict : dict, optional A dict containing additional information on the integration.

See Also -------- quad : 1-dimensional numerical integration dblquad, tplquad : double and triple integrals fixed_quad : fixed-order Gaussian quadrature quadrature : adaptive Gaussian quadrature

Examples -------- >>> from scipy import integrate >>> func = lambda x0,x1,x2,x3 : x0**2 + x1*x2 - x3**3 + np.sin(x0) + ( ... 1 if (x0-.2*x3-.5-.25*x1>0) else 0) >>> points = [lambda x1,x2,x3 : 0.2*x3 + 0.5 + 0.25*x1], [], [], [] >>> def opts0( *args, **kwargs): ... return 'points':[0.2*args[2] + 0.5 + 0.25*args[0]] >>> integrate.nquad(func, [0,1], [-1,1], [.13,.8], [-.15,1], ... opts=opts0,{},{},{}, full_output=True) (1.5267454070738633, 2.9437360001402324e-14, 'neval': 388962)

>>> scale = .1 >>> def func2(x0, x1, x2, x3, t0, t1): ... return x0*x1*x3**2 + np.sin(x2) + 1 + (1 if x0+t1*x1-t0>0 else 0) >>> def lim0(x1, x2, x3, t0, t1): ... return scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) - 1, ... scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) + 1 >>> def lim1(x2, x3, t0, t1): ... return scale * (t0*x2 + t1*x3) - 1, ... scale * (t0*x2 + t1*x3) + 1 >>> def lim2(x3, t0, t1): ... return scale * (x3 + t0**2*t1**3) - 1, ... scale * (x3 + t0**2*t1**3) + 1 >>> def lim3(t0, t1): ... return scale * (t0+t1) - 1, scale * (t0+t1) + 1 >>> def opts0(x1, x2, x3, t0, t1): ... return 'points' : [t0 - t1*x1] >>> def opts1(x2, x3, t0, t1): ... return {

}

>>> def opts2(x3, t0, t1): ... return {

}

>>> def opts3(t0, t1): ... return {

}

>>> integrate.nquad(func2, lim0, lim1, lim2, lim3, args=(0,0), ... opts=opts0, opts1, opts2, opts3) (25.066666666666666, 2.7829590483937256e-13)

Integrate a system of ordinary differential equations.

.. note:: For new code, use `scipy.integrate.solve_ivp` to solve a differential equation.

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, t, ...) or func(t, y, ...)

where y can be a vector.

.. note:: By default, the required order of the first two arguments of `func` are in the opposite order of the arguments in the system definition function used by the `scipy.integrate.ode` class and the function `scipy.integrate.solve_ivp`. To use a function with the signature ``func(t, y, ...)``, the argument `tfirst` must be set to ``True``.

Parameters ---------- func : callable(y, t, ...) or callable(t, y, ...) Computes the derivative of y at t. If the signature is ``callable(t, y, ...)``, then the argument `tfirst` must be set ``True``. 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. This sequence must be monotonically increasing or monotonically decreasing; repeated values are allowed. args : tuple, optional Extra arguments to pass to function. Dfun : callable(y, t, ...) or callable(t, y, ...) Gradient (Jacobian) of `func`. If the signature is ``callable(t, y, ...)``, then the argument `tfirst` must be set ``True``. col_deriv : bool, optional True if `Dfun` defines derivatives down columns (faster), otherwise `Dfun` should define derivatives across rows. full_output : bool, optional True if to return a dictionary of optional outputs as the second output printmessg : bool, optional Whether to print the convergence message tfirst: bool, optional If True, the first two arguments of `func` (and `Dfun`, if given) must ``t, y`` instead of the default ``y, t``.

.. versionadded:: 1.1.0

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, optional 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 rows contain the non-zero bands (starting with the lowest diagonal). Thus, the return matrix `jac` from `Dfun` should have shape ``(ml + mu + 1, len(y0))`` when ``ml >=0`` or ``mu >=0``. The data in `jac` must be stored such that ``jaci - j + mu, j`` holds the derivative of the `i`th equation with respect to the `j`th state variable. If `col_deriv` is True, the transpose of this `jac` must be returned. rtol, atol : float, optional 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, optional Vector of critical points (e.g. singularities) where integration care should be taken. h0 : float, (0: solver-determined), optional The step size to be attempted on the first step. hmax : float, (0: solver-determined), optional The maximum absolute step size allowed. hmin : float, (0: solver-determined), optional The minimum absolute step size allowed. ixpr : bool, optional Whether to generate extra printing at method switches. mxstep : int, (0: solver-determined), optional Maximum number of (internally defined) steps allowed for each integration point in t. mxhnil : int, (0: solver-determined), optional Maximum number of messages printed. mxordn : int, (0: solver-determined), optional Maximum order to be allowed for the non-stiff (Adams) method. mxords : int, (0: solver-determined), optional Maximum order to be allowed for the stiff (BDF) method.

See Also -------- solve_ivp : Solve an initial value problem for a system of ODEs. ode : a more object-oriented integrator based on VODE. quad : for finding the area under a curve.

Examples -------- The second order differential equation for the angle `theta` of a pendulum acted on by gravity with friction can be written::

theta''(t) + b*theta'(t) + c*sin(theta(t)) = 0

where `b` and `c` are positive constants, and a prime (') denotes a derivative. To solve this equation with `odeint`, we must first convert it to a system of first order equations. By defining the angular velocity ``omega(t) = theta'(t)``, we obtain the system::

theta'(t) = omega(t) omega'(t) = -b*omega(t) - c*sin(theta(t))

Let `y` be the vector `theta`, `omega`. We implement this system in python as:

>>> def pend(y, t, b, c): ... theta, omega = y ... dydt = omega, -b*omega - c*np.sin(theta) ... return dydt ...

We assume the constants are `b` = 0.25 and `c` = 5.0:

>>> b = 0.25 >>> c = 5.0

For initial conditions, we assume the pendulum is nearly vertical with `theta(0)` = `pi` - 0.1, and is initially at rest, so `omega(0)` = 0. Then the vector of initial conditions is

>>> y0 = np.pi - 0.1, 0.0

We will generate a solution at 101 evenly spaced samples in the interval 0 <= `t` <= 10. So our array of times is:

>>> t = np.linspace(0, 10, 101)

Call `odeint` to generate the solution. To pass the parameters `b` and `c` to `pend`, we give them to `odeint` using the `args` argument.

>>> from scipy.integrate import odeint >>> sol = odeint(pend, y0, t, args=(b, c))

The solution is an array with shape (101, 2). The first column is `theta(t)`, and the second is `omega(t)`. The following code plots both components.

>>> import matplotlib.pyplot as plt >>> plt.plot(t, sol:, 0, 'b', label='theta(t)') >>> plt.plot(t, sol:, 1, 'g', label='omega(t)') >>> plt.legend(loc='best') >>> plt.xlabel('t') >>> plt.grid() >>> plt.show()

Compute a definite integral.

Integrate func from `a` to `b` (possibly infinite interval) using a technique from the Fortran library QUADPACK.

Parameters ---------- func : function, scipy.LowLevelCallable A Python function or method to integrate. If `func` takes many arguments, it is integrated along the axis corresponding to the first argument.

If the user desires improved integration performance, then `f` may be a `scipy.LowLevelCallable` with one of the signatures::

double func(double x) double func(double x, void *user_data) double func(int n, double *xx) double func(int n, double *xx, void *user_data)

The ``user_data`` is the data contained in the `scipy.LowLevelCallable`. In the call forms with ``xx``, ``n`` is the length of the ``xx`` array which contains ``xx0 == x`` and the rest of the items are numbers contained in the ``args`` argument of quad.

In addition, certain ctypes call signatures are supported for backward compatibility, but those should not be used in new code. a : float Lower limit of integration (use -numpy.inf for -infinity). b : float Upper limit of integration (use numpy.inf for +infinity). args : tuple, optional Extra arguments to pass to `func`. full_output : int, optional Non-zero to return a dictionary of integration information. If non-zero, warning messages are also suppressed and the message is appended to the output tuple.

Returns ------- y : float The integral of func from `a` to `b`. abserr : float An estimate of the absolute error in the result. infodict : dict A dictionary containing additional information. Run scipy.integrate.quad_explain() for more information. message A convergence message. explain Appended only with 'cos' or 'sin' weighting and infinite integration limits, it contains an explanation of the codes in infodict'ierlst'

Other Parameters ---------------- epsabs : float or int, optional Absolute error tolerance. epsrel : float or int, optional Relative error tolerance. limit : float or int, optional An upper bound on the number of subintervals used in the adaptive algorithm. points : (sequence of floats,ints), optional A sequence of break points in the bounded integration interval where local difficulties of the integrand may occur (e.g., singularities, discontinuities). The sequence does not have to be sorted. Note that this option cannot be used in conjunction with ``weight``. weight : float or int, optional String indicating weighting function. Full explanation for this and the remaining arguments can be found below. wvar : optional Variables for use with weighting functions. wopts : optional Optional input for reusing Chebyshev moments. maxp1 : float or int, optional An upper bound on the number of Chebyshev moments. limlst : int, optional Upper bound on the number of cycles (>=3) for use with a sinusoidal weighting and an infinite end-point.

See Also -------- dblquad : double integral tplquad : triple integral nquad : n-dimensional integrals (uses `quad` recursively) fixed_quad : fixed-order Gaussian quadrature quadrature : adaptive Gaussian quadrature odeint : ODE integrator ode : ODE integrator simps : integrator for sampled data romb : integrator for sampled data scipy.special : for coefficients and roots of orthogonal polynomials

Notes -----

**Extra information for quad() inputs and outputs**

If full_output is non-zero, then the third output argument (infodict) is a dictionary with entries as tabulated below. For infinite limits, the range is transformed to (0,1) and the optional outputs are given with respect to this transformed range. Let M be the input argument limit and let K be infodict'last'. The entries are:

'neval' The number of function evaluations. 'last' The number, K, of subintervals produced in the subdivision process. 'alist' A rank-1 array of length M, the first K elements of which are the left end points of the subintervals in the partition of the integration range. 'blist' A rank-1 array of length M, the first K elements of which are the right end points of the subintervals. 'rlist' A rank-1 array of length M, the first K elements of which are the integral approximations on the subintervals. 'elist' A rank-1 array of length M, the first K elements of which are the moduli of the absolute error estimates on the subintervals. 'iord' A rank-1 integer array of length M, the first L elements of which are pointers to the error estimates over the subintervals with ``L=K`` if ``K<=M/2+2`` or ``L=M+1-K`` otherwise. Let I be the sequence ``infodict'iord'`` and let E be the sequence ``infodict'elist'``. Then ``EI[1], ..., EI[L]`` forms a decreasing sequence.

If the input argument points is provided (i.e. it is not None), the following additional outputs are placed in the output dictionary. Assume the points sequence is of length P.

'pts' A rank-1 array of length P+2 containing the integration limits and the break points of the intervals in ascending order. This is an array giving the subintervals over which integration will occur. 'level' A rank-1 integer array of length M (=limit), containing the subdivision levels of the subintervals, i.e., if (aa,bb) is a subinterval of ``(pts1, pts2)`` where ``pts0`` and ``pts2`` are adjacent elements of ``infodict'pts'``, then (aa,bb) has level l if ``|bb-aa| = |pts2-pts1| * 2**(-l)``. 'ndin' A rank-1 integer array of length P+2. After the first integration over the intervals (pts1, pts2), the error estimates over some of the intervals may have been increased artificially in order to put their subdivision forward. This array has ones in slots corresponding to the subintervals for which this happens.

**Weighting the integrand**

The input variables, *weight* and *wvar*, are used to weight the integrand by a select list of functions. Different integration methods are used to compute the integral with these weighting functions, and these do not support specifying break points. The possible values of weight and the corresponding weighting functions are.

========== =================================== ===================== ``weight`` Weight function used ``wvar`` ========== =================================== ===================== 'cos' cos(w*x) wvar = w 'sin' sin(w*x) wvar = w 'alg' g(x) = ((x-a)**alpha)*((b-x)**beta) wvar = (alpha, beta) 'alg-loga' g(x)*log(x-a) wvar = (alpha, beta) 'alg-logb' g(x)*log(b-x) wvar = (alpha, beta) 'alg-log' g(x)*log(x-a)*log(b-x) wvar = (alpha, beta) 'cauchy' 1/(x-c) wvar = c ========== =================================== =====================

wvar holds the parameter w, (alpha, beta), or c depending on the weight selected. In these expressions, a and b are the integration limits.

For the 'cos' and 'sin' weighting, additional inputs and outputs are available.

For finite integration limits, the integration is performed using a Clenshaw-Curtis method which uses Chebyshev moments. For repeated calculations, these moments are saved in the output dictionary:

'momcom' The maximum level of Chebyshev moments that have been computed, i.e., if ``M_c`` is ``infodict'momcom'`` then the moments have been computed for intervals of length ``|b-a| * 2**(-l)``, ``l=0,1,...,M_c``. 'nnlog' A rank-1 integer array of length M(=limit), containing the subdivision levels of the subintervals, i.e., an element of this array is equal to l if the corresponding subinterval is ``|b-a|* 2**(-l)``. 'chebmo' A rank-2 array of shape (25, maxp1) containing the computed Chebyshev moments. These can be passed on to an integration over the same interval by passing this array as the second element of the sequence wopts and passing infodict'momcom' as the first element.

If one of the integration limits is infinite, then a Fourier integral is computed (assuming w neq 0). If full_output is 1 and a numerical error is encountered, besides the error message attached to the output tuple, a dictionary is also appended to the output tuple which translates the error codes in the array ``info'ierlst'`` to English messages. The output information dictionary contains the following entries instead of 'last', 'alist', 'blist', 'rlist', and 'elist':

'lst' The number of subintervals needed for the integration (call it ``K_f``). 'rslst' A rank-1 array of length M_f=limlst, whose first ``K_f`` elements contain the integral contribution over the interval ``(a+(k-1)c, a+kc)`` where ``c = (2*floor(|w|) + 1) * pi / |w|`` and ``k=1,2,...,K_f``. 'erlst' A rank-1 array of length ``M_f`` containing the error estimate corresponding to the interval in the same position in ``infodict'rslist'``. 'ierlst' A rank-1 integer array of length ``M_f`` containing an error flag corresponding to the interval in the same position in ``infodict'rslist'``. See the explanation dictionary (last entry in the output tuple) for the meaning of the codes.

Examples -------- Calculate :math:`\int^4_0 x^2 dx` and compare with an analytic result

>>> from scipy import integrate >>> x2 = lambda x: x**2 >>> integrate.quad(x2, 0, 4) (21.333333333333332, 2.3684757858670003e-13) >>> print(4**3 / 3.) # analytical result 21.3333333333

Calculate :math:`\int^\infty_0 e^

x

}

dx`

>>> invexp = lambda x: np.exp(-x) >>> integrate.quad(invexp, 0, np.inf) (1.0, 5.842605999138044e-11)

>>> f = lambda x,a : a*x >>> y, err = integrate.quad(f, 0, 1, args=(1,)) >>> y 0.5 >>> y, err = integrate.quad(f, 0, 1, args=(3,)) >>> y 1.5

Calculate :math:`\int^1_0 x^2 + y^2 dx` with ctypes, holding y parameter as 1::

testlib.c => double func(int n, double argsn) return args[0]*args[0] + args[1]*args[1]; compile to library testlib.*

::

from scipy import integrate import ctypes lib = ctypes.CDLL('/home/.../testlib.*') #use absolute path lib.func.restype = ctypes.c_double lib.func.argtypes = (ctypes.c_int,ctypes.c_double) integrate.quad(lib.func,0,1,(1)) #(1.3333333333333333, 1.4802973661668752e-14) print((1.0**3/3.0 + 1.0) - (0.0**3/3.0 + 0.0)) #Analytic result # 1.3333333333333333

Be aware that pulse shapes and other sharp features as compared to the size of the integration interval may not be integrated correctly using this method. A simplified example of this limitation is integrating a y-axis reflected step function with many zero values within the integrals bounds.

>>> y = lambda x: 1 if x<=0 else 0 >>> integrate.quad(y, -1, 1) (1.0, 1.1102230246251565e-14) >>> integrate.quad(y, -1, 100) (1.0000000002199108, 1.0189464580163188e-08) >>> integrate.quad(y, -1, 10000) (0.0, 0.0)

Print extra information about integrate.quad() parameters and returns.

Parameters ---------- output : instance with 'write' method, optional Information about `quad` is passed to ``output.write()``. Default is ``sys.stdout``.

Returns ------- None

Adaptive integration of a vector-valued function.

Parameters ---------- f : callable Vector-valued function f(x) to integrate. a : float Initial point. b : float Final point. epsabs : float, optional Absolute tolerance. epsrel : float, optional Relative tolerance. norm : 'max', '2', optional Vector norm to use for error estimation. cache_size : int, optional Number of bytes to use for memoization. workers : int or map-like callable, optional If `workers` is an integer, part of the computation is done in parallel subdivided to this many tasks (using :class:`python:multiprocessing.pool.Pool`). Supply `-1` to use all cores available to the Process. Alternatively, supply a map-like callable, such as :meth:`python:multiprocessing.pool.Pool.map` for evaluating the population in parallel. This evaluation is carried out as ``workers(func, iterable)``. points : list, optional List of additional breakpoints. quadrature : 'gk21', 'gk15', 'trapz', optional Quadrature rule to use on subintervals. Options: 'gk21' (Gauss-Kronrod 21-point rule), 'gk15' (Gauss-Kronrod 15-point rule), 'trapz' (composite trapezoid rule). Default: 'gk21' for finite intervals and 'gk15' for (semi-)infinite full_output : bool, optional Return an additional ``info`` dictionary.

Returns ------- res : float, array-like Estimate for the result err : float Error estimate for the result in the given norm info : dict Returned only when ``full_output=True``. Info dictionary. Is an object with the attributes:

success : bool Whether integration reached target precision. status : int Indicator for convergence, success (0), failure (1), and failure due to rounding error (2). neval : int Number of function evaluations. intervals : ndarray, shape (num_intervals, 2) Start and end points of subdivision intervals. integrals : ndarray, shape (num_intervals, ...) Integral for each interval. Note that at most ``cache_size`` values are recorded, and the array may contains *nan* for missing items. errors : ndarray, shape (num_intervals,) Estimated integration error for each interval.

Notes ----- The algorithm mainly follows the implementation of QUADPACK's DQAG* algorithms, implementing global error control and adaptive subdivision.

The algorithm here has some differences to the QUADPACK approach:

Instead of subdividing one interval at a time, the algorithm subdivides N intervals with largest errors at once. This enables (partial) parallelization of the integration.

The logic of subdividing 'next largest' intervals first is then not implemented, and we rely on the above extension to avoid concentrating on 'small' intervals only.

The Wynn epsilon table extrapolation is not used (QUADPACK uses it for infinite intervals). This is because the algorithm here is supposed to work on vector-valued functions, in an user-specified norm, and the extension of the epsilon algorithm to this case does not appear to be widely agreed. For max-norm, using elementwise Wynn epsilon could be possible, but we do not do this here with the hope that the epsilon extrapolation is mainly useful in special cases.

References ---------- 1 R. Piessens, E. de Doncker, QUADPACK (1983).

Compute a definite integral using fixed-tolerance Gaussian quadrature.

Integrate `func` from `a` to `b` using Gaussian quadrature with absolute tolerance `tol`.

Parameters ---------- func : function A Python function or method to integrate. a : float Lower limit of integration. b : float Upper limit of integration. args : tuple, optional Extra arguments to pass to function. tol, rtol : float, optional Iteration stops when error between last two iterates is less than `tol` OR the relative change is less than `rtol`. maxiter : int, optional Maximum order of Gaussian quadrature. vec_func : bool, optional True or False if func handles arrays as arguments (is a 'vector' function). Default is True. miniter : int, optional Minimum order of Gaussian quadrature.

Returns ------- val : float Gaussian quadrature approximation (within tolerance) to integral. err : float Difference between last two estimates of the integral.

See also -------- romberg: adaptive Romberg quadrature fixed_quad: fixed-order Gaussian quadrature quad: adaptive quadrature using QUADPACK dblquad: double integrals tplquad: triple integrals romb: integrator for sampled data simps: integrator for sampled data cumtrapz: cumulative integration for sampled data ode: ODE integrator odeint: ODE integrator

Examples -------- >>> from scipy import integrate >>> f = lambda x: x**8 >>> integrate.quadrature(f, 0.0, 1.0) (0.11111111111111106, 4.163336342344337e-17) >>> print(1/9.0) # analytical result 0.1111111111111111

>>> integrate.quadrature(np.cos, 0.0, np.pi/2) (0.9999999999999536, 3.9611425250996035e-11) >>> np.sin(np.pi/2)-np.sin(0) # analytical result 1.0

Romberg integration using samples of a function.

Parameters ---------- y : array_like A vector of ``2**k + 1`` equally-spaced samples of a function. dx : float, optional The sample spacing. Default is 1. axis : int, optional The axis along which to integrate. Default is -1 (last axis). show : bool, optional When `y` is a single 1-D array, then if this argument is True print the table showing Richardson extrapolation from the samples. Default is False.

Returns ------- romb : ndarray The integrated result for `axis`.

See also -------- quad : adaptive quadrature using QUADPACK romberg : adaptive Romberg quadrature quadrature : adaptive Gaussian quadrature fixed_quad : fixed-order Gaussian quadrature dblquad : double integrals tplquad : triple integrals simps : integrators for sampled data cumtrapz : cumulative integration for sampled data ode : ODE integrators odeint : ODE integrators

Examples -------- >>> from scipy import integrate >>> x = np.arange(10, 14.25, 0.25) >>> y = np.arange(3, 12)

>>> integrate.romb(y) 56.0

>>> y = np.sin(np.power(x, 2.5)) >>> integrate.romb(y) -0.742561336672229

>>> integrate.romb(y, show=True) Richardson Extrapolation Table for Romberg Integration ==================================================================== -0.81576 4.63862 6.45674 -1.10581 -3.02062 -3.65245 -2.57379 -3.06311 -3.06595 -3.05664 -1.34093 -0.92997 -0.78776 -0.75160 -0.74256 ==================================================================== -0.742561336672229

Romberg integration of a callable function or method.

Returns the integral of `function` (a function of one variable) over the interval (`a`, `b`).

If `show` is 1, the triangular array of the intermediate results will be printed. If `vec_func` is True (default is False), then `function` is assumed to support vector arguments.

Parameters ---------- function : callable Function to be integrated. a : float Lower limit of integration. b : float Upper limit of integration.

Returns ------- results : float Result of the integration.

Other Parameters ---------------- args : tuple, optional Extra arguments to pass to function. Each element of `args` will be passed as a single argument to `func`. Default is to pass no extra arguments. tol, rtol : float, optional The desired absolute and relative tolerances. Defaults are 1.48e-8. show : bool, optional Whether to print the results. Default is False. divmax : int, optional Maximum order of extrapolation. Default is 10. vec_func : bool, optional Whether `func` handles arrays as arguments (i.e whether it is a 'vector' function). Default is False.

See Also -------- fixed_quad : Fixed-order Gaussian quadrature. quad : Adaptive quadrature using QUADPACK. dblquad : Double integrals. tplquad : Triple integrals. romb : Integrators for sampled data. simps : Integrators for sampled data. cumtrapz : Cumulative integration for sampled data. ode : ODE integrator. odeint : ODE integrator.

References ---------- .. 1 'Romberg's method' https://en.wikipedia.org/wiki/Romberg%27s_method

Examples -------- Integrate a gaussian from 0 to 1 and compare to the error function.

>>> from scipy import integrate >>> from scipy.special import erf >>> gaussian = lambda x: 1/np.sqrt(np.pi) * np.exp(-x**2) >>> result = integrate.romberg(gaussian, 0, 1, show=True) Romberg integration of <function vfunc at ...> from 0, 1

::

Steps StepSize Results 1 1.000000 0.385872 2 0.500000 0.412631 0.421551 4 0.250000 0.419184 0.421368 0.421356 8 0.125000 0.420810 0.421352 0.421350 0.421350 16 0.062500 0.421215 0.421350 0.421350 0.421350 0.421350 32 0.031250 0.421317 0.421350 0.421350 0.421350 0.421350 0.421350

The final result is 0.421350396475 after 33 function evaluations.

>>> print('%g %g' % (2*result, erf(1))) 0.842701 0.842701

Integrate y(x) using samples along the given axis and the composite Simpson's rule. If x is None, spacing of dx is assumed.

If there are an even number of samples, N, then there are an odd number of intervals (N-1), but Simpson's rule requires an even number of intervals. The parameter 'even' controls how this is handled.

Parameters ---------- y : array_like Array to be integrated. x : array_like, optional If given, the points at which `y` is sampled. dx : int, optional Spacing of integration points along axis of `y`. Only used when `x` is None. Default is 1. axis : int, optional Axis along which to integrate. Default is the last axis. even : str 'avg', 'first', 'last', optional 'avg' : Average two results:1) use the first N-2 intervals with a trapezoidal rule on the last interval and 2) use the last N-2 intervals with a trapezoidal rule on the first interval.

'first' : Use Simpson's rule for the first N-2 intervals with a trapezoidal rule on the last interval.

'last' : Use Simpson's rule for the last N-2 intervals with a trapezoidal rule on the first interval.

See Also -------- quad: adaptive quadrature using QUADPACK romberg: adaptive Romberg quadrature quadrature: adaptive Gaussian quadrature fixed_quad: fixed-order Gaussian quadrature dblquad: double integrals tplquad: triple integrals romb: integrators for sampled data cumtrapz: cumulative integration for sampled data ode: ODE integrators odeint: ODE integrators

Notes ----- For an odd number of samples that are equally spaced the result is exact if the function is a polynomial of order 3 or less. If the samples are not equally spaced, then the result is exact only if the function is a polynomial of order 2 or less.

Examples -------- >>> from scipy import integrate >>> x = np.arange(0, 10) >>> y = np.arange(0, 10)

>>> integrate.simps(y, x) 40.5

>>> y = np.power(x, 3) >>> integrate.simps(y, x) 1642.5 >>> integrate.quad(lambda x: x**3, 0, 9)0 1640.25

>>> integrate.simps(y, x, even='first') 1644.5

Solve a boundary-value problem for a system of ODEs.

This function numerically solves a first order system of ODEs subject to two-point boundary conditions::

dy / dx = f(x, y, p) + S * y / (x - a), a <= x <= b bc(y(a), y(b), p) = 0

Here x is a 1-dimensional independent variable, y(x) is a n-dimensional vector-valued function and p is a k-dimensional vector of unknown parameters which is to be found along with y(x). For the problem to be determined there must be n + k boundary conditions, i.e. bc must be (n + k)-dimensional function.

The last singular term in the right-hand side of the system is optional. It is defined by an n-by-n matrix S, such that the solution must satisfy S y(a) = 0. This condition will be forced during iterations, so it must not contradict boundary conditions. See 2_ for the explanation how this term is handled when solving BVPs numerically.

Problems in a complex domain can be solved as well. In this case y and p are considered to be complex, and f and bc are assumed to be complex-valued functions, but x stays real. Note that f and bc must be complex differentiable (satisfy Cauchy-Riemann equations 4_), otherwise you should rewrite your problem for real and imaginary parts separately. To solve a problem in a complex domain, pass an initial guess for y with a complex data type (see below).

Parameters ---------- fun : callable Right-hand side of the system. The calling signature is ``fun(x, y)``, or ``fun(x, y, p)`` if parameters are present. All arguments are ndarray: ``x`` with shape (m,), ``y`` with shape (n, m), meaning that ``y:, i`` corresponds to ``xi``, and ``p`` with shape (k,). The return value must be an array with shape (n, m) and with the same layout as ``y``. bc : callable Function evaluating residuals of the boundary conditions. The calling signature is ``bc(ya, yb)``, or ``bc(ya, yb, p)`` if parameters are present. All arguments are ndarray: ``ya`` and ``yb`` with shape (n,), and ``p`` with shape (k,). The return value must be an array with shape (n + k,). x : array_like, shape (m,) Initial mesh. Must be a strictly increasing sequence of real numbers with ``x0=a`` and ``x-1=b``. y : array_like, shape (n, m) Initial guess for the function values at the mesh nodes, i-th column corresponds to ``xi``. For problems in a complex domain pass `y` with a complex data type (even if the initial guess is purely real). p : array_like with shape (k,) or None, optional Initial guess for the unknown parameters. If None (default), it is assumed that the problem doesn't depend on any parameters. S : array_like with shape (n, n) or None Matrix defining the singular term. If None (default), the problem is solved without the singular term. fun_jac : callable or None, optional Function computing derivatives of f with respect to y and p. The calling signature is ``fun_jac(x, y)``, or ``fun_jac(x, y, p)`` if parameters are present. The return must contain 1 or 2 elements in the following order:

* df_dy : array_like with shape (n, n, m) where an element (i, j, q) equals to d f_i(x_q, y_q, p) / d (y_q)_j. * df_dp : array_like with shape (n, k, m) where an element (i, j, q) equals to d f_i(x_q, y_q, p) / d p_j.

Here q numbers nodes at which x and y are defined, whereas i and j number vector components. If the problem is solved without unknown parameters df_dp should not be returned.

If `fun_jac` is None (default), the derivatives will be estimated by the forward finite differences. bc_jac : callable or None, optional Function computing derivatives of bc with respect to ya, yb and p. The calling signature is ``bc_jac(ya, yb)``, or ``bc_jac(ya, yb, p)`` if parameters are present. The return must contain 2 or 3 elements in the following order:

* dbc_dya : array_like with shape (n, n) where an element (i, j) equals to d bc_i(ya, yb, p) / d ya_j. * dbc_dyb : array_like with shape (n, n) where an element (i, j) equals to d bc_i(ya, yb, p) / d yb_j. * dbc_dp : array_like with shape (n, k) where an element (i, j) equals to d bc_i(ya, yb, p) / d p_j.

If the problem is solved without unknown parameters dbc_dp should not be returned.

If `bc_jac` is None (default), the derivatives will be estimated by the forward finite differences. tol : float, optional Desired tolerance of the solution. If we define ``r = y' - f(x, y)`` where y is the found solution, then the solver tries to achieve on each mesh interval ``norm(r / (1 + abs(f)) < tol``, where ``norm`` is estimated in a root mean squared sense (using a numerical quadrature formula). Default is 1e-3. max_nodes : int, optional Maximum allowed number of the mesh nodes. If exceeded, the algorithm terminates. Default is 1000. verbose :

, 1, 2

, optional Level of algorithm's verbosity:

* 0 (default) : work silently. * 1 : display a termination report. * 2 : display progress during iterations. bc_tol : float, optional Desired absolute tolerance for the boundary condition residuals: `bc` value should satisfy ``abs(bc) < bc_tol`` component-wise. Equals to `tol` by default. Up to 10 iterations are allowed to achieve this tolerance.

Returns ------- Bunch object with the following fields defined: sol : PPoly Found solution for y as `scipy.interpolate.PPoly` instance, a C1 continuous cubic spline. p : ndarray or None, shape (k,) Found parameters. None, if the parameters were not present in the problem. x : ndarray, shape (m,) Nodes of the final mesh. y : ndarray, shape (n, m) Solution values at the mesh nodes. yp : ndarray, shape (n, m) Solution derivatives at the mesh nodes. rms_residuals : ndarray, shape (m - 1,) RMS values of the relative residuals over each mesh interval (see the description of `tol` parameter). niter : int Number of completed iterations. status : int Reason for algorithm termination:

* 0: The algorithm converged to the desired accuracy. * 1: The maximum number of mesh nodes is exceeded. * 2: A singular Jacobian encountered when solving the collocation system.

message : string Verbal description of the termination reason. success : bool True if the algorithm converged to the desired accuracy (``status=0``).

Notes ----- This function implements a 4-th order collocation algorithm with the control of residuals similar to 1_. A collocation system is solved by a damped Newton method with an affine-invariant criterion function as described in 3_.

Note that in 1_ integral residuals are defined without normalization by interval lengths. So their definition is different by a multiplier of h**0.5 (h is an interval length) from the definition used here.

.. versionadded:: 0.18.0

References ---------- .. 1 J. Kierzenka, L. F. Shampine, 'A BVP Solver Based on Residual Control and the Maltab PSE', ACM Trans. Math. Softw., Vol. 27, Number 3, pp. 299-316, 2001. .. 2 L.F. Shampine, P. H. Muir and H. Xu, 'A User-Friendly Fortran BVP Solver'. .. 3 U. Ascher, R. Mattheij and R. Russell 'Numerical Solution of Boundary Value Problems for Ordinary Differential Equations'. .. 4 `Cauchy-Riemann equations <https://en.wikipedia.org/wiki/Cauchy-Riemann_equations>`_ on Wikipedia.

Examples -------- In the first example we solve Bratu's problem::

y'' + k * exp(y) = 0 y(0) = y(1) = 0

for k = 1.

We rewrite the equation as a first order system and implement its right-hand side evaluation::

y1' = y2 y2' = -exp(y1)

>>> def fun(x, y): ... return np.vstack((y1, -np.exp(y0)))

Implement evaluation of the boundary condition residuals:

>>> def bc(ya, yb): ... return np.array(ya[0], yb[0])

Define the initial mesh with 5 nodes:

>>> x = np.linspace(0, 1, 5)

This problem is known to have two solutions. To obtain both of them we use two different initial guesses for y. We denote them by subscripts a and b.

>>> y_a = np.zeros((2, x.size)) >>> y_b = np.zeros((2, x.size)) >>> y_b0 = 3

Now we are ready to run the solver.

>>> from scipy.integrate import solve_bvp >>> res_a = solve_bvp(fun, bc, x, y_a) >>> res_b = solve_bvp(fun, bc, x, y_b)

Let's plot the two found solutions. We take an advantage of having the solution in a spline form to produce a smooth plot.

>>> x_plot = np.linspace(0, 1, 100) >>> y_plot_a = res_a.sol(x_plot)0 >>> y_plot_b = res_b.sol(x_plot)0 >>> import matplotlib.pyplot as plt >>> plt.plot(x_plot, y_plot_a, label='y_a') >>> plt.plot(x_plot, y_plot_b, label='y_b') >>> plt.legend() >>> plt.xlabel('x') >>> plt.ylabel('y') >>> plt.show()

We see that the two solutions have similar shape, but differ in scale significantly.

In the second example we solve a simple Sturm-Liouville problem::

y'' + k**2 * y = 0 y(0) = y(1) = 0

It is known that a non-trivial solution y = A * sin(k * x) is possible for k = pi * n, where n is an integer. To establish the normalization constant A = 1 we add a boundary condition::

y'(0) = k

Again we rewrite our equation as a first order system and implement its right-hand side evaluation::

y1' = y2 y2' = -k**2 * y1

>>> def fun(x, y, p): ... k = p0 ... return np.vstack((y1, -k**2 * y0))

Note that parameters p are passed as a vector (with one element in our case).

Implement the boundary conditions:

>>> def bc(ya, yb, p): ... k = p0 ... return np.array(ya[0], yb[0], ya[1] - k)

Setup the initial mesh and guess for y. We aim to find the solution for k = 2 * pi, to achieve that we set values of y to approximately follow sin(2 * pi * x):

>>> x = np.linspace(0, 1, 5) >>> y = np.zeros((2, x.size)) >>> y0, 1 = 1 >>> y0, 3 = -1

Run the solver with 6 as an initial guess for k.

>>> sol = solve_bvp(fun, bc, x, y, p=6)

We see that the found k is approximately correct:

>>> sol.p0 6.28329460046

And finally plot the solution to see the anticipated sinusoid:

>>> x_plot = np.linspace(0, 1, 100) >>> y_plot = sol.sol(x_plot)0 >>> plt.plot(x_plot, y_plot) >>> plt.xlabel('x') >>> plt.ylabel('y') >>> plt.show()

Solve an initial value problem for a system of ODEs.

This function numerically integrates a system of ordinary differential equations given an initial value::

dy / dt = f(t, y) y(t0) = y0

Here t is a one-dimensional independent variable (time), y(t) is an n-dimensional vector-valued function (state), and an n-dimensional vector-valued function f(t, y) determines the differential equations. The goal is to find y(t) approximately satisfying the differential equations, given an initial value y(t0)=y0.

Some of the solvers support integration in the complex domain, but note that for stiff ODE solvers, the right-hand side must be complex-differentiable (satisfy Cauchy-Riemann equations 11_). To solve a problem in the complex domain, pass y0 with a complex data type. Another option always available is to rewrite your problem for real and imaginary parts separately.

Parameters ---------- fun : callable Right-hand side of the system. The calling signature is ``fun(t, y)``. Here `t` is a scalar, and there are two options for the ndarray `y`: It can either have shape (n,); then `fun` must return array_like with shape (n,). Alternatively it can have shape (n, k); then `fun` must return an array_like with shape (n, k), i.e. each column corresponds to a single column in `y`. The choice between the two options is determined by `vectorized` argument (see below). The vectorized implementation allows a faster approximation of the Jacobian by finite differences (required for stiff solvers). t_span : 2-tuple of floats Interval of integration (t0, tf). The solver starts with t=t0 and integrates until it reaches t=tf. y0 : array_like, shape (n,) Initial state. For problems in the complex domain, pass `y0` with a complex data type (even if the initial value is purely real). method : string or `OdeSolver`, optional Integration method to use:

* 'RK45' (default): Explicit Runge-Kutta method of order 5(4) 1_. The error is controlled assuming accuracy of the fourth-order method, but steps are taken using the fifth-order accurate formula (local extrapolation is done). A quartic interpolation polynomial is used for the dense output 2_. Can be applied in the complex domain. * 'RK23': Explicit Runge-Kutta method of order 3(2) 3_. The error is controlled assuming accuracy of the second-order method, but steps are taken using the third-order accurate formula (local extrapolation is done). A cubic Hermite polynomial is used for the dense output. Can be applied in the complex domain. * 'DOP853': Explicit Runge-Kutta method of order 8 13_. Python implementation of the 'DOP853' algorithm originally written in Fortran 14_. A 7-th order interpolation polynomial accurate to 7-th order is used for the dense output. Can be applied in the complex domain. * 'Radau': Implicit Runge-Kutta method of the Radau IIA family of order 5 4_. The error is controlled with a third-order accurate embedded formula. A cubic polynomial which satisfies the collocation conditions is used for the dense output. * 'BDF': Implicit multi-step variable-order (1 to 5) method based on a backward differentiation formula for the derivative approximation 5_. The implementation follows the one described in 6_. A quasi-constant step scheme is used and accuracy is enhanced using the NDF modification. Can be applied in the complex domain. * 'LSODA': Adams/BDF method with automatic stiffness detection and switching 7_, 8_. This is a wrapper of the Fortran solver from ODEPACK.

Explicit Runge-Kutta methods ('RK23', 'RK45', 'DOP853') should be used for non-stiff problems and implicit methods ('Radau', 'BDF') for stiff problems 9_. Among Runge-Kutta methods, 'DOP853' is recommended for solving with high precision (low values of `rtol` and `atol`).

If not sure, first try to run 'RK45'. If it makes unusually many iterations, diverges, or fails, your problem is likely to be stiff and you should use 'Radau' or 'BDF'. 'LSODA' can also be a good universal choice, but it might be somewhat less convenient to work with as it wraps old Fortran code.

You can also pass an arbitrary class derived from `OdeSolver` which implements the solver. t_eval : array_like or None, optional Times at which to store the computed solution, must be sorted and lie within `t_span`. If None (default), use points selected by the solver. dense_output : bool, optional Whether to compute a continuous solution. Default is False. events : callable, or list of callables, optional Events to track. If None (default), no events will be tracked. Each event occurs at the zeros of a continuous function of time and state. Each function must have the signature ``event(t, y)`` and return a float. The solver will find an accurate value of `t` at which ``event(t, y(t)) = 0`` using a root-finding algorithm. By default, all zeros will be found. The solver looks for a sign change over each step, so if multiple zero crossings occur within one step, events may be missed. Additionally each `event` function might have the following attributes:

terminal: bool, optional Whether to terminate integration if this event occurs. Implicitly False if not assigned. direction: float, optional Direction of a zero crossing. If `direction` is positive, `event` will only trigger when going from negative to positive, and vice versa if `direction` is negative. If 0, then either direction will trigger event. Implicitly 0 if not assigned.

You can assign attributes like ``event.terminal = True`` to any function in Python. vectorized : bool, optional Whether `fun` is implemented in a vectorized fashion. Default is False. args : tuple, optional Additional arguments to pass to the user-defined functions. If given, the additional arguments are passed to all user-defined functions. So if, for example, `fun` has the signature ``fun(t, y, a, b, c)``, then `jac` (if given) and any event functions must have the same signature, and `args` must be a tuple of length 3. options Options passed to a chosen solver. All options available for already implemented solvers are listed below. first_step : float or None, optional Initial step size. Default is `None` which means that the algorithm should choose. max_step : float, optional Maximum allowed step size. Default is np.inf, i.e. the step size is not bounded and determined solely by the solver. rtol, atol : float or array_like, optional Relative and absolute tolerances. The solver keeps the local error estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a relative accuracy (number of correct digits). But if a component of `y` is approximately below `atol`, the error only needs to fall within the same `atol` threshold, and the number of correct digits is not guaranteed. If components of y have different scales, it might be beneficial to set different `atol` values for different components by passing array_like with shape (n,) for `atol`. Default values are 1e-3 for `rtol` and 1e-6 for `atol`. jac : array_like, sparse_matrix, callable or None, optional Jacobian matrix of the right-hand side of the system with respect to y, required by the 'Radau', 'BDF' and 'LSODA' method. The Jacobian matrix has shape (n, n) and its element (i, j) is equal to ``d f_i / d y_j``. There are three ways to define the Jacobian:

* If array_like or sparse_matrix, the Jacobian is assumed to be constant. Not supported by 'LSODA'. * If callable, the Jacobian is assumed to depend on both t and y; it will be called as ``jac(t, y)`` as necessary. For 'Radau' and 'BDF' methods, the return value might be a sparse matrix. * If None (default), the Jacobian will be approximated by finite differences.

It is generally recommended to provide the Jacobian rather than relying on a finite-difference approximation. jac_sparsity : array_like, sparse matrix or None, optional Defines a sparsity structure of the Jacobian matrix for a finite- difference approximation. Its shape must be (n, n). This argument is ignored if `jac` is not `None`. If the Jacobian has only few non-zero elements in *each* row, providing the sparsity structure will greatly speed up the computations 10_. A zero entry means that a corresponding element in the Jacobian is always zero. If None (default), the Jacobian is assumed to be dense. Not supported by 'LSODA', see `lband` and `uband` instead. lband, uband : int or None, optional Parameters defining the bandwidth of the Jacobian for the 'LSODA' method, i.e., ``jaci, j != 0 only for i - lband <= j <= i + uband``. Default is None. Setting these requires your jac routine to return the Jacobian in the packed format: the returned array must have ``n`` columns and ``uband + lband + 1`` rows in which Jacobian diagonals are written. Specifically ``jac_packeduband + i - j , j = jaci, j``. The same format is used in `scipy.linalg.solve_banded` (check for an illustration). These parameters can be also used with ``jac=None`` to reduce the number of Jacobian elements estimated by finite differences. min_step : float, optional The minimum allowed step size for 'LSODA' method. By default `min_step` is zero.

Returns ------- Bunch object with the following fields defined: t : ndarray, shape (n_points,) Time points. y : ndarray, shape (n, n_points) Values of the solution at `t`. sol : `OdeSolution` or None Found solution as `OdeSolution` instance; None if `dense_output` was set to False. t_events : list of ndarray or None Contains for each event type a list of arrays at which an event of that type event was detected. None if `events` was None. y_events : list of ndarray or None For each value of `t_events`, the corresponding value of the solution. None if `events` was None. nfev : int Number of evaluations of the right-hand side. njev : int Number of evaluations of the Jacobian. nlu : int Number of LU decompositions. status : int Reason for algorithm termination:

* -1: Integration step failed. * 0: The solver successfully reached the end of `tspan`. * 1: A termination event occurred.

message : string Human-readable description of the termination reason. success : bool True if the solver reached the interval end or a termination event occurred (``status >= 0``).

References ---------- .. 1 J. R. Dormand, P. J. Prince, 'A family of embedded Runge-Kutta formulae', Journal of Computational and Applied Mathematics, Vol. 6, No. 1, pp. 19-26, 1980. .. 2 L. W. Shampine, 'Some Practical Runge-Kutta Formulas', Mathematics of Computation,, Vol. 46, No. 173, pp. 135-150, 1986. .. 3 P. Bogacki, L.F. Shampine, 'A 3(2) Pair of Runge-Kutta Formulas', Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989. .. 4 E. Hairer, G. Wanner, 'Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems', Sec. IV.8. .. 5 `Backward Differentiation Formula <https://en.wikipedia.org/wiki/Backward_differentiation_formula>`_ on Wikipedia. .. 6 L. F. Shampine, M. W. Reichelt, 'THE MATLAB ODE SUITE', SIAM J. SCI. COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997. .. 7 A. C. Hindmarsh, 'ODEPACK, A Systematized Collection of ODE Solvers,' IMACS Transactions on Scientific Computation, Vol 1., pp. 55-64, 1983. .. 8 L. Petzold, 'Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations', SIAM Journal on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148, 1983. .. 9 `Stiff equation <https://en.wikipedia.org/wiki/Stiff_equation>`_ on Wikipedia. .. 10 A. Curtis, M. J. D. Powell, and J. Reid, 'On the estimation of sparse Jacobian matrices', Journal of the Institute of Mathematics and its Applications, 13, pp. 117-120, 1974. .. 11 `Cauchy-Riemann equations <https://en.wikipedia.org/wiki/Cauchy-Riemann_equations>`_ on Wikipedia. .. 12 `Lotka-Volterra equations <https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations>`_ on Wikipedia. .. 13 E. Hairer, S. P. Norsett G. Wanner, 'Solving Ordinary Differential Equations I: Nonstiff Problems', Sec. II. .. 14 `Page with original Fortran code of DOP853 <http://www.unige.ch/~hairer/software.html>`_.

Examples -------- Basic exponential decay showing automatically chosen time points.

>>> from scipy.integrate import solve_ivp >>> def exponential_decay(t, y): return -0.5 * y >>> sol = solve_ivp(exponential_decay, 0, 10, 2, 4, 8) >>> print(sol.t) 0. 0.11487653 1.26364188 3.06061781 4.81611105 6.57445806 8.33328988 10. >>> print(sol.y) [2. 1.88836035 1.06327177 0.43319312 0.18017253 0.07483045 0.03107158 0.01350781] [4. 3.7767207 2.12654355 0.86638624 0.36034507 0.14966091 0.06214316 0.02701561] [8. 7.5534414 4.25308709 1.73277247 0.72069014 0.29932181 0.12428631 0.05403123]

Specifying points where the solution is desired.

>>> sol = solve_ivp(exponential_decay, 0, 10, 2, 4, 8, ... t_eval=0, 1, 2, 4, 10) >>> print(sol.t) 0 1 2 4 10 >>> print(sol.y) [2. 1.21305369 0.73534021 0.27066736 0.01350938] [4. 2.42610739 1.47068043 0.54133472 0.02701876] [8. 4.85221478 2.94136085 1.08266944 0.05403753]

Cannon fired upward with terminal event upon impact. The ``terminal`` and ``direction`` fields of an event are applied by monkey patching a function. Here ``y0`` is position and ``y1`` is velocity. The projectile starts at position 0 with velocity +10. Note that the integration never reaches t=100 because the event is terminal.

>>> def upward_cannon(t, y): return y[1], -0.5 >>> def hit_ground(t, y): return y0 >>> hit_ground.terminal = True >>> hit_ground.direction = -1 >>> sol = solve_ivp(upward_cannon, 0, 100, 0, 10, events=hit_ground) >>> print(sol.t_events) array([40.]) >>> print(sol.t) 0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02 1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01

Use `dense_output` and `events` to find position, which is 100, at the apex of the cannonball's trajectory. Apex is not defined as terminal, so both apex and hit_ground are found. There is no information at t=20, so the sol attribute is used to evaluate the solution. The sol attribute is returned by setting ``dense_output=True``. Alternatively, the `y_events` attribute can be used to access the solution at the time of the event.

>>> def apex(t, y): return y1 >>> sol = solve_ivp(upward_cannon, 0, 100, 0, 10, ... events=(hit_ground, apex), dense_output=True) >>> print(sol.t_events) array([40.]), array([20.]) >>> print(sol.t) 0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02 1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01 >>> print(sol.sol(sol.t_events10)) 100. 0. >>> print(sol.y_events) array([[-5.68434189e-14, -1.00000000e+01]]), array([[1.00000000e+02, 1.77635684e-15]])

As an example of a system with additional parameters, we'll implement the Lotka-Volterra equations 12_.

>>> def lotkavolterra(t, z, a, b, c, d): ... x, y = z ... return a*x - b*x*y, -c*y + d*x*y ...

We pass in the parameter values a=1.5, b=1, c=3 and d=1 with the `args` argument.

>>> sol = solve_ivp(lotkavolterra, 0, 15, 10, 5, args=(1.5, 1, 3, 1), ... dense_output=True)

Compute a dense solution and plot it.

>>> t = np.linspace(0, 15, 300) >>> z = sol.sol(t) >>> import matplotlib.pyplot as plt >>> plt.plot(t, z.T) >>> plt.xlabel('t') >>> plt.legend('x', 'y', shadow=True) >>> plt.title('Lotka-Volterra System') >>> plt.show()

Compute a triple (definite) integral.

Return the triple integral of ``func(z, y, x)`` from ``x = a..b``, ``y = gfun(x)..hfun(x)``, and ``z = qfun(x,y)..rfun(x,y)``.

Parameters ---------- func : function A Python function or method of at least three variables in the order (z, y, x). a, b : float The limits of integration in x: `a` < `b` gfun : function or float The lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result or a float indicating a constant boundary curve. hfun : function or float The upper boundary curve in y (same requirements as `gfun`). qfun : function or float The lower boundary surface in z. It must be a function that takes two floats in the order (x, y) and returns a float or a float indicating a constant boundary surface. rfun : function or float The upper boundary surface in z. (Same requirements as `qfun`.) args : tuple, optional Extra arguments to pass to `func`. epsabs : float, optional Absolute tolerance passed directly to the innermost 1-D quadrature integration. Default is 1.49e-8. epsrel : float, optional Relative tolerance of the innermost 1-D integrals. Default is 1.49e-8.

Returns ------- y : float The resultant integral. abserr : float An estimate of the error.

See Also -------- quad: Adaptive quadrature using QUADPACK quadrature: Adaptive Gaussian quadrature fixed_quad: Fixed-order Gaussian quadrature dblquad: Double integrals nquad : N-dimensional integrals romb: Integrators for sampled data simps: Integrators for sampled data ode: ODE integrators odeint: ODE integrators scipy.special: For coefficients and roots of orthogonal polynomials

Examples --------

Compute the triple integral of ``x * y * z``, over ``x`` ranging from 1 to 2, ``y`` ranging from 2 to 3, ``z`` ranging from 0 to 1.

>>> from scipy import integrate >>> f = lambda z, y, x: x*y*z >>> integrate.tplquad(f, 1, 2, lambda x: 2, lambda x: 3, ... lambda x, y: 0, lambda x, y: 1) (1.8750000000000002, 3.324644794257407e-14)

Integrate along the given axis using the composite trapezoidal rule.

Integrate `y` (`x`) along given axis.

Parameters ---------- y : array_like Input array to integrate. x : array_like, optional The sample points corresponding to the `y` values. If `x` is None, the sample points are assumed to be evenly spaced `dx` apart. The default is None. dx : scalar, optional The spacing between sample points when `x` is None. The default is 1. axis : int, optional The axis along which to integrate.

Returns ------- trapz : float Definite integral as approximated by trapezoidal rule.

See Also -------- numpy.cumsum

Notes ----- Image 2_ illustrates trapezoidal rule -- y-axis locations of points will be taken from `y` array, by default x-axis distances between points will be 1.0, alternatively they can be provided with `x` array or with `dx` scalar. Return value will be equal to combined area under the red lines.

References ---------- .. 1 Wikipedia page: https://en.wikipedia.org/wiki/Trapezoidal_rule

.. 2 Illustration image: https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png

Examples -------- >>> np.trapz(1,2,3) 4.0 >>> np.trapz(1,2,3, x=4,6,8) 8.0 >>> np.trapz(1,2,3, dx=2) 8.0 >>> a = np.arange(6).reshape(2, 3) >>> a array([0, 1, 2], [3, 4, 5]) >>> np.trapz(a, axis=0) array(1.5, 2.5, 3.5) >>> np.trapz(a, axis=1) array(2., 8.)