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If given, the additional arguments are passed to all user-defined functions. So if, for example, " }, { "__type": "ParamRef", "__tag": 4071, "name": "fun" }, { "__type": "Text", "__tag": 4046, "value": " has the signature " }, { "__type": "InlineCode", "__tag": 4051, "value": "fun(t, y, a, b, c)" }, { "__type": "Text", "__tag": 4046, "value": ", then " }, { "__type": "InlineRole", "__tag": 4003, "value": "jac", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " (if given) and any event functions must have the same signature, and " }, { "__type": "ParamRef", "__tag": 4071, "name": "args" }, { "__type": "Text", "__tag": 4046, "value": " must be a tuple of length 3." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "**options", "annotation": "", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Options passed to a chosen solver. 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Default is " }, { "__type": "CrossRef", "__tag": 4002, "value": "None", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "builtins", "version": "?", "kind": "module", "path": "builtins:None" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " which means that the algorithm should choose." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "max_step" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Maximum allowed step size. Default is np.inf, i.e., the step size is not bounded and determined solely by the solver." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "rtol,atol" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Relative and absolute tolerances. The solver keeps the local error estimates less than " }, { "__type": "InlineCode", "__tag": 4051, "value": "atol + rtol * abs(y)" }, { "__type": "Text", "__tag": 4046, "value": ". Here " }, { "__type": "InlineRole", "__tag": 4003, "value": "rtol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " controls a relative accuracy (number of correct digits), while " }, { "__type": "InlineRole", "__tag": 4003, "value": "atol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " controls absolute accuracy (number of correct decimal places). To achieve the desired " }, { "__type": "InlineRole", "__tag": 4003, "value": "rtol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ", set " }, { "__type": "InlineRole", "__tag": 4003, "value": "atol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " to be smaller than the smallest value that can be expected from " }, { "__type": "InlineCode", "__tag": 4051, "value": "rtol * abs(y)" }, { "__type": "Text", "__tag": 4046, "value": " so that " }, { "__type": "InlineRole", "__tag": 4003, "value": "rtol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " dominates the allowable error. If " }, { "__type": "InlineRole", "__tag": 4003, "value": "atol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " is larger than " }, { "__type": "InlineCode", "__tag": 4051, "value": "rtol * abs(y)" }, { "__type": "Text", "__tag": 4046, "value": " the number of correct digits is not guaranteed. Conversely, to achieve the desired " }, { "__type": "InlineRole", "__tag": 4003, "value": "atol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " set " }, { "__type": "InlineRole", "__tag": 4003, "value": "rtol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " such that " }, { "__type": "InlineCode", "__tag": 4051, "value": "rtol * abs(y)" }, { "__type": "Text", "__tag": 4046, "value": " is always smaller than " }, { "__type": "InlineRole", "__tag": 4003, "value": "atol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ". If components of y have different scales, it might be beneficial to set different " }, { "__type": "InlineRole", "__tag": 4003, "value": "atol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " values for different components by passing array_like with shape (n,) for " }, { "__type": "InlineRole", "__tag": 4003, "value": "atol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ". Default values are 1e-3 for " }, { "__type": "InlineRole", "__tag": 4003, "value": "rtol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " and 1e-6 for " }, { "__type": "InlineRole", "__tag": 4003, "value": "atol", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "jac" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "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 " }, { "__type": "InlineCode", "__tag": 4051, "value": "d f_i / d y_j" }, { "__type": "Text", "__tag": 4046, "value": ". There are three ways to define the Jacobian:" } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If array_like or sparse_matrix, the Jacobian is assumed to be constant. Not supported by 'LSODA'." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If callable, the Jacobian is assumed to depend on both t and y; it will be called as " }, { "__type": "InlineCode", "__tag": 4051, "value": "jac(t, y)" }, { "__type": "Text", "__tag": 4046, "value": ", as necessary. Additional arguments have to be passed if " }, { "__type": "InlineCode", "__tag": 4051, "value": "args" }, { "__type": "Text", "__tag": 4046, "value": " is used (see documentation of " }, { "__type": "InlineCode", "__tag": 4051, "value": "args" }, { "__type": "Text", "__tag": 4046, "value": " argument). For 'Radau' and 'BDF' methods, the return value might be a sparse matrix." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If None (default), the Jacobian will be approximated by finite differences." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "It is generally recommended to provide the Jacobian rather than relying on a finite-difference approximation." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "jac_sparsity" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Defines a sparsity structure of the Jacobian matrix for a finite- difference approximation. Its shape must be (n, n). This argument is ignored if " }, { "__type": "InlineRole", "__tag": 4003, "value": "jac", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " is not " }, { "__type": "CrossRef", "__tag": 4002, "value": "None", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "builtins", "version": "?", "kind": "module", "path": "builtins:None" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ". If the Jacobian has only few non-zero elements in " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "each" } ] }, { "__type": "Text", "__tag": 4046, "value": " row, providing the sparsity structure will greatly speed up the computations " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "10" }, { "__type": "Text", "__tag": 4046, "value": ". 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 " }, { "__type": "InlineRole", "__tag": 4003, "value": "lband", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineRole", "__tag": 4003, "value": "uband", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " instead." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "lband,uband" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Parameters defining the bandwidth of the Jacobian for the 'LSODA' method, i.e., " }, { "__type": "InlineCode", "__tag": 4051, "value": "jac[i, j] != 0 only for i - lband <= j <= i + uband" }, { "__type": "Text", "__tag": 4046, "value": ". Default is None. Setting these requires your jac routine to return the Jacobian in the packed format: the returned array must have " }, { "__type": "InlineCode", "__tag": 4051, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " columns and " }, { "__type": "InlineCode", "__tag": 4051, "value": "uband + lband + 1" }, { "__type": "Text", "__tag": 4046, "value": " rows in which Jacobian diagonals are written. Specifically " }, { "__type": "InlineCode", "__tag": 4051, "value": "jac_packed[uband + i - j , j] = jac[i, j]" }, { "__type": "Text", "__tag": 4046, "value": ". The same format is used in " }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.linalg.solve_banded", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "?", "kind": "module", "path": "scipy.linalg._basic:solve_banded" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " (check for an illustration). These parameters can be also used with " }, { "__type": "InlineCode", "__tag": 4051, "value": "jac=None" }, { "__type": "Text", "__tag": 4046, "value": " to reduce the number of Jacobian elements estimated by finite differences." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "min_step" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The minimum allowed step size for 'LSODA' method. By default " }, { "__type": "InlineRole", "__tag": 4003, "value": "min_step", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " is zero." } ] } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This function numerically integrates a system of ordinary differential equations given an initial value " } ] }, { "__type": "Code", "__tag": 4050, "value": "dy / dt = f(t, y)\ny(t0) = y0", "execution_status": null, "out": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Here t is a 1-D independent variable (time), y(t) is an N-D vector-valued function (state), and an N-D 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." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "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 " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "11" }, { "__type": "Text", "__tag": 4046, "value": "). 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." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "build-install/usr/lib/python3.14/site-packages/scipy/integrate/_ivp/ivp.py", "item_line": 160, "item_type": "function", "aliases": [ "scipy.integrate.solve_ivp" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Basic exponential decay showing automatically chosen time points.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy.integrate import solve_ivp\ndef exponential_decay(t, y): return -0.5 * y\nsol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8])\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "print(sol.t)\nprint(sol.y)\n", "execution_status": "failure", "out": "[[2. 1.89 1.06 0.43 0.18 0.07 0.03 0.01]\n [4. 3.78 2.13 0.87 0.36 0.15 0.06 0.03]\n [8. 7.55 4.25 1.73 0.72 0.3 0.12 0.05]]\n" }, { "__type": "Text", "__tag": 4046, "value": "\nSpecifying points where the solution is desired.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8],\n t_eval=[0, 1, 2, 4, 10])\nprint(sol.t)\n", "execution_status": "success", "out": "[ 0 1 2 4 10]\n" }, { "__type": "Code", "__tag": 4050, "value": "print(sol.y)\n", "execution_status": "failure", "out": "[[2. 1.21 0.74 0.27 0.01]\n [4. 2.43 1.47 0.54 0.03]\n [8. 4.85 2.94 1.08 0.05]]\n" }, { "__type": "Text", "__tag": 4046, "value": "\nCannon fired upward with terminal event upon impact. The ``terminal`` and\n``direction`` fields of an event are applied by monkey patching a function.\nHere ``y[0]`` is position and ``y[1]`` is velocity. The projectile starts\nat position 0 with velocity +10. Note that the integration never reaches\nt=100 because the event is terminal.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def upward_cannon(t, y): return [y[1], -0.5]\ndef hit_ground(t, y): return y[0]\nhit_ground.terminal = True\nhit_ground.direction = -1\nsol = solve_ivp(upward_cannon, [0, 100], [0, 10], events=hit_ground)\nprint(sol.t_events)\n", "execution_status": "success", "out": "[array([40.])]\n" }, { "__type": "Code", "__tag": 4050, "value": "print(sol.t)\n", "execution_status": "failure", "out": "[0.00e+00 1.00e-04 1.10e-03 1.11e-02 1.11e-01 1.11e+00 1.11e+01 4.00e+01]\n" }, { "__type": "Text", "__tag": 4046, "value": "\nUse `dense_output` and `events` to find position, which is 100, at the apex\nof the cannonball's trajectory. Apex is not defined as terminal, so both\napex and hit_ground are found. There is no information at t=20, so the sol\nattribute is used to evaluate the solution. The sol attribute is returned\nby setting ``dense_output=True``. Alternatively, the `y_events` attribute\ncan be used to access the solution at the time of the event.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def apex(t, y): return y[1]\nsol = solve_ivp(upward_cannon, [0, 100], [0, 10],\n events=(hit_ground, apex), dense_output=True)\nprint(sol.t_events)\n", "execution_status": "success", "out": "[array([40.]), array([20.])]\n" }, { "__type": "Code", "__tag": 4050, "value": "print(sol.t)\nprint(sol.sol(sol.t_events[1][0]))\nprint(sol.y_events)\n", "execution_status": "failure", "out": "[array([[-5.68e-14, -1.00e+01]]), array([[1.00e+02, 1.78e-15]])]\n" }, { "__type": "Text", "__tag": 4046, "value": "\nAs an example of a system with additional parameters, we'll implement\nthe Lotka-Volterra equations [12]_.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def lotkavolterra(t, z, a, b, c, d):\n x, y = z\n return [a*x - b*x*y, -c*y + d*x*y]\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nWe pass in the parameter values a=1.5, b=1, c=3 and d=1 with the `args`\nargument.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sol = solve_ivp(lotkavolterra, [0, 15], [10, 5], args=(1.5, 1, 3, 1),\n dense_output=True)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nCompute a dense solution and plot it.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "t = np.linspace(0, 15, 300)\nz = sol.sol(t)\nimport matplotlib.pyplot as plt\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "plt.plot(t, z.T)\nplt.xlabel('t')\nplt.legend(['x', 'y'], shadow=True)\nplt.title('Lotka-Volterra System')\n", "execution_status": "failure", "out": "Text(0.5, 1.0, 'Lotka-Volterra System')\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success", "out": "" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "2.0.0.dev0+git20260907.41eeb59", "kind": "assets", "path": "fig-a177d8b243dc70f7.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nA couple examples of using solve_ivp to solve the differential\nequation ``y' = Ay`` with complex matrix ``A``.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "A = np.array([[-0.25 + 0.14j, 0, 0.33 + 0.44j],\n [0.25 + 0.58j, -0.2 + 0.14j, 0],\n [0, 0.2 + 0.4j, -0.1 + 0.97j]])\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nSolving an IVP with ``A`` from above and ``y`` as 3x1 vector:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def deriv_vec(t, y):\n return A @ y\nresult = solve_ivp(deriv_vec, [0, 25],\n np.array([10 + 0j, 20 + 0j, 30 + 0j]),\n t_eval=np.linspace(0, 25, 101))\nprint(result.y[:, 0])\n", "execution_status": "success", "out": "[10.+0.j 20.+0.j 30.+0.j]\n" }, { "__type": "Code", "__tag": 4050, "value": "print(result.y[:, -1])\n", "execution_status": "failure", "out": "[18.46+45.26j 10.02+36.23j -4.99+80.07j]\n" }, { "__type": "Text", "__tag": 4046, "value": "\nSolving an IVP with ``A`` from above with ``y`` as 3x3 matrix :\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def deriv_mat(t, y):\n return (A @ y.reshape(3, 3)).flatten()\ny0 = np.array([[2 + 0j, 3 + 0j, 4 + 0j],\n [5 + 0j, 6 + 0j, 7 + 0j],\n [9 + 0j, 34 + 0j, 78 + 0j]])\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "result = solve_ivp(deriv_mat, [0, 25], y0.flatten(),\n t_eval=np.linspace(0, 25, 101))\nprint(result.y[:, 0].reshape(3, 3))\n", "execution_status": "success", "out": "[[ 2.+0.j 3.+0.j 4.+0.j]\n [ 5.+0.j 6.+0.j 7.+0.j]\n [ 9.+0.j 34.+0.j 78.+0.j]]\n" }, { "__type": "Code", "__tag": 4050, "value": "print(result.y[:, -1].reshape(3, 3))\n", "execution_status": "failure", "out": "[[ 5.67 +12.08j 17.29 +31.03j 37.83 +63.25j]\n [ 3.4 +11.82j 21.33 +44.89j 53.18+103.8j ]\n [ -2.26 +22.19j -15.13 +70.2j -38.35+153.29j]]\n" } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "fun", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "t_span", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "y0", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "RK45" }, { "__type": "SigParam", "__tag": 4030, "name": "t_eval", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "dense_output", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "events", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "vectorized", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "args", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "options", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "VAR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "solve_ivp" }, "references": [ ".. 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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 `_ 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", " `_ on", " Wikipedia.", ".. [12] `Lotka-Volterra 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", " `_." ], "qa": "scipy.integrate._ivp.ivp:solve_ivp", "arbitrary": [], "local_refs": [ "**options", "args", "dense_output", "events", "fun", "method", "result", "t_eval", "t_span", "vectorized", "y0" ] }