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A cubic Hermite polynomial is used for the dense output." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Can be applied in the complex domain." } ] } ], "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/rk.py", "item_line": 183, "item_type": "class", "aliases": [ "scipy.integrate.RK23" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "This example demonstrates how to compute one period of a periodic\norbit in a two-dimensional dynamical system.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nimport scipy.integrate as itg\nimport matplotlib.pylab as plt\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nDefine a function that returns the right-hand side of a dynamical\nsystem with a stable periodic orbit.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def dynam_sys(t, x):\n dx1_dt = 5*(1-x[1]**2)*x[0] - x[1]\n dx2_dt = x[0]\n return [dx1_dt, dx2_dt]\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nThe system has an equilibrium at the origin. Create a solver\nobject with an initial state slightly away from the\nequilibrium.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "solver = itg.RK45(dynam_sys, 0, [0.25,0.25], 1E200, max_step=0.05)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nCreate an array in which to store the estimated solution.\nTo allow for concatenation of future states, initialize the array\nby reshaping the initial state.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "soly = solver.y.reshape(1, 2)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nTo find a point on the periodic orbit, run the solver\nfor ``2000`` integration steps. Store the estimated states in\n``soly``.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "for i in range(2000):\n solver.step()\n solyn = solver.y.reshape(1, 2)\n soly = np.concatenate((soly, solyn))\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nPlot ``soly`` with its first element in green and its final\nelement in magenta.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.plot(soly[:,0], soly[:,1])\nplt.plot(soly[0,0], soly[0,1], \"go\")\nplt.plot(soly[-1,0], soly[-1,1], \"mo\")\n", "execution_status": "failure", "out": "[]\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-dba818670c2cd6ee.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nThe plot shows that the last element in ``soly`` is located on a\nsegment of the periodic orbit where there is relatively little\nvariation in the estimated states. Create another solver and\ninitialize it with the last element in ``soly``.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "init = np.array([soly[-1,0], soly[-1,1]])\nPOsolver = itg.RK23(dynam_sys, 0, init, 5000, max_step=0.05)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nCreate a variable that tracks the distance of the current\nestimated state from the initial state. Run ``POsolver`` until\nthe current estimated state is within ``0.005`` units of the last\nestimated state.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "dist_init = 1\nPOsoly = POsolver.y.reshape(1, 2)\nwhile dist_init > 0.005:\n POsolver.step()\n POsolyn = POsolver.y.reshape(1, 2)\n POsoly = np.concatenate((POsoly, POsolyn))\n dist_init = np.linalg.norm(init-POsolyn)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nPlot ``POsoly`` with its initial element in green and its final\nelement in magenta.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.plot(POsoly[:,0], POsoly[:,1])\nplt.plot(POsoly[0,0], POsoly[0,1], \"go\")\nplt.plot(POsoly[-1,0], POsoly[-1,1], \"mo\")\n", "execution_status": "failure", "out": "[]\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-e33aad381e18a367.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nThe plot shows approximately one period of the orbit.\nThe final and initial states are so close that the final\nstate marker obscures the initial state marker." } ], "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": "t0", "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": "t_bound", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "max_step", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "inf" }, { "__type": "SigParam", "__tag": 4030, "name": "rtol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0.001" }, { "__type": "SigParam", "__tag": 4030, "name": "atol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "1e-06" }, { "__type": "SigParam", "__tag": 4030, "name": "vectorized", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "first_step", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "extraneous", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "VAR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "RK23" }, "references": [ ".. [1] P. Bogacki, L.F. Shampine, \"A 3(2) Pair of Runge-Kutta Formulas\",", " Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989." ], "qa": "scipy.integrate._ivp.rk:RK23", "arbitrary": [], "local_refs": [ "**extraneous", "atol", "direction", "first_step", "fun", "max_step", "n", "nfev", "njev", "nlu", "rtol", "status", "step_size", "t", "t0", "t_bound", "t_old", "vectorized", "y", "y0" ] }