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1) # only one real root at x = 1\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\n``fprime`` is not provided, use the secant method:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "root = optimize.newton(f, 1.5)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "root\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "root = optimize.newton(f, 1.5, fprime2=lambda x: 6 * x)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "root\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nOnly ``fprime`` is provided, use the Newton-Raphson method:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "root\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nBoth ``fprime2`` and ``fprime`` are provided, use Halley's method:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2,\n fprime2=lambda x: 6 * x)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "root\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nWhen we want to find roots for a set of related starting values and/or\nfunction parameters, we can provide both of those as an array of inputs:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "f = lambda x, a: x**3 - a\nfder = lambda x, a: 3 * x**2\nrng = np.random.default_rng()\nx = rng.standard_normal(100)\na = np.arange(-50, 50)\nvec_res = optimize.newton(f, x, fprime=fder, args=(a, ), maxiter=200)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThe above is the equivalent of solving for each value in ``(x, a)``\nseparately in a for-loop, just faster:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "loop_res = [optimize.newton(f, x0, fprime=fder, args=(a0,),\n maxiter=200)\n for x0, a0 in zip(x, a)]\nnp.allclose(vec_res, loop_res)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nPlot the results found for all values of ``a``:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "analytical_result = np.sign(a) * np.abs(a)**(1/3)\nfig, ax = plt.subplots()\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "ax.plot(a, analytical_result, 'o')\nax.plot(a, vec_res, '.')\nax.set_xlabel('$a$')\nax.set_ylabel('$x$ where $f(x, a)=0$')\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-8829e4c36b264aad.png" } } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "root", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._root:root" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "interface to root solvers for multi-input, multi-output functions" } ] } ], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "root_scalar", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._root_scalar:root_scalar" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "interface to root solvers for scalar functions" } ] } ], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "func", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "x0", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "fprime", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "args", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "()" }, { "__type": "SigParam", "__tag": 4030, "name": "tol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "1.48e-08" }, { "__type": "SigParam", "__tag": 4030, "name": "maxiter", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "50" }, { "__type": "SigParam", "__tag": 4030, "name": "fprime2", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "x1", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "rtol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0.0" }, { "__type": "SigParam", "__tag": 4030, "name": "full_output", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "disp", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "True" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "newton" }, "references": null, "qa": "scipy.optimize._zeros_py:newton", "arbitrary": [], "local_refs": [ "args", "converged", "disp", "fprime", "fprime2", "full_output", "func", "maxiter", "r", "root", "rtol", "tol", "x0", "x1", "zero_der" ] }