{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "z1pn", "annotation": "ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Location of nth zero of Y1'(z)" } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "y1z1pn", "annotation": "ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Value of derivative Y1(z1) for nth zero" } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compute nt zeros of Bessel derivative Y1'(z), and value at each zero." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "nt", "annotation": "int", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Number of zeros to return" } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "complex", "annotation": "bool, default False", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Set to False to return only the real zeros; set to True to return only the complex zeros with negative real part and positive imaginary part. Note that the complex conjugates of the latter are also zeros of the function, but are not returned by this routine." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The values are given by Y1(z1) at each z1 where Y1'(z1)=0." } ] } ], "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/special/_basic.py", "item_line": 731, "item_type": "function", "aliases": [ "scipy.special.y1p_zeros" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compute the first four roots of :math:`Y_1'` and the values of\n:math:`Y_1` at these roots.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy.special import y1p_zeros\ny1grad_roots, y1_values = y1p_zeros(4)\nwith np.printoptions(precision=5):\n print(f\"Y1' Roots: {y1grad_roots.real}\")\n print(f\"Y1 values: {y1_values.real}\")\n", "execution_status": "success", "out": "Y1' Roots: [ 3.68302 6.9415 10.1234 13.28576]\nY1 values: [ 0.41673 -0.30317 0.25091 -0.21897]\n" }, { "__type": "Text", "__tag": 4046, "value": "\n`y1p_zeros` can be used to calculate the extremal points of :math:`Y_1`\ndirectly. Here we plot :math:`Y_1` and the first four extrema.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import matplotlib.pyplot as plt\nfrom scipy.special import y1, yvp\ny1_roots, y1_values_at_roots = y1p_zeros(4)\nreal_roots = y1_roots.real\nxmax = 15\nx = np.linspace(0, xmax, 500)\nx[0] += 1e-15\nfig, ax = plt.subplots()\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "ax.plot(x, y1(x), label=r'$Y_1$')\nax.plot(x, yvp(1, x, 1), label=r\"$Y_1'$\")\nax.scatter(real_roots, np.zeros((4, )), s=30, c='r',\n label=r\"Roots of $Y_1'$\", zorder=5)\nax.scatter(real_roots, y1_values_at_roots.real, s=30, c='k',\n label=r\"Extrema of $Y_1$\", zorder=5)\nax.hlines(0, 0, xmax, color='k')\nax.set_ylim(-0.5, 0.5)\nax.set_xlim(0, xmax)\nax.legend(ncol=2, bbox_to_anchor=(1., 0.75))\n", "execution_status": "failure", "out": "\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.tight_layout()\nplt.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-3a077bcbfd0c886f.png" } } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "nt", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "complex", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "y1p_zeros" }, "references": [ ".. [1] Zhang, Shanjie and Jin, Jianming. \"Computation of Special", " Functions\", John Wiley and Sons, 1996, chapter 5.", " https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html" ], "qa": "scipy.special._basic:y1p_zeros", "arbitrary": [], "local_refs": [ "complex", "nt", "y1z1pn", "z1pn" ] }