{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Normally, the Newton-Cotes rules are used on smaller integration regions and a composite rule is used to return the total integral." } ] } ], "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": "an", "annotation": "ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "1-D array of weights to apply to the function at the provided sample positions." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "B", "annotation": "float", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Error coefficient." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Return weights and error coefficient for Newton-Cotes integration." } ] } ], "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": "rn", "annotation": "int", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "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 " }, { "__type": "ParamRef", "__tag": 4071, "name": "rn" }, { "__type": "Text", "__tag": 4046, "value": ". 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Then an " }, { "__type": "InlineMath", "__tag": 4057, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": "-point Newton-Cotes formula for the integral between " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_0" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_N" }, { "__type": "Text", "__tag": 4046, "value": " is:" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineMath", "__tag": 4057, "value": "\\int_{x_0}^{x_N} f(x)dx = \\Delta x \\sum_{i=0}^{N} a_i f(x_i) + B_N (\\Delta x)^{N+2} f^{N+1} (\\xi)" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\xi \\in [x_0,x_N]" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\Delta x = \\frac{x_N-x_0}{N}" }, { "__type": "Text", "__tag": 4046, "value": " is the average samples spacing." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If the samples are equally-spaced and N is even, then the error term is " }, { "__type": "InlineMath", "__tag": 4057, "value": "B_N (\\Delta x)^{N+3} f^{N+2}(\\xi)" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "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": "/scipy/integrate/_quadrature.py", "item_line": 966, "item_type": "function", "aliases": [ "scipy.integrate.newton_cotes" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compute the integral of sin(x) in [0, :math:`\\pi`]:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.integrate import newton_cotes\nimport numpy as np\ndef f(x):\n return np.sin(x)\na = 0\nb = np.pi\nexact = 2\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "for N in [2, 4, 6, 8, 10]:\n x = np.linspace(a, b, N + 1)\n an, B = newton_cotes(N, 1)\n dx = (b - a) / N\n quad = dx * np.sum(an * f(x))\n error = abs(quad - exact)\n print('{:2d} {:10.9f} {:.5e}'.format(N, quad, error))\n", "execution_status": "failure", "out": " 2 2.094395102 9.43951e-02\n 4 1.998570732 1.42927e-03\n 6 2.000017814 1.78136e-05\n 8 1.999999835 1.64725e-07\n10 2.000000001 1.14677e-09\n" } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "rn", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "equal", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "newton_cotes" }, "references": null, "qa": "scipy.integrate._quadrature:newton_cotes", "arbitrary": [], "local_refs": [ "B", "an", "equal", "rn" ] }