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See " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": " for the algorithm details." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Depending on the " }, { "__type": "ParamRef", "__tag": 4071, "name": "lower" }, { "__type": "Text", "__tag": 4046, "value": " keyword value, only lower or upper triangular part of the input array is referenced. 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Lower triangular (" }, { "__type": "InlineCode", "__tag": 4051, "value": "lower=True" }, { "__type": "Text", "__tag": 4046, "value": ") is the default." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "hermitian", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For complex-valued arrays, this defines whether " }, { "__type": "InlineCode", "__tag": 4051, "value": "A = A.conj().T" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineCode", "__tag": 4051, "value": "A = A.T" }, { "__type": "Text", "__tag": 4046, "value": " is assumed. 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Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This function returns a block diagonal matrix D consisting blocks of size at most 2x2 and also a possibly permuted unit lower triangular matrix " }, { "__type": "InlineCode", "__tag": 4051, "value": "L" }, { "__type": "Text", "__tag": 4046, "value": " such that the factorization " }, { "__type": "InlineCode", "__tag": 4051, "value": "A = L D L^H" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineCode", "__tag": 4051, "value": "A = L D L^T" }, { "__type": "Text", "__tag": 4046, "value": " holds. If " }, { "__type": "ParamRef", "__tag": 4071, "name": "lower" }, { "__type": "Text", "__tag": 4046, "value": " is False then (again possibly permuted) upper triangular matrices are returned as outer factors." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The permutation array can be used to triangularize the outer factors simply by a row shuffle, i.e., " }, { "__type": "InlineCode", "__tag": 4051, "value": "lu[perm, :]" }, { "__type": "Text", "__tag": 4046, "value": " is an upper/lower triangular matrix. This is also equivalent to multiplication with a permutation matrix " }, { "__type": "InlineCode", "__tag": 4051, "value": "P.dot(lu)" }, { "__type": "Text", "__tag": 4046, "value": ", where " }, { "__type": "InlineCode", "__tag": 4051, "value": "P" }, { "__type": "Text", "__tag": 4046, "value": " is a column-permuted identity matrix " }, { "__type": "InlineCode", "__tag": 4051, "value": "I[:, perm]" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Depending on the value of the boolean " }, { "__type": "ParamRef", "__tag": 4071, "name": "lower" }, { "__type": "Text", "__tag": 4046, "value": ", only upper or lower triangular part of the input array is referenced. Hence, a triangular matrix on entry would give the same result as if the full matrix is supplied." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The documentation is written assuming array arguments are of specified \"core\" shapes. However, array argument(s) of this function may have additional \"batch\" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see " }, { "__type": "InlineRole", "__tag": 4003, "value": "linalg_batch", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " for details. 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[1] J.R. Bunch, L. Kaufman, Some stable methods for calculating", " inertia and solving symmetric linear systems, Math. Comput. Vol.31,", " 1977. :doi:`10.2307/2005787`" ], "qa": "scipy.linalg._decomp_ldl:ldl", "arbitrary": [], "local_refs": [ "A", "check_finite", "d", "hermitian", "lower", "lu", "overwrite_a", "perm" ] }