{ "__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": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "", "annotation": "LinAlgError", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If SVD computation does not converge." } ] } ] } ] } ], "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": "U", "annotation": "ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Unitary matrix having left singular vectors as columns. Of shape " }, { "__type": "InlineCode", "__tag": 4051, "value": "(M, M)" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineCode", "__tag": 4051, "value": "(M, K)" }, { "__type": "Text", "__tag": 4046, "value": ", depending on " }, { "__type": "ParamRef", "__tag": 4071, "name": "full_matrices" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "s", "annotation": "ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The singular values, sorted in non-increasing order. Of shape (K,), with " }, { "__type": "InlineCode", "__tag": 4051, "value": "K = min(M, N)" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "Vh", "annotation": "ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Unitary matrix having right singular vectors as rows. Of shape " }, { "__type": "InlineCode", "__tag": 4051, "value": "(N, N)" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineCode", "__tag": 4051, "value": "(K, N)" }, { "__type": "Text", "__tag": 4046, "value": " depending on " }, { "__type": "ParamRef", "__tag": 4071, "name": "full_matrices" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "", "annotation": "For ``compute_uv=False``, only ``s`` is returned.", "desc": [] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Singular Value Decomposition." } ] } ], "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": "a", "annotation": "(M, N) array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Matrix to decompose." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "full_matrices", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If True (default), " }, { "__type": "CrossRef", "__tag": 4002, "value": "U", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "U" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "CrossRef", "__tag": 4002, "value": "Vh", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "Vh" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " are of shape " }, { "__type": "InlineCode", "__tag": 4051, "value": "(M, M)" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "(N, N)" }, { "__type": "Text", "__tag": 4046, "value": ". If False, the shapes are " }, { "__type": "InlineCode", "__tag": 4051, "value": "(M, K)" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "(K, N)" }, { "__type": "Text", "__tag": 4046, "value": ", where " }, { "__type": "InlineCode", "__tag": 4051, "value": "K = min(M, N)" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "compute_uv", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Whether to compute also " }, { "__type": "InlineCode", "__tag": 4051, "value": "U" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "Vh" }, { "__type": "Text", "__tag": 4046, "value": " in addition to " }, { "__type": "InlineCode", "__tag": 4051, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": ". Default is True." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "overwrite_a", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Whether to overwrite " }, { "__type": "ParamRef", "__tag": 4071, "name": "a" }, { "__type": "Text", "__tag": 4046, "value": "; may improve performance. Default is False." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "check_finite", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "lapack_driver", "annotation": "{'gesdd', 'gesvd'}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Whether to use the more efficient divide-and-conquer approach (" }, { "__type": "InlineCode", "__tag": 4051, "value": "'gesdd'" }, { "__type": "Text", "__tag": 4046, "value": ") or general rectangular approach (" }, { "__type": "InlineCode", "__tag": 4051, "value": "'gesvd'" }, { "__type": "Text", "__tag": 4046, "value": ") to compute the SVD. MATLAB and Octave use the " }, { "__type": "InlineCode", "__tag": 4051, "value": "'gesvd'" }, { "__type": "Text", "__tag": 4046, "value": " approach. Default is " }, { "__type": "InlineCode", "__tag": 4051, "value": "'gesdd'" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Factorizes the matrix " }, { "__type": "ParamRef", "__tag": 4071, "name": "a" }, { "__type": "Text", "__tag": 4046, "value": " into two unitary matrices " }, { "__type": "InlineCode", "__tag": 4051, "value": "U" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "Vh" }, { "__type": "Text", "__tag": 4046, "value": ", and a 1-D array " }, { "__type": "InlineCode", "__tag": 4051, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": " of singular values (real, non-negative) such that " }, { "__type": "InlineCode", "__tag": 4051, "value": "a == U @ S @ Vh" }, { "__type": "Text", "__tag": 4046, "value": ", where " }, { "__type": "InlineCode", "__tag": 4051, "value": "S" }, { "__type": "Text", "__tag": 4046, "value": " is a suitably shaped matrix of zeros with main diagonal " }, { "__type": "InlineCode", "__tag": 4051, "value": "s" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__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. Note that calls with zero-size batches are unsupported and will raise a " }, { "__type": "InlineCode", "__tag": 4051, "value": "ValueError" }, { "__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/linalg/_decomp_svd.py", "item_line": 16, "item_type": "function", "aliases": [ "scipy.linalg.svd" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import linalg\nrng = np.random.default_rng()\nm, n = 9, 6\na = rng.standard_normal((m, n)) + 1.j*rng.standard_normal((m, n))\nU, s, Vh = linalg.svd(a)\nU.shape, s.shape, Vh.shape\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nReconstruct the original matrix from the decomposition:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sigma = np.zeros((m, n))\nfor i in range(min(m, n)):\n sigma[i, i] = s[i]\na1 = np.dot(U, np.dot(sigma, Vh))\nnp.allclose(a, a1)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nAlternatively, use ``full_matrices=False`` (notice that the shape of\n``U`` is then ``(m, n)`` instead of ``(m, m)``):\n\n" }, { "__type": "Code", "__tag": 4050, "value": "U, s, Vh = linalg.svd(a, full_matrices=False)\nU.shape, s.shape, Vh.shape\nS = np.diag(s)\nnp.allclose(a, np.dot(U, np.dot(S, Vh)))\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "s2 = linalg.svd(a, compute_uv=False)\nnp.allclose(s, s2)\n", "execution_status": "success" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "diagsvd", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.linalg._decomp_svd:diagsvd" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Construct the Sigma matrix, given the vector s." } ] } ], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "svdvals", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "svdvals" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compute singular values of a matrix." } ] } ], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "a", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "full_matrices", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "True" }, { "__type": "SigParam", "__tag": 4030, "name": "compute_uv", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "True" }, { "__type": "SigParam", "__tag": 4030, "name": "overwrite_a", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "check_finite", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "True" }, { "__type": "SigParam", "__tag": 4030, "name": "lapack_driver", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "gesdd" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "svd" }, "references": null, "qa": "scipy.linalg._decomp_svd:svd", "arbitrary": [], "local_refs": [ "U", "Vh", "a", "check_finite", "compute_uv", "full_matrices", "lapack_driver", "overwrite_a", "s" ] }