{ "__type": "IngestedDoc", "__tag": 4010, "_content": {}, "_ordered_sections": [], "item_file": null, "item_line": null, "item_type": null, "aliases": [], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "see_also": [], "signature": null, "references": null, "qa": "user:how-to-index", "arbitrary": [ { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Admonition", "__tag": 4056, "kind": "seealso", "base_type": "note", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "seealso :ref:`basics.indexing`" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This page tackles common examples. For an in-depth look into indexing, refer to " }, { "__type": "CrossRef", "__tag": 4002, "value": "basics.indexing", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "How to index " }, { "__type": "InlineRole", "__tag": 4003, "value": "ndarrays <.ndarray>", "domain": null, "role": "class", "inventory": null } ], "level": 0, "target": "how-to-index" }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use " }, { "__type": "CrossRef", "__tag": 4002, "value": "basic-indexing", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": " features like " }, { "__type": "CrossRef", "__tag": 4002, "value": "slicing-and-striding", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": ", and " }, { "__type": "CrossRef", "__tag": 4002, "value": "dimensional-indexing-tools", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> a = np.arange(30).reshape(2, 3, 5)\n>>> a\narray([[[ 0, 1, 2, 3, 4],\n [ 5, 6, 7, 8, 9],\n [10, 11, 12, 13, 14]],\n\n [[15, 16, 17, 18, 19],\n [20, 21, 22, 23, 24],\n [25, 26, 27, 28, 29]]])\n>>> a[0, 2, :]\narray([10, 11, 12, 13, 14])\n>>> a[0, :, 3]\narray([ 3, 8, 13])", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that the output from indexing operations can have different shape from the original object. To preserve the original dimensions after indexing, you can use " }, { "__type": "InlineRole", "__tag": 4003, "value": "newaxis", "domain": null, "role": "func", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ". To use other such tools, refer to " }, { "__type": "CrossRef", "__tag": 4002, "value": "dimensional-indexing-tools", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> a[0, :, 3].shape\n(3,)\n>>> a[0, :, 3, np.newaxis].shape\n(3, 1)\n>>> a[0, :, 3, np.newaxis, np.newaxis].shape\n(3, 1, 1)", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Variables can also be used to index " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> y = 0\n>>> a[y, :, y+3]\narray([ 3, 8, 13])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Refer to " }, { "__type": "CrossRef", "__tag": 4002, "value": "dealing-with-variable-indices", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": " to see how to use " }, { "__type": "InlineCode", "__tag": 4051, "value": "python:slice" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineRole", "__tag": 4003, "value": "Ellipsis", "domain": "py", "role": "data", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " in your index variables." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Access specific/arbitrary rows and columns" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "To index columns, you have to index the last axis. Use " }, { "__type": "CrossRef", "__tag": 4002, "value": "dimensional-indexing-tools", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": " to get the desired number of dimensions " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> a = np.arange(24).reshape(2, 3, 4)\n>>> a\narray([[[ 0, 1, 2, 3],\n [ 4, 5, 6, 7],\n [ 8, 9, 10, 11]],\n\n [[12, 13, 14, 15],\n [16, 17, 18, 19],\n [20, 21, 22, 23]]])\n>>> a[..., 3]\narray([[ 3, 7, 11],\n [15, 19, 23]])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "To index specific elements in each column, make use of " }, { "__type": "CrossRef", "__tag": 4002, "value": "advanced-indexing", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": " as below " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> arr = np.arange(3*4).reshape(3, 4)\n>>> arr\narray([[ 0, 1, 2, 3],\n [ 4, 5, 6, 7],\n [ 8, 9, 10, 11]])\n>>> column_indices = [[1, 3], [0, 2], [2, 2]]\n>>> np.arange(arr.shape[0])\narray([0, 1, 2])\n>>> row_indices = np.arange(arr.shape[0])[:, np.newaxis]\n>>> row_indices\narray([[0],\n [1],\n [2]])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use the " }, { "__type": "InlineCode", "__tag": 4051, "value": "row_indices" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "column_indices" }, { "__type": "Text", "__tag": 4046, "value": " for advanced indexing " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> arr[row_indices, column_indices]\narray([[ 1, 3],\n [ 4, 6],\n [10, 10]])", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Index columns" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use " }, { "__type": "InlineRole", "__tag": 4003, "value": "take", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ". See also " }, { "__type": "InlineRole", "__tag": 4003, "value": "take_along_axis", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineRole", "__tag": 4003, "value": "put_along_axis", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> a = np.arange(30).reshape(2, 3, 5)\n>>> a\narray([[[ 0, 1, 2, 3, 4],\n [ 5, 6, 7, 8, 9],\n [10, 11, 12, 13, 14]],\n\n [[15, 16, 17, 18, 19],\n [20, 21, 22, 23, 24],\n [25, 26, 27, 28, 29]]])\n>>> np.take(a, [2, 3], axis=2)\narray([[[ 2, 3],\n [ 7, 8],\n [12, 13]],\n\n [[17, 18],\n [22, 23],\n [27, 28]]])\n>>> np.take(a, [2], axis=1)\narray([[[10, 11, 12, 13, 14]],\n\n [[25, 26, 27, 28, 29]]])", "execution_status": null } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Index along a specific axis" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use " }, { "__type": "CrossRef", "__tag": 4002, "value": "slicing-and-striding", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": " to access chunks of a large array " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> a = np.arange(100).reshape(10, 10)\n>>> a\narray([[ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9],\n [10, 11, 12, 13, 14, 15, 16, 17, 18, 19],\n [20, 21, 22, 23, 24, 25, 26, 27, 28, 29],\n [30, 31, 32, 33, 34, 35, 36, 37, 38, 39],\n [40, 41, 42, 43, 44, 45, 46, 47, 48, 49],\n [50, 51, 52, 53, 54, 55, 56, 57, 58, 59],\n [60, 61, 62, 63, 64, 65, 66, 67, 68, 69],\n [70, 71, 72, 73, 74, 75, 76, 77, 78, 79],\n [80, 81, 82, 83, 84, 85, 86, 87, 88, 89],\n [90, 91, 92, 93, 94, 95, 96, 97, 98, 99]])\n>>> a[2:5, 2:5]\narray([[22, 23, 24],\n [32, 33, 34],\n [42, 43, 44]])\n>>> a[2:5, 1:3]\narray([[21, 22],\n [31, 32],\n [41, 42]])\n>>> a[:5, :5]\narray([[ 0, 1, 2, 3, 4],\n [10, 11, 12, 13, 14],\n [20, 21, 22, 23, 24],\n [30, 31, 32, 33, 34],\n [40, 41, 42, 43, 44]])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The same thing can be done with advanced indexing in a slightly more complex way. Remember that " }, { "__type": "CrossRef", "__tag": 4002, "value": "advanced indexing creates a copy", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.copies" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": ": " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> a[np.arange(5)[:, None], np.arange(5)[None, :]]\narray([[ 0, 1, 2, 3, 4],\n [10, 11, 12, 13, 14],\n [20, 21, 22, 23, 24],\n [30, 31, 32, 33, 34],\n [40, 41, 42, 43, 44]])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "You can also use " }, { "__type": "InlineRole", "__tag": 4003, "value": "mgrid", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " to generate indices " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> indices = np.mgrid[0:6:2]\n>>> indices\narray([0, 2, 4])\n>>> a[:, indices]\narray([[ 0, 2, 4],\n [10, 12, 14],\n [20, 22, 24],\n [30, 32, 34],\n [40, 42, 44],\n [50, 52, 54],\n [60, 62, 64],\n [70, 72, 74],\n [80, 82, 84],\n [90, 92, 94]])", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Create subsets of larger matrices" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [], "title": [ { "__type": "Text", "__tag": 4046, "value": "Filter values" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use " }, { "__type": "InlineRole", "__tag": 4003, "value": "nonzero", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " to get a tuple of array indices of non-zero elements corresponding to every dimension " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> z = np.array([[1, 2, 3, 0], [0, 0, 5, 3], [4, 6, 0, 0]])\n>>> z\narray([[1, 2, 3, 0],\n [0, 0, 5, 3],\n [4, 6, 0, 0]])\n>>> np.nonzero(z)\n(array([0, 0, 0, 1, 1, 2, 2]), array([0, 1, 2, 2, 3, 0, 1]))", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use " }, { "__type": "InlineRole", "__tag": 4003, "value": "flatnonzero", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " to fetch indices of elements that are non-zero in the flattened version of the ndarray " } ] }, { "__type": "Code", "__tag": 4050, "value": " >>> np.flatnonzero(z)\n\tarray([0, 1, 2, 6, 7, 8, 9])", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Non-zero elements" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use " }, { "__type": "InlineRole", "__tag": 4003, "value": "where", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " to generate indices based on conditions and then use " }, { "__type": "CrossRef", "__tag": 4002, "value": "advanced-indexing", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Code", "__tag": 4050, "value": ">>> a = np.arange(30).reshape(2, 3, 5)\n>>> indices = np.where(a % 2 == 0)\n>>> indices\n(array([0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]), \narray([0, 0, 0, 1, 1, 2, 2, 2, 0, 0, 1, 1, 1, 2, 2]), \narray([0, 2, 4, 1, 3, 0, 2, 4, 1, 3, 0, 2, 4, 1, 3]))\n>>> a[indices]\narray([ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28])", "execution_status": null } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Or, use " }, { "__type": "CrossRef", "__tag": 4002, "value": "boolean-indexing", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "user:basics.indexing" }, "kind": "exists" }, { "__type": "Text", "__tag": 4046, "value": ": " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> a > 14\narray([[[False, False, False, False, False],\n [False, False, False, False, False],\n [False, False, False, False, False]],\n\n [[ True, True, True, True, True],\n [ True, True, True, True, True],\n [ True, True, True, True, True]]])\n>>> a[a > 14]\narray([15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29])", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Arbitrary conditions" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use assignment with filtering to replace desired values " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> p = np.arange(-10, 10).reshape(2, 2, 5)\n>>> p\narray([[[-10, -9, -8, -7, -6],\n [ -5, -4, -3, -2, -1]],\n\n [[ 0, 1, 2, 3, 4],\n [ 5, 6, 7, 8, 9]]])\n>>> q = p < 0\n>>> q\narray([[[ True, True, True, True, True],\n [ True, True, True, True, True]],\n\n [[False, False, False, False, False],\n [False, False, False, False, False]]])\n>>> p[q] = 0\n>>> p\narray([[[0, 0, 0, 0, 0],\n [0, 0, 0, 0, 0]],\n\n [[0, 1, 2, 3, 4],\n [5, 6, 7, 8, 9]]])", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Replace values after filtering" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use " }, { "__type": "InlineRole", "__tag": 4003, "value": "argmax", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineRole", "__tag": 4003, "value": "argmin", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ": " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> a = np.arange(30).reshape(2, 3, 5)\n>>> np.argmax(a)\n29\n>>> np.argmin(a)\n0", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Use the " }, { "__type": "InlineCode", "__tag": 4051, "value": "axis" }, { "__type": "Text", "__tag": 4046, "value": " keyword to get the indices of maximum and minimum values along a specific axis " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> np.argmax(a, axis=0)\narray([[1, 1, 1, 1, 1],\n [1, 1, 1, 1, 1],\n [1, 1, 1, 1, 1]])\n>>> np.argmax(a, axis=1)\narray([[2, 2, 2, 2, 2],\n [2, 2, 2, 2, 2]])\n>>> np.argmax(a, axis=2)\narray([[4, 4, 4],\n [4, 4, 4]])\n\n>>> np.argmin(a, axis=1)\narray([[0, 0, 0, 0, 0],\n [0, 0, 0, 0, 0]])\n>>> np.argmin(a, axis=2)\narray([[0, 0, 0],\n [0, 0, 0]])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Set " }, { "__type": "InlineCode", "__tag": 4051, "value": "keepdims" }, { "__type": "Text", "__tag": 4046, "value": " to " }, { "__type": "InlineCode", "__tag": 4051, "value": "True" }, { "__type": "Text", "__tag": 4046, "value": " to keep the axes which are reduced in the result as dimensions with size one " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> np.argmin(a, axis=2, keepdims=True)\narray([[[0],\n [0],\n [0]],\n\n [[0],\n [0],\n [0]]])\n>>> np.argmax(a, axis=1, keepdims=True)\narray([[[2, 2, 2, 2, 2]],\n\n [[2, 2, 2, 2, 2]]])", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "To get the indices of each maximum or minimum value for each (N-1)-dimensional array in an N-dimensional array, use " }, { "__type": "InlineRole", "__tag": 4003, "value": "reshape", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " to reshape the array to a 2D array, apply " }, { "__type": "InlineRole", "__tag": 4003, "value": "argmax", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineRole", "__tag": 4003, "value": "argmin", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " along " }, { "__type": "InlineCode", "__tag": 4051, "value": "axis=1" }, { "__type": "Text", "__tag": 4046, "value": " and use " }, { "__type": "InlineRole", "__tag": 4003, "value": "unravel_index", "domain": null, "role": "meth", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " to recover the index of the values per slice " } ] }, { "__type": "Code", "__tag": 4050, "value": ">>> x = np.arange(2*2*3).reshape(2, 2, 3) % 7 # 3D example array\n>>> x\narray([[[0, 1, 2],\n [3, 4, 5]],\n\n [[6, 0, 1],\n [2, 3, 4]]])\n>>> x_2d = np.reshape(x, (x.shape[0], -1))\n>>> indices_2d = np.argmax(x_2d, axis=1)\n>>> indices_2d\narray([5, 0])\n>>> np.unravel_index(indices_2d, x.shape[1:])\n(array([1, 0]), array([2, 0]))", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The first array returned contains the indices along axis 1 in the original array, the second array contains the indices along axis 2. The highest value in " }, { "__type": "InlineCode", "__tag": 4051, "value": "x[0]" }, { "__type": "Text", "__tag": 4046, "value": " is therefore " }, { "__type": "InlineCode", "__tag": 4051, "value": "x[0, 1, 2]" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Fetch indices of max/min values" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "It must be kept in mind that basic indexing produces " }, { "__type": "InlineCode", "__tag": 4051, "value": "views " }, { "__type": "Text", "__tag": 4046, "value": " and advanced indexing produces " }, { "__type": "InlineCode", "__tag": 4051, "value": "copies " }, { "__type": "Text", "__tag": 4046, "value": ", which are computationally less efficient. 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