301 lines
9.0 KiB
Python
301 lines
9.0 KiB
Python
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from sympy.core.containers import Dict
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from sympy.core.symbol import Dummy
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from sympy.utilities.iterables import is_sequence
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from sympy.utilities.misc import as_int, filldedent
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from .sparse import MutableSparseMatrix as SparseMatrix
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def _doktocsr(dok):
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"""Converts a sparse matrix to Compressed Sparse Row (CSR) format.
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Parameters
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==========
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A : contains non-zero elements sorted by key (row, column)
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JA : JA[i] is the column corresponding to A[i]
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IA : IA[i] contains the index in A for the first non-zero element
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of row[i]. Thus IA[i+1] - IA[i] gives number of non-zero
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elements row[i]. The length of IA is always 1 more than the
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number of rows in the matrix.
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Examples
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========
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>>> from sympy.matrices.sparsetools import _doktocsr
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>>> from sympy import SparseMatrix, diag
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>>> m = SparseMatrix(diag(1, 2, 3))
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>>> m[2, 0] = -1
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>>> _doktocsr(m)
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[[1, 2, -1, 3], [0, 1, 0, 2], [0, 1, 2, 4], [3, 3]]
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"""
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row, JA, A = [list(i) for i in zip(*dok.row_list())]
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IA = [0]*((row[0] if row else 0) + 1)
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for i, r in enumerate(row):
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IA.extend([i]*(r - row[i - 1])) # if i = 0 nothing is extended
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IA.extend([len(A)]*(dok.rows - len(IA) + 1))
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shape = [dok.rows, dok.cols]
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return [A, JA, IA, shape]
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def _csrtodok(csr):
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"""Converts a CSR representation to DOK representation.
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Examples
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========
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>>> from sympy.matrices.sparsetools import _csrtodok
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>>> _csrtodok([[5, 8, 3, 6], [0, 1, 2, 1], [0, 0, 2, 3, 4], [4, 3]])
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Matrix([
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[0, 0, 0],
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[5, 8, 0],
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[0, 0, 3],
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[0, 6, 0]])
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"""
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smat = {}
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A, JA, IA, shape = csr
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for i in range(len(IA) - 1):
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indices = slice(IA[i], IA[i + 1])
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for l, m in zip(A[indices], JA[indices]):
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smat[i, m] = l
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return SparseMatrix(*shape, smat)
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def banded(*args, **kwargs):
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"""Returns a SparseMatrix from the given dictionary describing
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the diagonals of the matrix. The keys are positive for upper
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diagonals and negative for those below the main diagonal. The
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values may be:
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* expressions or single-argument functions,
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* lists or tuples of values,
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* matrices
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Unless dimensions are given, the size of the returned matrix will
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be large enough to contain the largest non-zero value provided.
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kwargs
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======
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rows : rows of the resulting matrix; computed if
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not given.
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cols : columns of the resulting matrix; computed if
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not given.
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Examples
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========
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>>> from sympy import banded, ones, Matrix
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>>> from sympy.abc import x
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If explicit values are given in tuples,
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the matrix will autosize to contain all values, otherwise
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a single value is filled onto the entire diagonal:
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>>> banded({1: (1, 2, 3), -1: (4, 5, 6), 0: x})
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Matrix([
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[x, 1, 0, 0],
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[4, x, 2, 0],
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[0, 5, x, 3],
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[0, 0, 6, x]])
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A function accepting a single argument can be used to fill the
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diagonal as a function of diagonal index (which starts at 0).
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The size (or shape) of the matrix must be given to obtain more
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than a 1x1 matrix:
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>>> s = lambda d: (1 + d)**2
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>>> banded(5, {0: s, 2: s, -2: 2})
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Matrix([
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[1, 0, 1, 0, 0],
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[0, 4, 0, 4, 0],
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[2, 0, 9, 0, 9],
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[0, 2, 0, 16, 0],
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[0, 0, 2, 0, 25]])
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The diagonal of matrices placed on a diagonal will coincide
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with the indicated diagonal:
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>>> vert = Matrix([1, 2, 3])
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>>> banded({0: vert}, cols=3)
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Matrix([
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[1, 0, 0],
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[2, 1, 0],
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[3, 2, 1],
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[0, 3, 2],
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[0, 0, 3]])
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>>> banded(4, {0: ones(2)})
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Matrix([
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[1, 1, 0, 0],
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[1, 1, 0, 0],
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[0, 0, 1, 1],
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[0, 0, 1, 1]])
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Errors are raised if the designated size will not hold
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all values an integral number of times. Here, the rows
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are designated as odd (but an even number is required to
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hold the off-diagonal 2x2 ones):
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>>> banded({0: 2, 1: ones(2)}, rows=5)
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Traceback (most recent call last):
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...
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ValueError:
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sequence does not fit an integral number of times in the matrix
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And here, an even number of rows is given...but the square
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matrix has an even number of columns, too. As we saw
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in the previous example, an odd number is required:
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>>> banded(4, {0: 2, 1: ones(2)}) # trying to make 4x4 and cols must be odd
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Traceback (most recent call last):
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...
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ValueError:
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sequence does not fit an integral number of times in the matrix
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A way around having to count rows is to enclosing matrix elements
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in a tuple and indicate the desired number of them to the right:
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>>> banded({0: 2, 2: (ones(2),)*3})
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Matrix([
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[2, 0, 1, 1, 0, 0, 0, 0],
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[0, 2, 1, 1, 0, 0, 0, 0],
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[0, 0, 2, 0, 1, 1, 0, 0],
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[0, 0, 0, 2, 1, 1, 0, 0],
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[0, 0, 0, 0, 2, 0, 1, 1],
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[0, 0, 0, 0, 0, 2, 1, 1]])
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An error will be raised if more than one value
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is written to a given entry. Here, the ones overlap
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with the main diagonal if they are placed on the
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first diagonal:
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>>> banded({0: (2,)*5, 1: (ones(2),)*3})
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Traceback (most recent call last):
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...
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ValueError: collision at (1, 1)
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By placing a 0 at the bottom left of the 2x2 matrix of
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ones, the collision is avoided:
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>>> u2 = Matrix([
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... [1, 1],
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... [0, 1]])
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>>> banded({0: [2]*5, 1: [u2]*3})
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Matrix([
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[2, 1, 1, 0, 0, 0, 0],
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[0, 2, 1, 0, 0, 0, 0],
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[0, 0, 2, 1, 1, 0, 0],
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[0, 0, 0, 2, 1, 0, 0],
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[0, 0, 0, 0, 2, 1, 1],
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[0, 0, 0, 0, 0, 0, 1]])
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"""
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try:
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if len(args) not in (1, 2, 3):
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raise TypeError
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if not isinstance(args[-1], (dict, Dict)):
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raise TypeError
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if len(args) == 1:
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rows = kwargs.get('rows', None)
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cols = kwargs.get('cols', None)
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if rows is not None:
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rows = as_int(rows)
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if cols is not None:
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cols = as_int(cols)
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elif len(args) == 2:
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rows = cols = as_int(args[0])
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else:
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rows, cols = map(as_int, args[:2])
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# fails with ValueError if any keys are not ints
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_ = all(as_int(k) for k in args[-1])
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except (ValueError, TypeError):
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raise TypeError(filldedent(
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'''unrecognized input to banded:
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expecting [[row,] col,] {int: value}'''))
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def rc(d):
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# return row,col coord of diagonal start
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r = -d if d < 0 else 0
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c = 0 if r else d
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return r, c
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smat = {}
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undone = []
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tba = Dummy()
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# first handle objects with size
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for d, v in args[-1].items():
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r, c = rc(d)
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# note: only list and tuple are recognized since this
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# will allow other Basic objects like Tuple
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# into the matrix if so desired
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if isinstance(v, (list, tuple)):
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extra = 0
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for i, vi in enumerate(v):
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i += extra
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if is_sequence(vi):
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vi = SparseMatrix(vi)
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smat[r + i, c + i] = vi
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extra += min(vi.shape) - 1
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else:
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smat[r + i, c + i] = vi
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elif is_sequence(v):
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v = SparseMatrix(v)
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rv, cv = v.shape
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if rows and cols:
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nr, xr = divmod(rows - r, rv)
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nc, xc = divmod(cols - c, cv)
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x = xr or xc
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do = min(nr, nc)
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elif rows:
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do, x = divmod(rows - r, rv)
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elif cols:
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do, x = divmod(cols - c, cv)
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else:
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do = 1
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x = 0
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if x:
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raise ValueError(filldedent('''
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sequence does not fit an integral number of times
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in the matrix'''))
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j = min(v.shape)
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for i in range(do):
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smat[r, c] = v
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r += j
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c += j
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elif v:
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smat[r, c] = tba
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undone.append((d, v))
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s = SparseMatrix(None, smat) # to expand matrices
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smat = s.todok()
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# check for dim errors here
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if rows is not None and rows < s.rows:
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raise ValueError('Designated rows %s < needed %s' % (rows, s.rows))
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if cols is not None and cols < s.cols:
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raise ValueError('Designated cols %s < needed %s' % (cols, s.cols))
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if rows is cols is None:
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rows = s.rows
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cols = s.cols
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elif rows is not None and cols is None:
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cols = max(rows, s.cols)
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elif cols is not None and rows is None:
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rows = max(cols, s.rows)
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def update(i, j, v):
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# update smat and make sure there are
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# no collisions
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if v:
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if (i, j) in smat and smat[i, j] not in (tba, v):
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raise ValueError('collision at %s' % ((i, j),))
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smat[i, j] = v
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if undone:
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for d, vi in undone:
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r, c = rc(d)
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v = vi if callable(vi) else lambda _: vi
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i = 0
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while r + i < rows and c + i < cols:
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update(r + i, c + i, v(i))
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i += 1
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return SparseMatrix(rows, cols, smat)
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