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from .utilities import _iszero |
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def _columnspace(M, simplify=False): |
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"""Returns a list of vectors (Matrix objects) that span columnspace of ``M`` |
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Examples |
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======== |
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>>> from sympy import Matrix |
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>>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) |
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>>> M |
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Matrix([ |
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[ 1, 3, 0], |
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[-2, -6, 0], |
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[ 3, 9, 6]]) |
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>>> M.columnspace() |
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[Matrix([ |
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[ 1], |
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[-2], |
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[ 3]]), Matrix([ |
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[0], |
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[0], |
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[6]])] |
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See Also |
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======== |
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nullspace |
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rowspace |
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""" |
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reduced, pivots = M.echelon_form(simplify=simplify, with_pivots=True) |
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return [M.col(i) for i in pivots] |
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def _nullspace(M, simplify=False, iszerofunc=_iszero): |
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"""Returns list of vectors (Matrix objects) that span nullspace of ``M`` |
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Examples |
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======== |
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>>> from sympy import Matrix |
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>>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) |
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>>> M |
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Matrix([ |
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[ 1, 3, 0], |
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[-2, -6, 0], |
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[ 3, 9, 6]]) |
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>>> M.nullspace() |
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[Matrix([ |
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[-3], |
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[ 1], |
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[ 0]])] |
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See Also |
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======== |
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columnspace |
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rowspace |
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""" |
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reduced, pivots = M.rref(iszerofunc=iszerofunc, simplify=simplify) |
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free_vars = [i for i in range(M.cols) if i not in pivots] |
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basis = [] |
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for free_var in free_vars: |
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vec = [M.zero] * M.cols |
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vec[free_var] = M.one |
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for piv_row, piv_col in enumerate(pivots): |
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vec[piv_col] -= reduced[piv_row, free_var] |
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basis.append(vec) |
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return [M._new(M.cols, 1, b) for b in basis] |
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def _rowspace(M, simplify=False): |
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"""Returns a list of vectors that span the row space of ``M``. |
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Examples |
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======== |
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>>> from sympy import Matrix |
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>>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) |
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>>> M |
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Matrix([ |
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[ 1, 3, 0], |
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[-2, -6, 0], |
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[ 3, 9, 6]]) |
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>>> M.rowspace() |
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[Matrix([[1, 3, 0]]), Matrix([[0, 0, 6]])] |
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""" |
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reduced, pivots = M.echelon_form(simplify=simplify, with_pivots=True) |
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return [reduced.row(i) for i in range(len(pivots))] |
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def _orthogonalize(cls, *vecs, normalize=False, rankcheck=False): |
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"""Apply the Gram-Schmidt orthogonalization procedure |
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to vectors supplied in ``vecs``. |
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Parameters |
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========== |
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vecs |
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vectors to be made orthogonal |
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normalize : bool |
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If ``True``, return an orthonormal basis. |
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rankcheck : bool |
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If ``True``, the computation does not stop when encountering |
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linearly dependent vectors. |
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If ``False``, it will raise ``ValueError`` when any zero |
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or linearly dependent vectors are found. |
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Returns |
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======= |
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list |
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List of orthogonal (or orthonormal) basis vectors. |
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Examples |
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======== |
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>>> from sympy import I, Matrix |
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>>> v = [Matrix([1, I]), Matrix([1, -I])] |
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>>> Matrix.orthogonalize(*v) |
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[Matrix([ |
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[1], |
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[I]]), Matrix([ |
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[ 1], |
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[-I]])] |
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See Also |
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======== |
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MatrixBase.QRdecomposition |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process |
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""" |
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from .decompositions import _QRdecomposition_optional |
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if not vecs: |
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return [] |
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all_row_vecs = (vecs[0].rows == 1) |
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vecs = [x.vec() for x in vecs] |
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M = cls.hstack(*vecs) |
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Q, R = _QRdecomposition_optional(M, normalize=normalize) |
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if rankcheck and Q.cols < len(vecs): |
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raise ValueError("GramSchmidt: vector set not linearly independent") |
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ret = [] |
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for i in range(Q.cols): |
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if all_row_vecs: |
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col = cls(Q[:, i].T) |
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else: |
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col = cls(Q[:, i]) |
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ret.append(col) |
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return ret |
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