o 8Va8@sddlmZddlmZmZddlmZddlmZddl m Z ddl m Z m Z ddlmZdd lmZmZmZmZdd lmZmZGd d d e ZGd ddee ZdS))Callable)as_int is_sequence)Dict)SymPyDeprecationWarning) MatrixBase)MutableRepMatrix RepMatrix)_iszero)_liupc _row_structure_symbolic_cholesky_cholesky_sparse_LDLdecomposition_sparse)_lower_triangular_solve_sparse_upper_triangular_solve_sparsecseZdZdZefddZeddZddZdd Z d d Z d d Z ddZ ddZ ddZddZd*ddZd*ddZeedddZee dddZddZdd Zd+d"d#Zd+d$d%Zd&d'Zd(d)Zeje_eje_eje_eje_eje_eje_ZS), SparseMatrixa" A sparse matrix (a matrix with a large number of zero elements). Examples ======== >>> from sympy.matrices import SparseMatrix, ones >>> SparseMatrix(2, 2, range(4)) Matrix([ [0, 1], [2, 3]]) >>> SparseMatrix(2, 2, {(1, 1): 2}) Matrix([ [0, 0], [0, 2]]) A SparseMatrix can be instantiated from a ragged list of lists: >>> SparseMatrix([[1, 2, 3], [1, 2], [1]]) Matrix([ [1, 2, 3], [1, 2, 0], [1, 0, 0]]) For safety, one may include the expected size and then an error will be raised if the indices of any element are out of range or (for a flat list) if the total number of elements does not match the expected shape: >>> SparseMatrix(2, 2, [1, 2]) Traceback (most recent call last): ... ValueError: List length (2) != rows*columns (4) Here, an error is not raised because the list is not flat and no element is out of range: >>> SparseMatrix(2, 2, [[1, 2]]) Matrix([ [1, 2], [0, 0]]) But adding another element to the first (and only) row will cause an error to be raised: >>> SparseMatrix(2, 2, [[1, 2, 3]]) Traceback (most recent call last): ... ValueError: The location (0, 2) is out of designated range: (1, 1) To autosize the matrix, pass None for rows: >>> SparseMatrix(None, [[1, 2, 3]]) Matrix([[1, 2, 3]]) >>> SparseMatrix(None, {(1, 1): 1, (3, 3): 3}) Matrix([ [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 3]]) Values that are themselves a Matrix are automatically expanded: >>> SparseMatrix(4, 4, {(1, 1): ones(2)}) Matrix([ [0, 0, 0, 0], [0, 1, 1, 0], [0, 1, 1, 0], [0, 0, 0, 0]]) A ValueError is raised if the expanding matrix tries to overwrite a different element already present: >>> SparseMatrix(3, 3, {(0, 0): ones(2), (1, 1): 2}) Traceback (most recent call last): ... ValueError: collision at (1, 1) See Also ======== DenseMatrix MutableSparseMatrix ImmutableSparseMatrix c st|dkr"t|dtr"|dj}|dj}|d||fSit|dkr7|ddur7dd|dg}t|dkr|dd\}}||urPdurWnnd}}nd||fvratdt|dt|d}}t|dtr|d}d||fvrtd ||fddt |D}fd dt |D} |D]} | D]} || | } | j kr| | | f<qq||fSt|dt tfr3fd d } |dD]\\\}}}t|tr|D]\\} } }| || || |qqt|ttfr#j|fi|\}}D]\} } | || || | | fqqՈ |}| || |qnft|drtd d |dD }|sXj|dfi|\}}nA|d}t|||krptd t|||t |D]$} t |D]} || || } | } | j kr| | | f<qzqt|durÈ}|rtdd|Ddnd}|rtdd|Ddnd}n+D]&\} } | r| |ks| r| |krtd | | fd|dd|dq||fSt|dkrOt|dttfrO|d}d}t|D]0\} }t|ttfs|g}t|D]\} }|j kr4 || | f<q!t|t|}q|rFt|nd}|}||fStj|\}}}t |D]} t |D]} ||| | } | j krx| | | f<qbq\||fS)Nrrz*Pass rows=None and no cols for autosizing.z2{} and {} must be integers for this specification.cg|]}|qS_sympify.0iclsr7/usr/lib/python3/dist-packages/sympy/matrices/sparse.py z8SparseMatrix._handle_creation_inputs..crrr)rjrrrrr csR|r'||fvr|||fkrtd||f|||f|||f<dSdS)Nz)There is a collision at {} for {} and {}.) ValueErrorformat)rr!v)smatrrupdatesz4SparseMatrix._handle_creation_inputs..updatecss|]}t|VqdSN)rrrrr sz7SparseMatrix._handle_creation_inputs..zMThe length of the flat list ({}) does not match the specified size ({} * {}).cSsg|]\}}|qSrr)rr_rrrrcSsg|]\}}|qSrr)rr*crrrrr+z?The location {} is out of the designated range[{}, {}]x[{}, {}])len isinstancerrowscolstodokr"rrr#rangerZzerodictritemslisttuple_handle_creation_inputsranykeysmax enumeratesuper)rargskwargsr/r0r)r,opZ row_indicesZ col_indicesrr!valuer&r$Zvvr*ZflatZ flat_listr9rowZmat __class__)rr%rr7js                        "  "       z$SparseMatrix._handle_creation_inputscCstddddd|S)Nz+The private _smat attribute of SparseMatrixzthe .todok() methodiTz1.9)ZfeatureZ useinsteadZissueZdeprecated_since_version)rwarnr1selfrrr_smatszSparseMatrix._smatcKs(|j|dd|dt|dddS)NmethodLDL iszerofunctry_block_diagF)rHrJrK)invgetr )rFr>rrr _eval_inverses  zSparseMatrix._eval_inversecCsTt|stdi}|D]\}}||}|dkr |||<q||j|j|S)aaApply a function to each element of the matrix. Examples ======== >>> from sympy.matrices import SparseMatrix >>> m = SparseMatrix(2, 2, lambda i, j: i*2+j) >>> m Matrix([ [0, 1], [2, 3]]) >>> m.applyfunc(lambda i: 2*i) Matrix([ [0, 2], [4, 6]]) z`f` must be callable.r)callable TypeErrorr1r4_newr/r0)rFfZdokkr$Zfvrrr applyfuncszSparseMatrix.applyfunccCsddlm}||S)z,Returns an Immutable version of this Matrix.r)ImmutableSparseMatrix)Z immutablerU)rFrUrrr as_immutables zSparseMatrix.as_immutablecCt|S)aCReturns a mutable version of this matrix. Examples ======== >>> from sympy import ImmutableMatrix >>> X = ImmutableMatrix([[1, 2], [3, 4]]) >>> Y = X.as_mutable() >>> Y[1, 1] = 5 # Can set values in Y >>> Y Matrix([ [1, 2], [3, 5]]) )MutableSparseMatrixrErrr as_mutable szSparseMatrix.as_mutablecs*fddttdddDS)aReturns a column-sorted list of non-zero elements of the matrix. Examples ======== >>> from sympy.matrices import SparseMatrix >>> a=SparseMatrix(((1, 2), (3, 4))) >>> a Matrix([ [1, 2], [3, 4]]) >>> a.CL [(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)] See Also ======== sympy.matrices.sparse.SparseMatrix.row_list cg|] }t||fqSrr6rrSrErrrEz)SparseMatrix.col_list..cSs tt|Sr')r5reversedrSrrrEs z'SparseMatrix.col_list..key)sortedr5r1r9rErrErcol_list1s*zSparseMatrix.col_listcCs t|S)z2Returns the number of non-zero elements in Matrix.)r-r1rErrrnnzGs zSparseMatrix.nnzcs&fddtdddDS)aReturns a row-sorted list of non-zero elements of the matrix. Examples ======== >>> from sympy.matrices import SparseMatrix >>> a = SparseMatrix(((1, 2), (3, 4))) >>> a Matrix([ [1, 2], [3, 4]]) >>> a.RL [(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)] See Also ======== sympy.matrices.sparse.SparseMatrix.col_list crZrr[r\rErrr_r]z)SparseMatrix.row_list..cSrWr')r5r_rrrr``sz'SparseMatrix.row_list..ra)rcr1r9rErrErrow_listKs zSparseMatrix.row_listcCs||S)z"Scalar element-wise multiplicationr)rFZscalarrrrscalar_multiplybszSparseMatrix.scalar_multiplyrIcCs|j}||j|d||S)aReturn the least-square fit to the data. By default the cholesky_solve routine is used (method='CH'); other methods of matrix inversion can be used. To find out which are available, see the docstring of the .inv() method. Examples ======== >>> from sympy.matrices import SparseMatrix, Matrix, ones >>> A = Matrix([1, 2, 3]) >>> B = Matrix([2, 3, 4]) >>> S = SparseMatrix(A.row_join(B)) >>> S Matrix([ [1, 2], [2, 3], [3, 4]]) If each line of S represent coefficients of Ax + By and x and y are [2, 3] then S*xy is: >>> r = S*Matrix([2, 3]); r Matrix([ [ 8], [13], [18]]) But let's add 1 to the middle value and then solve for the least-squares value of xy: >>> xy = S.solve_least_squares(Matrix([8, 14, 18])); xy Matrix([ [ 5/3], [10/3]]) The error is given by S*xy - r: >>> S*xy - r Matrix([ [1/3], [1/3], [1/3]]) >>> _.norm().n(2) 0.58 If a different xy is used, the norm will be higher: >>> xy += ones(2, 1)/10 >>> (S*xy - r).norm().n(2) 1.5 rH)TrL)rFrhsrHtrrrsolve_least_squaresfs6z SparseMatrix.solve_least_squarescCsD|js|j|jkr td|j|jkrtddS|j|d|S)zReturn solution to self*soln = rhs using given inversion method. 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