ó ~9­\c@ s`ddlmZmZddlZddlmZddlmZddlm Z m Z m Z ddl m Z ddlmZmZddlmZdd lmZdd lmZdd lmZdd lmZmZdd lmZmZddlm Z m!Z!ddl"m#Z$ddl%m&Z&ddl'm(Z(d„Z)d„Z*de fd„ƒYZ+d„Z,de+e fd„ƒYZ-e-Z.Z/e0d„Z1e0d„Z2d„Z3d„Z4d„Z5e&ddfƒd „ƒZ6e7d!„Z8d"„Z9d#„Z:e;d$„Z<gd%„Z=d&„Z>d'„Z?d(„Z@eAd)d*eAe;d+eAd,„ZBd-d.„ZCd/„ZDdS(0iÿÿÿÿ(tdivisiontprint_functionN(t SympifyError(tBasic(t is_sequencetrangetreduce(tExpr(t count_opst expand_mul(tS(tSymbol(tsympify(tsqrt(tcostsin(ta2idxtclassof(t MatrixBaset ShapeError(tsimplify(tdoctest_depends_on(t filldedentcC s|jS(sReturns True if x is zero.(tis_zero(tx((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt_iszeroscC s8t|ƒt|ƒkr"||kSt|ƒt|ƒkS(s—Compares the elements of a list/tuple `a` and a list/tuple `b`. `_compare_sequence((1,2), [1, 2])` is True, whereas `(1,2) == [1, 2]` is False(ttypettuple(tatb((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt_compare_sequences t DenseMatrixcB sÎeZeZdZdZd„Zd„Zd„Ze d„Z d„Z d„Z d„Z d „Zd „Zd „Zd „Zd „Zd„Ze d„Zd„Zd„Zd„Zd„Zed„ZRS(g…ëQ¸$@icC s¤t|ƒ}t|ddƒ}t|ddƒ}d||fkrFtS||krVtSt|tƒrxt|j|jƒSt|tƒr t|jt|ƒjƒSdS(Ntshape( R tgetattrtNonetFalset isinstancetMatrixRt_matR(tselftothert self_shapet other_shape((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt__eq__-s  cC sàt|tƒr±|\}}y/|j|ƒ\}}|j||j|SWqÜttfk r­t|tƒry|j s’t|tƒr|j r|dkt ksè||j dkt ksè|dkt ksè||j dkt kr÷t dƒ‚nddl m }||||ƒSt|tƒrBtt|jƒƒ|}nt|ƒrQn |g}t|tƒr…tt|jƒƒ|}nt|ƒr”n |g}|j||ƒSXn+t|tƒrË|j|S|jt|ƒSdS(s™Return portion of self defined by key. If the key involves a slice then a list will be returned (if key is a single slice) or a matrix (if key was a tuple involving a slice). Examples ======== >>> from sympy import Matrix, I >>> m = Matrix([ ... [1, 2 + I], ... [3, 4 ]]) If the key is a tuple that doesn't involve a slice then that element is returned: >>> m[1, 0] 3 When a tuple key involves a slice, a matrix is returned. Here, the first column is selected (all rows, column 0): >>> m[:, 0] Matrix([ [1], [3]]) If the slice is not a tuple then it selects from the underlying list of elements that are arranged in row order and a list is returned if a slice is involved: >>> m[0] 1 >>> m[::2] [1, 3] iisindex out of boundaryiÿÿÿÿ(t MatrixElementN(R$Rtkey2ijR&tcolst TypeErrort IndexErrorRt is_numbertTrueR t ValueErrort"sympy.matrices.expressions.matexprR,tslicetlistRtrowsRtextractR(R'tkeytitjR,((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt __getitem__:s4$ 2++     cC s tƒ‚dS(N(tNotImplementedError(R'R9tvalue((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt __setitem__€sc  sçt|j|jƒ‰|rx¼t|jƒD]â‰xltˆƒD]^‰dˆˆˆft|ˆˆft‡‡‡fd†tˆƒDƒƒƒˆˆˆfWt|ˆˆft‡‡fd†tˆƒDƒƒƒ}|jtkr÷tdƒ‚nt|ƒˆˆˆfsc3 s3|])}ˆˆ|fˆˆ|fjƒVqdS(N(R@(RARB(RCR:(s3lib/python2.7/site-packages/sympy/matrices/dense.pys ’ss Matrix must be positive-definitec3 s-|]#}ˆˆ|fˆˆ|fVqdS(N((RARB(RCR:R;(s3lib/python2.7/site-packages/sympy/matrices/dense.pys šsc3 s#|]}ˆˆ|fdVqdS(iN((RARB(RCR:(s3lib/python2.7/site-packages/sympy/matrices/dense.pys œs( tzerosR7RR tsumt is_positiveR#R3R t_new(R't hermitiantLii2((RCR:R;s3lib/python2.7/site-packages/sympy/matrices/dense.pyt _choleskyƒs$!;)87c s%ˆjˆjˆj‡‡fd†ƒS(slHelper function of function diagonal_solve, without the error checks, to be used privately. c sˆ||fˆ||fS(N((R:R;(trhsR'(s3lib/python2.7/site-packages/sympy/matrices/dense.pyt£t(RGR7R.(R'RK((RKR's3lib/python2.7/site-packages/sympy/matrices/dense.pyt_diagonal_solveŸscC sZgt|j|jƒD]\}}||^q}t||ƒj|j|j|dtƒS(Ntcopy(tzipR&RRGR7R.R#(R'R(RRtmat((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt _eval_add¥s2c sf|j‰|j‰‡‡fd†|Dƒ}|jt|ƒtˆƒt‡fd†|DƒƒdtƒS(Nc3 s*|] }ˆD]}|ˆ|Vq qdS(N((RAR:R;(R.tcolsList(s3lib/python2.7/site-packages/sympy/matrices/dense.pys ®sc3 s|]}ˆ|VqdS(N((RAR:(RQ(s3lib/python2.7/site-packages/sympy/matrices/dense.pys °sRO(R&R.RGtlenR6R#(R'trowsListRStindices((R.RSRQs3lib/python2.7/site-packages/sympy/matrices/dense.pyt _eval_extract«s   c s¢ddlm}|j|j}}|j|j}}||}|j}|j} tjg|| } |jdkr€|jdkr€|j‰|j‰xçtt| ƒƒD]Ð} | | | | } } t|| || dƒ}t| ||ƒ}‡‡fd†t ||ƒDƒ}y||Œ| | Èsc3 s'|]\}}ˆ|ˆ|VqdS(N((RARR(RQRY(s3lib/python2.7/site-packages/sympy/matrices/dense.pys ÑscS s||S(N((RR((s3lib/python2.7/site-packages/sympy/matrices/dense.pyRLÒRMRO(tsympyRXR7R.R tZeroR&RRTRPR/RRRRGR#(R'R(RXt self_rowst self_colst other_rowst other_colst other_lent new_mat_rowst new_mat_colstnew_matR:trowtcolt row_indicest col_indicestvec((RQRYs3lib/python2.7/site-packages/sympy/matrices/dense.pyt_eval_matrix_mul²s*     ""!cC sZgt|j|jƒD]\}}||^q}t||ƒj|j|j|dtƒS(NRO(RPR&RRGR7R.R#(R'R(RRRQ((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt_eval_matrix_mul_elementwiseÕs2c K sddlm}|jddƒ}|jdtƒ}|jdtƒr’|jƒ}g}x-|D]%}|j|jd|d|ƒƒq_W||ŒS|jƒ}|dkr¿|j d|ƒ} nN|dkrà|j d|ƒ} n-|dkr|j d|ƒ} n t d ƒ‚|j | ƒS( sÞReturn the matrix inverse using the method indicated (default is Gauss elimination). kwargs ====== method : ('GE', 'LU', or 'ADJ') iszerofunc try_block_diag Notes ===== According to the ``method`` keyword, it calls the appropriate method: GE .... inverse_GE(); default LU .... inverse_LU() ADJ ... inverse_ADJ() According to the ``try_block_diag`` keyword, it will try to form block diagonal matrices using the method get_diag_blocks(), invert these individually, and then reconstruct the full inverse matrix. Note, the GE and LU methods may require the matrix to be simplified before it is inverted in order to properly detect zeros during pivoting. In difficult cases a custom zero detection function can be provided by setting the ``iszerosfunc`` argument to a function that should return True if its argument is zero. The ADJ routine computes the determinant and uses that to detect singular matrices in addition to testing for zeros on the diagonal. See Also ======== inverse_LU inverse_GE inverse_ADJ iÿÿÿÿ(tdiagtmethodtGEt iszerofuncttry_block_diagtLUtADJsInversion method unrecognized(tsympy.matricesRktgetRR#tget_diag_blockstappendtinvt as_mutablet inverse_GEt inverse_LUt inverse_ADJR3RG( R'tkwargsRkRlRntblockstrtblocktMtrv((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt _eval_inverseÙs$'  #      cC s?g|jD]}||^q }|j|j|j|dtƒS(NRO(R&RGR7R.R#(R'R(RRQ((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt_eval_scalar_muls cC s?g|jD]}||^q }|j|j|j|dtƒS(NRO(R&RGR7R.R#(R'R(RRQ((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt_eval_scalar_rmuls cC sKt|jƒ}|j}gt|jƒD]}||||d|!^q(S(Ni(R6R&R.RR7(R'RQR.R:((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt _eval_tolist s c  s t|j|jƒ‰t|jƒ‰|r'xÀt|jƒD]æ‰xotˆƒD]a‰dˆˆˆft|ˆˆft‡‡‡‡fd†tˆƒDƒƒƒˆˆˆf3sc3 sA|]7}ˆˆ|fˆˆ|fjƒˆ||fVqdS(N(R@(RARB(R…RCR:(s3lib/python2.7/site-packages/sympy/matrices/dense.pys 5ss Matrix must be positive-definitec3 s;|]1}ˆˆ|fˆˆ|fˆ||fVqdS(N((RARB(R…RCR:R;(s3lib/python2.7/site-packages/sympy/matrices/dense.pys <sc3 s1|]'}ˆˆ|fdˆ||fVqdS(iN((RARB(R…RCR:(s3lib/python2.7/site-packages/sympy/matrices/dense.pys =s( RDR7teyeRR RERFR#R3RG(R'RH((R…RCR:R;s3lib/python2.7/site-packages/sympy/matrices/dense.pyt_LDLdecomposition%s"$;6!8Dc sÌtˆj|jƒ‰x§t|jƒD]–‰xtˆjƒD]|‰ˆˆˆfdkrftdƒ‚n|ˆˆft‡‡‡‡fd†tˆƒDƒƒˆˆˆfˆˆˆfJs(RDR7R.RR/RERG(R'RK((RˆR:R;R's3lib/python2.7/site-packages/sympy/matrices/dense.pyt_lower_triangular_solve@s%4c sÜtˆj|jƒ‰x·t|jƒD]¦‰xttˆjƒƒD]†‰ˆˆˆfdkrltdƒ‚n|ˆˆft‡‡‡‡fd†tˆdˆjƒDƒƒˆˆˆfˆˆˆfVsi(RDR7R.RtreversedR3RERG(R'RK((RˆR:R;R's3lib/python2.7/site-packages/sympy/matrices/dense.pyt_upper_triangular_solveNs%>cC sNddlm}|jr5|jr5|j|jƒƒS|j|j|jgƒS(s4Returns an Immutable version of this Matrix i(tImmutableDenseMatrix(t immutableRŒR7R.RGttolist(R'tcls((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt as_immutableZscC s t|ƒS(sBReturns 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]]) (R%(R'((s3lib/python2.7/site-packages/sympy/matrices/dense.pyRwbsc C sßt|ddƒ}t|ddƒ}d||fkr:tS||krJtSt}xˆt|jƒD]w}xnt|jƒD]]}|||fj|||f|ƒ}|tkr²tS|tk rv|tkrv|}qvqvWq`W|S(sWApplies ``equals`` to corresponding elements of the matrices, trying to prove that the elements are equivalent, returning True if they are, False if any pair is not, and None (or the first failing expression if failing_expression is True) if it cannot be decided if the expressions are equivalent or not. This is, in general, an expensive operation. Examples ======== >>> from sympy.matrices import Matrix >>> from sympy.abc import x >>> from sympy import cos >>> A = Matrix([x*(x - 1), 0]) >>> B = Matrix([x**2 - x, 0]) >>> A == B False >>> A.simplify() == B.simplify() True >>> A.equals(B) True >>> A.equals(2) False See Also ======== sympy.core.expr.equals R N(R!R"R#R2RR7R.tequals( R'R(tfailing_expressionR)R*R€R:R;tans((s3lib/python2.7/site-packages/sympy/matrices/dense.pyR‘ss & (t__name__t __module__R#t is_MatrixExprt _op_priorityt_class_priorityR+R<R?R2RJRNRRRWRiRjRR‚RƒR„R‡R‰R‹RRwR‘(((s3lib/python2.7/site-packages/sympy/matrices/dense.pyR&s, F      #  ?       cC swt|dtƒr|jƒSt|tƒr/|St|dƒrs|jƒ}t|jƒdkrit |ƒSt |ƒS|S(s0Return a matrix as a Matrix, otherwise return x.t is_Matrixt __array__i( R!R#RwR$RthasattrRšRTR R R%(RR((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt_force_mutable¡s    tMutableDenseMatrixcB s¡eZd„Zed„ƒZd„Zd„Zd„Zd„Zd„Z d„Z d„Z d „Z d „Z d „Zd „Zd eeed„Zd„ZRS(cO s|j||ŽS(N(RG(RtargsR{((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt__new__°scO s |jdtƒtkrKt|ƒdkr9tdƒ‚n|\}}}n'|j||Ž\}}}t|ƒ}tj|ƒ}||_ ||_ ||_ |S(NROisA'copy=False' requires a matrix be initialized as rows,cols,[list]( RsR2R#RTR/t_handle_creation_inputsR6tobjectRŸR7R.R&(RRžR{R7R.t flat_listR'((s3lib/python2.7/site-packages/sympy/matrices/dense.pyRG³s    cC sL|j||ƒ}|dk rH|\}}}||j||j|>> from sympy import Matrix, I, zeros, ones >>> m = Matrix(((1, 2+I), (3, 4))) >>> m Matrix([ [1, 2 + I], [3, 4]]) >>> m[1, 0] = 9 >>> m Matrix([ [1, 2 + I], [9, 4]]) >>> m[1, 0] = [[0, 1]] To replace row r you assign to position r*m where m is the number of columns: >>> M = zeros(4) >>> m = M.cols >>> M[3*m] = ones(1, m)*2; M Matrix([ [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2, 2, 2, 2]]) And to replace column c you can assign to position c: >>> M[2] = ones(m, 1)*4; M Matrix([ [0, 0, 4, 0], [0, 0, 4, 0], [0, 0, 4, 0], [2, 2, 4, 2]]) N(t_setitemR"R&R.(R'R9R>R€R:R;((s3lib/python2.7/site-packages/sympy/matrices/dense.pyR?Ås( cC s |jƒS(N(RO(R'((s3lib/python2.7/site-packages/sympy/matrices/dense.pyRwòscC s||j ks||jkrAtd||j|jfƒ‚nx6t|jdddƒD]}|j|||j=q[W|jd8_dS(s0Delete the given column. Examples ======== >>> from sympy.matrices import eye >>> M = eye(3) >>> M.col_del(1) >>> M Matrix([ [1, 0], [0, 0], [0, 1]]) See Also ======== col row_del s/Index out of range: 'i=%s', valid -%s <= i < %siiÿÿÿÿN(R.R0RR7R&(R'R:R;((s3lib/python2.7/site-packages/sympy/matrices/dense.pytcol_delõs  cC sggtt|j|d|j…tt|jƒƒƒƒD]}||Œ^q8|j|d|j…>> from sympy.matrices import eye >>> M = eye(3) >>> M.col_op(1, lambda v, i: v + 2*M[i, 0]); M Matrix([ [1, 2, 0], [0, 1, 0], [0, 0, 1]]) See Also ======== col row_op N(R6RPR&R.RR7(R'R;tftt((s3lib/python2.7/site-packages/sympy/matrices/dense.pytcol_opscC sVxOtd|jƒD];}|||f|||f|||f<|||f>> from sympy.matrices import Matrix >>> M = Matrix([[1, 0], [1, 0]]) >>> M Matrix([ [1, 0], [1, 0]]) >>> M.col_swap(0, 1) >>> M Matrix([ [0, 1], [0, 1]]) See Also ======== col row_swap iN(RR7(R'R:R;RB((s3lib/python2.7/site-packages/sympy/matrices/dense.pytcol_swap'scC s;t|ƒs%tdt|ƒƒ‚n|j|t|ƒƒS(srCopy in elements from a list. Parameters ========== key : slice The section of this matrix to replace. value : iterable The iterable to copy values from. Examples ======== >>> from sympy.matrices import eye >>> I = eye(3) >>> I[:2, 0] = [1, 2] # col >>> I Matrix([ [1, 0, 0], [2, 1, 0], [0, 0, 1]]) >>> I[1, :2] = [[3, 4]] >>> I Matrix([ [1, 0, 0], [3, 4, 0], [0, 0, 1]]) See Also ======== copyin_matrix s,`value` must be an ordered iterable, not %s.(RR/Rt copyin_matrixR%(R'R9R>((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt copyin_listBs" c C sº|j|ƒ\}}}}|j}||||}} ||| fkr`ttdƒƒ‚nxSt|jƒD]B} x9t|jƒD](} || | f|| || |f>> from sympy.matrices import Matrix, eye >>> M = Matrix([[0, 1], [2, 3], [4, 5]]) >>> I = eye(3) >>> I[:3, :2] = M >>> I Matrix([ [0, 1, 0], [2, 3, 0], [4, 5, 1]]) >>> I[0, 1] = M >>> I Matrix([ [0, 0, 1], [2, 2, 3], [4, 4, 5]]) See Also ======== copyin_list sXThe Matrix `value` doesn't have the same dimensions as the in sub-Matrix given by `key`.N(t key2boundsR RRRR7R.( R'R9R>trlotrhitclotchiR tdrtdcR:R;((s3lib/python2.7/site-packages/sympy/matrices/dense.pyR©hs# cC s|gt|ƒ|_dS(snFill the matrix with the scalar value. See Also ======== zeros ones N(RTR&(R'R>((s3lib/python2.7/site-packages/sympy/matrices/dense.pytfill—s cC s||j ks||jkrAtd||j|jfƒ‚n|dkr]||j7}n|j||j|d|j5|jd8_dS(s#Delete the given row. Examples ======== >>> from sympy.matrices import eye >>> M = eye(3) >>> M.row_del(1) >>> M Matrix([ [1, 0, 0], [0, 0, 1]]) See Also ======== row col_del s1Index out of range: 'i = %s', valid -%s <= i < %siiN(R7R0R&R.(R'R:((s3lib/python2.7/site-packages/sympy/matrices/dense.pytrow_del¢s cC sy||j}|j|||j!}gt|tt|jƒƒƒD]\}}|||ƒ^qC|j|||j+dS(s³In-place operation on row ``i`` using two-arg functor whose args are interpreted as ``(self[i, j], j)``. Examples ======== >>> from sympy.matrices import eye >>> M = eye(3) >>> M.row_op(1, lambda v, j: v + 2*M[0, j]); M Matrix([ [1, 0, 0], [2, 1, 0], [0, 0, 1]]) See Also ======== row zip_row_op col_op N(R.R&RPR6R(R'R:R¥ti0triRR;((s3lib/python2.7/site-packages/sympy/matrices/dense.pytrow_op¾s cC sVxOtd|jƒD];}|||f|||f|||f<|||f>> from sympy.matrices import Matrix >>> M = Matrix([[0, 1], [1, 0]]) >>> M Matrix([ [0, 1], [1, 0]]) >>> M.row_swap(0, 1) >>> M Matrix([ [1, 0], [0, 1]]) See Also ======== row col_swap iN(RR.(R'R:R;RB((s3lib/python2.7/site-packages/sympy/matrices/dense.pytrow_swapØsg333333û?c C sVxOtt|jƒƒD]8}t|j|d|d|d|d|ƒ|j|>> from sympy.matrices import eye >>> M = eye(3) >>> M.zip_row_op(1, 0, lambda v, u: v + 2*u); M Matrix([ [1, 0, 0], [2, 1, 0], [0, 0, 1]]) See Also ======== row row_op col_op N(R.R&RP( R'R:RBR¥R´tk0RµtrkRty((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt zip_row_ops   (R”R•RŸt classmethodRGR?RwR¤R§R¨RªR©R²R³R¶R·RR#RRÀ(((s3lib/python2.7/site-packages/sympy/matrices/dense.pyR¯s  -     & /   cC sPddlm}|t|ƒ|ƒ}x$t|ƒD]\}}|||>> from sympy import pi >>> from sympy.matrices import rot_axis3 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_axis3(theta) Matrix([ [ 1/2, sqrt(3)/2, 0], [-sqrt(3)/2, 1/2, 0], [ 0, 0, 1]]) If we rotate by pi/2 (90 degrees): >>> rot_axis3(pi/2) Matrix([ [ 0, 1, 0], [-1, 0, 0], [ 0, 0, 1]]) See Also ======== rot_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis rot_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis ii(iii(RRR%(tthetatcttsttlil((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt rot_axis3Hs #     cC sDt|ƒ}t|ƒ}|d| fd|d|ff}t|ƒS(sëReturns a rotation matrix for a rotation of theta (in radians) about the 2-axis. Examples ======== >>> from sympy import pi >>> from sympy.matrices import rot_axis2 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_axis2(theta) Matrix([ [ 1/2, 0, -sqrt(3)/2], [ 0, 1, 0], [sqrt(3)/2, 0, 1/2]]) If we rotate by pi/2 (90 degrees): >>> rot_axis2(pi/2) Matrix([ [0, 0, -1], [0, 1, 0], [1, 0, 0]]) See Also ======== rot_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis rot_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis ii(iii(RRR%(RËRÌRÍRÎ((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt rot_axis2ss #   cC sDt|ƒ}t|ƒ}dd||fd| |ff}t|ƒS(sëReturns a rotation matrix for a rotation of theta (in radians) about the 1-axis. Examples ======== >>> from sympy import pi >>> from sympy.matrices import rot_axis1 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_axis1(theta) Matrix([ [1, 0, 0], [0, 1/2, sqrt(3)/2], [0, -sqrt(3)/2, 1/2]]) If we rotate by pi/2 (90 degrees): >>> rot_axis1(pi/2) Matrix([ [1, 0, 0], [0, 0, 1], [0, -1, 0]]) See Also ======== rot_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis rot_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis ii(iii(RRR%(RËRÌRÍRÎ((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt rot_axis1žs #   tmodulesRÃcK srddlm}m}||dtƒ}xC||ƒD]5}td|djtt|ƒƒf|||>> from sympy import symarray >>> symarray('', 3) [_0 _1 _2] If you want multiple symarrays to contain distinct symbols, you *must* provide unique prefixes: >>> a = symarray('', 3) >>> b = symarray('', 3) >>> a[0] == b[0] True >>> a = symarray('a', 3) >>> b = symarray('b', 3) >>> a[0] == b[0] False Creating symarrays with a prefix: >>> symarray('a', 3) [a_0 a_1 a_2] For more than one dimension, the shape must be given as a tuple: >>> symarray('a', (2, 3)) [[a_0_0 a_0_1 a_0_2] [a_1_0 a_1_1 a_1_2]] >>> symarray('a', (2, 3, 2)) [[[a_0_0_0 a_0_0_1] [a_0_1_0 a_0_1_1] [a_0_2_0 a_0_2_1]] [[a_1_0_0 a_1_0_1] [a_1_1_0 a_1_1_1] [a_1_2_0 a_1_2_1]]] For setting assumptions of the underlying Symbols: >>> [s.is_real for s in symarray('a', 2, real=True)] [True, True] iÿÿÿÿ(RÂtndindexRÆs%s_%st_(RÃRÂRÓR¡R tjointmaptstr(tprefixR R{RÂRÓtarrtindex((s3lib/python2.7/site-packages/sympy/matrices/dense.pytsymarrayÉs A"c stddlm}tttˆƒƒ‰|s@‡‡fd†}n‡‡fd†}tˆƒ}||||ƒjƒS(sYGiven linear difference operator L of order 'k' and homogeneous equation Ly = 0 we want to compute kernel of L, which is a set of 'k' sequences: a(n), b(n), ... z(n). Solutions of L are linearly independent iff their Casoratian, denoted as C(a, b, ..., z), do not vanish for n = 0. Casoratian is defined by k x k determinant:: + a(n) b(n) . . . z(n) + | a(n+1) b(n+1) . . . z(n+1) | | . . . . | | . . . . | | . . . . | + a(n+k-1) b(n+k-1) . . . z(n+k-1) + It proves very useful in rsolve_hyper() where it is applied to a generating set of a recurrence to factor out linearly dependent solutions and return a basis: >>> from sympy import Symbol, casoratian, factorial >>> n = Symbol('n', integer=True) Exponential and factorial are linearly independent: >>> casoratian([2**n, factorial(n)], n) != 0 True i(R%c sˆ|jˆˆ|ƒS(N(tsubs(R:R;(tntseqs(s3lib/python2.7/site-packages/sympy/matrices/dense.pyRL9RMc sˆ|jˆ|ƒS(N(RÜ(R:R;(RÝRÞ(s3lib/python2.7/site-packages/sympy/matrices/dense.pyRL;RM(tdenseR%R6RÖR RTtdet(RÞRÝtzeroR%R¥RB((RÝRÞs3lib/python2.7/site-packages/sympy/matrices/dense.pyt casoratians cO s ddlm}|j||ŽS(s`Create square identity matrix n x n See Also ======== diag zeros ones i(R%(RßR%R†(RžR{R%((s3lib/python2.7/site-packages/sympy/matrices/dense.pyR†Bs c sEddlm‰‡fd†‰‡fd†|Dƒ}ˆj||ŽS(sÅCreate a sparse, diagonal matrix from a list of diagonal values. Notes ===== When arguments are matrices they are fitted in resultant matrix. The returned matrix is a mutable, dense matrix. To make it a different type, send the desired class for keyword ``cls``. Examples ======== >>> from sympy.matrices import diag, Matrix, ones >>> diag(1, 2, 3) Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> diag(*[1, 2, 3]) Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) The diagonal elements can be matrices; diagonal filling will continue on the diagonal from the last element of the matrix: >>> from sympy.abc import x, y, z >>> a = Matrix([x, y, z]) >>> b = Matrix([[1, 2], [3, 4]]) >>> c = Matrix([[5, 6]]) >>> diag(a, 7, b, c) Matrix([ [x, 0, 0, 0, 0, 0], [y, 0, 0, 0, 0, 0], [z, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0], [0, 0, 1, 2, 0, 0], [0, 0, 3, 4, 0, 0], [0, 0, 0, 0, 5, 6]]) When diagonal elements are lists, they will be treated as arguments to Matrix: >>> diag([1, 2, 3], 4) Matrix([ [1, 0], [2, 0], [3, 0], [0, 4]]) >>> diag([[1, 2, 3]], 4) Matrix([ [1, 2, 3, 0], [0, 0, 0, 4]]) A given band off the diagonal can be made by padding with a vertical or horizontal "kerning" vector: >>> hpad = ones(0, 2) >>> vpad = ones(2, 0) >>> diag(vpad, 1, 2, 3, hpad) + diag(hpad, 4, 5, 6, vpad) Matrix([ [0, 0, 4, 0, 0], [0, 0, 0, 5, 0], [1, 0, 0, 0, 6], [0, 2, 0, 0, 0], [0, 0, 3, 0, 0]]) The type is mutable by default but can be made immutable by setting the ``mutable`` flag to False: >>> type(diag(1)) >>> from sympy.matrices import ImmutableMatrix >>> type(diag(1, cls=ImmutableMatrix)) See Also ======== eye i(R%c s*t|ƒr&t|tƒ r&ˆ|ƒS|S(N(RR$R(RÉ(R%(s3lib/python2.7/site-packages/sympy/matrices/dense.pyt normalize¬s c3 s|]}ˆ|ƒVqdS(N((RARÉ(Rã(s3lib/python2.7/site-packages/sympy/matrices/dense.pys °s(RßR%Rk(tvaluesR{((R%Rãs3lib/python2.7/site-packages/sympy/matrices/dense.pyRkQsWcC sËg}t|ƒ}xxt|ƒD]j}||}x/t|ƒD]!}|||j||ƒ8}q<W|jƒs|tdƒ‚n|j|ƒqW|rÇx1tt|ƒƒD]}||jƒ||} x5t | d|ƒD] } || | f|| | f>> from sympy import Function, hessian, pprint >>> from sympy.abc import x, y >>> f = Function('f')(x, y) >>> g1 = Function('g')(x, y) >>> g2 = x**2 + 3*y >>> pprint(hessian(f, (x, y), [g1, g2])) [ d d ] [ 0 0 --(g(x, y)) --(g(x, y)) ] [ dx dy ] [ ] [ 0 0 2*x 3 ] [ ] [ 2 2 ] [d d d ] [--(g(x, y)) 2*x ---(f(x, y)) -----(f(x, y))] [dx 2 dy dx ] [ dx ] [ ] [ 2 2 ] [d d d ] [--(g(x, y)) 3 -----(f(x, y)) ---(f(x, y)) ] [dy dy dx 2 ] [ dy ] References ========== https://en.wikipedia.org/wiki/Hessian_matrix See Also ======== sympy.matrices.mutable.Matrix.jacobian wronskian is)`varlist` must be a column or row vector.is `len(varlist)` must not be zero.s*Improper variable list in hessian functiontdiffs'Function `f` (%s) is not differentiable(R$RR RR.tTRŽRRTR3R!RDRÄRRì( R¥tvarlistt constraintsRÝRÉtNRéRBtgR:R;((s3lib/python2.7/site-packages/sympy/matrices/dense.pythessianËs8,       ):"cC s&ddlm}|jd|d|ƒS(sû Create a Jordan block: Examples ======== >>> from sympy.matrices import jordan_cell >>> from sympy.abc import x >>> jordan_cell(x, 4) Matrix([ [x, 1, 0, 0], [0, x, 1, 0], [0, 0, x, 1], [0, 0, 0, x]]) i(R%tsizet eigenvalue(RßR%t jordan_block(teigenvalRÝR%((s3lib/python2.7/site-packages/sympy/matrices/dense.pyt jordan_cellscC s |j|ƒS(sReturn the Hadamard product (elementwise product) of A and B >>> from sympy.matrices import matrix_multiply_elementwise >>> from sympy.matrices import Matrix >>> A = Matrix([[0, 1, 2], [3, 4, 5]]) >>> B = Matrix([[1, 10, 100], [100, 10, 1]]) >>> matrix_multiply_elementwise(A, B) Matrix([ [ 0, 10, 200], [300, 40, 5]]) See Also ======== __mul__ (tmultiply_elementwise(tAtB((s3lib/python2.7/site-packages/sympy/matrices/dense.pytmatrix_multiply_elementwise-scO sBd|kr"|jdƒ|d>> from sympy.matrices import randMatrix >>> randMatrix(3) # doctest:+SKIP [25, 45, 27] [44, 54, 9] [23, 96, 46] >>> randMatrix(3, 2) # doctest:+SKIP [87, 29] [23, 37] [90, 26] >>> randMatrix(3, 3, 0, 2) # doctest:+SKIP [0, 2, 0] [2, 0, 1] [0, 0, 1] >>> randMatrix(3, symmetric=True) # doctest:+SKIP [85, 26, 29] [26, 71, 43] [29, 43, 57] >>> A = randMatrix(3, seed=1) >>> B = randMatrix(3, seed=2) >>> A == B # doctest:+SKIP False >>> A == randMatrix(3, seed=1) True >>> randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP [77, 70, 0], [70, 0, 0], [ 0, 0, 88] c sˆjˆˆƒS(N(trandint(R:R;(tmaxtmintprng(s3lib/python2.7/site-packages/sympy/matrices/dense.pyRLŠRMids4For symmetric matrices, r must equal c, but %i != %iN(R"trandomtRandomR%RGtintR R[R&tshuffleR3RDRtsampleRTRÿ(R}RüRRtseedt symmetrictpercentRRÉtzR:R;tijR>((RRRs3lib/python2.7/site-packages/sympy/matrices/dense.pyt randMatrixTs*0  $   ; )"tbareissc s‹ddlm}x1tdtˆƒƒD]}tˆ|ƒˆ|sT   ÿ| ÿs    + + +M ,  d  M   L !