a RG5d@sdZddlmZddlmZmZddlm Z ddl m Z ddl m Z dd lmZmZmZmZmZmZdd lmZdd lmZdd lmZdd lmZmZddZGdddZdS)a Module for the DomainMatrix class. A DomainMatrix represents a matrix with elements that are in a particular Domain. Each DomainMatrix internally wraps a DDM which is used for the lower-level operations. The idea is that the DomainMatrix class provides the convenience routines for converting between Expr and the poly domains as well as unifying matrices with different domains. )reduce)UnionTuple_sympify)Domainconstruct_domain)DMNonSquareMatrixError DMShapeError DMDomainError DMFormatErrorDMBadInputError DMNotAField)DDM)SDM) DomainScalar)ZZEXRAWcCs t||S)aConvenient alias for DomainMatrix.from_list Examples ======= >>> from sympy import ZZ >>> from sympy.polys.matrices import DM >>> DM([[1, 2], [3, 4]], ZZ) DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ) See also ======= DomainMatrix.from_list ) DomainMatrix from_list)rowsdomainr]/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/matrices/domainmatrix.pyDM"srcseZdZUdZeeefed<ee e fed<e ed<ddddZ d d Z d d Z d dZddZddZefddZeddZeddZeddZedddZeddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+Zed,d-Zeddd.d/Zddd0d1Z d2d3Z!d4d5Z"d6d7Z#d8d9Z$d:d;Z%dd?Z'e(d@dAZ)e(dBdCZ*e(dDdEZ+e(dFdGZ,dHdIZ-dJdKZ.dLdMZ/ddNdOZ0dPdQZ1dRdSZ2dTdUZ3dVdWZ4dXdYZ5dZd[Z6d\d]Z7d^d_Z8d`daZ9dbdcZ:dddeZ;dfdgZdldmZ?dndoZ@dpdqZAdrdsZBdtduZCdvdwZDdxdyZEdzd{ZFd|d}ZGd~dZHddZIddZJddZKddZLddZMddZNeddZOedddZPeddddZQeddZRddZSddZTZUS)ra Associate Matrix with :py:class:`~.Domain` Explanation =========== DomainMatrix uses :py:class:`~.Domain` for its internal representation which makes it more faster for many common operations than current SymPy Matrix class, but this advantage makes it not entirely compatible with Matrix. DomainMatrix could be found analogous to numpy arrays with "dtype". In the DomainMatrix, each matrix has a domain such as :ref:`ZZ` or :ref:`QQ(a)`. Examples ======== Creating a DomainMatrix from the existing Matrix class: >>> from sympy import Matrix >>> from sympy.polys.matrices import DomainMatrix >>> Matrix1 = Matrix([ ... [1, 2], ... [3, 4]]) >>> A = DomainMatrix.from_Matrix(Matrix1) >>> A DomainMatrix({0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}, (2, 2), ZZ) Driectly forming a DomainMatrix: >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> A DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ) See Also ======== DDM SDM Domain Poly repshaperNfmtcCst|ttfrtdnt|tr>|j|||ntdSr/)rof_typer&r%intrsetitemNotImplementedError)r1r7valuer8r9rrr __setitem__s  zDomainMatrix.__setitem__cs<t|ttfstdt|}||_|j|_|j|_|S)a Create a new DomainMatrix efficiently from DDM/SDM. Examples ======== Create a :py:class:`~.DomainMatrix` with an dense internal representation as :py:class:`~.DDM`: >>> from sympy.polys.domains import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.ddm import DDM >>> drep = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> dM = DomainMatrix.from_rep(drep) >>> dM DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ) Create a :py:class:`~.DomainMatrix` with a sparse internal representation as :py:class:`~.SDM`: >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import ZZ >>> drep = SDM({0:{1:ZZ(1)},1:{0:ZZ(2)}}, (2, 2), ZZ) >>> dM = DomainMatrix.from_rep(drep) >>> dM DomainMatrix({0: {1: 1}, 1: {0: 2}}, (2, 2), ZZ) Parameters ========== rep: SDM or DDM The internal sparse or dense representation of the matrix. Returns ======= DomainMatrix A :py:class:`~.DomainMatrix` wrapping *rep*. Notes ===== This takes ownership of rep as its internal representation. If rep is being mutated elsewhere then a copy should be provided to ``from_rep``. Only minimal verification or checking is done on *rep* as this is supposed to be an efficient internal routine. z rep should be of type DDM or SDM) r%rrr&superr.rrr)r,rr1 __class__rrr+s2 zDomainMatrix.from_repcsJt|}|sdn t|d}fddfdd|D}t|||fS)aS Convert a list of lists into a DomainMatrix Parameters ========== rows: list of lists Each element of the inner lists should be either the single arg, or tuple of args, that would be passed to the domain constructor in order to form an element of the domain. See examples. Returns ======= DomainMatrix containing elements defined in rows Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy import FF, QQ, ZZ >>> A = DomainMatrix.from_list([[1, 0, 1], [0, 0, 1]], ZZ) >>> A DomainMatrix([[1, 0, 1], [0, 0, 1]], (2, 3), ZZ) >>> B = DomainMatrix.from_list([[1, 0, 1], [0, 0, 1]], FF(7)) >>> B DomainMatrix([[1 mod 7, 0 mod 7, 1 mod 7], [0 mod 7, 0 mod 7, 1 mod 7]], (2, 3), GF(7)) >>> C = DomainMatrix.from_list([[(1, 2), (3, 1)], [(1, 4), (5, 1)]], QQ) >>> C DomainMatrix([[1/2, 3], [1/4, 5]], (2, 2), QQ) See Also ======== from_list_sympy rcst|tr|S|Sr/)r%tuple)errr z(DomainMatrix.from_list..csg|]}fdd|DqS)csg|] }|qSrr).0rJconvrr !rMz5DomainMatrix.from_list...rrNrowrOrrrQ!rMz*DomainMatrix.from_list..)lenr)r,rrnrowsncols domain_rowsr)rPrrrs ' zDomainMatrix.from_listc svt||ksJtfdd|Ds*Jdd|D}|j|fi|\}fddt|D}t||f|S)a Convert a list of lists of Expr into a DomainMatrix using construct_domain Parameters ========== nrows: number of rows ncols: number of columns rows: list of lists Returns ======= DomainMatrix containing elements of rows Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.abc import x, y, z >>> A = DomainMatrix.from_list_sympy(1, 3, [[x, y, z]]) >>> A DomainMatrix([[x, y, z]], (1, 3), ZZ[x,y,z]) See Also ======== sympy.polys.constructor.construct_domain, from_dict_sympy c3s|]}t|kVqdSr/)rTrRrVrr FrMz/DomainMatrix.from_list_sympy..cSsg|]}|D] }t|q qSrrrNrSitemrrrrQHrMz0DomainMatrix.from_list_sympy..cs&g|]fddtDqS)csg|]}|qSrr)rNc) items_domainrVrrrrQLrMz;DomainMatrix.from_list_sympy...)range)rNr]rV)r^rrQLrM)rTall get_domainr_r)r,rUrVrkwargs items_sympyrrWrr`rfrom_list_sympy%s zDomainMatrix.from_list_sympyc  stfdd|Dstdtfdd|Ds@tddd|D}|j|fi|\}}d}i} |D]2\} } i| | <| D]} ||| | | <|d 7}qqxt| f|S) a Parameters ========== nrows: number of rows ncols: number of cols elemsdict: dict of dicts containing non-zero elements of the DomainMatrix Returns ======= DomainMatrix containing elements of elemsdict Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.abc import x,y,z >>> elemsdict = {0: {0:x}, 1:{1: y}, 2: {2: z}} >>> A = DomainMatrix.from_dict_sympy(3, 3, elemsdict) >>> A DomainMatrix({0: {0: x}, 1: {1: y}, 2: {2: z}}, (3, 3), ZZ[x,y,z]) See Also ======== from_list_sympy c3s&|]}d|koknVqdSrNr)rNr^)rUrrrYprMz/DomainMatrix.from_dict_sympy..zRow out of rangec3s0|](}|D]}d|ko knVq qdSrfr)rNrSr\rXrrrYrrMzColumn out of rangecSs"g|]}|D] }t|qqSr)valuesrrZrrrrQurMz0DomainMatrix.from_dict_sympy..rr )rarrgrbitemsr) r,rUrVZ elemsdictrcrdrr]idxZ items_dictr8rSr9r)rVrUrfrom_dict_sympyPs  zDomainMatrix.from_dict_sympyr"cKsH|dkr(|jg|j|Ri|S|jg|j|Ri|S)a Convert Matrix to DomainMatrix Parameters ========== M: Matrix Returns ======= Returns DomainMatrix with identical elements as M Examples ======== >>> from sympy import Matrix >>> from sympy.polys.matrices import DomainMatrix >>> M = Matrix([ ... [1.0, 3.4], ... [2.4, 1]]) >>> A = DomainMatrix.from_Matrix(M) >>> A DomainMatrix({0: {0: 1.0, 1: 3.4}, 1: {0: 2.4, 1: 1.0}}, (2, 2), RR) We can keep internal representation as ddm using fmt='dense' >>> from sympy import Matrix, QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix.from_Matrix(Matrix([[QQ(1, 2), QQ(3, 4)], [QQ(0, 1), QQ(0, 1)]]), fmt='dense') >>> A.rep [[1/2, 3/4], [0, 0]] See Also ======== Matrix r#)rertolistrjZtodod)r,Mr!rcrrr from_Matrixs( zDomainMatrix.from_MatrixcKst|fi|\}}||fSr/r )r,rdrcKZitems_KrrrrbszDomainMatrix.get_domaincCs||jSr/)r+rcopyr1rrrroszDomainMatrix.copycCs"|dur|S||j|S)a Change the domain of DomainMatrix to desired domain or field Parameters ========== K : Represents the desired domain or field. Alternatively, ``None`` may be passed, in which case this method just returns a copy of this DomainMatrix. Returns ======= DomainMatrix DomainMatrix with the desired domain or field Examples ======== >>> from sympy import ZZ, ZZ_I >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> A.convert_to(ZZ_I) DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ_I) N)ror+r convert_tor1rnrrrrqszDomainMatrix.convert_tocCs |tSr/)rqrrprrrr=szDomainMatrix.to_sympycCs|j}||S)a Returns a DomainMatrix with the appropriate field Returns ======= DomainMatrix DomainMatrix with the appropriate field Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> A.to_field() DomainMatrix([[1, 2], [3, 4]], (2, 2), QQ) )r get_fieldrqrrrrrto_fields zDomainMatrix.to_fieldcCs"|jjdkr|S|t|jS)a{ Return a sparse DomainMatrix representation of *self*. Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy import QQ >>> A = DomainMatrix([[1, 0],[0, 2]], (2, 2), QQ) >>> A.rep [[1, 0], [0, 2]] >>> B = A.to_sparse() >>> B.rep {0: {0: 1}, 1: {1: 2}} r")rr!r+rZfrom_ddmrprrr to_sparses zDomainMatrix.to_sparsecCs"|jjdkr|S|t|jS)a Return a dense DomainMatrix representation of *self*. Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy import QQ >>> A = DomainMatrix({0: {0: 1}, 1: {1: 2}}, (2, 2), QQ) >>> A.rep {0: {0: 1}, 1: {1: 2}} >>> B = A.to_dense() >>> B.rep [[1, 0], [0, 2]] r#)rr!r+rr)rprrrto_dense s zDomainMatrix.to_densecsBdd|D}t|dkr|Stdd|tfdd|DS)z#Convert matrices to a common domaincSsh|] }|jqSrrKrNmatrixrrr $rMz-DomainMatrix._unify_domain..r cSs ||Sr/)unify)xyrrrrL'rMz,DomainMatrix._unify_domain..c3s|]}|VqdSr/)rqrwrKrrrY(rMz-DomainMatrix._unify_domain..)rTrrI)r,matricesdomainsrrKr _unify_domain!s  zDomainMatrix._unify_domaincGs^dd|D}t|dkr|S|dkr8tdd|DS|dkrRtdd|DStd d S) zConvert matrices to the same format. If all matrices have the same format, then return unmodified. Otherwise convert both to the preferred format given as *fmt* which should be 'dense' or 'sparse'. cSsh|] }|jjqSr)rr!rwrrrry2rMz*DomainMatrix._unify_fmt..r r"css|]}|VqdSr/)rurwrrrrY6rMz*DomainMatrix._unify_fmt..r#css|]}|VqdSr/)rvrwrrrrY8rMr$N)rTrIr*)r,r!r}formatsrrr _unify_fmt*s zDomainMatrix._unify_fmtcGs0|f|}tj|}|dur,tj|d|i}|S)aL Unifies the domains and the format of self and other matrices. Parameters ========== others : DomainMatrix fmt: string 'dense', 'sparse' or `None` (default) The preferred format to convert to if self and other are not already in the same format. If `None` or not specified then no conversion if performed. Returns ======= Tuple[DomainMatrix] Matrices with unified domain and format Examples ======== Unify the domain of DomainMatrix that have different domains: >>> from sympy import ZZ, QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ) >>> B = DomainMatrix([[QQ(1, 2), QQ(2)]], (1, 2), QQ) >>> Aq, Bq = A.unify(B) >>> Aq DomainMatrix([[1, 2]], (1, 2), QQ) >>> Bq DomainMatrix([[1/2, 2]], (1, 2), QQ) Unify the format (dense or sparse): >>> A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ) >>> B = DomainMatrix({0:{0: ZZ(1)}}, (2, 2), ZZ) >>> B.rep {0: {0: 1}} >>> A2, B2 = A.unify(B, fmt='dense') >>> B2.rep [[1, 0], [0, 0]] See Also ======== convert_to, to_dense, to_sparse Nr!)rrr)r1r!othersr}rrrrz<s 5  zDomainMatrix.unifycs<ddlm}j}fdd|D}|gj|RS)a  Convert DomainMatrix to Matrix Returns ======= Matrix MutableDenseMatrix for the DomainMatrix Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> A.to_Matrix() Matrix([ [1, 2], [3, 4]]) See Also ======== from_Matrix r)MutableDenseMatrixcs"g|]}|D]}j|q qSr)rr=)rNrSrJrprrrQrMz*DomainMatrix.to_Matrix..)sympy.matrices.denserrto_listr)r1rZelemlistZelements_sympyrrpr to_Matrixws  zDomainMatrix.to_MatrixcCs |jSr/)rrrprrrrszDomainMatrix.to_listcCs |jSr/)r to_list_flatrprrrrszDomainMatrix.to_list_flatcCs |jSr/)rto_dokrprrrrszDomainMatrix.to_dokcCsdt|j|j|jfS)NzDomainMatrix(%s, %r, %r))strrrrrprrr__repr__szDomainMatrix.__repr__cCs||jS)zMatrix transpose of ``self``)r+r transposerprrrrszDomainMatrix.transposecs"j\}fddt|DS)Ncs(g|] }tD]}||fjqqSr)r_element)rNr8r9colsr1rrrQrMz%DomainMatrix.flat..)rr_)r1rrrrflats zDomainMatrix.flatcCs |jSr/)ris_zero_matrixrprrrrszDomainMatrix.is_zero_matrixcCs |jS)z~ Says whether this matrix is upper-triangular. True can be returned even if the matrix is not square. )ris_upperrprrrrszDomainMatrix.is_uppercCs |jS)z~ Says whether this matrix is lower-triangular. True can be returned even if the matrix is not square. )ris_lowerrprrrrszDomainMatrix.is_lowercCs|jd|jdkS)Nrr )rrprrr is_squareszDomainMatrix.is_squarecCs|\}}t|Sr/)rrefrT)r1rpivotsrrrranks zDomainMatrix.rankcGs0|j|ddi^}}t|jjdd|DS)a\Horizontally stack the given matrices. Parameters ========== B: DomainMatrix Matrices to stack horizontally. Returns ======= DomainMatrix DomainMatrix by stacking horizontally. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> B = DomainMatrix([[ZZ(5), ZZ(6)], [ZZ(7), ZZ(8)]], (2, 2), ZZ) >>> A.hstack(B) DomainMatrix([[1, 2, 5, 6], [3, 4, 7, 8]], (2, 4), ZZ) >>> C = DomainMatrix([[ZZ(9), ZZ(10)], [ZZ(11), ZZ(12)]], (2, 2), ZZ) >>> A.hstack(B, C) DomainMatrix([[1, 2, 5, 6, 9, 10], [3, 4, 7, 8, 11, 12]], (2, 6), ZZ) See Also ======== unify r!r#css|] }|jVqdSr/rrNZBkrrrrYrMz&DomainMatrix.hstack..)rzrr+rhstackABrrrrs#zDomainMatrix.hstackcGs0|j|ddi^}}t|jjdd|DS)abVertically stack the given matrices. Parameters ========== B: DomainMatrix Matrices to stack vertically. Returns ======= DomainMatrix DomainMatrix by stacking vertically. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> B = DomainMatrix([[ZZ(5), ZZ(6)], [ZZ(7), ZZ(8)]], (2, 2), ZZ) >>> A.vstack(B) DomainMatrix([[1, 2], [3, 4], [5, 6], [7, 8]], (4, 2), ZZ) >>> C = DomainMatrix([[ZZ(9), ZZ(10)], [ZZ(11), ZZ(12)]], (2, 2), ZZ) >>> A.vstack(B, C) DomainMatrix([[1, 2], [3, 4], [5, 6], [7, 8], [9, 10], [11, 12]], (6, 2), ZZ) See Also ======== unify r!r#css|] }|jVqdSr/rrrrrrYrMz&DomainMatrix.vstack..)rzrr+rvstackrrrrrs#zDomainMatrix.vstackcCs"|dur|j}||j||Sr/)rr+r applyfunc)r1funcrrrrrszDomainMatrix.applyfunccCs*t|tstS|j|dd\}}||SNr#r )r%rNotImplementedrzaddrrrr__add__s zDomainMatrix.__add__cCs*t|tstS|j|dd\}}||Sr)r%rrrzsubrrrr__sub__!s zDomainMatrix.__sub__cCs|Sr/)negrrrr__neg__'szDomainMatrix.__neg__cCsft|tr&|j|dd\}}||S||jvr:||St|tr^||\}}||jStSdS)zA * Br#r N) r%rrzmatmulr scalarmulrrrrrrr__mul__*s      zDomainMatrix.__mul__cCs@||jvr||St|tr8||\}}||jStSdSr/)r rscalarmulr%rrzrrrrrr__rmul__7s     zDomainMatrix.__rmul__cCst|tstS||S)zA ** n)r%rArpow)rr;rrr__pow__@s zDomainMatrix.__pow__cCsz|j|jkr&d|j||jf}t|||krHd|j||jf}t||jj|jjkrvd|jj||jjf}t|dS)NzDomain mismatch: %s %s %szShape mismatch: %s %s %szFormat mismatch: %s %s %s)rrrr rr!r)aopbashapeZbshaper-rrr_checkFs zDomainMatrix._checkcCs(|d||j|j||j|jS)a Adds two DomainMatrix matrices of the same Domain Parameters ========== A, B: DomainMatrix matrices to add Returns ======= DomainMatrix DomainMatrix after Addition Raises ====== DMShapeError If the dimensions of the two DomainMatrix are not equal ValueError If the domain of the two DomainMatrix are not same Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> B = DomainMatrix([ ... [ZZ(4), ZZ(3)], ... [ZZ(2), ZZ(1)]], (2, 2), ZZ) >>> A.add(B) DomainMatrix([[5, 5], [5, 5]], (2, 2), ZZ) See Also ======== sub, matmul +)rrr+rrrrrrrQs.zDomainMatrix.addcCs(|d||j|j||j|jS)a Subtracts two DomainMatrix matrices of the same Domain Parameters ========== A, B: DomainMatrix matrices to substract Returns ======= DomainMatrix DomainMatrix after Substraction Raises ====== DMShapeError If the dimensions of the two DomainMatrix are not equal ValueError If the domain of the two DomainMatrix are not same Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> B = DomainMatrix([ ... [ZZ(4), ZZ(3)], ... [ZZ(2), ZZ(1)]], (2, 2), ZZ) >>> A.sub(B) DomainMatrix([[-3, -1], [1, 3]], (2, 2), ZZ) See Also ======== add, matmul -)rrr+rrrrrrrs.zDomainMatrix.subcCs||jS)a Returns the negative of DomainMatrix Parameters ========== A : Represents a DomainMatrix Returns ======= DomainMatrix DomainMatrix after Negation Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> A.neg() DomainMatrix([[-1, -2], [-3, -4]], (2, 2), ZZ) )r+rrrrrrrszDomainMatrix.negcCs||j|S)aW Performs term by term multiplication for the second DomainMatrix w.r.t first DomainMatrix. Returns a DomainMatrix whose rows are list of DomainMatrix matrices created after term by term multiplication. Parameters ========== A, B: DomainMatrix matrices to multiply term-wise Returns ======= DomainMatrix DomainMatrix after term by term multiplication Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> B = DomainMatrix([ ... [ZZ(1), ZZ(1)], ... [ZZ(0), ZZ(1)]], (2, 2), ZZ) >>> A.mul(B) DomainMatrix([[DomainMatrix([[1, 1], [0, 1]], (2, 2), ZZ), DomainMatrix([[2, 2], [0, 2]], (2, 2), ZZ)], [DomainMatrix([[3, 3], [0, 3]], (2, 2), ZZ), DomainMatrix([[4, 4], [0, 4]], (2, 2), ZZ)]], (2, 2), ZZ) See Also ======== matmul )r+rmulrrrrrrs*zDomainMatrix.mulcCs||j|Sr/)r+rrmulrrrrrszDomainMatrix.rmulcCs0|d||jd|jd||j|jS)a Performs matrix multiplication of two DomainMatrix matrices Parameters ========== A, B: DomainMatrix to multiply Returns ======= DomainMatrix DomainMatrix after multiplication Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> B = DomainMatrix([ ... [ZZ(1), ZZ(1)], ... [ZZ(0), ZZ(1)]], (2, 2), ZZ) >>> A.matmul(B) DomainMatrix([[1, 3], [3, 7]], (2, 2), ZZ) See Also ======== mul, pow, add, sub *r r)rrr+rrrrrrrs&zDomainMatrix.matmulcCsL||jjkrt|j|jS||jjkr0|S|r>||S||SdSr/) rzerorzerosronerorr)rlamdareverserrr _scalarmul*s   zDomainMatrix._scalarmulcCs|j|ddS)NFrrrrrrrr4szDomainMatrix.scalarmulcCs|j|ddS)NTrrrrrrr7szDomainMatrix.rscalarmulcCs$|j|jksJ||j|jSr/)rr+rmul_elementwiserrrrr:szDomainMatrix.mul_elementwisecCszt|tst|r"tt|t}t|ts0tS||\}}|j|j j krTt |j|j j krj|S| d|jS)z Method for Scalar Divisonr )r%rArr@rrrtrzrrrZeroDivisionErrorrrrrrr __truediv__>s zDomainMatrix.__truediv__cCs|j\}}||krtd|dkr,tdnR|dkrB|||jS|dkrN|S|ddkrj|||dS||d}||SdS)a Computes A**n Parameters ========== A : DomainMatrix n : exponent for A Returns ======= DomainMatrix DomainMatrix on computing A**n Raises ====== NotImplementedError if n is negative. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(1)], ... [ZZ(0), ZZ(1)]], (2, 2), ZZ) >>> A.pow(2) DomainMatrix([[1, 2], [0, 1]], (2, 2), ZZ) See Also ======== matmul zPower of a nonsquare matrixrzNegative powersr rN)rr rCeyer)rr;rUrVZsqrtAnrrrrNs)    zDomainMatrix.powcCs |j\}}||ksJ|jS)azCompute the strongly connected components of a DomainMatrix Explanation =========== A square matrix can be considered as the adjacency matrix for a directed graph where the row and column indices are the vertices. In this graph if there is an edge from vertex ``i`` to vertex ``j`` if ``M[i, j]`` is nonzero. This routine computes the strongly connected components of that graph which are subsets of the rows and columns that are connected by some nonzero element of the matrix. The strongly connected components are useful because many operations such as the determinant can be computed by working with the submatrices corresponding to each component. Examples ======== Find the strongly connected components of a matrix: >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> M = DomainMatrix([[ZZ(1), ZZ(0), ZZ(2)], ... [ZZ(0), ZZ(3), ZZ(0)], ... [ZZ(4), ZZ(6), ZZ(5)]], (3, 3), ZZ) >>> M.scc() [[1], [0, 2]] Compute the determinant from the components: >>> MM = M.to_Matrix() >>> MM Matrix([ [1, 0, 2], [0, 3, 0], [4, 6, 5]]) >>> MM[[1], [1]] Matrix([[3]]) >>> MM[[0, 2], [0, 2]] Matrix([ [1, 2], [4, 5]]) >>> MM.det() -9 >>> MM[[1], [1]].det() * MM[[0, 2], [0, 2]].det() -9 The components are given in reverse topological order and represent a permutation of the rows and columns that will bring the matrix into block lower-triangular form: >>> MM[[1, 0, 2], [1, 0, 2]] Matrix([ [3, 0, 0], [0, 1, 2], [6, 4, 5]]) Returns ======= List of lists of integers Each list represents a strongly connected component. See also ======== sympy.matrices.matrices.MatrixBase.strongly_connected_components sympy.utilities.iterables.strongly_connected_components )rrscc)r1rrrrrrsG  zDomainMatrix.scccCs0|jjstd|j\}}||t|fS)ae Returns reduced-row echelon form and list of pivots for the DomainMatrix Returns ======= (DomainMatrix, list) reduced-row echelon form and list of pivots for the DomainMatrix Raises ====== ValueError If the domain of DomainMatrix not a Field Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [QQ(2), QQ(-1), QQ(0)], ... [QQ(-1), QQ(2), QQ(-1)], ... [QQ(0), QQ(0), QQ(2)]], (3, 3), QQ) >>> rref_matrix, rref_pivots = A.rref() >>> rref_matrix DomainMatrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], (3, 3), QQ) >>> rref_pivots (0, 1, 2) See Also ======== convert_to, lu Not a field)ris_Fieldrrrr+rI)r1Zrref_ddmrrrrrs&zDomainMatrix.rrefcCs6|jjstd|\}}|j\}}|t||S)a Returns the columnspace for the DomainMatrix Returns ======= DomainMatrix The columns of this matrix form a basis for the columnspace. Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [QQ(1), QQ(-1)], ... [QQ(2), QQ(-2)]], (2, 2), QQ) >>> A.columnspace() DomainMatrix([[1], [2]], (2, 1), QQ) r)rrrrrr?r_r1rrrrrrr columnspaces   zDomainMatrix.columnspacecCs>|jjstd|\}}|j\}}|tt|t|S)a Returns the rowspace for the DomainMatrix Returns ======= DomainMatrix The rows of this matrix form a basis for the rowspace. Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [QQ(1), QQ(-1)], ... [QQ(2), QQ(-2)]], (2, 2), QQ) >>> A.rowspace() DomainMatrix([[1, -1]], (1, 2), QQ) r)rrrrrr?r_rTrrrrrowspaces   zDomainMatrix.rowspacecCs$|jjstd||jdS)a Returns the nullspace for the DomainMatrix Returns ======= DomainMatrix The rows of this matrix form a basis for the nullspace. Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [QQ(1), QQ(-1)], ... [QQ(2), QQ(-2)]], (2, 2), QQ) >>> A.nullspace() DomainMatrix([[1, 1]], (1, 2), QQ) rr)rrrr+r nullspacerprrrr4szDomainMatrix.nullspacecCs:|jjstd|j\}}||kr&t|j}||S)a# Finds the inverse of the DomainMatrix if exists Returns ======= DomainMatrix DomainMatrix after inverse Raises ====== ValueError If the domain of DomainMatrix not a Field DMNonSquareMatrixError If the DomainMatrix is not a not Square DomainMatrix Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [QQ(2), QQ(-1), QQ(0)], ... [QQ(-1), QQ(2), QQ(-1)], ... [QQ(0), QQ(0), QQ(2)]], (3, 3), QQ) >>> A.inv() DomainMatrix([[2/3, 1/3, 1/6], [1/3, 2/3, 1/3], [0, 0, 1/2]], (3, 3), QQ) See Also ======== neg r)rrrrr rinvr+)r1r:r;rrrrrNs%  zDomainMatrix.invcCs |j\}}||krt|jS)a Returns the determinant of a Square DomainMatrix Returns ======= S.Complexes determinant of Square DomainMatrix Raises ====== ValueError If the domain of DomainMatrix not a Field Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> A.det() -2 )rr rdetr1r:r;rrrr{s zDomainMatrix.detcCs6|jjstd|j\}}}|||||fS)a% Returns Lower and Upper decomposition of the DomainMatrix Returns ======= (L, U, exchange) L, U are Lower and Upper decomposition of the DomainMatrix, exchange is the list of indices of rows exchanged in the decomposition. Raises ====== ValueError If the domain of DomainMatrix not a Field Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [QQ(1), QQ(-1)], ... [QQ(2), QQ(-2)]], (2, 2), QQ) >>> A.lu() (DomainMatrix([[1, 0], [2, 1]], (2, 2), QQ), DomainMatrix([[1, -1], [0, 0]], (2, 2), QQ), []) See Also ======== lu_solve r)rrrrlur+)r1LUZswapsrrrrs"zDomainMatrix.lucCsD|jd|jdkrtd|jjs,td|j|j}||S)am Solver for DomainMatrix x in the A*x = B Parameters ========== rhs : DomainMatrix B Returns ======= DomainMatrix x in A*x = B Raises ====== DMShapeError If the DomainMatrix A and rhs have different number of rows ValueError If the domain of DomainMatrix A not a Field Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [QQ(1), QQ(2)], ... [QQ(3), QQ(4)]], (2, 2), QQ) >>> B = DomainMatrix([ ... [QQ(1), QQ(1)], ... [QQ(0), QQ(1)]], (2, 2), QQ) >>> A.lu_solve(B) DomainMatrix([[-2, -1], [3/2, 1]], (2, 2), QQ) See Also ======== lu rShaper)rr rrrrlu_solver+)r1rhssolrrrrs -zDomainMatrix.lu_solvec Cs|jd|jdkrtd|j|jks0|jjs8td||}|\}}||j }|ddddfj \}}||}||fS)Nrrr) rr rrrrrr+r particularr) rrZAaugZArrefrrZ nullspace_repZ nonpivotsrrrr_solves   zDomainMatrix._solvecCs$|j\}}||krtd|jS)a Returns the coefficients of the characteristic polynomial of the DomainMatrix. These elements will be domain elements. The domain of the elements will be same as domain of the DomainMatrix. Returns ======= list coefficients of the characteristic polynomial Raises ====== DMNonSquareMatrixError If the DomainMatrix is not a not Square DomainMatrix Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> A.charpoly() [1, -5, -2] z not square)rr rcharpolyrrrrrs zDomainMatrix.charpolycCs$t|tr||f}|t||S)a Return identity matrix of size n Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy import QQ >>> DomainMatrix.eye(3, QQ) DomainMatrix({0: {0: 1}, 1: {1: 1}, 2: {2: 1}}, (3, 3), QQ) )r%rAr+rrr,rrrrrr*s zDomainMatrix.eyecCs,|durt|}||f}|t|||S)a3 Return diagonal matrix with entries from ``diagonal``. Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy import ZZ >>> DomainMatrix.diag([ZZ(5), ZZ(6)], ZZ) DomainMatrix({0: {0: 5}, 1: {1: 6}}, (2, 2), ZZ) N)rTr+rdiag)r,diagonalrrNrrrr<szDomainMatrix.diagcCs|t||S)a%Returns a zero DomainMatrix of size shape, belonging to the specified domain Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy import QQ >>> DomainMatrix.zeros((2, 3), QQ) DomainMatrix({}, (2, 3), QQ) )r+rr)r,rrr!rrrrOs zDomainMatrix.zeroscCs|t||S)a9Returns a DomainMatrix of 1s, of size shape, belonging to the specified domain Examples ======== >>> from sympy.polys.matrices import DomainMatrix >>> from sympy import QQ >>> DomainMatrix.ones((2,3), QQ) DomainMatrix([[1, 1, 1], [1, 1, 1]], (2, 3), QQ) )r+ronesrrrrr^s zDomainMatrix.onescCs*t|t|stS|j|jko(|j|jkS)a Checks for two DomainMatrix matrices to be equal or not Parameters ========== A, B: DomainMatrix to check equality Returns ======= Boolean True for equal, else False Raises ====== NotImplementedError If B is not a DomainMatrix Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> A = DomainMatrix([ ... [ZZ(1), ZZ(2)], ... [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> B = DomainMatrix([ ... [ZZ(1), ZZ(1)], ... [ZZ(0), ZZ(1)]], (2, 2), ZZ) >>> A.__eq__(A) True >>> A.__eq__(B) False )r%typerrrrrrr__eq__ms'zDomainMatrix.__eq__cCs2|j|jkrdS|j|jkr*||\}}||kS)NF)rrrzrrrrunify_eqs   zDomainMatrix.unify_eq)r")N)N)V__name__ __module__ __qualname____doc__tUnionrr__annotations__tTuplerArr.r3r<r>r?rE classmethodr+rrerjrmrbrorqr=rtrurvrrrzrrrrrrrpropertyrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr __classcell__rrrGrr5s 0&  : - * 1 , " ;#    &&    21,) 8K+-"'4$   +rN) r functoolsrtypingrrrrsympy.core.sympifyrr~r constructorr exceptionsr r rrrrZddmrZsdmrZ domainscalarrsympy.polys.domainsrrrrrrrrs