\h%SrSSKJrJrJrJrJr SSKJr SSK J r SSK J r SSK Jr SSKJrJrJr SS KJr SS KJr \ S :waS S /r"SS\5rSrSrSrSrSrSrSr Sr!Sr"Sr#Sr$Sr%g)z Module for the SDM class. )addnegpossubmul) defaultdict) GROUND_TYPES)doctest_depends_on)_strongly_connected_components)DMBadInputError DMDomainError DMShapeError)QQ)DDMflintz SDM.to_dfmzSDM.to_dfm_or_ddmc^\rSrSrSrSrSrSrU4SjrSr Sr Sr S r S r S r\S 5rS r\S5r\S5rSrSr\S5rSr\S5rSr\S5rSr\S5rSrSrSrSr \!"S/S9S5r"\!"S/S9S 5r#\S!5r$\S"5r%\S#5r&\SNS$j5r'S%r(S&r)S'r*S(r+S)r,S*r-S+r.S,r/S-r0S.r1S/r2S0r3S1r4S2r5S3r6S4r7S5r8S6r9S7r:S8r;S9rS<r?S=r@SNS>jrAS?rBS@rCSArDSBrESCrFSDrGSErHSFrISGrJSHrK\L"SISJ54SKjrM\L"SISJ54SLjrNSMrOU=rP$)OSDMacSparse matrix based on polys domain elements This is a dict subclass and is a wrapper for a dict of dicts that supports basic matrix arithmetic +, -, *, **. In order to create a new :py:class:`~.SDM`, a dict of dicts mapping non-zero elements to their corresponding row and column in the matrix is needed. We also need to specify the shape and :py:class:`~.Domain` of our :py:class:`~.SDM` object. We declare a 2x2 :py:class:`~.SDM` matrix belonging to QQ domain as shown below. The 2x2 Matrix in the example is .. math:: A = \left[\begin{array}{ccc} 0 & \frac{1}{2} \\ 0 & 0 \end{array} \right] >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> elemsdict = {0:{1:QQ(1, 2)}} >>> A = SDM(elemsdict, (2, 2), QQ) >>> A {0: {1: 1/2}} We can manipulate :py:class:`~.SDM` the same way as a Matrix class >>> from sympy import ZZ >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ) >>> B = SDM({0:{0: ZZ(3)}, 1:{1:ZZ(4)}}, (2, 2), ZZ) >>> A + B {0: {0: 3, 1: 2}, 1: {0: 1, 1: 4}} Multiplication >>> A*B {0: {1: 8}, 1: {0: 3}} >>> A*ZZ(2) {0: {1: 4}, 1: {0: 2}} sparseFc>^^[TU]U5 U=Ul=uUlUlummX0l[ U4SjU55(d [S5e[ U4SjUR555(d [S5eg)Nc3N># UHnSUs=:*=(a T:Os v M g7frN).0rms P/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/matrices/sdm.py SDM.__init__..Ss,t!1::A::ts"%zRow out of rangec3`># UH#oHnSUs=:*=(a T:Os v M M% g7frr)rrowcns rrr Us%D#11::A:::s+.zColumn out of range) super__init__shaperowscolsdomainallr values)self elemsdictr'r*rr$ __class__s @@rr& SDM.__init__Nsy #388 8)TY DAq ,t,,,!"45 5DDDD!"78 8EcXU$![as URup4U*Us=::aU:aKO OHU*Us=::aU:a:O O7XU-X$-s$![a URRss$f=f[ S5ef=fNzindex out of range)KeyErrorr'r*zero IndexError)r-ijrr$s rgetitem SDM.getitemXs 771:  7::DArQ{{rQ{{,A;qu--,;;+++,!!566 7s- :B AB A72B6A77Bc6URupEU*Us=::aU:aO OU*Us=::aU:d O [S5eX-X%-p!U(a X0UU'gURUS5nUbXb U(dX ggg![a X#0X'gf=f![a gf=fr3)r'r6r4get)r-r7r8valuerr$rowis rsetitem SDM.setitemeszza ! a ! 12 2uae1  %"Q 88At$D$  G   %* % s$ A6(B 6BB BBcURup4[U5Un[U5Un0nUR5HcupX;dM U R5V V s0sHupX;dM URU 5U _M n n n U (dMPXURU5'Me UR U[ U5[ U54UR 5$s sn n fN)r'rangeitemsindexnewlenr*) r-slice1slice2rr$ricisdmr7r"r8es r extract_sliceSDM.extract_slicezszz 1Xf  1Xf jjlFAw25))+I+$!~rxx{A~+I3'* $ # xxc"gs2w/== Js  C+CcU(aU(aU(d0UR[U5[U54UR5$URup4U*[ U5s=::a[ U5s=::aU:d O [ S5eU*[ U5s=::a[ U5s=::aU:d O [ S5e[[5n[[5n[U5HupxXXU-RU5 M [U5HupXjU-RU 5 M [U5n [U5n Un 0nXR5-H\nXn0nXR5-Hn XnXjHn UUU 'M M U(dM?XXHnUR5X'M M^ URU[U5[U54UR5$)NzRow index out of rangezColumn index out of range)zerosrGr*r'minmaxr6rlist enumerateappendsetkeyscopyrF)r-r(r)rr$rowmapcolmapi2i1j2j1rowsetcolsetsdm1sdm2row1row2row1_j1s rextract SDM.extracts$::s4y#d)4dkkB Bzzc$i03t90q056 6c$i03t90q089 9T"T"oFB 6N ! !" %&oFB 6N ! !" %&VV99;&B8DDyy{*( *B&DH%+t *B#yy{DH%'xxs4y#d)4dkkBBr1c/nUR5HDup#SRSUR555nURU<SU<S35 MF SSRU5-$)N, c38# UHupU<SU<3v M g7f)z: Nr)rr8elems rrSDM.__str__..s Q['!Q!5[sz: {}z{%s})rDjoinrV)r-rowsstrr7r"elemsstrs r__str__ SDM.__str__sZjjlFAyy QSYY[ QQH NNH5 6# '***r1c[U5Rn[RU5nU<SU<SUR<SUR <S3$)N(rj))type__name__dict__repr__r'r*)r-clsr(s rrz SDM.__repr__s64j!!}}T"#&djj$++FFr1cU"XU5$)a Parameters ========== sdm: A dict of dicts for non-zero elements in SDM shape: tuple representing dimension of SDM domain: Represents :py:class:`~.Domain` of SDM Returns ======= An :py:class:`~.SDM` object Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> elemsdict = {0:{1: QQ(2)}} >>> A = SDM.new(elemsdict, (2, 2), QQ) >>> A {0: {1: 2}} r)r{rLr'r*s rrFSDM.news63v&&r1cUR5VVs0sHupXR5_M nnnURX0RUR5$s snnf)z Returns the copy of a :py:class:`~.SDM` object Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> elemsdict = {0:{1:QQ(2)}, 1:{}} >>> A = SDM(elemsdict, (2, 2), QQ) >>> B = A.copy() >>> B {0: {1: 2}, 1: {}} )rDrYrFr'r*)Ar7AiAcs rrYSDM.copysF )* 2 uqal  2uuR!((++3sAc^^ ^ Uunm [T5U:Xa[U 4SjT55(d [S5eUU 4Sjm U 4Sj[U55nUVVs0sHupgU(dMXg_M nnnU"XU5$s snnf)aJ Create :py:class:`~.SDM` object from a list of lists. Parameters ========== ddm: list of lists containing domain elements shape: Dimensions of :py:class:`~.SDM` matrix domain: Represents :py:class:`~.Domain` of :py:class:`~.SDM` object Returns ======= :py:class:`~.SDM` containing elements of ddm Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> ddm = [[QQ(1, 2), QQ(0)], [QQ(0), QQ(3, 4)]] >>> A = SDM.from_list(ddm, (2, 2), QQ) >>> A {0: {0: 1/2}, 1: {1: 3/4}} See Also ======== to_list from_list_flat from_dok from_ddm c3@># UHn[U5T:Hv M g7frB)rG)rr"r$s rr SDM.from_list..s%Csc#h!msszInconsistent row-list/shapecr>[T5Vs0sHnTUU(dMUTUU_M sn$s snfrB)rC)r7r8ddmr$s rSDM.from_list..s3qGASVAYKAc!fQiKGGs4 4c36># UHoT"U54v M g7frBr)rr7getrows rrrs2AVAYs)rGr+r rC) r{rr'r*rirowsr7r"rLrr$s ` @@r from_list SDM.from_listswN1CA #%Cs%C"C"C!"?@ @G2q2$)1E&!SvqvE13v&&2s  B/BcNURXRUR5$)a Create :py:class:`~.SDM` from a :py:class:`~.DDM`. Examples ======== >>> from sympy.polys.matrices.ddm import DDM >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> ddm = DDM( [[QQ(1, 2), 0], [0, QQ(3, 4)]], (2, 2), QQ) >>> A = SDM.from_ddm(ddm) >>> A {0: {0: 1/2}, 1: {1: 3/4}} >>> SDM.from_ddm(ddm).to_ddm() == ddm True See Also ======== to_ddm from_list from_list_flat from_dok )rr'r*)r{rs rfrom_ddm SDM.from_ddms4}}S))SZZ88r1cURupURRn[U5Vs/sHoC/U-PM nnUR 5H%upgUR 5H upXUU'M M' U$s snf)z Convert a :py:class:`~.SDM` object to a list of lists. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> elemsdict = {0:{1:QQ(2)}, 1:{}} >>> A = SDM(elemsdict, (2, 2), QQ) >>> A.to_list() [[0, 2], [0, 0]] )r'r*r5rCrD) Mrr$r5_rr7r"r8rMs rto_list SDM.to_list5ss wwxx}}#(8,8avz8,ggiFA Aq $  -sA>cURupURRnU/X--nUR5H'upVUR5HupxXXR-U-'M M) U$)aY Convert :py:class:`~.SDM` to a flat list. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0:{1:QQ(2)}, 1:{0: QQ(3)}}, (2, 2), QQ) >>> A.to_list_flat() [0, 2, 3, 0] >>> A == A.from_list_flat(A.to_list_flat(), A.shape, A.domain) True See Also ======== from_list_flat to_list to_dok to_ddm )r'r*r5rD) rrr$r5flatr7r"r8rMs r to_list_flatSDM.to_list_flatMsb.wwxx}}vggiFA  !QS1W $  r1cUupE[U5XE-:wa [S5e[[5n[ U5H"upxU(dM[ Xu5upXU U 'M$ U"XbU5$)aj Create :py:class:`~.SDM` from a flat list of elements. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM.from_list_flat([QQ(0), QQ(2), QQ(0), QQ(0)], (2, 2), QQ) >>> A {0: {1: 2}} >>> A == A.from_list_flat(A.to_list_flat(), A.shape, A.domain) True See Also ======== to_list_flat from_list from_dok from_ddm zInconsistent flat-list shape)rGr rryrUdivmod) r{elementsr'r*rr$rLinjelementr7r8s rfrom_list_flatSDM.from_list_flatlsj0 x=AE !!"@A A$%h/LCwc~#Aq 03v&&r1cUR5n[U5n[UR55nX R4nX44$)a Convert :class:`SDM` to a flat list of nonzero elements and data. Explanation =========== This is used to operate on a list of the elements of a matrix and then reconstruct a modified matrix with elements in the same positions using :meth:`from_flat_nz`. Zero elements are omitted from the list. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0:{1:QQ(2)}, 1:{0: QQ(3)}}, (2, 2), QQ) >>> elements, data = A.to_flat_nz() >>> elements [2, 3] >>> A == A.from_flat_nz(elements, data, A.domain) True See Also ======== from_flat_nz to_list_flat sympy.polys.matrices.ddm.DDM.to_flat_nz sympy.polys.matrices.domainmatrix.DomainMatrix.to_flat_nz )to_doktuplerTr,r')rdokindicesrdatas r to_flat_nzSDM.to_flat_nzs<>hhj* %!~r1cVUupE[[XA55nURXeU5$)z Reconstruct a :class:`~.SDM` after calling :meth:`to_flat_nz`. See :meth:`to_flat_nz` for explanation. See Also ======== to_flat_nz from_list_flat sympy.polys.matrices.ddm.DDM.from_flat_nz sympy.polys.matrices.domainmatrix.DomainMatrix.from_flat_nz )ryzipfrom_dok)r{rrr*rr'rs r from_flat_nzSDM.from_flat_nzs+3w)*||C//r1ctUR5VVs0sHupXR5_M snn$s snnf)a@ Convert to dictionary of dictionaries (dod) format. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0: {1: QQ(2)}, 1: {0: QQ(3)}}, (2, 2), QQ) >>> A.to_dod() {0: {1: 2}, 1: {0: 3}} See Also ======== from_dod sympy.polys.matrices.domainmatrix.DomainMatrix.to_dod )rDrY)rr7r"s rto_dod SDM.to_dods,&-.GGI6I&!88: I666s4c[[5nUR5H.upVUR5HupxU(dMXUU'M M0 U"XBU5$)a Create :py:class:`~.SDM` from dictionary of dictionaries (dod) format. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> dod = {0: {1: QQ(2)}, 1: {0: QQ(3)}} >>> A = SDM.from_dod(dod, (2, 2), QQ) >>> A {0: {1: 2}, 1: {0: 3}} >>> A == SDM.from_dod(A.to_dod(), A.shape, A.domain) True See Also ======== to_dod sympy.polys.matrices.domainmatrix.DomainMatrix.to_dod rryrD) r{dodr'r*rLr7r"r8rMs rfrom_dod SDM.from_dodsQ.$iikFA 1 !F1I$"3v&&r1c UR5VVVVs0sH#upUR5H up4X4U_M M% snnnn$s snnnnf)a Convert to dictionary of keys (dok) format. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0: {1: QQ(2)}, 1: {0: QQ(3)}}, (2, 2), QQ) >>> A.to_dok() {(0, 1): 2, (1, 0): 3} See Also ======== from_dok to_list to_list_flat to_ddm rDrr7r"r8rMs rr SDM.to_doks:*)* J faciikda k JJJs*A c[[5nUR5HuupVnU(dMXtUU'M U"XBU5$)a} Create :py:class:`~.SDM` from dictionary of keys (dok) format. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> dok = {(0, 1): QQ(2), (1, 0): QQ(3)} >>> A = SDM.from_dok(dok, (2, 2), QQ) >>> A {0: {1: 2}, 1: {0: 3}} >>> A == SDM.from_dok(A.to_dok(), A.shape, A.domain) True See Also ======== to_dok from_list from_list_flat from_ddm r)r{rr'r*rLr7r8rMs rr SDM.from_doksC2$IFQAqAq %3v&&r1c#n# UR5HnUR5ShvN M gN 7f)z Iterate over the nonzero values of a :py:class:`~.SDM` matrix. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0: {1: QQ(2)}, 1: {0: QQ(3)}}, (2, 2), QQ) >>> list(A.iter_values()) [2, 3] N)r,)rr"s r iter_valuesSDM.iter_values/s)88:Czz| # # #s '53 5c#~# UR5H%upUR5H up4X4U4v M M' g7f)aN Iterate over indices and values of the nonzero elements. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0: {1: QQ(2)}, 1: {0: QQ(3)}}, (2, 2), QQ) >>> list(A.iter_items()) [((0, 1), 2), ((1, 0), 3)] See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.iter_items Nrrs r iter_itemsSDM.iter_items@s6$ggiFA fai$ s;=c`[UR5URUR5$)z Convert a :py:class:`~.SDM` object to a :py:class:`~.DDM` object Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0:{1:QQ(2)}, 1:{}}, (2, 2), QQ) >>> A.to_ddm() [[0, 2], [0, 0]] )rrr'r*rs rto_ddm SDM.to_ddmVs!199;22r1cU$)z5 Convert to :py:class:`~.SDM` format (returns self). rrs rto_sdm SDM.to_sdmfs r1r) ground_typesc>UR5R5$)aH Convert a :py:class:`~.SDM` object to a :py:class:`~.DFM` object Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0:{1:QQ(2)}, 1:{}}, (2, 2), QQ) >>> A.to_dfm() [[0, 2], [0, 0]] See Also ======== to_ddm to_dfm_or_ddm sympy.polys.matrices.domainmatrix.DomainMatrix.to_dfm )rto_dfmrs rr SDM.to_dfmls*xxz  ""r1c>UR5R5$)a Convert to :py:class:`~.DFM` if possible, else :py:class:`~.DDM`. Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0:{1:QQ(2)}, 1:{}}, (2, 2), QQ) >>> A.to_dfm_or_ddm() [[0, 2], [0, 0]] >>> type(A.to_dfm_or_ddm()) # depends on the ground types See Also ======== to_ddm to_dfm sympy.polys.matrices.domainmatrix.DomainMatrix.to_dfm_or_ddm )r to_dfm_or_ddmrs rrSDM.to_dfm_or_ddms.xxz''))r1cU"0X5$)aQ Returns a :py:class:`~.SDM` of size shape, belonging to the specified domain In the example below we declare a matrix A where, .. math:: A := \left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right] >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM.zeros((2, 3), QQ) >>> A {} r)r{r'r*s rrQ SDM.zeross*2u%%r1cURnUupE[[[U5U/U-55n[U5Vs0sHowUR 5_M nnU"XU5$s snfrB)oneryrrCrY) r{r'r*rrr$r"r7rLs ronesSDM.oness_jj3uQx#q)*&+Ah/h#((*}h/3v&&0sA%c[U[5(aXpCOUup4URn[[ X455Vs0sHofXe0_M nnU"XsU4U5$s snf)z Returns a identity :py:class:`~.SDM` matrix of dimensions size x size, belonging to the specified domain Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> I = SDM.eye((2, 2), QQ) >>> I {0: {0: 1}, 1: {1: 1}} ) isinstanceintrrCrR)r{r'r*r(r)rr7rLs reyeSDM.eyes_" eS ! !$JDjj$)#d/$:;$:q1({$:;3t f--%8TQAyqa&y%8>3v&&?s A A cp[U5nURXRSSS2UR5$)z Returns the transpose of a :py:class:`~.SDM` matrix Examples ======== >>> from sympy.polys.matrices.sdm import SDM >>> from sympy import QQ >>> A = SDM({0:{1:QQ(2)}, 1:{}}, (2, 2), QQ) >>> A.transpose() {1: {0: 2}} N) sdm_transposerFr'r*)rMTs r transpose SDM.transposes/1 uuR211r1c[U[5(d[$URUR:wa'[ SUR<SUR<35eUR U5$)NMatrix size mismatch: z + )rrNotImplementedr'rrrBs r__add__ SDM.__add__J!S!!! ! WW !''177ST TuuQxr1c[U[5(d[$URUR:wa'[ SUR<SUR<35eUR U5$)Nrz - )rrrr'rrrs r__sub__ SDM.__sub__rr1c"UR5$rB)rrs r__neg__ SDM.__neg__s uuwr1c[U[5(aURU5$XR;aUR U5$[ $)zA * B)rrmatmulr*rrrs r__mul__ SDM.__mul__s9 a  88A;  ((]558O! !r1cNXR;aURU5$[$rB)r*rmulr)abs r__rmul__ SDM.__rmul__s =66!9 ! !r1cURUR:wa[eURup#URupEX4:wa[e[ XURX%5nUR XbU4UR5$)a Performs matrix multiplication of two SDM matrices Parameters ========== A, B: SDM to multiply Returns ======= SDM SDM after multiplication Raises ====== DomainError If domain of A does not match with that of B Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ) >>> B = SDM({0:{0:ZZ(2), 1:ZZ(3)}, 1:{0:ZZ(4)}}, (2, 2), ZZ) >>> A.matmul(B) {0: {0: 8}, 1: {0: 2, 1: 3}} )r*rr'r sdm_matmulrF)rrrr$n2oCs rr SDM.matmul sgB 88qxx  ww 7  qQXXq ,uuQA))r1cp^[UU4Sj5nURX RUR5$)z Multiplies each element of A with a scalar b Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ) >>> A.mul(ZZ(3)) {0: {1: 6}, 1: {0: 3}} c>UT-$rBraijrs rrSDM.mul..Es Ar1 unop_dictrFr'r*rrCsdms ` rrSDM.mul7s+-.uuT77AHH--r1cp^[UU4Sj5nURX RUR5$)Nc>TU-$rBrr s rrSDM.rmul..Is #r1r rs ` rrSDM.rmulHs)-.uuT77AHH--r1c*^URUR:wa[eURUR:wa[eURRmU4Sjn[ X[ X"5nURX0RUR5$)Nc>T$rBr)rMr5s rr%SDM.mul_elementwise..Rs$r1)r*rr'rr5 binop_dictrrF)rrfzerorr5s @rmul_elementwiseSDM.mul_elementwiseLsh 88qxx   77agg  xx}}!U2uuT77AHH--r1c[X[[[5nURX RUR 5$)a Adds two :py:class:`~.SDM` matrices Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ) >>> B = SDM({0:{0: ZZ(3)}, 1:{1:ZZ(4)}}, (2, 2), ZZ) >>> A.add(B) {0: {0: 3, 1: 2}, 1: {0: 1, 1: 4}} )rrrrFr'r*rrrs rrSDM.addV- !S#.uuT77AHH--r1c[X[[[5nUR X R UR 5$)a# Subtracts two :py:class:`~.SDM` matrices Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ) >>> B = SDM({0:{0: ZZ(3)}, 1:{1:ZZ(4)}}, (2, 2), ZZ) >>> A.sub(B) {0: {0: -3, 1: 2}, 1: {0: 1, 1: -4}} )rrrrrFr'r*rs rrSDM.subirr1cn[U[5nURXRUR5$)z Returns the negative of a :py:class:`~.SDM` matrix Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ) >>> A.neg() {0: {1: -2}, 1: {0: -1}} )r rrFr'r*)rrs rrSDM.neg|s)C uuT77AHH--r1c^^URmTT:XaUR5$[UUU4Sj5nURX RT5$)a Converts the :py:class:`~.Domain` of a :py:class:`~.SDM` matrix to K Examples ======== >>> from sympy import ZZ, QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ) >>> A.convert_to(QQ) {0: {1: 2}, 1: {0: 1}} c(>TRUT5$rB) convert_from)rMKKolds rr SDM.convert_to..sANN1d$;r1)r*rYr rFr')rr'Akr(s ` @r convert_toSDM.convert_tosBxx 9668O q; <uuR!$$r1cP[[[UR555$)a!Number of non-zero elements in the :py:class:`~.SDM` matrix. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{1: ZZ(2)}, 1:{0:ZZ(1)}}, (2, 2), ZZ) >>> A.nnz() 2 See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.nnz )summaprGr,rs rnnzSDM.nnzs"3sAHHJ'((r1c URupX:Xde[U5nUVs0sHoD[URU/55_M nn[ X55$s snf)a#Strongly connected components of a square matrix *A*. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0: ZZ(2)}, 1:{1:ZZ(1)}}, (2, 2), ZZ) >>> A.scc() [[0], [1]] See also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.scc )r'rCrTr<r )rr(r)VrEmaps rsccSDM.sccsX"WW || $K/01q!4a %%q1-a662s%Acn[U5upnURXRUR5U4$)a Returns reduced-row echelon form and list of pivots for the :py:class:`~.SDM` Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(2), 1:QQ(4)}}, (2, 2), QQ) >>> A.rref() ({0: {0: 1, 1: 2}}, [0]) ) sdm_irrefrFr'r*)rrpivotsrs rrrefSDM.rrefs/!| 1uuQ*F22r1cURn[X5up#nURX RUR5nXSU4$)a Returns reduced-row echelon form (RREF) with denominator and pivots. Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(2), 1:QQ(4)}}, (2, 2), QQ) >>> A.rref_den() ({0: {0: 1, 1: 2}}, 1, [0]) )r* sdm_rref_denrFr')rr' A_rref_sdmdenomr9A_rrefs rrref_den SDM.rref_dens? HH$0$6! 6z77AHH5f$$r1cZUR5R5R5$)z Returns inverse of a matrix A Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ) >>> A.inv() {0: {0: -2, 1: 1}, 1: {0: 3/2, 1: -1/2}} )rinvrrs rrDSDM.invs# $$&--//r1c>UR5R5$)z Returns determinant of A Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ) >>> A.det() -2 )rdetrs rrGSDM.dets* $$&&r1cUR5R5upnURU5URU5U4$)a Returns LU decomposition for a matrix A Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ) >>> A.lu() ({0: {0: 1}, 1: {0: 3, 1: 1}}, {0: {0: 1, 1: 2}, 1: {1: -2}}, []) )rlur)rLUswapss rrJSDM.lus6hhjmmo ezz!}ajjmU22r1cUR5R5upUR5nUR5nX44$)z QR decomposition for SDM (Sparse Domain Matrix). Returns: - Q: Orthogonal matrix as a SDM. - R: Upper triangular matrix as a SDM. )rqrr)r-ddm_qddm_rQRs rrPSDM.qr,s7{{}'')  LLN LLNt r1czURUR5RUR555$)a Uses LU decomposition to solve Ax = b, Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ) >>> b = SDM({0:{0:QQ(1)}, 1:{0:QQ(2)}}, (2, 1), QQ) >>> A.lu_solve(b) {1: {0: 1/2}} )rrlu_solve)rrs rrW SDM.lu_solve9s* zz!((*--ahhj9::r1cUR5R5upp4UR5nUR5nUR5nUR5nXVXx4$)zx Fraction free LU decomposition of SDM. Uses DDM implementation. See Also ======== sympy.polys.matrices.ddm.DDM.fflu )rfflur) r-ddm_pddm_lddm_dddm_uPrKDrLs rrZSDM.ffluKsV&*%7%7%9%>%>%@"e LLN LLN LLN LLNQzr1cURSnURRn[U5up4n[ X2XU5upg[ [ U55n[U5U4nURXhUR5U4$)a( Nullspace of a :py:class:`~.SDM` matrix A. The domain of the matrix must be a field. It is better to use the :meth:`~.DomainMatrix.nullspace` method rather than this method which is otherwise no longer used. Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0: QQ(2), 1: QQ(4)}}, (2, 2), QQ) >>> A.nullspace() ({0: {0: -2, 1: 1}}, [1]) See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.nullspace The preferred way to get the nullspace of a matrix. r ) r'r*rr8sdm_nullspace_from_rrefryrUrGrF) rncolsrrr9nzcolsr' nonpivotsr's r nullspace SDM.nullspace]sq4 hhll%aL6.qufM  1 QuuQqxx()33r1cPURup#URnUc'[[[UR 555nU(d'UR X34U5[[U554$[U5U:XaURSU4U5/4$USUSnURU5(ae[U5n[[5nUR5H2upU R5HupXzRX45 M M4 /n /n [U5HDn X;aM U RU 5 X0nXzH upU *XU'M U RU5 MF [![#U 55nUR%U[U 5U4U5nUU 4$)a Returns nullspace for a :py:class:`~.SDM` matrix ``A`` in RREF. The domain of the matrix can be any domain. The matrix must already be in reduced row echelon form (RREF). Examples ======== >>> from sympy import QQ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0: QQ(2), 1: QQ(4)}}, (2, 2), QQ) >>> A_rref, pivots = A.rref() >>> A_null, nonpivots = A_rref.nullspace_from_rref(pivots) >>> A_null {0: {0: -2, 1: 1}} >>> pivots [0] >>> nonpivots [1] See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.nullspace The higher-level function that would usually be called instead of calling this one directly. sympy.polys.matrices.domainmatrix.DomainMatrix.nullspace_from_rref The higher-level direct equivalent of this function. sympy.polys.matrices.ddm.DDM.nullspace_from_rref The equivalent function for dense :py:class:`~.DDM` matrices. r)r'r*sortedr/rRr,rrTrCrGrQis_zerorWrrDrVryrUrF)rr9rr$r' pivot_val pivots_set nonzero_colsr7rr8AijbasisrfveciprLA_nulls rnullspace_from_rrefSDM.nullspace_from_rrefsJww HH >CQXXZ01F55!#T%(^3 3 [A 77Aq61%r) )aDO 99Y''''[ #4( WWYEA((*&&x0% qA   Q .C'?#&$2J+ LL 9U#$sSZOQ/ ""r1cURSn[U5up#n[X!U5nU(aSU0O0nURUSUS- 4UR5$)Nr r)r'r8sdm_particular_from_rrefrFr*)rrdrr9rer_reps r particularSDM.particularsU %aL6 $Qv 6qebuuS1eAg,11r1c[UR55nURup4URnUHnURupxXs:XdeURU:XdeUR 5H?upUR U S5n U c0=X)'n U R 5H upXX-'M MA XH- nM UR X#U4UR5$)a'Horizontally stacks :py:class:`~.SDM` matrices. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(4)}}, (2, 2), ZZ) >>> B = SDM({0: {0: ZZ(5), 1: ZZ(6)}, 1: {0: ZZ(7), 1: ZZ(8)}}, (2, 2), ZZ) >>> A.hstack(B) {0: {0: 1, 1: 2, 2: 5, 3: 6}, 1: {0: 3, 1: 4, 2: 7, 3: 8}} >>> C = SDM({0: {0: ZZ(9), 1: ZZ(10)}, 1: {0: ZZ(11), 1: ZZ(12)}}, (2, 2), ZZ) >>> A.hstack(B, C) {0: {0: 1, 1: 2, 2: 5, 3: 6, 4: 9, 5: 10}, 1: {0: 3, 1: 4, 2: 7, 3: 8, 4: 11, 5: 12}} N)ryrYr'r*rDr<rF)rrAnewr(r)r*BkBkrowsBkcolsr7Bkirr8Bkijs rhstack SDM.hstacks$AFFH~WW BXXNF> !>99& &&((*XXa&:#%%DGb"yy{GA#'qxL + % NDuuT$<22r1cJ[UR55nURup4URnUHMnURupxX:XdeURU:XdeUR 5H upXX-'M X7- nMO UR X#U4UR5$)aCVertically stacks :py:class:`~.SDM` matrices. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices.sdm import SDM >>> A = SDM({0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(4)}}, (2, 2), ZZ) >>> B = SDM({0: {0: ZZ(5), 1: ZZ(6)}, 1: {0: ZZ(7), 1: ZZ(8)}}, (2, 2), ZZ) >>> A.vstack(B) {0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}, 2: {0: 5, 1: 6}, 3: {0: 7, 1: 8}} >>> C = SDM({0: {0: ZZ(9), 1: ZZ(10)}, 1: {0: ZZ(11), 1: ZZ(12)}}, (2, 2), ZZ) >>> A.vstack(B, C) {0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}, 2: {0: 5, 1: 6}, 3: {0: 7, 1: 8}, 4: {0: 9, 1: 10}, 5: {0: 11, 1: 12}} )ryrYr'r*rDrF) rrr|r(r)r*r}r~rr7rs rvstack SDM.vstacks$AFFH~WW BXXNF> !>99& &&((*!$QX% NDuuT$<22r1cUR5VVVVs0sH0up4X4R5VVs0sH upVXQ"U5_M snn_M2 nnnnnURXpRU5$s snnfs snnnnfrB)rDrFr')r-funcr*r7r"r8rMrLs r applyfunc SDM.applyfunc#s]EIZZ\R\61q))+6+$!1d1g:+66\RxxZZ007RsA4 A.A4 .A4 cURnURup#[XU5nUR/US--nUR 5H upgXuU'M U$)ar Returns the coefficients of the characteristic polynomial of the :py:class:`~.SDM` matrix. These elements will be domain elements. The domain of the elements will be same as domain of the :py:class:`~.SDM`. Examples ======== >>> from sympy import QQ, Symbol >>> from sympy.polys.matrices.sdm import SDM >>> from sympy.polys import Poly >>> A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ) >>> A.charpoly() [1, -5, -2] We can create a polynomial using the coefficients using :py:class:`~.Poly` >>> x = Symbol('x') >>> p = Poly(A.charpoly(), x, domain=A.domain) >>> p Poly(x**2 - 5*x - 2, x, domain='QQ') r )r*r'sdm_berkr5rD)rr'r$rpdictplistr7pis rcharpoly SDM.charpoly'sV2 HHwwq!AE"[[]EA!H# r1cU(+$)z0 Says whether this matrix has all zero entries. rr-s ris_zero_matrixSDM.is_zero_matrixHs xr1cB[SUR555$)zf Says whether this matrix is upper-triangular. True can be returned even if the matrix is not square. c3B# UHupUH o1U:*v M M g7frBrrr7r"r8s rrSDM.is_upper..SB fac6c6 r+rDrs ris_upper SDM.is_upperN B BBBr1cB[SUR555$)zf Says whether this matrix is lower-triangular. True can be returned even if the matrix is not square. c3B# UHupUH o1U:v M M g7frBrrs rrSDM.is_lower..Zrrrrs ris_lower SDM.is_lowerUrr1cB[SUR555$)z^ Says whether this matrix is diagonal. True can be returned even if the matrix is not square. c3B# UHupUH o1U:Hv M M g7frBrrs rr"SDM.is_diagonal..arrrrs r is_diagonalSDM.is_diagonal\rr1cURupURRnUR5VVs/sHupEXB:dM UR XC5PM snn$s snnf)z/ Returns the diagonal of the matrix as a list. )r'r*r5rDr<)r-rr$r5r7r"s rr SDM.diagonalcsNzz{{/3zz|E|VQqu  |EEEs A A cXUR5RUS9R5$)zA Returns the LLL-reduced basis for the :py:class:`~.SDM` matrix. delta)rlllr)rrs rrSDM.lllks( $$5$188::r1cUR5RUS9up#UR5UR54$)z: Returns the LLL-reduced basis and transformation matrix. r)r lll_transformr)rrreduced transforms rrSDM.lll_transformqs<__.<<5<I~~!1!1!333r1)r)r*r(r'rB)Qrx __module__ __qualname____firstlineno____doc__fmtis_DFMis_DDMr&r9r?rNrgrrrz classmethodrFrYrrrrrrrrrrrrrrrr rrrQrrrrrrrrrrrrrrrrr+r0r5r:rArDrGrJrPrWrZrgrtryrrrrrrrrrrrr__static_attributes__ __classcell__)r/s@rrrsl.` C F F9 7$* >$CL+G ''8,&,','\9960>''B#J00$7*'':K.''<$" ,3  gY/#0#,gY/*0*0&&,''..0'' 2$"" (*T."...&.&.$%()&7.3$%(0"'.3$ ;$$ 4DU#n2#3J3B1B CCCF1X; "!Qx44r1rc[U5[U5pe0nXV-HnXXp0n [U 5[U 5pX-HnU"XX5nU(dMXU'M X- HnU"X5nU(dMXU'M X- HnU"X5nU(dMXU'M U (dMXU'M XV- HEnXn 0n U R5HunnU"U5nU(dMXU'M U (dMAXU'MG Xe- HEnXn 0n U R5HunnU"U5nU(dMXU'M U (dMAXU'MG U$rB)rWrD)rrfabfafbAnzBnzrr7rBiCiAnziBnzir8CijroBijs rrrysj1vs1v A YqtB Wc"gdAbeRU#Cs1ARU)Cs1ARU)Cs1 2aD#&Y T hhjFAsS'Cs1! 2aDY T hhjFAsS'Cs1! 2aD Hr1c0nUR5HBup40nUR5HupgU"U5nU(dMXU'M U(dM>XRU'MD U$rBr) rfrr7rrr8rors rr r sZ A hhjFAC&Cs1! 2aD Hr1c0nUR5H&up#UR5H upEXQUU'M M( U$![a X%0X'M)f=frB)rDr4)rrr7Mir8Mijs rrrs\ BhhjFA !1a! I ! !s?AAc|^^URUU4SjTR5TR5-55$)Nc3:># UHnTUTU-v M g7frBr)rr8rrs rrsdm_dotvec..s:&91!&9s)r.rX)rrr's`` r sdm_dotvecrs) 55:affh&9: ::r1cn0nUR5HupE[XQU5nU(dMXcU'M U$rB)rDr)rrr'rr7rrs r sdm_matvecmulrs8 A q ! 2aD Hr1cUR(a [XX#U5$0n[U5nUR5Hupx0n [U5n X-Hon Xn XR5HRupU R U S5nUb'XU--nU(aXU 'M.U R U 5 MAX-nU(dMNXU 'MT Mq U (dMXU'M U$rB)is_EXRAWsdm_matmul_exrawrWrDr<pop)rrr'rrrB_knzr7rrAi_knzkAikr8Bkjrs rrrs" zzaA.. A FE RA%C$**,ffQo?c /C #1q )Cs #1'  2aD%& Hr1c xURn0n[U5nUR5GH*up[[5n [U 5n X-Hzn Xn X]-U:Xa2XR5HupXR X-5 M MA[ U5H*nXR XU RX5-5 M, M| X- H7n XYU -nUU:wdM[ U5HnXR U5 M M9 0nU R5H%unnURU5nU(dM UUU'M' U(dGM&UXh'GM- UR5Hun nUR5HupX_-U:wdM[ U5HnURU05RX5n X:XdM+URU05nURX5X--nUU:wa UUU'UXh'MdURUS5 U(aUXh'MURUS5 M M M U$rB) r5rWrDrrTrVrCr<r.r)rrr'rrr5rrr7rCi_listrrrr8rzAikrCij_listrr}s rrrs 66D A FEd#RA%CzT!djjlFAJ%%ci0+qAJ%%caDHHQ,=&=>" AQ%>> from sympy import QQ >>> from sympy.polys.matrices.sdm import sdm_irref >>> A = {0: {0: QQ(1), 1: QQ(2)}, 1: {0: QQ(3), 1: QQ(4)}} >>> Arref, pivots, _ = sdm_irref(A) >>> Arref {0: {0: 1}, 1: {1: 1}} >>> pivots [0, 1] The analogous calculation with :py:class:`~.MutableDenseMatrix` would be >>> from sympy import Matrix >>> M = Matrix([[1, 2], [3, 4]]) >>> Mrref, pivots = M.rref() >>> Mrref Matrix([ [1, 0], [0, 1]]) >>> pivots (0, 1) Notes ===== The cost of this algorithm is determined purely by the nonzero elements of the matrix. No part of the cost of any step in this algorithm depends on the number of rows or columns in the matrix. No step depends even on the number of nonzero rows apart from the primary loop over those rows. The implementation is much faster than ddm_rref for sparse matrices. In fact at the time of writing it is also (slightly) faster than the dense implementation even if the input is a fully dense matrix so it seems to be faster in all cases. The elements of the matrix should support exact division with ``/``. For example elements of any domain that is a field (e.g. ``QQ``) should be fine. No attempt is made to handle inexact arithmetic. See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.rref The higher-level function that would normally be used to call this routine. sympy.polys.matrices.dense.ddm_irref The dense equivalent of this routine. sdm_rref_den Fraction-free version of this routine. c3@# UHoR5v M g7frB)rY)rrs rrsdm_irref..s3 "GGII skeyrrr ) rjr,rRrWrrrDremoverrGrUry)rArows pivot_row_mapreduced_pivotsnonreduced_pivotsnonzero_columnsrr8roAjAinzAjnzrrAijinvlr*AkjAknzAklr9r$p pivot2rowr#sr7r(r:s rr8r88s[v 3 3 =E MUN"#&O  YY[$&88: I:1Hfaf: I#SW,A!B%Cr7Dr7D[ ! FF1I KKN[ecqEk)qEFF1I !-$  Gea2w bA EVOE!$$Q+A!BQ%Cr7DD[ 1"&&q)! FF1I KKND[ecBEk)BqEFF1Iv'*11!4!2w!|""1%!((+),, r7a<   q !  ! !! $6#&**1-M %VN6 7F"+F"34"3$!QA"3I4@O@U@U@WX@W1q3A9Q<33@WOX&, -fM! fD -  $ D  ((WJN53X -s04 M!M!$M' M2M-(M27M9-M2c U(d0UR/4$[U5S:Xa6UR5un[U5nX#nSUR 50XC/4$UR (a UR nO URn[[UR556upg0n0n UR5n U R5n /n Sn Sn[U[S9nUGHnUR5VVs0sHup4X:;dM X4_M nnn0nXR5-HznURU5nXUnUR5HKunnURU5nUc UU-UU'M$UUU--nU(aUUU'M:URU5 MM M| [U5n[U5nU =(d URnUU- H nUU*UU'M UU- HnUUU-UU'M UU-H0nUUU-UU- nU(aUUU'MURU5 M2 U(dGM[[U5nURU5n[U R55GH*unnU UnUU;a2UR5HunnUU-nUb U"UU5nUUU'M MDURU5n[U5n[U5nUU- H#nU*UU-UU'UcMU"UUU5UU'M% UU- H"nUUU-UU'UcMU"UUU5UU'M$ UU-H?nUUU-UUU-- nU(aUb U"UU5nUUU'M.URU5 MA U(aGMU RU5 UUU'GM- [U 5nU R!U5 U(aUX'OUX'UR#U5(d U cUn OX-n UbU"X5n U nGM U c URn 0UEU En U R5VVs0sH unnUU_M n!nn[%U 5VVs/sH unnU!UU4PM n"nn[[U"56un#n$[U#5n#[%U$5H unnXU#U'M ['[%U$55n%U%U U#4$s snnfs snnfs snnf)a Return the reduced row echelon form (RREF) of A with denominator. The RREF is computed using fraction-free Gauss-Jordan elimination. Explanation =========== The algorithm used is the fraction-free version of Gauss-Jordan elimination described as FFGJ in [1]_. Here it is modified to handle zero or missing pivots and to avoid redundant arithmetic. This implementation is also optimized for sparse matrices. The domain $K$ must support exact division (``K.exquo``) but does not need to be a field. This method is suitable for most exact rings and fields like :ref:`ZZ`, :ref:`QQ` and :ref:`QQ(a)`. In the case of :ref:`QQ` or :ref:`K(x)` it might be more efficient to clear denominators and use :ref:`ZZ` or :ref:`K[x]` instead. For inexact domains like :ref:`RR` and :ref:`CC` use ``ddm_irref`` instead. Examples ======== >>> from sympy.polys.matrices.sdm import sdm_rref_den >>> from sympy.polys.domains import ZZ >>> A = {0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(4)}} >>> A_rref, den, pivots = sdm_rref_den(A, ZZ) >>> A_rref {0: {0: -2}, 1: {1: -2}} >>> den -2 >>> pivots [0, 1] See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.rref_den Higher-level interface to ``sdm_rref_den`` that would usually be used instead of calling this function directly. sympy.polys.matrices.sdm.sdm_rref_den The ``SDM`` method that uses this function. sdm_irref Computes RREF using field division. ddm_irref_den The dense version of this algorithm. References ========== .. [1] Fraction-free algorithms for linear and polynomial equations. George C. Nakos , Peter R. Turner , Robert M. Williams. https://dl.acm.org/doi/10.1145/271130.271133 r rNr)rrGr,rRrYis_ExactexquoquorrjrDrXrr<rWrTrVis_onerUry)&rr'rr8rorr rows_in_ordercol_to_row_reducedcol_to_row_unreducedr unreduced A_rref_rowsr?divisorA_rows Ai_cancelrrAjk Aik_cancelAi_nz Ai_cancel_nzdrpkr*rrrAk_nzr7 col_to_row row_to_colA_rref_rows_colr9r@r>s& rr=r=sz AEE2 Q1hhj GeBGGIS)) zzF1779-.A %%'G$))+IK EG Ms +F$&88: B:1Afaf: B WWY&A&&)C!45B((*3&]]1- %#&9IaL!+cCi!7J!'1 ! ! a(%'&B9~ NQUU%Aq\MBqE&%AqEAIBqE&%AQ%!)il*C1q &  GffQi.4467EBQB{!hhjFAs)C*#C1BqE ) &&)CGEGEU]1 1&!"Q%1BqE# U]be 1&!"Q%1BqE# U]BqEkC"Q%K/*#C1BqEFF1I#2$((,)*"2&S8V  2 &' #$%  !xx}}}   %)E}@ }@&?*>?J#-#3#3#56#541a!Q$#5J68A+8NO8Nuq" 1 r*8NOO&12NFF &\F6"26!9 #i'(J uf $$[CD7Os8 R2R2&R8R>c[[[U55[U5- 5n/nUH=nXq0nURUS5Hn X U*XU 'M UR U5 M? Xe4$)z%Get nullspace from A which is in RREFr)rjrWrCr<rV) rrrdr9rnrfr'r8Kjr7s rrcrc sss5<(3v;67I A W!!!R(AT!WHBayM)   <r1c0n[U5H-upEXRUS- S5nUcM!X`UU- X5'M/ U$)z1Get a particular solution from A which is in RREFr N)rUr<)rrdr9r_r7r8Ains rrwrwsK A&!dhhuQw% ?1a=AD" Hr1c ,URnURnUS:XaSU0$US:Xa3SU0nURS05RSU5=n(aU*US'UR00[[54upxpUR 5H]upU R 5HDupU (aU (aXU S- U S- 'M"U (a XU S- 'M2U (a XU S- 'MBUnMF M_ U n[ XU5nXG*U*/n[SUS-5H6n [XU5nU(d O"[ XU5nURU*5 M8 U(a-US(d#UR5 U(a US(dM#[XS- U5nUSSS2n0n[[U5[[U5[U5-US-55HBn [ [UU [U5- S-55n[ UUU5=n(dM=UUU 'MD U$)a Berkowitz algorithm for computing the characteristic polynomial. Explanation =========== The Berkowitz algorithm is a division-free algorithm for computing the characteristic polynomial of a matrix over any commutative ring using only arithmetic in the coefficient ring. This implementation is for sparse matrices represented in a dict-of-dicts format (like :class:`SDM`). Examples ======== >>> from sympy import Matrix >>> from sympy.polys.matrices.sdm import sdm_berk >>> from sympy.polys.domains import ZZ >>> M = {0: {0: ZZ(1), 1:ZZ(2)}, 1: {0:ZZ(3), 1:ZZ(4)}} >>> sdm_berk(M, 2, ZZ) {0: 1, 1: -5, 2: -2} >>> Matrix([[1, 2], [3, 4]]).charpoly() PurePoly(lambda**2 - 5*lambda - 2, lambda, domain='ZZ') See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.charpoly The high-level interface to this function. sympy.polys.matrices.dense.ddm_berk The dense version of this function. References ========== .. [1] https://en.wikipedia.org/wiki/Samuelson%E2%80%93Berkowitz_algorithm rr rrN)r5rr<rryrDrrCrrVrrrRrSrGrU)rr$r'r5rrM00rrTrrr7rr8rAnCRCTvalsRAnCqTqTiTqis rrr#sJ 66D %%CAv3x aC%%2,""1d+ +3 +tE!HRT!22JA!hhjFAQ!!A#qs !A#!A#!. C A! B "rcNE 1ac]AA& !!$ dU  b   b  aCA" $B$KE B 3q63s1vc%j0!A#6 7 )E1SZ<>2 3RA& &3 &BqE8 Ir1N)&roperatorrrrrr collectionsrsympy.external.gmpyr sympy.utilities.decoratorr sympy.utilities.iterablesr exceptionsr rrsympy.polys.domainsrrr__doctest_skip__ryrrr rrrrrr8r=rcrwrrr1rr)s -,#,8DDD"7$&9:]4$]4@++ \  ; ) X= @})@P%f  rr1