a CCCfKm@sdZddlZddlZddlmZddlmZmZm Z m Z ddgZ GdddZ Gdd d e Z Gd d d e ZGd d d e ZdddZGddde ZGddde ZGddde ZGddde ZGddde ZGdddeZGddde ZddZdS) aWAbstract linear algebra library. This module defines a class hierarchy that implements a kind of "lazy" matrix representation, called the ``LinearOperator``. It can be used to do linear algebra with extremely large sparse or structured matrices, without representing those explicitly in memory. Such matrices can be added, multiplied, transposed, etc. As a motivating example, suppose you want have a matrix where almost all of the elements have the value one. The standard sparse matrix representation skips the storage of zeros, but not ones. By contrast, a LinearOperator is able to represent such matrices efficiently. First, we need a compact way to represent an all-ones matrix:: >>> import numpy as np >>> from scipy.sparse.linalg._interface import LinearOperator >>> class Ones(LinearOperator): ... def __init__(self, shape): ... super().__init__(dtype=None, shape=shape) ... def _matvec(self, x): ... return np.repeat(x.sum(), self.shape[0]) Instances of this class emulate ``np.ones(shape)``, but using a constant amount of storage, independent of ``shape``. The ``_matvec`` method specifies how this linear operator multiplies with (operates on) a vector. We can now add this operator to a sparse matrix that stores only offsets from one:: >>> from scipy.sparse.linalg._interface import aslinearoperator >>> from scipy.sparse import csr_matrix >>> offsets = csr_matrix([[1, 0, 2], [0, -1, 0], [0, 0, 3]]) >>> A = aslinearoperator(offsets) + Ones(offsets.shape) >>> A.dot([1, 2, 3]) array([13, 4, 15]) The result is the same as that given by its dense, explicitly-stored counterpart:: >>> (np.ones(A.shape, A.dtype) + offsets.toarray()).dot([1, 2, 3]) array([13, 4, 15]) Several algorithms in the ``scipy.sparse`` library are able to operate on ``LinearOperator`` instances. N)issparse)isshape isintlikeasmatrixis_pydata_spmatrixLinearOperatoraslinearoperatorcseZdZdZdZdZfddZddZdd Zd d Z d d Z ddZ ddZ ddZ ddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,d-Zd.d/Zd0d1Zd2d3Zd4d5ZeeZ d6d7Z!ee!Z"d8d9Z#d:d;Z$Z%S)`_. Examples -------- >>> import numpy as np >>> from scipy.sparse.linalg import LinearOperator >>> def mv(v): ... return np.array([2*v[0], 3*v[1]]) ... >>> A = LinearOperator((2,2), matvec=mv) >>> A <2x2 _CustomLinearOperator with dtype=float64> >>> A.matvec(np.ones(2)) array([ 2., 3.]) >>> A * np.ones(2) array([ 2., 3.]) NcsX|turttSt|}t|jtjkrPt|jtjkrPtjdt dd|SdS)NzMLinearOperator subclass should implement at least one of _matvec and _matmat.r )category stacklevel) rsuper__new___CustomLinearOperatortype_matvec_matmatwarningswarnRuntimeWarning)clsargskwargsobj __class__Z/var/www/html/django/DPS/env/lib/python3.9/site-packages/scipy/sparse/linalg/_interface.pyr s  zLinearOperator.__new__cCsB|durt|}t|}t|s2td|d||_||_dS)zInitialize this LinearOperator. To be called by subclasses. ``dtype`` may be None; ``shape`` should be convertible to a length-2 tuple. Nzinvalid shape z (must be 2-d))npdtypetupler ValueErrorshape)selfrr!rrr__init__s zLinearOperator.__init__cCs2|jdur.t|jd}t||j|_dS)zCCalled from subclasses at the end of the __init__ routine. N)rrZzerosr!asarraymatvec)r"vrrr _init_dtypes zLinearOperator._init_dtypecstfdd|jDS)zDefault matrix-matrix multiplication handler. Falls back on the user-defined _matvec method, so defining that will define matrix multiplication (though in a very suboptimal way). csg|]}|ddqSr$)r&reshape.0colr"rr z*LinearOperator._matmat..)rhstackTr"Xrr/rrszLinearOperator._matmatcCs||ddS)ayDefault matrix-vector multiplication handler. If self is a linear operator of shape (M, N), then this method will be called on a shape (N,) or (N, 1) ndarray, and should return a shape (M,) or (M, 1) ndarray. This default implementation falls back on _matmat, so defining that will define matrix-vector multiplication as well. r$r*)matmatr+r"xrrrrs zLinearOperator._matveccCst|}|j\}}|j|fkr6|j|dfkr6td||}t|tjrVt|}n t|}|j dkrv| |}n |j dkr| |d}ntd|S)axMatrix-vector multiplication. Performs the operation y=A*x where A is an MxN linear operator and x is a column vector or 1-d array. Parameters ---------- x : {matrix, ndarray} An array with shape (N,) or (N,1). Returns ------- y : {matrix, ndarray} A matrix or ndarray with shape (M,) or (M,1) depending on the type and shape of the x argument. Notes ----- This matvec wraps the user-specified matvec routine or overridden _matvec method to ensure that y has the correct shape and type. r*dimension mismatchr z/invalid shape returned by user-defined matvec()) r asanyarrayr!r r isinstancematrixrr%ndimr+r"r8MNyrrrr&s         zLinearOperator.matveccCst|}|j\}}|j|fkr6|j|dfkr6td||}t|tjrVt|}n t|}|j dkrv| |}n |j dkr| |d}ntd|S)aAdjoint matrix-vector multiplication. Performs the operation y = A^H * x where A is an MxN linear operator and x is a column vector or 1-d array. Parameters ---------- x : {matrix, ndarray} An array with shape (M,) or (M,1). Returns ------- y : {matrix, ndarray} A matrix or ndarray with shape (N,) or (N,1) depending on the type and shape of the x argument. Notes ----- This rmatvec wraps the user-specified rmatvec routine or overridden _rmatvec method to ensure that y has the correct shape and type. r*r9r z0invalid shape returned by user-defined rmatvec()) rr:r!r _rmatvecr;r<rr%r=r+r>rrrrmatvecs         zLinearOperator.rmatveccCs&t|jtjkrtn |j|SdS)z6Default implementation of _rmatvec; defers to adjoint.N)r_adjointrNotImplementedErrorHr&r7rrrrB+szLinearOperator._rmatvecc Cst|st|st|}|jdkr6td|jd|jd|jdkrbtd|jd|jz||}Wn@ty}z(t|st|rt d|WYd }~n d }~00t |tj rt |}|S) aPMatrix-matrix multiplication. Performs the operation y=A*X where A is an MxN linear operator and X dense N*K matrix or ndarray. Parameters ---------- X : {matrix, ndarray} An array with shape (N,K). Returns ------- Y : {matrix, ndarray} A matrix or ndarray with shape (M,K) depending on the type of the X argument. Notes ----- This matmat wraps any user-specified matmat routine or overridden _matmat method to ensure that y has the correct type. r z$expected 2-d ndarray or matrix, not z-drr*dimension mismatch: , zdUnable to multiply a LinearOperator with a sparse matrix. Wrap the matrix in aslinearoperator first.N) rrrr:r=r r!r Exception TypeErrorr;r<rr"r5Yerrrr63s&   zLinearOperator.matmatc Cst|st|st|}|jdkr2td|j|jd|jdkr^td|jd|jz||}Wn@ty}z(t|st|rt d|WYd}~n d}~00t |tj rt |}|S)a;Adjoint matrix-matrix multiplication. Performs the operation y = A^H * x where A is an MxN linear operator and x is a column vector or 1-d array, or 2-d array. The default implementation defers to the adjoint. Parameters ---------- X : {matrix, ndarray} A matrix or 2D array. Returns ------- Y : {matrix, ndarray} A matrix or 2D array depending on the type of the input. Notes ----- This rmatmat wraps the user-specified rmatmat routine. r z(expected 2-d ndarray or matrix, not %d-drrGrHzfUnable to multiply a LinearOperator with a sparse matrix. Wrap the matrix in aslinearoperator() first.N) rrrr:r=r r!_rmatmatrIrJr;r<rrKrrrrmatmatbs*   zLinearOperator.rmatmatcs:tjtjkr*tfdd|jDSj|SdS)z@Default implementation of _rmatmat defers to rmatvec or adjoint.csg|]}|ddqSr))rCr+r,r/rrr0r1z+LinearOperator._rmatmat..N)rrDrrr2r3rFr6r4rr/rrNszLinearOperator._rmatmatcCs||SNrr7rrr__call__szLinearOperator.__call__cCs ||SrP)dotr7rrr__mul__szLinearOperator.__mul__cCs t|stdt|d|S)Nz.Can only divide a linear operator by a scalar.g?)risscalarr _ScaledLinearOperatorr"otherrrr __truediv__s zLinearOperator.__truediv__cCst|trt||St|r(t||St|sBt|sBt|}|j dksd|j dkrn|j ddkrn| |S|j dkr| |St d|dS)arMatrix-matrix or matrix-vector multiplication. Parameters ---------- x : array_like 1-d or 2-d array, representing a vector or matrix. Returns ------- Ax : array 1-d or 2-d array (depending on the shape of x) that represents the result of applying this linear operator on x. r*r +expected 1-d or 2-d array or matrix, got %rN)r;r_ProductLinearOperatorrrTrUrrr%r=r!r&r6r r7rrrrRs     "   zLinearOperator.dotcCst|rtd||SNz0Scalar operands are not allowed, use '*' instead)rrTr rSrVrrr __matmul__s zLinearOperator.__matmul__cCst|rtd||Sr[)rrTr __rmul__rVrrr __rmatmul__s zLinearOperator.__rmatmul__cCs"t|rt||S||SdSrP)rrTrU_rdotr7rrrr]s  zLinearOperator.__rmul__cCst|trt||St|r(t||St|sBt|sBt|}|j dksd|j dkrt|j ddkrt|j |j j S|j dkr|j |j j Std|dS)aMatrix-matrix or matrix-vector multiplication from the right. Parameters ---------- x : array_like 1-d or 2-d array, representing a vector or matrix. Returns ------- xA : array 1-d or 2-d array (depending on the shape of x) that represents the result of applying this linear operator on x from the right. Notes ----- This is copied from dot to implement right multiplication. r*r rrYN)r;rrZrrTrUrrr%r=r!r3r&r6r r7rrrr_s     " zLinearOperator._rdotcCst|rt||StSdSrP)rrT_PowerLinearOperatorNotImplemented)r"prrr__pow__s  zLinearOperator.__pow__cCst|trt||StSdSrP)r;r_SumLinearOperatorrar7rrr__add__s  zLinearOperator.__add__cCs t|dS)Nr$)rUr/rrr__neg__szLinearOperator.__neg__cCs || SrP)rer7rrr__sub__ szLinearOperator.__sub__cCs<|j\}}|jdurd}ndt|j}d|||jj|fS)Nzunspecified dtypezdtype=z<%dx%d %s with %s>)r!rstrr__name__)r"r?r@dtrrr__repr__ s   zLinearOperator.__repr__cCs|S)aHermitian adjoint. Returns the Hermitian adjoint of self, aka the Hermitian conjugate or Hermitian transpose. For a complex matrix, the Hermitian adjoint is equal to the conjugate transpose. Can be abbreviated self.H instead of self.adjoint(). Returns ------- A_H : LinearOperator Hermitian adjoint of self. )rDr/rrradjointszLinearOperator.adjointcCs|S)zTranspose this linear operator. Returns a LinearOperator that represents the transpose of this one. Can be abbreviated self.T instead of self.transpose(). ) _transposer/rrr transpose'szLinearOperator.transposecCst|S)z6Default implementation of _adjoint; defers to rmatvec.)_AdjointLinearOperatorr/rrrrD1szLinearOperator._adjointcCst|S)z? Default implementation of _transpose; defers to rmatvec + conj)_TransposedLinearOperatorr/rrrrm5szLinearOperator._transpose)&ri __module__ __qualname____doc__r=Z__array_ufunc__r r#r(rrr&rCrBr6rOrNrQrSrXrRr\r^r]r_rcrerfrgrkrlpropertyrFrnr3rDrm __classcell__rrrrr7sBV   ///. % csReZdZdZdfdd ZfddZddZd d Zfd d Zd dZ Z S)rz>Linear operator defined in terms of user-specified operations.Ncs8t||d|_||_||_||_||_|dS)Nr)r r#r"_CustomLinearOperator__matvec_impl#_CustomLinearOperator__rmatvec_impl#_CustomLinearOperator__rmatmat_impl"_CustomLinearOperator__matmat_implr()r"r!r&rCr6rrOrrrr#=sz_CustomLinearOperator.__init__cs$|jdur||St|SdSrP)ryr rr4rrrrJs  z_CustomLinearOperator._matmatcCs ||SrP)rvr7rrrrPsz_CustomLinearOperator._matveccCs |j}|durtd||S)Nzrmatvec is not defined)rwrE)r"r8funcrrrrBSsz_CustomLinearOperator._rmatveccs$|jdur||St|SdSrP)rxr rNr4rrrrNYs  z_CustomLinearOperator._rmatmatcCs.t|jd|jdf|j|j|j|j|jdS)Nr*r)r!r&rCr6rOr)rr!rwrvrxryrr/rrrrD_sz_CustomLinearOperator._adjoint)NNNN) rirqrrrsr#rrrBrNrDrurrrrr:s  rcs@eZdZdZfddZddZddZdd Zd d ZZ S) roz$Adjoint of arbitrary Linear Operatorcs8|jd|jdf}tj|j|d||_|f|_dSNr*r)rr!r!r r#rArr"r}r!rrrr#ksz_AdjointLinearOperator.__init__cCs |j|SrP)r}rBr7rrrrqsz_AdjointLinearOperator._matveccCs |j|SrP)r}rr7rrrrBtsz_AdjointLinearOperator._rmatveccCs |j|SrP)r}rNr7rrrrwsz_AdjointLinearOperator._matmatcCs |j|SrP)r}rr7rrrrNzsz_AdjointLinearOperator._rmatmat rirqrrrsr#rrBrrNrurrrrrohs  rocs@eZdZdZfddZddZddZdd Zd d ZZ S) rpz*Transposition of arbitrary Linear Operatorcs8|jd|jdf}tj|j|d||_|f|_dSr{r|r~rrrr#sz"_TransposedLinearOperator.__init__cCst|jt|SrP)rconjr}rBr7rrrrsz!_TransposedLinearOperator._matveccCst|jt|SrP)rrr}rr7rrrrBsz"_TransposedLinearOperator._rmatveccCst|jt|SrP)rrr}rNr7rrrrsz!_TransposedLinearOperator._matmatcCst|jt|SrP)rrr}rr7rrrrNsz"_TransposedLinearOperator._rmatmatrrrrrrp}s  rpcCs>|dur g}|D]"}|durt|dr||jqtj|S)Nr)hasattrappendrrZ result_type) operatorsZdtypesrrrr _get_dtypes rcsDeZdZfddZddZddZddZd d Zd d ZZ S) rdcsdt|trt|tstd|j|jkr>td|d|d||f|_tt||g|jdS)N)both operands have to be a LinearOperatorz cannot add  and : shape mismatch)r;rr r!rr r#rr"r}Brrrr#s   z_SumLinearOperator.__init__cCs |jd||jd|SNrr*rr&r7rrrrsz_SumLinearOperator._matveccCs |jd||jd|SrrrCr7rrrrBsz_SumLinearOperator._rmatveccCs |jd||jd|SrrrOr7rrrrNsz_SumLinearOperator._rmatmatcCs |jd||jd|Srrr6r7rrrrsz_SumLinearOperator._matmatcCs|j\}}|j|jSrPrrFrrrrrDs z_SumLinearOperator._adjoint rirqrrr#rrBrNrrDrurrrrrds  rdcsDeZdZfddZddZddZddZd d Zd d ZZ S) rZcszt|trt|tstd|jd|jdkrFtd|d|dtt||g|jd|jdf||f|_dS)Nrr*rzcannot multiply rr)r;rr r!r r#rrrrrrr#s z_ProductLinearOperator.__init__cCs|jd|jd|Srrr7rrrrsz_ProductLinearOperator._matveccCs|jd|jd|SNr*rrr7rrrrBsz_ProductLinearOperator._rmatveccCs|jd|jd|Srrr7rrrrNsz_ProductLinearOperator._rmatmatcCs|jd|jd|Srrr7rrrrsz_ProductLinearOperator._matmatcCs|j\}}|j|jSrPrrrrrrDs z_ProductLinearOperator._adjointrrrrrrZs  rZcsDeZdZfddZddZddZddZd d Zd d ZZ S) rUcspt|tstdt|s$tdt|tr@|j\}}||}t|gt|g}t ||j ||f|_dS)NLinearOperator expected as Azscalar expected as alpha) r;rr rrTrUrrrr r#r!)r"r}alphaZalpha_originalrrrrr#s    z_ScaledLinearOperator.__init__cCs|jd|jd|Srrr7rrrrsz_ScaledLinearOperator._matveccCs t|jd|jd|Sr)rrrrCr7rrrrBsz_ScaledLinearOperator._rmatveccCs t|jd|jd|Sr)rrrrOr7rrrrNsz_ScaledLinearOperator._rmatmatcCs|jd|jd|Srrr7rrrrsz_ScaledLinearOperator._matmatcCs|j\}}|jt|SrP)rrFrr)r"r}rrrrrDs z_ScaledLinearOperator._adjointrrrrrrUs  rUcsLeZdZfddZddZddZddZd d Zd d Zd dZ Z S)r`csnt|tstd|jd|jdkr2td|t|rB|dkrJtdtt|g|j||f|_dS)Nrrr*z&square LinearOperator expected, got %rz"non-negative integer expected as p) r;rr r!rr r#rrr"r}rbrrrr#s  z_PowerLinearOperator.__init__cCs.tj|dd}t|jdD] }||}q|S)NT)copyr*)rarrayranger)r"Zfunr8resirrr_powers z_PowerLinearOperator._powercCs||jdj|SNr)rrr&r7rrrrsz_PowerLinearOperator._matveccCs||jdj|Sr)rrrCr7rrrrB sz_PowerLinearOperator._rmatveccCs||jdj|Sr)rrrOr7rrrrN sz_PowerLinearOperator._rmatmatcCs||jdj|Sr)rrr6r7rrrrsz_PowerLinearOperator._matmatcCs|j\}}|j|SrPrrrrrrDs z_PowerLinearOperator._adjoint) rirqrrr#rrrBrNrrDrurrrrr`s r`cs,eZdZfddZddZddZZS)MatrixLinearOperatorcs*t|j|j||_d|_|f|_dSrP)r r#rr!r}_MatrixLinearOperator__adjr)r"r}rrrr#szMatrixLinearOperator.__init__cCs |j|SrP)r}rRr4rrrrszMatrixLinearOperator._matmatcCs|jdurt||_|jSrP)r_AdjointMatrixOperatorr/rrrrD"s  zMatrixLinearOperator._adjoint)rirqrrr#rrDrurrrrrs rc@s(eZdZddZeddZddZdS)rcCs6|jj|_||_|f|_|jd|jdf|_dSr)r}r3r_AdjointMatrixOperator__adjointrr!)r"rlrrrr#(sz_AdjointMatrixOperator.__init__cCs|jjSrP)rrr/rrrr.sz_AdjointMatrixOperator.dtypecCs|jSrP)rr/rrrrD2sz_AdjointMatrixOperator._adjointN)rirqrrr#rtrrDrrrrr's rcsFeZdZdfdd ZddZddZdd Zd d Zd d ZZ S)IdentityOperatorNcst||dSrP)r r#)r"r!rrrrr#7szIdentityOperator.__init__cCs|SrPrr7rrrr:szIdentityOperator._matveccCs|SrPrr7rrrrB=szIdentityOperator._rmatveccCs|SrPrr7rrrrN@szIdentityOperator._rmatmatcCs|SrPrr7rrrrCszIdentityOperator._matmatcCs|SrPrr/rrrrDFszIdentityOperator._adjoint)Nrrrrrr6s rcCst|tr|St|tjs&t|tjrP|jdkr8tdtt|}t |St |s`t |rht |St |drt |drd}d}d}t |dr|j }t |dr|j}t |dr|j}t|j|j|||d Std dS) aReturn A as a LinearOperator. 'A' may be any of the following types: - ndarray - matrix - sparse matrix (e.g. csr_matrix, lil_matrix, etc.) - LinearOperator - An object with .shape and .matvec attributes See the LinearOperator documentation for additional information. Notes ----- If 'A' has no .dtype attribute, the data type is determined by calling :func:`LinearOperator.matvec()` - set the .dtype attribute to prevent this call upon the linear operator creation. Examples -------- >>> import numpy as np >>> from scipy.sparse.linalg import aslinearoperator >>> M = np.array([[1,2,3],[4,5,6]], dtype=np.int32) >>> aslinearoperator(M) <2x3 MatrixLinearOperator with dtype=int32> r zarray must have ndim <= 2r!r&NrCrOr)rCrOrztype not understood)r;rrZndarrayr<r=r Z atleast_2dr%rrrrrCrOrr!r&rJ)r}rCrOrrrrrJs.      )N)rsrnumpyrZ scipy.sparserZscipy.sparse._sputilsrrrr__all__rrrorprrdrZrUr`rrrrrrrrs,, . !#