o `^hq@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) aUAbstract 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_array >>> offsets = csr_array([[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=int8> >>> A.matvec(np.ones(2)) array([ 2., 3.]) >>> A @ np.ones(2) array([ 2., 3.]) NcsT|tur ttSt|}t|jtjkr(t|jtjkr(tjdt dd|S)NzMLinearOperator subclass should implement at least one of _matvec and _matmat.r )category stacklevel) rsuper__new___CustomLinearOperatortype_matvec_matmatwarningswarnRuntimeWarning)clsargskwargsobj __class__\/home/air/shanriGPT/back/venv/lib/python3.10/site-packages/scipy/sparse/linalg/_interface.pyr s  zLinearOperator.__new__cCsB|dur t|}t|}t|std|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__cCsf|jdur1tj|jdtjd}z t||}Wnty*tt|_YdSw|j|_dSdS)aDetermine the dtype by executing `matvec` on an `int8` test vector. In `np.promote_types` hierarchy, the type `int8` is the smallest, so we call `matvec` on `int8` and use the promoted dtype of the output to set the default `dtype` of the `LinearOperator`. We assume that `matmat`, `rmatvec`, and `rmatmat` would result in the same dtype of the output given an `int8` input as `matvec`. Called from subclasses at the end of the __init__ routine. N)r) rrzerosr!int8asarraymatvec OverflowErrorint)r"vmatvec_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). cg|] }|ddqSr$)r(reshape.0colr"rr z*LinearOperator._matmat..)rhstackTr"Xrr5rrszLinearOperator._matmatcCs||ddS)ayDefault matrix-vector multiplication handler. If self is a linear operator of shape (M, N), then this method will be called on a shape (N,) or (N, 1) ndarray, and should return a shape (M,) or (M, 1) ndarray. This default implementation falls back on _matmat, so defining that will define matrix-vector multiplication as well. r$r0)matmatr1r"xrrrrs zLinearOperator._matveccCst|}|j\}}|j|fkr|j|dfkrtd||}t|tjr+t|}nt|}|j dkr<| |}|S|j dkrI| |d}|Std)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. r0dimension mismatchr z/invalid shape returned by user-defined matvec()) r asanyarrayr!r r isinstancematrixrr'ndimr1r"r>MNyrrrr(         zLinearOperator.matveccCst|}|j\}}|j|fkr|j|dfkrtd||}t|tjr+t|}nt|}|j dkr<| |}|S|j dkrI| |d}|Std)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. r0r?r z0invalid shape returned by user-defined rmatvec()) rr@r!r _rmatvecrArBrr'rCr1rDrrrrmatvecrHzLinearOperator.rmatveccCs t|jtjkr t|j|S)z6Default implementation of _rmatvec; defers to adjoint.)r_adjointrNotImplementedErrorHr(r=rrrrIAs zLinearOperator._rmatvecc Cst|s t|s t|}|jdkrtd|jd|jd|jdkr1td|jd|jz||}WntyR}zt|sHt|rMt d|d }~wwt |tj r]t |}|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-drr0dimension mismatch: , zdUnable to multiply a LinearOperator with a sparse matrix. Wrap the matrix in aslinearoperator first.N) rrrr@rCr r!r Exception TypeErrorrArBrr"r;Yerrrr<Is*   zLinearOperator.matmatc Cst|s t|s t|}|jdkrtd|j|jd|jdkr/td|jd|jz||}WntyP}zt|sFt|rKt d|d}~wwt |tj r[t |}|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-drrNrOzfUnable to multiply a LinearOperator with a sparse matrix. Wrap the matrix in aslinearoperator() first.N) rrrr@rCr r!_rmatmatrPrQrArBrrRrrrrmatmatxs.   zLinearOperator.rmatmatcs6tjtjkrtfdd|jDSj|S)z@Default implementation of _rmatmat defers to rmatvec or adjoint.cr.r/)rJr1r2r5rrr6r7z+LinearOperator._rmatmat..)rrKrrr8r9rMr<r:rr5rrUs zLinearOperator._rmatmatcCs||SNrr=rrr__call__szLinearOperator.__call__cC ||SrW)dotr=rrr__mul__ zLinearOperator.__mul__cCs t|s tdt|d|S)Nz.Can only divide a linear operator by a scalar.g?)risscalarr _ScaledLinearOperatorr"otherrrr __truediv__s zLinearOperator.__truediv__cCst|tr t||St|rt||St|s!t|s!t|}|j dks2|j dkr7|j ddkr7| |S|j dkrA| |St d|)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. r0r )expected 1-d or 2-d array or matrix, got )rAr_ProductLinearOperatorrr]r^rrr'rCr!r(r<r r=rrrrZs     "   zLinearOperator.dotcCt|r td||SNz0Scalar operands are not allowed, use '*' instead)rr]r r[r_rrr __matmul__  zLinearOperator.__matmul__cCrdre)rr]r __rmul__r_rrr __rmatmul__rgzLinearOperator.__rmatmul__cCst|r t||S||SrW)rr]r^_rdotr=rrrrhs   zLinearOperator.__rmul__cCst|tr t||St|rt||St|s!t|s!t|}|j dks2|j dkr:|j ddkr:|j |j j S|j dkrG|j |j j Std|)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. r0r rrb)rArrcrr]r^rrr'rCr!r9r(r<r r=rrrrjs     " zLinearOperator._rdotcCst|r t||StSrW)rr]_PowerLinearOperatorNotImplemented)r"prrr__pow__  zLinearOperator.__pow__cCst|tr t||StSrW)rAr_SumLinearOperatorrlr=rrr__add__rozLinearOperator.__add__cCs t|dS)Nr$)r^r5rrr__neg__r\zLinearOperator.__neg__cCs || SrW)rqr=rrr__sub__ zLinearOperator.__sub__cCs<|j\}}|jdur d}ndt|j}d|||jj|fS)Nzunspecified dtypezdtype=z<%dx%d %s with %s>)r!rstrr__name__)r"rErFdtrrr__repr__ s  zLinearOperator.__repr__cC|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. )rKr5rrradjoint)szLinearOperator.adjointcCry)zTranspose this linear operator. Returns a LinearOperator that represents the transpose of this one. Can be abbreviated self.T instead of self.transpose(). ) _transposer5rrr transpose;szLinearOperator.transposecCt|S)z6Default implementation of _adjoint; defers to rmatvec.)_AdjointLinearOperatorr5rrrrKEzLinearOperator._adjointcCr})z? 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'A' may be any of the following types: - ndarray - matrix - sparse array (e.g. csr_array, lil_array, 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(NrJrVr)rJrVrztype not understood)rArrndarrayrBrCr atleast_2dr'rrrrrJrVrr!r(rQ)rrJrVrrrrr_s.      rW)rrnumpyr scipy.sparserscipy.sparse._sputilsrrrr__all__rrr~rrrprcr^rkrrrrrrrrs., .  "#