o Eb@sdZddlmmZddlZedZddZ ddZ dd Z d d Z d d Z ddZddZddZddZddZddZddZddZddZdqd!d"Zdqd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.Zd/d0Zd1d2Zd3d4Z d5d6Z!d7d8Z"d9d:Z#d;d<Z$d=d>Z%d?d@Z&dAdBZ'dCdDZ(dEdFZ)dGdHZ*dIdJZ+dKdLZ,dMdNZ-dOdPZ.dQdRZ/dqdSdTZ0dqdUdVZ1dWdXZ2dYdZZ3d[d\Z4d]d^Z5d_d`Z6dadbZ7dcddZ8dedfZ9dgdhZ:didjZ;dkdlZdS)rz0 Direct wrappers for Fortran `id_dist` backend. Nznonzero return codecCs0t|}|jjr|jdd}|St|}|S)z6 Same as np.asfortranarray, but ensure a copy Forder)npZasarrayflags f_contiguouscopyasfortranarray)Ar E/usr/lib/python3/dist-packages/scipy/linalg/_interpolative_backend.py_asfortranarray_copy(s   r cC t|S)a  Generate standard uniform pseudorandom numbers via a very efficient lagged Fibonacci method. :param n: Number of pseudorandom numbers to generate. :type n: int :return: Pseudorandom numbers. :rtype: :class:`numpy.ndarray` )_idid_srand)nr r r r8s rcCst|}t|dS)z Initialize seed values for :func:`id_srand` (any appropriately random numbers will do). :param t: Array of 55 seed values. :type t: :class:`numpy.ndarray` N)rr r id_srandi)tr r r rHs rcCs tdS)z5 Reset seed values to their original values. N)r id_srandor r r r rUs rcCt|||S)a| Transform real vector via a composition of Rokhlin's random transform, random subselection, and an FFT. In contrast to :func:`idd_sfrm`, this routine works best when the length of the transformed vector is the power-of-two integer output by :func:`idd_frmi`, or when the length is not specified but instead determined a posteriori from the output. The returned transformed vector is randomly permuted. :param n: Greatest power-of-two integer satisfying `n <= x.size` as obtained from :func:`idd_frmi`; `n` is also the length of the output vector. :type n: int :param w: Initialization array constructed by :func:`idd_frmi`. :type w: :class:`numpy.ndarray` :param x: Vector to be transformed. :type x: :class:`numpy.ndarray` :return: Transformed vector. :rtype: :class:`numpy.ndarray` )ridd_frmrwxr r r r`rcCt||||S)a Transform real vector via a composition of Rokhlin's random transform, random subselection, and an FFT. In contrast to :func:`idd_frm`, this routine works best when the length of the transformed vector is known a priori. :param l: Length of transformed vector, satisfying `l <= n`. :type l: int :param n: Greatest power-of-two integer satisfying `n <= x.size` as obtained from :func:`idd_sfrmi`. :type n: int :param w: Initialization array constructed by :func:`idd_sfrmi`. :type w: :class:`numpy.ndarray` :param x: Vector to be transformed. :type x: :class:`numpy.ndarray` :return: Transformed vector. :rtype: :class:`numpy.ndarray` )ridd_sfrmlrrrr r r r}rcCr)aC Initialize data for :func:`idd_frm`. :param m: Length of vector to be transformed. :type m: int :return: Greatest power-of-two integer `n` satisfying `n <= m`. :rtype: int :return: Initialization array to be used by :func:`idd_frm`. :rtype: :class:`numpy.ndarray` )ridd_frmimr r r r  r cC t||S)a Initialize data for :func:`idd_sfrm`. :param l: Length of output transformed vector. :type l: int :param m: Length of the vector to be transformed. :type m: int :return: Greatest power-of-two integer `n` satisfying `n <= m`. :rtype: int :return: Initialization array to be used by :func:`idd_sfrm`. :rtype: :class:`numpy.ndarray` )r idd_sfrmirr"r r r r% r%cCZt|}t||\}}}|jd}|jd|||j|||fdd}|||fS)a Compute ID of a real matrix to a specified relative precision. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Rank of ID. :rtype: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` Nrr)r riddp_idshapeTravelreshapeepsr kidxrnormsrprojr r r r*  , r*cCVt|}t||\}}|jd}|jd|||j|||fdd}||fS)aQ Compute ID of a real matrix to a specified rank. :param A: Matrix. :type A: :class:`numpy.ndarray` :param k: Rank of ID. :type k: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` r)Nrr)r riddr_idr+r,r-r.r r1r2r3rr4r r r r7  ,r7cC8t|}|jdkrt|||S|ddt|fS)as Reconstruct matrix from real ID. :param B: Skeleton matrix. :type B: :class:`numpy.ndarray` :param idx: Column index array. :type idx: :class:`numpy.ndarray` :param proj: Interpolation coefficients. :type proj: :class:`numpy.ndarray` :return: Reconstructed matrix. :rtype: :class:`numpy.ndarray` rN)rr sizer idd_reconidargsortBr2r4r r r r<  r<cCr$)a6 Reconstruct interpolation matrix from real ID. :param idx: Column index array. :type idx: :class:`numpy.ndarray` :param proj: Interpolation coefficients. :type proj: :class:`numpy.ndarray` :return: Interpolation matrix. :rtype: :class:`numpy.ndarray` )r idd_reconintr2r4r r r rA rAcCt|}t|||S)aN Reconstruct skeleton matrix from real ID. :param A: Original matrix. :type A: :class:`numpy.ndarray` :param k: Rank of ID. :type k: int :param idx: Column index array. :type idx: :class:`numpy.ndarray` :return: Skeleton matrix. :rtype: :class:`numpy.ndarray` )rr r idd_copycolsr r1r2r r r rE% rEcC2t|}t|||\}}}}|rt|||fS)a Convert real ID to SVD. :param B: Skeleton matrix. :type B: :class:`numpy.ndarray` :param idx: Column index array. :type idx: :class:`numpy.ndarray` :param proj: Interpolation coefficients. :type proj: :class:`numpy.ndarray` :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` )rr r idd_id2svd_RETCODE_ERRORr?r2r4UVSierr r r rI?  rIcCt|||||\}}|S)a Estimate spectral norm of a real matrix by the randomized power method. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matvect: Function to apply the matrix transpose to a vector, with call signature `y = matvect(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvect: function :param matvec: Function to apply the matrix to a vector, with call signature `y = matvec(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec: function :param its: Number of power method iterations. :type its: int :return: Spectral norm estimate. :rtype: float )r idd_snorm)r"rmatvectmatvecitssnormvr r r rSbrSc Ct|||||||S)a0 Estimate spectral norm of the difference of two real matrices by the randomized power method. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matvect: Function to apply the transpose of the first matrix to a vector, with call signature `y = matvect(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvect: function :param matvect2: Function to apply the transpose of the second matrix to a vector, with call signature `y = matvect2(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvect2: function :param matvec: Function to apply the first matrix to a vector, with call signature `y = matvec(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec: function :param matvec2: Function to apply the second matrix to a vector, with call signature `y = matvec2(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec2: function :param its: Number of power method iterations. :type its: int :return: Spectral norm estimate of matrix difference. :rtype: float )r idd_diffsnorm)r"rrTZmatvect2rUmatvec2rVr r r r['r[cC0t|}t||\}}}}|rt|||fS)a Compute SVD of a real matrix to a specified rank. :param A: Matrix. :type A: :class:`numpy.ndarray` :param k: Rank of SVD. :type k: int :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` )rr riddr_svdrJr r1rLrMrNrOr r r r_  r_c Ct|}|j\}}t||\}}}}}} | rt||d|||dj||fdd} ||d|||dj||fdd} ||d||d} | | | fS)a Compute SVD of a real matrix to a specified relative precision. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` r)rr)rr r+riddp_svdrJr. r0r r"rr1iUiViSrrOrLrMrNr r r rc  ** rcc Cst|}|j\}}t|\}}tj|d|d|ddd}t||||\}}}|d|||j|||fdd}|||fS)a Compute ID of a real matrix to a specified relative precision using random sampling. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Rank of ID. :rtype: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` r)rrN)rr r+r emptyriddp_aidr. r0r r"rn2rr4r1r2r r r rks   "& rkcCsZt|}|j\}}t|\}}tj|||d|ddd}t||||\}}|S)ae Estimate rank of a real matrix to a specified relative precision using random sampling. The output rank is typically about 8 higher than the actual rank. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Rank estimate. :rtype: int r)rr)rr r+r rjr idd_estrankr0r r"rrmrrar1r r r rns   "rncCst|}|j\}}t|\}}tjtt||dd|d|ddt||dd|d|ddd}t||||\}}} } }} | rMt ||d|||dj ||fdd} || d| ||dj ||fdd} || d| |d}| | |fS)a Compute SVD of a real matrix to a specified relative precision using random sampling. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` r)rirr) rr r+rr rjmaxmin iddp_asvdrJr.r0r r"rrmZwinitrr1rerfrgrOrLrMrNr r r rv-s  4** rvcCs~tj|dd|t||ddd}t|||||\}}}}|dkr't|d|||j|||fdd}|||fS)a Compute ID of a real matrix to a specified relative precision using random matrix-vector multiplication. :param eps: Relative precision. :type eps: float :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matvect: Function to apply the matrix transpose to a vector, with call signature `y = matvect(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvect: function :return: Rank of ID. :rtype: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` r)rirrrN)rrjruriddp_ridrJr.)r0r"rrTr4r1r2rOr r r rxWs (& rxcC"t||||\}}}|rt|S)aQ Estimate rank of a real matrix to a specified relative precision using random matrix-vector multiplication. :param eps: Relative precision. :type eps: float :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matvect: Function to apply the matrix transpose to a vector, with call signature `y = matvect(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvect: function :return: Rank estimate. :rtype: int )r idd_findrankrJ)r0r"rrTr1rprOr r r rz}rzcCt|||||\}}}}} } | rt| |d|||dj||fdd} | |d|||dj||fdd} | |d||d} | | | fS)a Compute SVD of a real matrix to a specified relative precision using random matrix-vector multiplication. :param eps: Relative precision. :type eps: float :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matvect: Function to apply the matrix transpose to a vector, with call signature `y = matvect(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvect: function :param matvec: Function to apply the matrix to a vector, with call signature `y = matvec(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec: function :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` r)rr)r iddp_rsvdrJr.)r0r"rrTrUr1rerfrgrrOrLrMrNr r r r}#** r}cCxt|}|j\}}t|||}t|||\}}||kr-tj|||fddd}||fS|j|||fdd}||fS)ag Compute ID of a real matrix to a specified rank using random sampling. :param A: Matrix. :type A: :class:`numpy.ndarray` :param k: Rank of ID. :type k: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` Zfloat64rZdtyperr)rr r+ iddr_aidiriddr_aidrjr.r r1r"rrr2r4r r r r   rcCr)aO Initialize array for :func:`iddr_aid`. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param k: Rank of ID. :type k: int :return: Initialization array to be used by :func:`iddr_aid`. :rtype: :class:`numpy.ndarray` )rrr"rr1r r r rrc Cst|}|j\}}tjd|d|d|d|d|dddd}t|||}||d |j<t|||\}}}} | d krEt|||fS) a Compute SVD of a real matrix to a specified rank using random sampling. :param A: Matrix. :type A: :class:`numpy.ndarray` :param k: Rank of SVD. :type k: int :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` rirsdrrNr) rr r+rjrr;r iddr_asvdrJ r r1r"rrZw_rLrMrNrOr r r rs  :  rcCBt||||\}}|d|||j|||fdd}||fS)a Compute ID of a real matrix to a specified rank using random matrix-vector multiplication. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matvect: Function to apply the matrix transpose to a vector, with call signature `y = matvect(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvect: function :param k: Rank of ID. :type k: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` Nrr)riddr_ridr.)r"rrTr1r2r4r r r r)&rc Cs0t|||||\}}}}|dkrt|||fS)a Compute SVD of a real matrix to a specified rank using random matrix-vector multiplication. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matvect: Function to apply the matrix transpose to a vector, with call signature `y = matvect(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvect: function :param matvec: Function to apply the matrix to a vector, with call signature `y = matvec(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec: function :param k: Rank of SVD. :type k: int :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` r)r iddr_rsvdrJ) r"rrTrUr1rLrMrNrOr r r rMs# rcCr)a Transform complex vector via a composition of Rokhlin's random transform, random subselection, and an FFT. In contrast to :func:`idz_sfrm`, this routine works best when the length of the transformed vector is the power-of-two integer output by :func:`idz_frmi`, or when the length is not specified but instead determined a posteriori from the output. The returned transformed vector is randomly permuted. :param n: Greatest power-of-two integer satisfying `n <= x.size` as obtained from :func:`idz_frmi`; `n` is also the length of the output vector. :type n: int :param w: Initialization array constructed by :func:`idz_frmi`. :type w: :class:`numpy.ndarray` :param x: Vector to be transformed. :type x: :class:`numpy.ndarray` :return: Transformed vector. :rtype: :class:`numpy.ndarray` )ridz_frmrr r r rzrrcCr)a Transform complex vector via a composition of Rokhlin's random transform, random subselection, and an FFT. In contrast to :func:`idz_frm`, this routine works best when the length of the transformed vector is known a priori. :param l: Length of transformed vector, satisfying `l <= n`. :type l: int :param n: Greatest power-of-two integer satisfying `n <= x.size` as obtained from :func:`idz_sfrmi`. :type n: int :param w: Initialization array constructed by :func:`idd_sfrmi`. :type w: :class:`numpy.ndarray` :param x: Vector to be transformed. :type x: :class:`numpy.ndarray` :return: Transformed vector. :rtype: :class:`numpy.ndarray` )ridz_sfrmrr r r rrrcCr)aC Initialize data for :func:`idz_frm`. :param m: Length of vector to be transformed. :type m: int :return: Greatest power-of-two integer `n` satisfying `n <= m`. :rtype: int :return: Initialization array to be used by :func:`idz_frm`. :rtype: :class:`numpy.ndarray` )ridz_frmir!r r r rr#rcCr$)a Initialize data for :func:`idz_sfrm`. :param l: Length of output transformed vector. :type l: int :param m: Length of the vector to be transformed. :type m: int :return: Greatest power-of-two integer `n` satisfying `n <= m`. :rtype: int :return: Initialization array to be used by :func:`idz_sfrm`. :rtype: :class:`numpy.ndarray` )r idz_sfrmir&r r r rr'rcCr()a Compute ID of a complex matrix to a specified relative precision. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Rank of ID. :rtype: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` r)Nrr)r ridzp_idr+r,r-r.r/r r r rr5rcCr6)aT Compute ID of a complex matrix to a specified rank. :param A: Matrix. :type A: :class:`numpy.ndarray` :param k: Rank of ID. :type k: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` r)Nrr)r ridzr_idr+r,r-r.r8r r r rr9rcCr:)av Reconstruct matrix from complex ID. :param B: Skeleton matrix. :type B: :class:`numpy.ndarray` :param idx: Column index array. :type idx: :class:`numpy.ndarray` :param proj: Interpolation coefficients. :type proj: :class:`numpy.ndarray` :return: Reconstructed matrix. :rtype: :class:`numpy.ndarray` rN)rr r;r idz_reconidr=r>r r r rr@rcCr$)a9 Reconstruct interpolation matrix from complex ID. :param idx: Column index array. :type idx: :class:`numpy.ndarray` :param proj: Interpolation coefficients. :type proj: :class:`numpy.ndarray` :return: Interpolation matrix. :rtype: :class:`numpy.ndarray` )r idz_reconintrBr r r r-rCrcCrD)aQ Reconstruct skeleton matrix from complex ID. :param A: Original matrix. :type A: :class:`numpy.ndarray` :param k: Rank of ID. :type k: int :param idx: Column index array. :type idx: :class:`numpy.ndarray` :return: Skeleton matrix. :rtype: :class:`numpy.ndarray` )rr r idz_copycolsrFr r r r?rGrcCrH)a Convert complex ID to SVD. :param B: Skeleton matrix. :type B: :class:`numpy.ndarray` :param idx: Column index array. :type idx: :class:`numpy.ndarray` :param proj: Interpolation coefficients. :type proj: :class:`numpy.ndarray` :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` )rr r idz_id2svdrJrKr r r rYrPrcCrR)a Estimate spectral norm of a complex matrix by the randomized power method. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matveca: Function to apply the matrix adjoint to a vector, with call signature `y = matveca(x)`, where `x` and `y` are the input and output vectors, respectively. :type matveca: function :param matvec: Function to apply the matrix to a vector, with call signature `y = matvec(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec: function :param its: Number of power method iterations. :type its: int :return: Spectral norm estimate. :rtype: float )r idz_snorm)r"rmatvecarUrVrWrXr r r r|rYrc CrZ)a/ Estimate spectral norm of the difference of two complex matrices by the randomized power method. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matveca: Function to apply the adjoint of the first matrix to a vector, with call signature `y = matveca(x)`, where `x` and `y` are the input and output vectors, respectively. :type matveca: function :param matveca2: Function to apply the adjoint of the second matrix to a vector, with call signature `y = matveca2(x)`, where `x` and `y` are the input and output vectors, respectively. :type matveca2: function :param matvec: Function to apply the first matrix to a vector, with call signature `y = matvec(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec: function :param matvec2: Function to apply the second matrix to a vector, with call signature `y = matvec2(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec2: function :param its: Number of power method iterations. :type its: int :return: Spectral norm estimate of matrix difference. :rtype: float )r idz_diffsnorm)r"rrZmatveca2rUr\rVr r r rr]rcCr^)a Compute SVD of a complex matrix to a specified rank. :param A: Matrix. :type A: :class:`numpy.ndarray` :param k: Rank of SVD. :type k: int :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` )rr ridzr_svdrJr`r r r rrarc Crb)a Compute SVD of a complex matrix to a specified relative precision. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` r)rr)rr r+ridzp_svdrJr.rdr r r rrhrc Cst|}|j\}}t|\}}tj|d|d|dddd}t||||\}}}|d|||j|||fdd}|||fS)a Compute ID of a complex matrix to a specified relative precision using random sampling. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Rank of ID. :rtype: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` rir) complex128rrNr)rr r+rrjridzp_aidr.rlr r r r s   $& rcCs\t|}|j\}}t|\}}tj|||d|dddd}t||||\}}|S)ah Estimate rank of a complex matrix to a specified relative precision using random sampling. The output rank is typically about 8 higher than the actual rank. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Rank estimate. :rtype: int r)rrr)rr r+rrjr idz_estrankror r r r)s   $rcCst|}|j\}}t|\}}tjtt||dd|d|ddt||dd|d|dtjdd}t ||||\}}} } }} | rOt ||d|||dj ||fdd } || d| ||dj ||fdd } || d| |d}| | |fS) a Compute SVD of a complex matrix to a specified relative precision using random sampling. :param eps: Relative precision. :type eps: float :param A: Matrix. :type A: :class:`numpy.ndarray` :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` r)rqrr rirrr) rr r+rrrjrtrur idzp_asvdrJr.rwr r r rGs  4** rcCs~tj|dd|t||dtjdd}t|||||\}}}}|r't|d|||j|||fdd}|||fS)a Compute ID of a complex matrix to a specified relative precision using random matrix-vector multiplication. :param eps: Relative precision. :type eps: float :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matveca: Function to apply the matrix adjoint to a vector, with call signature `y = matveca(x)`, where `x` and `y` are the input and output vectors, respectively. :type matveca: function :return: Rank of ID. :rtype: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` r)rirrNr)rrjrurridzp_ridrJr.)r0r"rrr4r1r2rOr r r rqs& rcCry)aR Estimate rank of a complex matrix to a specified relative precision using random matrix-vector multiplication. :param eps: Relative precision. :type eps: float :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matveca: Function to apply the matrix adjoint to a vector, with call signature `y = matveca(x)`, where `x` and `y` are the input and output vectors, respectively. :type matveca: function :return: Rank estimate. :rtype: int )r idz_findrankrJ)r0r"rrr1rprOr r r rr{rcCr|)a Compute SVD of a complex matrix to a specified relative precision using random matrix-vector multiplication. :param eps: Relative precision. :type eps: float :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matveca: Function to apply the matrix adjoint to a vector, with call signature `y = matveca(x)`, where `x` and `y` are the input and output vectors, respectively. :type matveca: function :param matvec: Function to apply the matrix to a vector, with call signature `y = matvec(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec: function :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` r)rr)r idzp_rsvdrJr.)r0r"rrrUr1rerfrgrrOrLrMrNr r r rr~rcCr)aj Compute ID of a complex matrix to a specified rank using random sampling. :param A: Matrix. :type A: :class:`numpy.ndarray` :param k: Rank of ID. :type k: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` rrrr)rr r+ idzr_aidiridzr_aidrjr.rr r r rrrcCr)aO Initialize array for :func:`idzr_aid`. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param k: Rank of ID. :type k: int :return: Initialization array to be used by :func:`idzr_aid`. :rtype: :class:`numpy.ndarray` )rrrr r r rrrc Cst|}|j\}}tjd|d|d|d|d|dd|ddd d }t|||}||d |j<t|||\}}}} | rHt|||fS) a Compute SVD of a complex matrix to a specified rank using random sampling. :param A: Matrix. :type A: :class:`numpy.ndarray` :param k: Rank of SVD. :type k: int :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` rirrr ZrrrN) rr r+rjrr;r idzr_asvdrJrr r r r!s  6  rcCr)a Compute ID of a complex matrix to a specified rank using random matrix-vector multiplication. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matveca: Function to apply the matrix adjoint to a vector, with call signature `y = matveca(x)`, where `x` and `y` are the input and output vectors, respectively. :type matveca: function :param k: Rank of ID. :type k: int :return: Column index array. :rtype: :class:`numpy.ndarray` :return: Interpolation coefficients. :rtype: :class:`numpy.ndarray` Nrr)ridzr_ridr.)r"rrr1r2r4r r r rGrrc Cs,t|||||\}}}}|rt|||fS)a Compute SVD of a complex matrix to a specified rank using random matrix-vector multiplication. :param m: Matrix row dimension. :type m: int :param n: Matrix column dimension. :type n: int :param matveca: Function to apply the matrix adjoint to a vector, with call signature `y = matveca(x)`, where `x` and `y` are the input and output vectors, respectively. :type matveca: function :param matvec: Function to apply the matrix to a vector, with call signature `y = matvec(x)`, where `x` and `y` are the input and output vectors, respectively. :type matvec: function :param k: Rank of SVD. :type k: int :return: Left singular vectors. :rtype: :class:`numpy.ndarray` :return: Right singular vectors. :rtype: :class:`numpy.ndarray` :return: Singular values. :rtype: :class:`numpy.ndarray` )r idzr_rsvdrJ) r"rrrUr1rLrMrNrOr r r rks# r)rQ)?__doc__Zscipy.linalg._interpolativeZlinalgZ_interpolativerZnumpyr RuntimeErrorrJr rrrrrr r%r*r7r<rArErIrSr[r_rcrkrnrvrxrzr}rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r r r st   # .$*&"0$$- # .$*("0& $