U md@sdZddlmmZddlZedZddZ ddZ dd Z d d Z d d Z ddZddZddZddZddZddZddZddZddZdqd!d"Zdrd#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/dsdSdTZ0dtdUdVZ1dWdXZ2dYdZZ3d[d\Z4d]d^Z5d_d`Z6dadbZ7dcddZ8dedfZ9dgdhZ:didjZ;dkdlZdS)uz0 Direct wrappers for Fortran `id_dist` backend. Nznonzero return codecCs.t|}|jjr |jdd}n t|}|S)z6 Same as np.asfortranarray, but ensure a copy Forder)npZasarrayflags f_contiguouscopyasfortranarray)Ar \/home/sam/Atlas/atlas_env/lib/python3.8/site-packages/scipy/linalg/_interpolative_backend.py_asfortranarray_copy(s   r cCs 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 rUsrcCst|||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`srcCst||||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}srcCs t|S)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 rsrcCs 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_sfrmirrr r r rsrcCsZt|}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!s  ,r!cCsVt|}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 r(r)r*rr+r r r r,s  ,r,cCs<t|}|jdkr"t|||S|ddt|fSdS)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_reconidargsortBr)r+r r r r/s  r/cCs t||S)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_reconintr)r+r r r r3sr3cCst|}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 r(r)r r r r5%s r5cCs2t|}t|||\}}}}|r(t|||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_ERRORr2r)r+UVSierr r r r7?s  r7cCst|||||\}}|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)rrmatvectmatvecitssnormvr r r r?bsr?c Cst|||||||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)rrr@Zmatvect2rAmatvec2rBr r r rEs'rEcCs0t|}t||\}}}}|r&t|||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_svdr8r r(r:r;r<r=r r r rGs  rGc Cst|}|j\}}t||\}}}}}} | r4t||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_svdr8r% r'r rrr(iUiViSrr=r:r;r<r r r rIs  **rIc 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"remptyriddp_aidr% r'r rrn2rr+r(r)r r r rPs   "&rPcCsZt|}|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"rrOr idd_estrankr'r rrrRrrar(r r r rSs    "rScCst|}|j\}}t|\}}tjtt||dd|d|ddt||dd|d|ddd}t||||\}}} } }} | rt ||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 rNrr) rr r"rrrOmaxmin iddp_asvdr8r%r'r rrrRZwinitrr(rKrLrMr=r:r;r<r r r r[-s   4**r[cCs~tj|dd|t||ddd}t|||||\}}}}|dkrNt|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 rNrrrN)rrOrZriddp_ridr8r%)r'rrr@r+r(r)r=r r r r]Ws (&r]cCs"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_findrankr8)r'rrr@r(rUr=r r r r^}sr^cCst|||||\}}}}} } | r&t| |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_rsvdr8r%)r'rrr@rAr(rKrLrMrr=r:r;r<r r r r_s#**r_cCsrt|}|j\}}t|||}t|||\}}||krTtj|||fddd}n|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_aidrOr%r r(rrrr)r+r r r rbs   rbcCst|||S)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` )rrarrr(r r r rasrac Cst|}|j\}}tjd|d|d|d|d|dddd}t|||}||d |j<t|||\}}}} | d krt|||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` rNrXdrrNr) rr r"rOrar.r iddr_asvdr8 r r(rrrZw_r:r;r<r=r r r ris  : ricCsBt||||\}}|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%)rrr@r(r)r+r r r rk)s&rkc Cs0t|||||\}}}}|dkr&t|||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_rsvdr8) rrr@rAr(r:r;r<r=r r r rlMs#rlcCst|||S)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 rmzsrmcCst||||S)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 rnsrncCs t|S)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_frmirr r r rosrocCs t||S)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_sfrmirr r r rpsrpcCsZt|}t||\}}}|jd}|jd|||j|||fdd}|||fS)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 rqs  ,rqcCsVt|}t||\}}|jd}|jd|||j|||fdd}||fS)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%r-r r r rrs  ,rrcCs<t|}|jdkr"t|||S|ddt|fSdS)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_reconidr0r1r r r rss  rscCs t||S)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_reconintr4r r r rt-srtcCst|}t|||S)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_copycolsr6r r r ru?s rucCs2t|}t|||\}}}}|r(t|||fS)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_id2svdr8r9r r r rvYs  rvcCst|||||\}}|S)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)rrmatvecarArBrCrDr r r rw|srwc Cst|||||||S)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)rrrxZmatveca2rArFrBr r r rys'rycCs0t|}t||\}}}}|r&t|||fS)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_svdr8rHr r r rzs  rzc Cst|}|j\}}t||\}}}}}} | r4t||d|||dj||fdd} ||d|||dj||fdd} ||d||d} | | | fS)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_svdr8r%rJr r r r{s  **r{c 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` rNr complex128rr`Nr)rr r"rorOridzp_aidr%rQr r r r} s   $&r}cCs\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 r|rr`)rr r"rorOr idz_estrankrTr r r r~)s    $r~cCst|}|j\}}t|\}}tjtt||dd|d|ddt||dd|d|dtjdd}t ||||\}}} } }} | rt ||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 rVrW rNrr`r) rr r"rrorOrYrZr| idzp_asvdr8r%r\r r r rGs"  4**rcCs~tj|dd|t||dtjdd}t|||||\}}}}|rNt|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 rNrr`Nr)rrOrZr|ridzp_ridr8r%)r'rrrxr+r(r)r=r r r rqs&rcCs"t||||\}}}|rt|S)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_findrankr8)r'rrrxr(rUr=r r r rsrcCst|||||\}}}}} } | r&t| |d|||dj||fdd} | |d|||dj||fdd} | |d||d} | | | fS)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_rsvdr8r%)r'rrrxrAr(rKrLrMrr=r:r;r<r r r rs#**rcCsrt|}|j\}}t|||}t|||\}}||krTtj|||fddd}n|j|||fdd}||fS)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` r|rr`r)rr r" idzr_aidiridzr_aidrOr%rcr r r rs   rcCst|||S)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` )rrrdr r r rsrc Cst|}|j\}}tjd|d|d|d|d|dd|ddd d }t|||}||d |j<t|||\}}}} | rt|||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` rNrfrgr Zr|rr`N) rr r"rOrr.r idzr_asvdr8rjr r r r!s  6 rcCsBt||||\}}|d|||j|||fdd}||fS)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%)rrrxr(r)r+r r r rGs&rc Cs,t|||||\}}}}|r"t|||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_rsvdr8) rrrxrAr(r:r;r<r=r r r rks#r)r>)r>)r>)r>)?__doc__Zscipy.linalg._interpolativeZlinalgZ_interpolativernumpyr RuntimeErrorr8r rrrrrrrr!r,r/r3r5r7r?rErGrIrPrSr[r]r^r_rbrarirkrlrmrnrorprqrrrsrtrurvrwryrzr{r}r~rrrrrrrrrr r r r sr  # .$*&"0$$-# .$*("0&$