UgdZddlmcmZddlZedZdZ dZ dZ dZ dZ d Zd Zd Zd Zd ZdZdZdZdZd;dZd;dZdZdZdZdZdZdZdZdZ dZ!dZ"dZ#d Z$d!Z%d"Z&d#Z'd$Z(d%Z)d&Z*d'Z+d(Z,d)Z-d*Z.d+Z/d;d,Z0d;d-Z1d.Z2d/Z3d0Z4d1Z5d2Z6d3Z7d4Z8d5Z9d6Z:d7Z;d8ZdS)r>v$ QA+a##KC  A 399;;x1Q3x ( (!QqS ( = =D 9rctj|}|jdkrtj|||S|ddtj|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_reconidargsortBr9r;s rrCrCO$ !A y1}}q#t,,,BJsOO#$$rc,tj||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_reconintr9r;s rrIrI  C & &&rcVtj|}tj|||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 r8r9s rrMrM%)$ !A  Aq# & &&rc|tj|}tj|||\}}}}|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_ERRORrFr9r;UVSiers rrQrQ?H0 !A>!S$//LAq!S  a7Nrc<tj|||||\}}|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)rmatvectmatvecitssnormvs rr[r[b$8}Q7FC88HE1 Lrc 6tj|||||||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)rr\matvect2r]matvec2r^s rrcrc"N  Q7Hfgs K KKrcztj|}tj||\}}}}|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_svdrRr r8rTrUrVrWs rrhrhF* !A<1%%LAq!S  a7Nrcltj|}|j\}}tj||\}}}}}} | rt ||dz |||zzdz ||fd} ||dz |||zzdz ||fd} ||dz ||zdz } | | | 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` r0rr)rr r2riddp_svdrRr5 r7r r)rr8iUiViSrrWrTrUrVs rrlrl* !A 7DAqLa00Ar2r1c  "Q$r!A#vax-  !Qs 33A "Q$r!A#vax-  !Qs 33A "Q$r!tAv+A a7Nrc@tj|}|j\}}t|\}}tj|d|zdzz|zdzd}t j||||\}}}|d|||z z|||z fd}|||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` r0rrN)rr r2r'emptyriddp_aidr5 r7r r)rn2rr;r8r9s rrurus, !A 7DAq QKKEB 8AqtaxL2%) 5 5 5D<Q400LAsD AaC> ! !1ac(# ! 6 6D c4<rctj|}|j\}}t|\}}tj||z|dz|dzzzd}t j||||\}}|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 r0rr)rr r2r'rtr idd_estrankr7r r)rrwrrar8s rryrysv$ !A 7DAq QKKEB !B$!a%"q&)) 5 5 5B OCAr * *EAr Hrc ltj|}|j\}}tj|\}}tjt t||dzd|zd|zzdzzdt||dzzzd|zdz|dzzd}tj||||\}}} } }} | rt||dz |||zzdz  ||fd} || dz | ||zzdz  ||fd} || dz | |zdz }| | |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` r0rsrr) rr r2rr'rtmaxmin iddp_asvdrRr5r7r r)rrwwinitrr8rnrorprWrTrUrVs rrr-sb, !A 7DAq QIB  SAYY]QqS1Q3Y] +bQAo = qS1WrAv      A M#q%;;Ar2r1c  "Q$r!A#vax-  !Qs 33A "Q$r!A#vax-  !Qs 33A "Q$r!tAv+A a7Nrctj|dzd|zt||dzzzd}tj|||||\}}}}|dkrt |d|||z z|||z fd}|||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` r0rsrrrN)rrtrriddp_ridrRr5)r7r)rr\r;r8r9rWs rrrWs< 8AEAaCQQ//s ; ; ;D S!Q>>AsD# axx AaC> ! !1ac(# ! 6 6D c4<rcNtj||||\}}}|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_findrankrR)r7r)rr\r8r{rWs rrr}30!#q!W55JAr3  Hrc6tj|||||\}}}}} } | rt| |dz |||zzdz ||fd} | |dz |||zzdz ||fd} | |dz ||zdz } | | | 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` r0rr)r iddp_rsvdrRr5)r7r)rr\r]r8rnrorprrWrTrUrVs rrrF M#q!WfEEAr2r1c  "Q$r!A#vax-  !Qs 33A "Q$r!A#vax-  !Qs 33A "Q$r!tAv+A a7Nrctj|}|j\}}t|||}t j|||\}}||krtj|||z fdd}n||||z fd}||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` float64rdtyperr)rr r2 iddr_aidiriddr_aidrtr5r r8r)rrr9r;s rrrs$ !A 7DAq!QA Q1%%ICAvvxAaC ===||Q!HC|00 9rc.tj|||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` )rrr)rr8s rrr$ =Aq ! !!rc@tj|}|j\}}tjd|zdz|zd|zdz|zzd|dzzzdzd}t |||}||d |j<t j|||\}}}} | 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` rsrdrrNr) rr r2rtrrBr iddr_asvdrR r r8r)rrw_rTrUrVrWs rrrs* !A 7DAq !A#(A1r1 ,r!Q$w6 ! !1ac(# ! 6 6D 9rc`tj|||||\}}}}|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_rsvdrR) r)rr\r]r8rTrUrVrWs rrrMs>F=Aw::LAq!S axx a7Nrc.tj|||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_frmrs rrrzr rc0tj||||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_sfrmr#s rrrr%rc*tj|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_frmir(s rrrr*rc,tj||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_sfrmir-s rrrr.rct|}tj||\}}}|jd}|jd|||z z|||z fd}|||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` r0Nrr)rridzp_idr2r3r4r5r6s rrrr<rct|}tj||\}}|jd}|jd|||z z|||z fd}||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` r0Nrr)rridzr_idr2r3r4r5r?s rrrr@rctj|}|jdkrtj|||S|ddtj|fS)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 rBr idz_reconidrDrEs rrrrGrc,tj||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_reconintrJs rrr-rKrcVtj|}tj|||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_copycolsrNs rrr?rOrc|tj|}tj|||\}}}}|rt|||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_id2svdrRrSs rrrYrXrc<tj|||||\}}|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)r)rmatvecar]r^r_r`s rrr|rarc 6tj|||||||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)r)rrmatveca2r]rer^s rrrrfrcztj|}tj||\}}}}|rt|||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_svdrRris rrrrjrcltj|}|j\}}tj||\}}}}}} | rt ||dz |||zzdz ||fd} ||dz |||zzdz ||fd} ||dz ||zdz } | | | 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` r0rr)rr r2ridzp_svdrRr5rms rrrrqrcBtj|}|j\}}t|\}}tj|d|zdzz|zdzdd}t j||||\}}}|d|||z z|||z fd}|||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` rsr0 complex128rrNr)rr r2rrtridzp_aidr5rvs rrr s, !A 7DAq QKKEB 8AqtaxL2%)S I I ID<Q400LAsD AaC> ! !1ac(# ! 6 6D c4<rctj|}|j\}}t|\}}tj||z|dz|dzzzdd}t j||||\}}|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 r0rrr)rr r2rrtr idz_estrankrzs rrr)sx$ !A 7DAq QKKEB !B$!a%"q&))S I I IB OCAr * *EAr Hrc tj|}|j\}}tj|\}}tjt t||dzd|zd|zzdzzdt||dzzzd|zdz|dzztjd}tj ||||\}}} } }} | rt||dz |||zzdz  ||fd } || dz | ||zzdz  ||fd } || dz | |zdz }| | |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` r0r}r~ rsrrr) rr r2rrrtrrr idzp_asvdrRr5rs rrrGsf, !A 7DAq QIB  SAYY]QqS1Q3Y^ ,qQA~ = qS1WrAv   m3 ( ( (A M#q%;;Ar2r1c  "Q$r!A#vax-  !Qs 33A "Q$r!A#vax-  !Qs 33A "Q$r!tAv+A a7Nrc(tj|dzd|zt||dzzztjd}t j|||||\}}}}|rt |d|||z z|||z fd}|||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` r0rsrrNr)rrtrrridzp_ridrRr5)r7r)rrr;r8r9rWs rrrqs< 8 A!SAYY]##m3 ( ( (D S!Q>>AsD#  AaC> ! !1ac(# ! 6 6D c4<rcNtj||||\}}}|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_findrankrR)r7r)rrr8r{rWs rrrrrc6tj|||||\}}}}} } | rt| |dz |||zzdz ||fd} | |dz |||zzdz ||fd} | |dz ||zdz } | | | 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` r0rr)r idzp_rsvdrRr5)r7r)rrr]r8rnrorprrWrTrUrVs rrrrrctj|}|j\}}t|||}t j|||\}}||krtj|||z fdd}n||||z fd}||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` rrrr)rr r2 idzr_aidiridzr_aidrtr5rs rrrs$ !A 7DAq!QA Q1%%ICAvvxAaC C@@@||Q!HC|00 9rc.tj|||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` )rrrs rrrrrcFtj|}|j\}}tjd|zdz|zd|zdz|zzd|dzzzd|zzdzdd }t |||}||d |j<t j|||\}}}} | 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` rsrrr ZrrrN) rr r2rtrrBr idzr_asvdrRrs rrr!s* !A 7DAq  1r1 !b!|#a1f,r!t3b8# ' ' 'A 1a  BAhrwhK=Aq))LAq!S  a7Nrctj||||\}}|d|||z z|||z fd}||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_ridr5)r)rrr8r9r;s rrrGrrcXtj|||||\}}}}|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_rsvdrR) r)rrr]r8rTrUrVrWs rrrks=F=Aw::LAq!S  a7Nr)rY)?__doc__scipy.linalg._interpolativelinalg_interpolativernumpyr RuntimeErrorrRrrrrrr"r'r,r1r>rCrIrMrQr[rcrhrlruryrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrs<*))))))))344            :$$$:$282%%%2'''$'''4F@'L'L'L'L\8H>   <###T###L   D)))`:"""2HH&&&Z   :$$$:$282%%%2'''$'''4F@'L'L'L'L\8H>   <###T%%%P   D)))`:"""2LH&&&&&r