ó ʽ÷Xc@`s¦dZddlmZmZmZdgZddlmZmZm Z ddl m Z ddl m Z ddlmZdd lmZd d d d ded „ZdS(s¢ Copyright (C) 2010 David Fong and Michael Saunders LSMR uses an iterative method. 07 Jun 2010: Documentation updated 03 Jun 2010: First release version in Python David Chin-lung Fong clfong@stanford.edu Institute for Computational and Mathematical Engineering Stanford University Michael Saunders saunders@stanford.edu Systems Optimization Laboratory Dept of MS&E, Stanford University. i(tdivisiontprint_functiontabsolute_importtlsmr(tzerostinftyt atleast_1d(tnorm(tsqrt(taslinearoperatori(t _sym_orthoggíµ ÷Æ°>g„×—AcL C`s¬t|ƒ}t|ƒ}|jdkr6|jƒ}nd(}d } d } d } d } |j\} }t| |gƒ}|d)krŠ|}n|rñtdƒtdƒtd| |fƒtd|ƒtd||fƒtd||fƒn|}t|ƒ}t |ƒ}d }|d krMd||}|j |ƒ}t|ƒ}n|d krjd||}nd }||}|}d}d}d}d }|j ƒ}t |ƒ}t |ƒ}|}d }d} d }!d }"d }#d }$||}%d }&d}'t |%ƒ}(d})d }*|}+d },d }-|d kr?d|}-n|}.||}/|/d krŽ|rrt|d ƒn||,||.|/|(|)|*fS|rtdƒt| | ƒd}0||}1d||d f}2d|.|/f}3d|0|1f}4tdj |2|3|4gƒƒnxð||kr|d}|j|ƒ||}t|ƒ}|d kr§d||}|j |ƒ||}t|ƒ}|d kr§d||}q§nt||ƒ\}5}6}7|}8t|7|ƒ\}9}:}|:|};|9|}|}<|#}=||}>||}?t|||;ƒ\}}}||}#| |}||>||8|<|}||#|||}||;||}|5|}@|6 |}A|9|@}B|: |@}|"}Ct| |>ƒ\}D}E}F|E|}"|D|} |E ||D|B}|=|C|!|F}!|#|"|!| }G|$|A|A}$t |$||Gd||ƒ}.|%||}%t |%ƒ}(|%||}%t|&|<ƒ}&|dkršt|'|<ƒ}'nt|&|?ƒt|'|?ƒ})t|ƒ}/t|ƒ}*|.|+}0|(|.d krù|/|(|.}1nt}1d|)}H|0d|(|*|+}I|||(|*|+}J||krJd},nd|Hdkrcd},nd|1dkr|d},nd|Idkr•d},n|H|-krªd},n|1|kr¿d},n|0|JkrÔd},n|rí|dksN|d ksN||d ksN|d d ksN|Hd!|-ksN|1d!|ksN|0d!|JksN|,d krí| | krzd } tdƒt| | ƒn| d} d||d f}2d|.|/f}3d|0|1f}4d"|(|)f}Ktdj |2|3|4|Kgƒƒqín|,d krPqqW|rtdƒtd#ƒt||,ƒtd$|,|.fƒtd%|(|/fƒtd&||)fƒtd'|*ƒt|2|3ƒt|4|Kƒn||,||.|/|(|)|*fS(*sìIterative solver for least-squares problems. lsmr solves the system of linear equations ``Ax = b``. If the system is inconsistent, it solves the least-squares problem ``min ||b - Ax||_2``. A is a rectangular matrix of dimension m-by-n, where all cases are allowed: m = n, m > n, or m < n. B is a vector of length m. The matrix A may be dense or sparse (usually sparse). Parameters ---------- A : {matrix, sparse matrix, ndarray, LinearOperator} Matrix A in the linear system. b : array_like, shape (m,) Vector b in the linear system. damp : float Damping factor for regularized least-squares. `lsmr` solves the regularized least-squares problem:: min ||(b) - ( A )x|| ||(0) (damp*I) ||_2 where damp is a scalar. If damp is None or 0, the system is solved without regularization. atol, btol : float, optional Stopping tolerances. `lsmr` continues iterations until a certain backward error estimate is smaller than some quantity depending on atol and btol. Let ``r = b - Ax`` be the residual vector for the current approximate solution ``x``. If ``Ax = b`` seems to be consistent, ``lsmr`` terminates when ``norm(r) <= atol * norm(A) * norm(x) + btol * norm(b)``. Otherwise, lsmr terminates when ``norm(A^{T} r) <= atol * norm(A) * norm(r)``. If both tolerances are 1.0e-6 (say), the final ``norm(r)`` should be accurate to about 6 digits. (The final x will usually have fewer correct digits, depending on ``cond(A)`` and the size of LAMBDA.) If `atol` or `btol` is None, a default value of 1.0e-6 will be used. Ideally, they should be estimates of the relative error in the entries of A and B respectively. For example, if the entries of `A` have 7 correct digits, set atol = 1e-7. This prevents the algorithm from doing unnecessary work beyond the uncertainty of the input data. conlim : float, optional `lsmr` terminates if an estimate of ``cond(A)`` exceeds `conlim`. For compatible systems ``Ax = b``, conlim could be as large as 1.0e+12 (say). For least-squares problems, `conlim` should be less than 1.0e+8. If `conlim` is None, the default value is 1e+8. Maximum precision can be obtained by setting ``atol = btol = conlim = 0``, but the number of iterations may then be excessive. maxiter : int, optional `lsmr` terminates if the number of iterations reaches `maxiter`. The default is ``maxiter = min(m, n)``. For ill-conditioned systems, a larger value of `maxiter` may be needed. show : bool, optional Print iterations logs if ``show=True``. Returns ------- x : ndarray of float Least-square solution returned. istop : int istop gives the reason for stopping:: istop = 0 means x=0 is a solution. = 1 means x is an approximate solution to A*x = B, according to atol and btol. = 2 means x approximately solves the least-squares problem according to atol. = 3 means COND(A) seems to be greater than CONLIM. = 4 is the same as 1 with atol = btol = eps (machine precision) = 5 is the same as 2 with atol = eps. = 6 is the same as 3 with CONLIM = 1/eps. = 7 means ITN reached maxiter before the other stopping conditions were satisfied. itn : int Number of iterations used. normr : float ``norm(b-Ax)`` normar : float ``norm(A^T (b - Ax))`` norma : float ``norm(A)`` conda : float Condition number of A. normx : float ``norm(x)`` Notes ----- .. versionadded:: 0.11.0 References ---------- .. [1] D. C.-L. Fong and M. A. Saunders, "LSMR: An iterative algorithm for sparse least-squares problems", SIAM J. Sci. Comput., vol. 33, pp. 2950-2971, 2011. http://arxiv.org/abs/1006.0758 .. [2] LSMR Software, http://web.stanford.edu/group/SOL/software/lsmr/ is:The exact solution is x = 0 s:Ax - b is small enough, given atol, btol s:The least-squares solution is good enough, given atol s:The estimate of cond(Abar) has exceeded conlim s:Ax - b is small enough for this machine s:The least-squares solution is good enough for this machines:Cond(Abar) seems to be too large for this machine s:The iteration limit has been reached s( itn x(1) norm r norm Ars% compatible LS norm A cond Aiit s2LSMR Least-squares solution of Ax = b s'The matrix A has %8g rows and %8g colssdamp = %20.14e s,atol = %8.2e conlim = %8.2e s'btol = %8.2e maxiter = %8g g}Ô%­I²Ts %6g %12.5es %10.3e %10.3es %8.1e %8.1etiiiiiii(i gš™™™™™ñ?s %8.1e %8.1es LSMR finishedsistop =%8g normr =%8.1es! normA =%8.1e normAr =%8.1esitn =%8g condA =%8.1es normx =%8.1e(s:The exact solution is x = 0 s:Ax - b is small enough, given atol, btol s:The least-squares solution is good enough, given atol s:The estimate of cond(Abar) has exceeded conlim s:Ax - b is small enough for this machine s:The least-squares solution is good enough for this machines:Cond(Abar) seems to be too large for this machine s:The iteration limit has been reached N(R RtndimtsqueezetshapetmintNonetprintRRtrmatvectcopyRtjointmatvecR tmaxtabsR(LtAtbtdamptatoltbtoltconlimtmaxitertshowtmsgthdg1thdg2tpfreqtpcounttmtntminDimtutbetatvtalphatitntzetabartalphabartrhotrhobartcbartsbarththbartxtbetaddtbetadtrhodoldt tautildeoldt thetatildetzetatdtnormA2tmaxrbartminrbartnormAtcondAtnormxtnormbtistoptctoltnormrtnormarttest1ttest2tstr1tstr2tstr3tchattshattalphahattrhooldtctstthetanewt rhobaroldtzetaoldtthetabartrhotempt betaacutet betachecktbetahatt thetatildeoldt ctildeoldt stildeoldt rhotildeoldttaudttest3tt1trtoltstr4((s>/tmp/pip-build-7oUkmx/scipy/scipy/sparse/linalg/isolve/lsmr.pyRs`k                                                          (      %    N(t__doc__t __future__RRRt__all__tnumpyRRRt numpy.linalgRtmathRtscipy.sparse.linalg.interfaceR tlsqrR RtFalseR(((s>/tmp/pip-build-7oUkmx/scipy/scipy/sparse/linalg/isolve/lsmr.pyts