+ 0i"=bRtR.t^RIHtHtHtHt^RIHt^RI H t ^RI H t ^RI HtR RltR#) a 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. lsmr)zerosinf atleast_1d result_type)norm)sqrt)aslinearoperator) _sym_orthoNc \V4p\V4pVP^8dVP4pR!p Rp Rp ^p ^p VPwr\ W.4pVfTpVf\ W\4pM\ WV\4pV'da\R4\R4\RV RV R24\RVR R 24\R VR R VR R 24\RVR RVR R 24Tp\V4pVf\VV4pVP4pM<\VP44pVVPV4, p\V4pV^8d.^V, V,pVPV4p\V4pM\VV4p^pV^8d^V, V,p^pVV,pTp^p^p^p^pVP4p\VV4p Tp!^p"^p#^p$^p%^p&^p'VV,p(^p)Rp*\V(4p+^p,^p-^p.^p/V^8d ^V, p/Tp0VV,p1V1^8Xd%V'd\V ^,4VV.VV0V1V+V,V-3#V^8Xd^VR"&VV.VV0V1V+V,V-3#V'dc\R4\W4^p2VV, p3VR RV^,R 2p4RV0R RV1R 2p5RV2R RV3R 2p6\RPV4V5V6.44VV8EdXV^,pVV),pVVPV4, p\V4pV^8dUV^V, ,pVV),pVVPV4, p\V4pV^8dV^V, ,p\!VV4wp7p8p9Tp:\!V9V4wp;pT&p?VV,p@VV,pA\!VV,V=4wpppVV,p&V)V,pV V@V,V:V>,, ),p V V, p VV&VV,, V ,, pVV=V, ),pVV, pV7V!,pBV8)V!,pCV;VB,pDV<)VB,p!T%pE\!V#V@4wpFpGpHVGV,p%VFV,p#VG)V",VFVD,,p"V?VEV$,, VH, p$V&V%V$,, V#, pIV'VCVC,,p'\V'V"VI, ^,,V!V!,,4p0V(VV,,p(\V(4p+V(VV,,p(\#V)V>4p)V^8d \ V*V>4p*\#V)XA4\ V*VA4, p,\%V4p1\V4p-V0V, p2V+V0,^8wdV1V+V0,, p3M\&p3^V,, pJV2^V+V-,V, ,, pKWCV+,V-,V, ,pLVV8d^p.^XJ,^8:d^p.^V3,^8:d^p.^XK,^8:d^p.XJV/8:d^p.V3V8:d^p.V2XL8:d^p.V'dV^(8:gUV^ 8:gNVV^ , 8g@V^ ,^8Xg2XJRV/,8:g$V3RV,8:gV2RXL,8:gV.^8wdtW8d^p \R4\W4V ^,p VR RV^,R 2p4RV0R RV1R 2p5RV2R RV3R 2p6RV+R RV,R 2pM\RPV4V5V6VM.44V.^8gEK^V'd\R4\R4\V V.,4\RV.R RV0R 24\RV+R RV1R 24\RVR RV,R 24\R V-R 24\X4X54\X6XM4VV.VV0V1V+V,V-3#)#aGIterative 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 : {sparse array, ndarray, LinearOperator} Matrix A in the linear system. Alternatively, ``A`` can be a linear operator which can produce ``Ax`` and ``A^H x`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. 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. Default is 0. 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^H r) <= atol * norm(A) * norm(r)``. If both tolerances are 1.0e-6 (default), 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. Default is 1e8. 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. Default is False. show : bool, optional Print iterations logs if ``show=True``. Default is False. x0 : array_like, shape (n,), optional Initial guess of ``x``, if None zeros are used. Default is None. .. versionadded:: 1.0.0 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. If x0 was given, then x=x0 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^H (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. :arxiv:`1006.0758` .. [2] LSMR Software, https://web.stanford.edu/group/SOL/software/lsmr/ Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import lsmr >>> A = csc_array([[1., 0.], [1., 1.], [0., 1.]], dtype=float) The first example has the trivial solution ``[0, 0]`` >>> b = np.array([0., 0., 0.], dtype=float) >>> x, istop, itn, normr = lsmr(A, b)[:4] >>> istop 0 >>> x array([0., 0.]) The stopping code ``istop=0`` returned indicates that a vector of zeros was found as a solution. The returned solution `x` indeed contains ``[0., 0.]``. The next example has a non-trivial solution: >>> b = np.array([1., 0., -1.], dtype=float) >>> x, istop, itn, normr = lsmr(A, b)[:4] >>> istop 1 >>> x array([ 1., -1.]) >>> itn 1 >>> normr 4.440892098500627e-16 As indicated by ``istop=1``, `lsmr` found a solution obeying the tolerance limits. The given solution ``[1., -1.]`` obviously solves the equation. The remaining return values include information about the number of iterations (`itn=1`) and the remaining difference of left and right side of the solved equation. The final example demonstrates the behavior in the case where there is no solution for the equation: >>> b = np.array([1., 0.01, -1.], dtype=float) >>> x, istop, itn, normr = lsmr(A, b)[:4] >>> istop 2 >>> x array([ 1.00333333, -0.99666667]) >>> A.dot(x)-b array([ 0.00333333, -0.00333333, 0.00333333]) >>> normr 0.005773502691896255 `istop` indicates that the system is inconsistent and thus `x` is rather an approximate solution to the corresponding least-squares problem. `normr` contains the minimal distance that was found. z( itn x(1) norm r norm Arz% compatible LS norm A cond A z2LSMR Least-squares solution of Ax = b zThe matrix A has z rows and z columnszdamp = z20.14e zatol = z8.2ez conlim = zbtol = z maxiter = 8gg}Ô%IT6gz12.5ez10.3ez z8.1eg?z LSMR finishedzistop =z normr =z normA =z normAr =zitn =z condA =z normx =)z9The exact solution is x = 0, or x = x0, if x0 was given z:Ax - b is small enough, given atol, btol z:The least-squares solution is good enough, given atol z:The estimate of cond(Abar) has exceeded conlim z:Ax - b is small enough for this machine z:The least-squares solution is good enough for this machinez:Cond(Abar) seems to be too large for this machine z:The iteration limit has been reached )r rndimsqueezeshapeminrfloatprintrrcopymatvecrmatvecrjoinr maxabsr)NAbdampatolbtolconlimmaxitershowx0msghdg1hdg2pfreqpcountmnminDimdtypeunormbxbetavalphaitnzetabaralphabarrhorhobarcbarsbarhhbarbetaddbetadrhodold tautildeold thetatildezetadnormA2maxrbarminrbarnormAcondAnormxistopctolnormrnormartest1test2str1str2str3chatshatalphahatrhooldcsthetanew rhobaroldzetaoldthetabarrhotemp betaacute betacheckbetahat thetatildeold ctildeold stildeold rhotildeoldtaudtest3t1rtolstr4sN&&&&&&&&& ^/var/www/html/photoedit/myenv/lib/python3.14/site-packages/scipy/sparse/linalg/_isolve/lsmr.pyrrs1P A1 Avvz IIK IC 6D 2D E F 77DA![F zA%(A"e,  c  CD !!Jqc:; V}B'( T{"Vi%7899   dcFl#t ++ 3  Q 6M EFN i-i # ,6w,I) 9k' f$ e#i'&99 !<< K zK//7: I% %Q%$,**Vf_<=$+%V %%-'gy) 7'9-GGW%GW(== WQ   EMa eem,EEE  a%%-%// 0UlU*U22 '>E u9>E u9>E r6Q;E D=E D=E D=E RSBYC7R<,?bA 5C$J#6t##*)< ?F#J$%!b1Q4,/5-q7E$rngחANFN)__doc____all__numpyrrrr numpy.linalgrmathrscipy.sparse.linalg._interfacer scipy.sparse.linalg._isolve.lsqrr rrrmrlrvs+$ (55;7I=rm