(ph"=dSrS/rSSKJrJrJrJr SSKJr SSK J r SSK J r SSK Jr S S jrg) 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  [U5n[U5nURS:aUR5nSn Sn Sn Sn Sn URup[ X/5nUcUnUc[ X[5nO[ XU[5nU(a`[S5 [S5 [S US US 35 [S US S35 [SUSSUSS35 [SUSSUSS35 Un[U5nUc[UU5nUR5nO8[UR55nUURU5- n[U5nUS:a%SU- U-nURU5n[U5nO[UU5nSnUS:aSU- U-nSnUU-nUnSnSnSnSnUR5n[UU5n Un!Sn"Sn#Sn$Sn%Sn&Sn'UU-n(Sn)Sn*[U(5n+Sn,Sn-Sn.Sn/US:aSU- n/Un0UU-n1U1S:XaU(a[U S5 UU.UU0U1U+U,U-4$US:XaSUS'UU.UU0U1U+U,U-4$U(aZ[S5 [X5 Sn2UU- n3USSUSS3n4SU0SSU1S3n5SU2SSU3S3n6[SRU4U5U6/55 UU:GaUS-nUU*-nUURU5- n[U5nUS:a;USU- -nUU*-nUURU5- n[U5nUS:aUSU- -n[!UU5un7n8n9Un:[!U9U5un;nU&n?UU-n@UU-nA[!UU-U=5unnnUU-n&U*U-nU U@U-U:U>-- *-n U U- n UU&UU-- U -- nUU=U- *-nUU- nU7U!-nBU8*U!-nCU;UB-nDU<*UB-n!U%nE[!U#U@5unFnGnHUGU-n%UFU-n#UG*U"-UFUD--n"U?UEU$-- UH- n$U&U%U$-- U#- nIU'UCUC--n'[U'U"UI- S--U!U!--5n0U(UU--n([U(5n+U(UU--n([#U)U>5n)US:a [ U*U>5n*[#U)WA5[ U*UA5- n,[%U5n1[U5n-U0U- n2U+U0-S:wa U1U+U0-- n3O[&n3SU,- nJU2SU+U--U- -- nKXCU+-U--U- -nLUU:aSn.SWJ-S::aSn.SU3-S::aS n.SWK-S::aS!n.WJU/::aS"n.U3U::aSn.U2WL::aSn.U(aUS#::d9US$::d3UUS$- :d*US$-S:Xd!WJS%U/-::dU3S%U-::dU2S%WL-::dU.S:wajX:aSn [S5 [X5 U S-n USSUSS3n4SU0SSU1S3n5SU2SSU3S3n6SU+SSU,S3nM[SRU4U5U6UM/55 U.S:aO UU:aGMU(a[S5 [S&5 [U U.5 [S'U.SS(U0S35 [S)U+SS*U1S35 [S+USS,U,S35 [S-U-S35 [W4W55 [W6WM5 UU.UU0U1U+U,U-4$).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. )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 z( itn x(1) norm r norm Arz% compatible LS norm A cond Ar 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 z8.1e( g?z LSMR finishedzistop =z normr =z normA =z normAr =zitn =z condA =z normx =)r rndimsqueezeshapeminrfloatprintrrcopymatvecrmatvecr joinr maxabsr)NAbdampatolbtolconlimmaxitershowx0msghdg1hdg2pfreqpcountmnminDimdtypeunormbxbetavalphaitnzetabaralphabarrhorhobarcbarsbarhhbarbetaddbetadrhodold tautildeold thetatildezetadnormA2maxrbarminrbarnormAcondAnormxistopctolnormrnormartest1test2str1str2str3chatshatalphahatrhooldcsthetanew rhobaroldzetaoldthetabarrhotemp betaacute betacheckbetahat thetatildeold ctildeold stildeold rhotildeoldtaudtest3t1rtolstr4sN S/var/www/html/venv/lib/python3.13/site-packages/scipy/sparse/linalg/_isolve/lsmr.pyrrs<P 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$rygחANFN)__doc____all__numpyrrrr numpy.linalgrmathr scipy.sparse.linalg._interfacer scipy.sparse.linalg._isolve.lsqrr rrrxrwrs3$ (55;77:&*I=rx