470 lines
15 KiB
Python
470 lines
15 KiB
Python
"""
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Copyright (C) 2010 David Fong and Michael Saunders
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LSMR uses an iterative method.
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07 Jun 2010: Documentation updated
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03 Jun 2010: First release version in Python
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David Chin-lung Fong clfong@stanford.edu
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Institute for Computational and Mathematical Engineering
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Stanford University
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Michael Saunders saunders@stanford.edu
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Systems Optimization Laboratory
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Dept of MS&E, Stanford University.
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"""
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from __future__ import division, print_function, absolute_import
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__all__ = ['lsmr']
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from numpy import zeros, infty, atleast_1d
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from numpy.linalg import norm
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from math import sqrt
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from scipy.sparse.linalg.interface import aslinearoperator
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from .lsqr import _sym_ortho
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def lsmr(A, b, damp=0.0, atol=1e-6, btol=1e-6, conlim=1e8,
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maxiter=None, show=False, x0=None):
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"""Iterative solver for least-squares problems.
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lsmr solves the system of linear equations ``Ax = b``. If the system
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is inconsistent, it solves the least-squares problem ``min ||b - Ax||_2``.
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A is a rectangular matrix of dimension m-by-n, where all cases are
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allowed: m = n, m > n, or m < n. B is a vector of length m.
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The matrix A may be dense or sparse (usually sparse).
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Parameters
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----------
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A : {matrix, sparse matrix, ndarray, LinearOperator}
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Matrix A in the linear system.
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b : array_like, shape (m,)
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Vector b in the linear system.
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damp : float
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Damping factor for regularized least-squares. `lsmr` solves
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the regularized least-squares problem::
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min ||(b) - ( A )x||
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||(0) (damp*I) ||_2
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where damp is a scalar. If damp is None or 0, the system
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is solved without regularization.
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atol, btol : float, optional
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Stopping tolerances. `lsmr` continues iterations until a
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certain backward error estimate is smaller than some quantity
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depending on atol and btol. Let ``r = b - Ax`` be the
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residual vector for the current approximate solution ``x``.
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If ``Ax = b`` seems to be consistent, ``lsmr`` terminates
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when ``norm(r) <= atol * norm(A) * norm(x) + btol * norm(b)``.
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Otherwise, lsmr terminates when ``norm(A^{T} r) <=
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atol * norm(A) * norm(r)``. If both tolerances are 1.0e-6 (say),
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the final ``norm(r)`` should be accurate to about 6
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digits. (The final x will usually have fewer correct digits,
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depending on ``cond(A)`` and the size of LAMBDA.) If `atol`
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or `btol` is None, a default value of 1.0e-6 will be used.
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Ideally, they should be estimates of the relative error in the
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entries of A and B respectively. For example, if the entries
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of `A` have 7 correct digits, set atol = 1e-7. This prevents
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the algorithm from doing unnecessary work beyond the
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uncertainty of the input data.
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conlim : float, optional
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`lsmr` terminates if an estimate of ``cond(A)`` exceeds
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`conlim`. For compatible systems ``Ax = b``, conlim could be
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as large as 1.0e+12 (say). For least-squares problems,
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`conlim` should be less than 1.0e+8. If `conlim` is None, the
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default value is 1e+8. Maximum precision can be obtained by
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setting ``atol = btol = conlim = 0``, but the number of
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iterations may then be excessive.
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maxiter : int, optional
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`lsmr` terminates if the number of iterations reaches
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`maxiter`. The default is ``maxiter = min(m, n)``. For
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ill-conditioned systems, a larger value of `maxiter` may be
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needed.
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show : bool, optional
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Print iterations logs if ``show=True``.
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x0 : array_like, shape (n,), optional
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Initial guess of x, if None zeros are used.
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.. versionadded:: 1.0.0
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Returns
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-------
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x : ndarray of float
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Least-square solution returned.
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istop : int
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istop gives the reason for stopping::
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istop = 0 means x=0 is a solution. If x0 was given, then x=x0 is a
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solution.
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= 1 means x is an approximate solution to A*x = B,
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according to atol and btol.
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= 2 means x approximately solves the least-squares problem
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according to atol.
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= 3 means COND(A) seems to be greater than CONLIM.
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= 4 is the same as 1 with atol = btol = eps (machine
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precision)
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= 5 is the same as 2 with atol = eps.
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= 6 is the same as 3 with CONLIM = 1/eps.
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= 7 means ITN reached maxiter before the other stopping
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conditions were satisfied.
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itn : int
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Number of iterations used.
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normr : float
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``norm(b-Ax)``
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normar : float
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``norm(A^T (b - Ax))``
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norma : float
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``norm(A)``
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conda : float
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Condition number of A.
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normx : float
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``norm(x)``
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Notes
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-----
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.. versionadded:: 0.11.0
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References
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----------
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.. [1] D. C.-L. Fong and M. A. Saunders,
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"LSMR: An iterative algorithm for sparse least-squares problems",
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SIAM J. Sci. Comput., vol. 33, pp. 2950-2971, 2011.
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http://arxiv.org/abs/1006.0758
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.. [2] LSMR Software, http://web.stanford.edu/group/SOL/software/lsmr/
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Examples
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--------
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>>> from scipy.sparse import csc_matrix
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>>> from scipy.sparse.linalg import lsmr
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>>> A = csc_matrix([[1., 0.], [1., 1.], [0., 1.]], dtype=float)
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The first example has the trivial solution `[0, 0]`
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>>> b = np.array([0., 0., 0.], dtype=float)
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>>> x, istop, itn, normr = lsmr(A, b)[:4]
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>>> istop
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0
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>>> x
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array([ 0., 0.])
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The stopping code `istop=0` returned indicates that a vector of zeros was
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found as a solution. The returned solution `x` indeed contains `[0., 0.]`.
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The next example has a non-trivial solution:
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>>> b = np.array([1., 0., -1.], dtype=float)
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>>> x, istop, itn, normr = lsmr(A, b)[:4]
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>>> istop
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1
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>>> x
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array([ 1., -1.])
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>>> itn
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1
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>>> normr
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4.440892098500627e-16
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As indicated by `istop=1`, `lsmr` found a solution obeying the tolerance
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limits. The given solution `[1., -1.]` obviously solves the equation. The
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remaining return values include information about the number of iterations
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(`itn=1`) and the remaining difference of left and right side of the solved
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equation.
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The final example demonstrates the behavior in the case where there is no
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solution for the equation:
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>>> b = np.array([1., 0.01, -1.], dtype=float)
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>>> x, istop, itn, normr = lsmr(A, b)[:4]
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>>> istop
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2
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>>> x
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array([ 1.00333333, -0.99666667])
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>>> A.dot(x)-b
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array([ 0.00333333, -0.00333333, 0.00333333])
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>>> normr
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0.005773502691896255
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`istop` indicates that the system is inconsistent and thus `x` is rather an
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approximate solution to the corresponding least-squares problem. `normr`
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contains the minimal distance that was found.
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"""
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A = aslinearoperator(A)
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b = atleast_1d(b)
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if b.ndim > 1:
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b = b.squeeze()
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msg = ('The exact solution is x = 0, or x = x0, if x0 was given ',
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'Ax - b is small enough, given atol, btol ',
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'The least-squares solution is good enough, given atol ',
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'The estimate of cond(Abar) has exceeded conlim ',
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'Ax - b is small enough for this machine ',
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'The least-squares solution is good enough for this machine',
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'Cond(Abar) seems to be too large for this machine ',
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'The iteration limit has been reached ')
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hdg1 = ' itn x(1) norm r norm A''r'
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hdg2 = ' compatible LS norm A cond A'
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pfreq = 20 # print frequency (for repeating the heading)
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pcount = 0 # print counter
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m, n = A.shape
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# stores the num of singular values
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minDim = min([m, n])
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if maxiter is None:
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maxiter = minDim
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if show:
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print(' ')
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print('LSMR Least-squares solution of Ax = b\n')
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print('The matrix A has %8g rows and %8g cols' % (m, n))
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print('damp = %20.14e\n' % (damp))
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print('atol = %8.2e conlim = %8.2e\n' % (atol, conlim))
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print('btol = %8.2e maxiter = %8g\n' % (btol, maxiter))
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u = b
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normb = norm(b)
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if x0 is None:
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x = zeros(n)
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beta = normb.copy()
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else:
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x = atleast_1d(x0)
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u = u - A.matvec(x)
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beta = norm(u)
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if beta > 0:
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u = (1 / beta) * u
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v = A.rmatvec(u)
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alpha = norm(v)
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else:
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v = zeros(n)
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alpha = 0
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if alpha > 0:
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v = (1 / alpha) * v
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# Initialize variables for 1st iteration.
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itn = 0
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zetabar = alpha * beta
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alphabar = alpha
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rho = 1
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rhobar = 1
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cbar = 1
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sbar = 0
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h = v.copy()
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hbar = zeros(n)
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# Initialize variables for estimation of ||r||.
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betadd = beta
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betad = 0
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rhodold = 1
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tautildeold = 0
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thetatilde = 0
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zeta = 0
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d = 0
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# Initialize variables for estimation of ||A|| and cond(A)
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normA2 = alpha * alpha
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maxrbar = 0
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minrbar = 1e+100
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normA = sqrt(normA2)
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condA = 1
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normx = 0
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# Items for use in stopping rules, normb set earlier
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istop = 0
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ctol = 0
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if conlim > 0:
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ctol = 1 / conlim
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normr = beta
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# Reverse the order here from the original matlab code because
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# there was an error on return when arnorm==0
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normar = alpha * beta
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if normar == 0:
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if show:
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print(msg[0])
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return x, istop, itn, normr, normar, normA, condA, normx
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if show:
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print(' ')
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print(hdg1, hdg2)
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test1 = 1
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test2 = alpha / beta
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str1 = '%6g %12.5e' % (itn, x[0])
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str2 = ' %10.3e %10.3e' % (normr, normar)
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str3 = ' %8.1e %8.1e' % (test1, test2)
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print(''.join([str1, str2, str3]))
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# Main iteration loop.
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while itn < maxiter:
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itn = itn + 1
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# Perform the next step of the bidiagonalization to obtain the
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# next beta, u, alpha, v. These satisfy the relations
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# beta*u = a*v - alpha*u,
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# alpha*v = A'*u - beta*v.
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u = A.matvec(v) - alpha * u
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beta = norm(u)
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if beta > 0:
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u = (1 / beta) * u
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v = A.rmatvec(u) - beta * v
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alpha = norm(v)
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if alpha > 0:
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v = (1 / alpha) * v
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# At this point, beta = beta_{k+1}, alpha = alpha_{k+1}.
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# Construct rotation Qhat_{k,2k+1}.
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chat, shat, alphahat = _sym_ortho(alphabar, damp)
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# Use a plane rotation (Q_i) to turn B_i to R_i
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rhoold = rho
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c, s, rho = _sym_ortho(alphahat, beta)
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thetanew = s*alpha
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alphabar = c*alpha
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# Use a plane rotation (Qbar_i) to turn R_i^T to R_i^bar
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rhobarold = rhobar
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zetaold = zeta
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thetabar = sbar * rho
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rhotemp = cbar * rho
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cbar, sbar, rhobar = _sym_ortho(cbar * rho, thetanew)
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zeta = cbar * zetabar
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zetabar = - sbar * zetabar
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# Update h, h_hat, x.
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hbar = h - (thetabar * rho / (rhoold * rhobarold)) * hbar
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x = x + (zeta / (rho * rhobar)) * hbar
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h = v - (thetanew / rho) * h
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# Estimate of ||r||.
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# Apply rotation Qhat_{k,2k+1}.
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betaacute = chat * betadd
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betacheck = -shat * betadd
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# Apply rotation Q_{k,k+1}.
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betahat = c * betaacute
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betadd = -s * betaacute
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# Apply rotation Qtilde_{k-1}.
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# betad = betad_{k-1} here.
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thetatildeold = thetatilde
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ctildeold, stildeold, rhotildeold = _sym_ortho(rhodold, thetabar)
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thetatilde = stildeold * rhobar
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rhodold = ctildeold * rhobar
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betad = - stildeold * betad + ctildeold * betahat
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# betad = betad_k here.
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# rhodold = rhod_k here.
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tautildeold = (zetaold - thetatildeold * tautildeold) / rhotildeold
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taud = (zeta - thetatilde * tautildeold) / rhodold
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d = d + betacheck * betacheck
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normr = sqrt(d + (betad - taud)**2 + betadd * betadd)
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# Estimate ||A||.
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normA2 = normA2 + beta * beta
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normA = sqrt(normA2)
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normA2 = normA2 + alpha * alpha
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# Estimate cond(A).
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maxrbar = max(maxrbar, rhobarold)
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if itn > 1:
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minrbar = min(minrbar, rhobarold)
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condA = max(maxrbar, rhotemp) / min(minrbar, rhotemp)
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# Test for convergence.
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# Compute norms for convergence testing.
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normar = abs(zetabar)
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normx = norm(x)
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# Now use these norms to estimate certain other quantities,
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# some of which will be small near a solution.
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test1 = normr / normb
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if (normA * normr) != 0:
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test2 = normar / (normA * normr)
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else:
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test2 = infty
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test3 = 1 / condA
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t1 = test1 / (1 + normA * normx / normb)
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rtol = btol + atol * normA * normx / normb
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# The following tests guard against extremely small values of
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# atol, btol or ctol. (The user may have set any or all of
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# the parameters atol, btol, conlim to 0.)
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# The effect is equivalent to the normAl tests using
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# atol = eps, btol = eps, conlim = 1/eps.
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if itn >= maxiter:
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istop = 7
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if 1 + test3 <= 1:
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istop = 6
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if 1 + test2 <= 1:
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istop = 5
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if 1 + t1 <= 1:
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istop = 4
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# Allow for tolerances set by the user.
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if test3 <= ctol:
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istop = 3
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if test2 <= atol:
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istop = 2
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if test1 <= rtol:
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istop = 1
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# See if it is time to print something.
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if show:
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if (n <= 40) or (itn <= 10) or (itn >= maxiter - 10) or \
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(itn % 10 == 0) or (test3 <= 1.1 * ctol) or \
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(test2 <= 1.1 * atol) or (test1 <= 1.1 * rtol) or \
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(istop != 0):
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if pcount >= pfreq:
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pcount = 0
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print(' ')
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print(hdg1, hdg2)
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pcount = pcount + 1
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str1 = '%6g %12.5e' % (itn, x[0])
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str2 = ' %10.3e %10.3e' % (normr, normar)
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str3 = ' %8.1e %8.1e' % (test1, test2)
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str4 = ' %8.1e %8.1e' % (normA, condA)
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print(''.join([str1, str2, str3, str4]))
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if istop > 0:
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break
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# Print the stopping condition.
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if show:
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print(' ')
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print('LSMR finished')
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print(msg[istop])
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print('istop =%8g normr =%8.1e' % (istop, normr))
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print(' normA =%8.1e normAr =%8.1e' % (normA, normar))
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print('itn =%8g condA =%8.1e' % (itn, condA))
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print(' normx =%8.1e' % (normx))
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print(str1, str2)
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print(str3, str4)
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return x, istop, itn, normr, normar, normA, condA, normx
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