363 lines
10 KiB
Python
363 lines
10 KiB
Python
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from __future__ import division, print_function, absolute_import
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from numpy import sqrt, inner, finfo, zeros
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from numpy.linalg import norm
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from .utils import make_system
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__all__ = ['minres']
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def minres(A, b, x0=None, shift=0.0, tol=1e-5, maxiter=None,
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M=None, callback=None, show=False, check=False):
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"""
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Use MINimum RESidual iteration to solve Ax=b
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MINRES minimizes norm(A*x - b) for a real symmetric matrix A. Unlike
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the Conjugate Gradient method, A can be indefinite or singular.
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If shift != 0 then the method solves (A - shift*I)x = b
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Parameters
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----------
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A : {sparse matrix, dense matrix, LinearOperator}
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The real symmetric N-by-N matrix of the linear system
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b : {array, matrix}
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Right hand side of the linear system. Has shape (N,) or (N,1).
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Returns
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-------
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x : {array, matrix}
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The converged solution.
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info : integer
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Provides convergence information:
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0 : successful exit
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>0 : convergence to tolerance not achieved, number of iterations
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<0 : illegal input or breakdown
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Other Parameters
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----------------
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x0 : {array, matrix}
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Starting guess for the solution.
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tol : float
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Tolerance to achieve. The algorithm terminates when the relative
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residual is below `tol`.
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maxiter : integer
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Maximum number of iterations. Iteration will stop after maxiter
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steps even if the specified tolerance has not been achieved.
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M : {sparse matrix, dense matrix, LinearOperator}
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Preconditioner for A. The preconditioner should approximate the
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inverse of A. Effective preconditioning dramatically improves the
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rate of convergence, which implies that fewer iterations are needed
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to reach a given error tolerance.
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callback : function
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User-supplied function to call after each iteration. It is called
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as callback(xk), where xk is the current solution vector.
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References
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----------
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Solution of sparse indefinite systems of linear equations,
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C. C. Paige and M. A. Saunders (1975),
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SIAM J. Numer. Anal. 12(4), pp. 617-629.
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https://web.stanford.edu/group/SOL/software/minres/
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This file is a translation of the following MATLAB implementation:
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https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip
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"""
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A, M, x, b, postprocess = make_system(A, M, x0, b)
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matvec = A.matvec
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psolve = M.matvec
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first = 'Enter minres. '
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last = 'Exit minres. '
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n = A.shape[0]
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if maxiter is None:
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maxiter = 5 * n
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msg = [' beta2 = 0. If M = I, b and x are eigenvectors ', # -1
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' beta1 = 0. The exact solution is x = 0 ', # 0
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' A solution to Ax = b was found, given rtol ', # 1
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' A least-squares solution was found, given rtol ', # 2
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' Reasonable accuracy achieved, given eps ', # 3
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' x has converged to an eigenvector ', # 4
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' acond has exceeded 0.1/eps ', # 5
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' The iteration limit was reached ', # 6
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' A does not define a symmetric matrix ', # 7
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' M does not define a symmetric matrix ', # 8
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' M does not define a pos-def preconditioner '] # 9
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if show:
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print(first + 'Solution of symmetric Ax = b')
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print(first + 'n = %3g shift = %23.14e' % (n,shift))
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print(first + 'itnlim = %3g rtol = %11.2e' % (maxiter,tol))
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print()
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istop = 0
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itn = 0
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Anorm = 0
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Acond = 0
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rnorm = 0
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ynorm = 0
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xtype = x.dtype
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eps = finfo(xtype).eps
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x = zeros(n, dtype=xtype)
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# Set up y and v for the first Lanczos vector v1.
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# y = beta1 P' v1, where P = C**(-1).
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# v is really P' v1.
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y = b
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r1 = b
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y = psolve(b)
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beta1 = inner(b,y)
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if beta1 < 0:
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raise ValueError('indefinite preconditioner')
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elif beta1 == 0:
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return (postprocess(x), 0)
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beta1 = sqrt(beta1)
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if check:
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# are these too strict?
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# see if A is symmetric
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w = matvec(y)
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r2 = matvec(w)
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s = inner(w,w)
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t = inner(y,r2)
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z = abs(s - t)
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epsa = (s + eps) * eps**(1.0/3.0)
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if z > epsa:
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raise ValueError('non-symmetric matrix')
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# see if M is symmetric
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r2 = psolve(y)
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s = inner(y,y)
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t = inner(r1,r2)
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z = abs(s - t)
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epsa = (s + eps) * eps**(1.0/3.0)
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if z > epsa:
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raise ValueError('non-symmetric preconditioner')
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# Initialize other quantities
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oldb = 0
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beta = beta1
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dbar = 0
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epsln = 0
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qrnorm = beta1
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phibar = beta1
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rhs1 = beta1
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rhs2 = 0
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tnorm2 = 0
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ynorm2 = 0
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cs = -1
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sn = 0
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w = zeros(n, dtype=xtype)
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w2 = zeros(n, dtype=xtype)
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r2 = r1
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if show:
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print()
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print()
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print(' Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|')
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while itn < maxiter:
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itn += 1
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s = 1.0/beta
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v = s*y
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y = matvec(v)
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y = y - shift * v
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if itn >= 2:
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y = y - (beta/oldb)*r1
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alfa = inner(v,y)
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y = y - (alfa/beta)*r2
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r1 = r2
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r2 = y
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y = psolve(r2)
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oldb = beta
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beta = inner(r2,y)
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if beta < 0:
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raise ValueError('non-symmetric matrix')
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beta = sqrt(beta)
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tnorm2 += alfa**2 + oldb**2 + beta**2
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if itn == 1:
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if beta/beta1 <= 10*eps:
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istop = -1 # Terminate later
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# tnorm2 = alfa**2 ??
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gmax = abs(alfa)
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gmin = gmax
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# Apply previous rotation Qk-1 to get
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# [deltak epslnk+1] = [cs sn][dbark 0 ]
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# [gbar k dbar k+1] [sn -cs][alfak betak+1].
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oldeps = epsln
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delta = cs * dbar + sn * alfa # delta1 = 0 deltak
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gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
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epsln = sn * beta # epsln2 = 0 epslnk+1
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dbar = - cs * beta # dbar 2 = beta2 dbar k+1
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root = norm([gbar, dbar])
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Arnorm = phibar * root
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# Compute the next plane rotation Qk
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gamma = norm([gbar, beta]) # gammak
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gamma = max(gamma, eps)
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cs = gbar / gamma # ck
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sn = beta / gamma # sk
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phi = cs * phibar # phik
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phibar = sn * phibar # phibark+1
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# Update x.
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denom = 1.0/gamma
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w1 = w2
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w2 = w
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w = (v - oldeps*w1 - delta*w2) * denom
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x = x + phi*w
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# Go round again.
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gmax = max(gmax, gamma)
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gmin = min(gmin, gamma)
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z = rhs1 / gamma
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ynorm2 = z**2 + ynorm2
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rhs1 = rhs2 - delta*z
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rhs2 = - epsln*z
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# Estimate various norms and test for convergence.
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Anorm = sqrt(tnorm2)
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ynorm = sqrt(ynorm2)
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epsa = Anorm * eps
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epsx = Anorm * ynorm * eps
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epsr = Anorm * ynorm * tol
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diag = gbar
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if diag == 0:
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diag = epsa
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qrnorm = phibar
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rnorm = qrnorm
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test1 = rnorm / (Anorm*ynorm) # ||r|| / (||A|| ||x||)
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test2 = root / Anorm # ||Ar|| / (||A|| ||r||)
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# Estimate cond(A).
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# In this version we look at the diagonals of R in the
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# factorization of the lower Hessenberg matrix, Q * H = R,
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# where H is the tridiagonal matrix from Lanczos with one
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# extra row, beta(k+1) e_k^T.
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Acond = gmax/gmin
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# See if any of the stopping criteria are satisfied.
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# In rare cases, istop is already -1 from above (Abar = const*I).
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if istop == 0:
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t1 = 1 + test1 # These tests work if tol < eps
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t2 = 1 + test2
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if t2 <= 1:
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istop = 2
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if t1 <= 1:
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istop = 1
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if itn >= maxiter:
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istop = 6
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if Acond >= 0.1/eps:
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istop = 4
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if epsx >= beta:
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istop = 3
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# if rnorm <= epsx : istop = 2
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# if rnorm <= epsr : istop = 1
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if test2 <= tol:
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istop = 2
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if test1 <= tol:
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istop = 1
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# See if it is time to print something.
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prnt = False
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if n <= 40:
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prnt = True
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if itn <= 10:
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prnt = True
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if itn >= maxiter-10:
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prnt = True
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if itn % 10 == 0:
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prnt = True
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if qrnorm <= 10*epsx:
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prnt = True
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if qrnorm <= 10*epsr:
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prnt = True
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if Acond <= 1e-2/eps:
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prnt = True
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if istop != 0:
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prnt = True
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if show and prnt:
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str1 = '%6g %12.5e %10.3e' % (itn, x[0], test1)
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str2 = ' %10.3e' % (test2,)
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str3 = ' %8.1e %8.1e %8.1e' % (Anorm, Acond, gbar/Anorm)
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print(str1 + str2 + str3)
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if itn % 10 == 0:
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print()
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if callback is not None:
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callback(x)
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if istop != 0:
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break # TODO check this
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if show:
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print()
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print(last + ' istop = %3g itn =%5g' % (istop,itn))
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print(last + ' Anorm = %12.4e Acond = %12.4e' % (Anorm,Acond))
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print(last + ' rnorm = %12.4e ynorm = %12.4e' % (rnorm,ynorm))
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print(last + ' Arnorm = %12.4e' % (Arnorm,))
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print(last + msg[istop+1])
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if istop == 6:
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info = maxiter
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else:
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info = 0
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return (postprocess(x),info)
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if __name__ == '__main__':
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from scipy import ones, arange
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from scipy.linalg import norm
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from scipy.sparse import spdiags
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n = 10
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residuals = []
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def cb(x):
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residuals.append(norm(b - A*x))
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# A = poisson((10,),format='csr')
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A = spdiags([arange(1,n+1,dtype=float)], [0], n, n, format='csr')
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M = spdiags([1.0/arange(1,n+1,dtype=float)], [0], n, n, format='csr')
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A.psolve = M.matvec
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b = 0*ones(A.shape[0])
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x = minres(A,b,tol=1e-12,maxiter=None,callback=cb)
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# x = cg(A,b,x0=b,tol=1e-12,maxiter=None,callback=cb)[0]
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