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