# scipy/scipy

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 # Copyright (C) 2009, Pauli Virtanen # Distributed under the same license as Scipy. import numpy as np from scipy.linalg import get_blas_funcs from utils import make_system __all__ = ['lgmres'] def norm2(q): q = np.asarray(q) nrm2 = get_blas_funcs('nrm2', dtype=q.dtype) return nrm2(q) def lgmres(A, b, x0=None, tol=1e-5, maxiter=1000, M=None, callback=None, inner_m=30, outer_k=3, outer_v=None, store_outer_Av=True): """ Solve a matrix equation using the LGMRES algorithm. The LGMRES algorithm [BJM]_ [BPh]_ is designed to avoid some problems in the convergence in restarted GMRES, and often converges in fewer iterations. Parameters ---------- A : {sparse matrix, dense matrix, LinearOperator} The real or complex N-by-N matrix of the linear system. b : {array, matrix} Right hand side of the linear system. Has shape (N,) or (N,1). x0 : {array, matrix} Starting guess for the solution. tol : float Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below `tol`. maxiter : int 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. inner_m : int, optional Number of inner GMRES iterations per each outer iteration. outer_k : int, optional Number of vectors to carry between inner GMRES iterations. According to [BJM]_, good values are in the range of 1...3. However, note that if you want to use the additional vectors to accelerate solving multiple similar problems, larger values may be beneficial. outer_v : list of tuples, optional List containing tuples ``(v, Av)`` of vectors and corresponding matrix-vector products, used to augment the Krylov subspace, and carried between inner GMRES iterations. The element ``Av`` can be `None` if the matrix-vector product should be re-evaluated. This parameter is modified in-place by `lgmres`, and can be used to pass "guess" vectors in and out of the algorithm when solving similar problems. store_outer_Av : bool, optional Whether LGMRES should store also A*v in addition to vectors `v` in the `outer_v` list. Default is True. Returns ------- x : array or matrix The converged solution. info : int Provides convergence information: - 0 : successful exit - >0 : convergence to tolerance not achieved, number of iterations - <0 : illegal input or breakdown Notes ----- The LGMRES algorithm [BJM]_ [BPh]_ is designed to avoid the slowing of convergence in restarted GMRES, due to alternating residual vectors. Typically, it often outperforms GMRES(m) of comparable memory requirements by some measure, or at least is not much worse. Another advantage in this algorithm is that you can supply it with 'guess' vectors in the `outer_v` argument that augment the Krylov subspace. If the solution lies close to the span of these vectors, the algorithm converges faster. This can be useful if several very similar matrices need to be inverted one after another, such as in Newton-Krylov iteration where the Jacobian matrix often changes little in the nonlinear steps. References ---------- .. [BJM] A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). .. [BPh] A.H. Baker, PhD thesis, University of Colorado (2003). http://amath.colorado.edu/activities/thesis/allisonb/Thesis.ps """ from scipy.linalg.basic import lstsq A,M,x,b,postprocess = make_system(A,M,x0,b) if not np.isfinite(b).all(): raise ValueError("RHS must contain only finite numbers") matvec = A.matvec psolve = M.matvec if outer_v is None: outer_v = [] axpy, dot, scal = None, None, None b_norm = norm2(b) if b_norm == 0: b_norm = 1 for k_outer in xrange(maxiter): r_outer = matvec(x) - b # -- callback if callback is not None: callback(x) # -- determine input type routines if axpy is None: if np.iscomplexobj(r_outer) and not np.iscomplexobj(x): x = x.astype(r_outer.dtype) axpy, dot, scal = get_blas_funcs(['axpy', 'dot', 'scal'], (x, r_outer)) # -- check stopping condition r_norm = norm2(r_outer) if r_norm < tol * b_norm or r_norm < tol: break # -- inner LGMRES iteration vs0 = -psolve(r_outer) inner_res_0 = norm2(vs0) if inner_res_0 == 0: rnorm = norm2(r_outer) raise RuntimeError("Preconditioner returned a zero vector; " "|v| ~ %.1g, |M v| = 0" % rnorm) vs0 = scal(1.0/inner_res_0, vs0) hs = [] vs = [vs0] ws = [] y = None for j in xrange(1, 1 + inner_m + len(outer_v)): # -- Arnoldi process: # # Build an orthonormal basis V and matrices W and H such that # A W = V H # Columns of W, V, and H are stored in `ws`, `vs` and `hs`. # # The first column of V is always the residual vector, `vs0`; # V has *one more column* than the other of the three matrices. # # The other columns in V are built by feeding in, one # by one, some vectors `z` and orthonormalizing them # against the basis so far. The trick here is to # feed in first some augmentation vectors, before # starting to construct the Krylov basis on `v0`. # # It was shown in [BJM]_ that a good choice (the LGMRES choice) # for these augmentation vectors are the `dx` vectors obtained # from a couple of the previous restart cycles. # # Note especially that while `vs0` is always the first # column in V, there is no reason why it should also be # the first column in W. (In fact, below `vs0` comes in # W only after the augmentation vectors.) # # The rest of the algorithm then goes as in GMRES, one # solves a minimization problem in the smaller subspace # spanned by W (range) and V (image). # # XXX: Below, I'm lazy and use `lstsq` to solve the # small least squares problem. Performance-wise, this # is in practice acceptable, but it could be nice to do # it on the fly with Givens etc. # # ++ evaluate v_new = None if j < len(outer_v) + 1: z, v_new = outer_v[j-1] elif j == len(outer_v) + 1: z = vs0 else: z = vs[-1] if v_new is None: v_new = psolve(matvec(z)) else: # Note: v_new is modified in-place below. Must make a # copy to ensure that the outer_v vectors are not # clobbered. v_new = v_new.copy() # ++ orthogonalize hcur = [] for v in vs: alpha = dot(v, v_new) hcur.append(alpha) v_new = axpy(v, v_new, v.shape[0], -alpha) # v_new -= alpha*v hcur.append(norm2(v_new)) if hcur[-1] == 0: # Exact solution found; bail out. # Zero basis vector (v_new) in the least-squares problem # does no harm, so we can just use the same code as usually; # it will give zero (inner) residual as a result. bailout = True else: bailout = False v_new = scal(1.0/hcur[-1], v_new) vs.append(v_new) hs.append(hcur) ws.append(z) # XXX: Ugly: should implement the GMRES iteration properly, # with Givens rotations and not using lstsq. Instead, we # spare some work by solving the LSQ problem only every 5 # iterations. if not bailout and j % 5 != 1 and j < inner_m + len(outer_v) - 1: continue # -- GMRES optimization problem hess = np.zeros((j+1, j), x.dtype) e1 = np.zeros((j+1,), x.dtype) e1[0] = inner_res_0 for q in xrange(j): hess[:(q+2),q] = hs[q] y, resids, rank, s = lstsq(hess, e1) inner_res = norm2(np.dot(hess, y) - e1) # -- check for termination if inner_res < tol * inner_res_0: break # -- GMRES terminated: eval solution dx = ws[0]*y[0] for w, yc in zip(ws[1:], y[1:]): dx = axpy(w, dx, dx.shape[0], yc) # dx += w*yc # -- Store LGMRES augmentation vectors nx = norm2(dx) if store_outer_Av: q = np.dot(hess, y) ax = vs[0]*q[0] for v, qc in zip(vs[1:], q[1:]): ax = axpy(v, ax, ax.shape[0], qc) outer_v.append((dx/nx, ax/nx)) else: outer_v.append((dx/nx, None)) # -- Retain only a finite number of augmentation vectors while len(outer_v) > outer_k: del outer_v[0] # -- Apply step x += dx else: # didn't converge ... return postprocess(x), maxiter return postprocess(x), 0
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