|
# Copyright (C) 2009, Pauli Virtanen <pav@iki.fi> # Distributed under the same license as Scipy.
inner_m=30, outer_k=3, outer_v=None, store_outer_Av=True, prepend_outer_v=False, atol=None): """ Solve a matrix equation using the LGMRES algorithm.
The LGMRES algorithm [1]_ [2]_ 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, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is `tol`.
.. warning::
The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : int, optional 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}, optional 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, optional 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 [1]_, 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. prepend_outer_v : bool, optional Whether to put outer_v augmentation vectors before Krylov iterates. In standard LGMRES, prepend_outer_v=False.
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 [1]_ [2]_ 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 ---------- .. [1] A.H. Baker and E.R. Jessup and T. Manteuffel, "A Technique for Accelerating the Convergence of Restarted GMRES", SIAM J. Matrix Anal. Appl. 26, 962 (2005). .. [2] A.H. Baker, "On Improving the Performance of the Linear Solver restarted GMRES", PhD thesis, University of Colorado (2003).
Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import lgmres >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> b = np.array([2, 4, -1], dtype=float) >>> x, exitCode = lgmres(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True """ 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")
if atol is None: warnings.warn("scipy.sparse.linalg.lgmres called without specifying `atol`. " "The default value will change in the future. To preserve " "current behavior, set ``atol=tol``.", category=DeprecationWarning, stacklevel=2) atol = tol
matvec = A.matvec psolve = M.matvec
if outer_v is None: outer_v = []
axpy, dot, scal = None, None, None nrm2 = get_blas_funcs('nrm2', [b])
b_norm = nrm2(b) ptol_max_factor = 1.0
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, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (x, r_outer))
# -- check stopping condition r_norm = nrm2(r_outer) if r_norm <= max(atol, tol * b_norm): break
# -- inner LGMRES iteration v0 = -psolve(r_outer) inner_res_0 = nrm2(v0)
if inner_res_0 == 0: rnorm = nrm2(r_outer) raise RuntimeError("Preconditioner returned a zero vector; " "|v| ~ %.1g, |M v| = 0" % rnorm)
v0 = scal(1.0/inner_res_0, v0)
ptol = min(ptol_max_factor, max(atol, tol*b_norm)/r_norm)
try: Q, R, B, vs, zs, y, pres = _fgmres(matvec, v0, inner_m, lpsolve=psolve, atol=ptol, outer_v=outer_v, prepend_outer_v=prepend_outer_v) y *= inner_res_0 if not np.isfinite(y).all(): # Overflow etc. in computation. There's no way to # recover from this, so we have to bail out. raise LinAlgError() except LinAlgError: # Floating point over/underflow, non-finite result from # matmul etc. -- report failure. return postprocess(x), k_outer + 1
# Inner loop tolerance control if pres > ptol: ptol_max_factor = min(1.0, 1.5 * ptol_max_factor) else: ptol_max_factor = max(1e-16, 0.25 * ptol_max_factor)
# -- GMRES terminated: eval solution dx = zs[0]*y[0] for w, yc in zip(zs[1:], y[1:]): dx = axpy(w, dx, dx.shape[0], yc) # dx += w*yc
# -- Store LGMRES augmentation vectors nx = nrm2(dx) if nx > 0: if store_outer_Av: q = Q.dot(R.dot(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 |