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 [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 : float, optional Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below `tol`. 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. 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, SIAM J. Matrix Anal. Appl. 26, 962 (2005). .. [2] A.H. Baker, PhD thesis, University of Colorado (2003). http://amath.colorado.edu/activities/thesis/allisonb/Thesis.ps """ 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 nrm2 = get_blas_funcs('nrm2', [b]) b_norm = nrm2(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, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (x, r_outer)) trtrs = get_lapack_funcs('trtrs', (x, r_outer)) # -- check stopping condition r_norm = nrm2(r_outer) if r_norm <= tol * b_norm or r_norm <= tol: break # -- inner LGMRES iteration vs0 = -psolve(r_outer) inner_res_0 = nrm2(vs0) if inner_res_0 == 0: rnorm = nrm2(r_outer) raise RuntimeError("Preconditioner returned a zero vector; " "|v| ~ %.1g, |M v| = 0" % rnorm) vs0 = scal(1.0/inner_res_0, vs0) vs = [vs0] ws = [] y = None # H is stored in QR factorized form Q = np.ones((1, 1), dtype=vs0.dtype) R = np.zeros((1, 0), dtype=vs0.dtype) eps = np.finfo(vs0.dtype).eps 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). # # ++ 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 = np.zeros(j+1, dtype=Q.dtype) for i, v in enumerate(vs): alpha = dot(v, v_new) hcur[i] = alpha v_new = axpy(v, v_new, v.shape[0], -alpha) # v_new -= alpha*v hcur[-1] = nrm2(v_new) with np.errstate(over='ignore', divide='ignore'): # Careful with denormals alpha = 1/hcur[-1] if np.isfinite(alpha): v_new = scal(alpha, v_new) else: # v_new either zero (solution in span of previous # vectors) or we have nans. If we already have # previous vectors in R, we can discard the current # vector and bail out. if j > 1: j -= 1 break vs.append(v_new) ws.append(z) # -- GMRES optimization problem # Add new column to H=Q*R, padding other columns with zeros Q2 = np.zeros((j+1, j+1), dtype=Q.dtype, order='F') Q2[:j,:j] = Q Q2[j,j] = 1 R2 = np.zeros((j+1, j-1), dtype=R.dtype, order='F') R2[:j,:] = R Q, R = qr_insert(Q2, R2, hcur, j-1, which='col', overwrite_qru=True, check_finite=False) # Transformed least squares problem # || Q R y - inner_res_0 * e_1 ||_2 = min! # Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0] # Residual is immediately known inner_res = abs(Q[0,-1]) * inner_res_0 # -- check for termination if inner_res <= tol * inner_res_0: break # -- Get the LSQ problem solution y, info = trtrs(R[:j,:j], Q[0,:j].conj()) if info != 0: # Zero diagonal -> exact solution, but we handled that above raise RuntimeError("QR solution failed") y *= inner_res_0 if not np.isfinite(y).all(): # Floating point over/underflow, non-finite result from # matmul etc. -- report failure. return postprocess(x), k_outer + 1 # -- 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 = 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
def _fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=(), outer_v=(), prepend_outer_v=False): """ FGMRES Arnoldi process, with optional projection or augmentation Parameters ---------- matvec : callable Operation A*x v0 : ndarray Initial vector, normalized to nrm2(v0) == 1 m : int Number of GMRES rounds atol : float Absolute tolerance for early exit lpsolve : callable Left preconditioner L rpsolve : callable Right preconditioner R CU : list of (ndarray, ndarray) Columns of matrices C and U in GCROT outer_v : list of ndarrays Augmentation vectors in LGMRES prepend_outer_v : bool, optional Whether augmentation vectors come before or after Krylov iterates Raises ------ LinAlgError If nans encountered Returns ------- Q, R : ndarray QR decomposition of the upper Hessenberg H=QR B : ndarray Projections corresponding to matrix C vs : list of ndarray Columns of matrix V zs : list of ndarray Columns of matrix Z y : ndarray Solution to ||H y - e_1||_2 = min! res : float The final (preconditioned) residual norm """ if lpsolve is None: lpsolve = lambda x: x if rpsolve is None: rpsolve = lambda x: x axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (v0, )) vs = [v0] zs = [] y = None res = np.nan m = m + len(outer_v) # Orthogonal projection coefficients B = np.zeros((len(cs), m), dtype=v0.dtype) # H is stored in QR factorized form Q = np.ones((1, 1), dtype=v0.dtype) R = np.zeros((1, 0), dtype=v0.dtype) eps = np.finfo(v0.dtype).eps breakdown = False # FGMRES Arnoldi process for j in range(m): # L A Z = C B + V H if prepend_outer_v and j < len(outer_v): z, w = outer_v[j] elif prepend_outer_v and j == len(outer_v): z = rpsolve(v0) w = None elif not prepend_outer_v and j >= m - len(outer_v): z, w = outer_v[j - (m - len(outer_v))] else: z = rpsolve(vs[-1]) w = None if w is None: w = lpsolve(matvec(z)) else: # w is clobbered below w = w.copy() w_norm = nrm2(w) # GCROT projection: L A -> (1 - C C^H) L A # i.e. orthogonalize against C for i, c in enumerate(cs): alpha = dot(c, w) B[i, j] = alpha w = axpy(c, w, c.shape[0], -alpha) # w -= alpha*c # Orthogonalize against V hcur = np.zeros(j + 2, dtype=Q.dtype) for i, v in enumerate(vs): alpha = dot(v, w) hcur[i] = alpha w = axpy(v, w, v.shape[0], -alpha) # w -= alpha*v hcur[i + 1] = nrm2(w) with np.errstate(over='ignore', divide='ignore'): # Careful with denormals alpha = 1 / hcur[-1] if np.isfinite(alpha): w = scal(alpha, w) if not (hcur[-1] > eps * w_norm): # w essentially in the span of previous vectors, # or we have nans. Bail out after updating the QR # solution. breakdown = True vs.append(w) zs.append(z) # Arnoldi LSQ problem # Add new column to H=Q@R, padding other columns with zeros Q2 = np.zeros((j + 2, j + 2), dtype=Q.dtype, order='F') Q2[:j + 1, :j + 1] = Q Q2[j + 1, j + 1] = 1 R2 = np.zeros((j + 2, j), dtype=R.dtype, order='F') R2[:j + 1, :] = R Q, R = qr_insert(Q2, R2, hcur, j, which='col', overwrite_qru=True, check_finite=False) # Transformed least squares problem # || Q R y - inner_res_0 * e_1 ||_2 = min! # Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0] # Residual is immediately known res = abs(Q[0, -1]) # Check for termination if res < atol or breakdown: break if not np.isfinite(R[j, j]): # nans encountered, bail out raise LinAlgError() # -- Get the LSQ problem solution # The problem is triangular, but the condition number may be # bad (or in case of breakdown the last diagonal entry may be # zero), so use lstsq instead of trtrs. y, _, _, _, = lstsq(R[:j + 1, :j + 1], Q[0, :j + 1].conj()) B = B[:, :j + 1] return Q, R, B, vs, zs, y, res
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 [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 : float, optional Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below `tol`. 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. 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, SIAM J. Matrix Anal. Appl. 26, 962 (2005). .. [2] A.H. Baker, PhD thesis, University of Colorado (2003). http://amath.colorado.edu/activities/thesis/allisonb/Thesis.ps """ 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 nrm2 = get_blas_funcs('nrm2', [b]) b_norm = nrm2(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, nrm2 = get_blas_funcs( ['axpy', 'dot', 'scal', 'nrm2'], (x, r_outer)) trtrs = get_lapack_funcs('trtrs', (x, r_outer)) # -- check stopping condition r_norm = nrm2(r_outer) if r_norm <= tol * b_norm or r_norm <= tol: break # -- inner LGMRES iteration vs0 = -psolve(r_outer) inner_res_0 = nrm2(vs0) if inner_res_0 == 0: rnorm = nrm2(r_outer) raise RuntimeError("Preconditioner returned a zero vector; " "|v| ~ %.1g, |M v| = 0" % rnorm) vs0 = scal(1.0 / inner_res_0, vs0) vs = [vs0] ws = [] y = None # H is stored in QR factorized form Q = np.ones((1, 1), dtype=vs0.dtype) R = np.zeros((1, 0), dtype=vs0.dtype) eps = np.finfo(vs0.dtype).eps 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). # # ++ 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 = np.zeros(j + 1, dtype=Q.dtype) for i, v in enumerate(vs): alpha = dot(v, v_new) hcur[i] = alpha v_new = axpy(v, v_new, v.shape[0], -alpha) # v_new -= alpha*v hcur[-1] = nrm2(v_new) with np.errstate(over='ignore', divide='ignore'): # Careful with denormals alpha = 1 / hcur[-1] if np.isfinite(alpha): v_new = scal(alpha, v_new) else: # v_new either zero (solution in span of previous # vectors) or we have nans. If we already have # previous vectors in R, we can discard the current # vector and bail out. if j > 1: j -= 1 break vs.append(v_new) ws.append(z) # -- GMRES optimization problem # Add new column to H=Q*R, padding other columns with zeros Q2 = np.zeros((j + 1, j + 1), dtype=Q.dtype, order='F') Q2[:j, :j] = Q Q2[j, j] = 1 R2 = np.zeros((j + 1, j - 1), dtype=R.dtype, order='F') R2[:j, :] = R Q, R = qr_insert(Q2, R2, hcur, j - 1, which='col', overwrite_qru=True, check_finite=False) # Transformed least squares problem # || Q R y - inner_res_0 * e_1 ||_2 = min! # Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0] # Residual is immediately known inner_res = abs(Q[0, -1]) * inner_res_0 # -- check for termination if inner_res <= tol * inner_res_0: break # -- Get the LSQ problem solution y, info = trtrs(R[:j, :j], Q[0, :j].conj()) if info != 0: # Zero diagonal -> exact solution, but we handled that above raise RuntimeError("QR solution failed") y *= inner_res_0 if not np.isfinite(y).all(): # Floating point over/underflow, non-finite result from # matmul etc. -- report failure. return postprocess(x), k_outer + 1 # -- 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 = 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
def _fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=(), outer_v=(), prepend_outer_v=False): """ FGMRES Arnoldi process, with optional projection or augmentation Parameters ---------- matvec : callable Operation A*x v0 : ndarray Initial vector, normalized to nrm2(v0) == 1 m : int Number of GMRES rounds atol : float Absolute tolerance for early exit lpsolve : callable Left preconditioner L rpsolve : callable Right preconditioner R CU : list of (ndarray, ndarray) Columns of matrices C and U in GCROT outer_v : list of ndarrays Augmentation vectors in LGMRES prepend_outer_v : bool, optional Whether augmentation vectors come before or after Krylov iterates Raises ------ LinAlgError If nans encountered Returns ------- Q, R : ndarray QR decomposition of the upper Hessenberg H=QR B : ndarray Projections corresponding to matrix C vs : list of ndarray Columns of matrix V zs : list of ndarray Columns of matrix Z y : ndarray Solution to ||H y - e_1||_2 = min! res : float The final (preconditioned) residual norm """ if lpsolve is None: lpsolve = lambda x: x if rpsolve is None: rpsolve = lambda x: x axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (v0,)) vs = [v0] zs = [] y = None res = np.nan m = m + len(outer_v) # Orthogonal projection coefficients B = np.zeros((len(cs), m), dtype=v0.dtype) # H is stored in QR factorized form Q = np.ones((1, 1), dtype=v0.dtype) R = np.zeros((1, 0), dtype=v0.dtype) eps = np.finfo(v0.dtype).eps breakdown = False # FGMRES Arnoldi process for j in xrange(m): # L A Z = C B + V H if prepend_outer_v and j < len(outer_v): z, w = outer_v[j] elif prepend_outer_v and j == len(outer_v): z = rpsolve(v0) w = None elif not prepend_outer_v and j >= m - len(outer_v): z, w = outer_v[j - (m - len(outer_v))] else: z = rpsolve(vs[-1]) w = None if w is None: w = lpsolve(matvec(z)) else: # w is clobbered below w = w.copy() w_norm = nrm2(w) # GCROT projection: L A -> (1 - C C^H) L A # i.e. orthogonalize against C for i, c in enumerate(cs): alpha = dot(c, w) B[i,j] = alpha w = axpy(c, w, c.shape[0], -alpha) # w -= alpha*c # Orthogonalize against V hcur = np.zeros(j+2, dtype=Q.dtype) for i, v in enumerate(vs): alpha = dot(v, w) hcur[i] = alpha w = axpy(v, w, v.shape[0], -alpha) # w -= alpha*v hcur[i+1] = nrm2(w) with np.errstate(over='ignore', divide='ignore'): # Careful with denormals alpha = 1/hcur[-1] if np.isfinite(alpha): w = scal(alpha, w) if not (hcur[-1] > eps * w_norm): # w essentially in the span of previous vectors, # or we have nans. Bail out after updating the QR # solution. breakdown = True vs.append(w) zs.append(z) # Arnoldi LSQ problem # Add new column to H=Q*R, padding other columns with zeros Q2 = np.zeros((j+2, j+2), dtype=Q.dtype, order='F') Q2[:j+1,:j+1] = Q Q2[j+1,j+1] = 1 R2 = np.zeros((j+2, j), dtype=R.dtype, order='F') R2[:j+1,:] = R Q, R = qr_insert(Q2, R2, hcur, j, which='col', overwrite_qru=True, check_finite=False) # Transformed least squares problem # || Q R y - inner_res_0 * e_1 ||_2 = min! # Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0] # Residual is immediately known res = abs(Q[0,-1]) # Check for termination if res < atol or breakdown: break if not np.isfinite(R[j,j]): # nans encountered, bail out raise LinAlgError() # -- Get the LSQ problem solution # The problem is triangular, but the condition number may be # bad (or in case of breakdown the last diagonal entry may be # zero), so use lstsq instead of trtrs. y, _, _, _, = lstsq(R[:j+1,:j+1], Q[0,:j+1].conj()) B = B[:,:j+1] return Q, R, B, vs, zs, y, res
def _fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=(), outer_v=(), prepend_outer_v=False): if lpsolve is None: def lpsolve(x): return x if rpsolve is None: def rpsolve(x): return x axpy, dot, scal, nrm2 = get_blas_funcs( ['axpy', 'dot', 'scal', 'nrm2'], (v0,)) vs = [v0] zs = [] y = None m = m + len(outer_v) B = np.zeros((len(cs), m), dtype=v0.dtype) Q = np.ones((1, 1), dtype=v0.dtype) R = np.zeros((1, 0), dtype=v0.dtype) eps = np.finfo(v0.dtype).eps breakdown = False for j in xrange(m): if prepend_outer_v and j < len(outer_v): z, w = outer_v[j] elif prepend_outer_v and j == len(outer_v): z = rpsolve(v0) w = None elif not prepend_outer_v and j >= m - len(outer_v): z, w = outer_v[j - (m - len(outer_v))] else: z = rpsolve(vs[-1]) w = None if w is None: w = lpsolve(matvec(z)) else: w = w.copy() w_norm = nrm2(w) for i, c in enumerate(cs): alpha = dot(c, w) B[i, j] = alpha w = axpy(c, w, c.shape[0], -alpha) hcur = np.zeros(j + 2, dtype=Q.dtype) for i, v in enumerate(vs): alpha = dot(v, w) hcur[i] = alpha w = axpy(v, w, v.shape[0], -alpha) hcur[i + 1] = nrm2(w) with np.errstate(over='ignore', divide='ignore'): alpha = 1 / hcur[-1] if np.isfinite(alpha): w = scal(alpha, w) if not (hcur[-1] > eps * w_norm): breakdown = True vs.append(w) zs.append(z) Q2 = np.zeros((j + 2, j + 2), dtype=Q.dtype, order='F') Q2[:j + 1, :j + 1] = Q Q2[j + 1, j + 1] = 1 R2 = np.zeros((j + 2, j), dtype=R.dtype, order='F') R2[:j + 1, :] = R Q, R = qr_insert(Q2, R2, hcur, j, which='col', overwrite_qru=True, check_finite=False) res = abs(Q[0, -1]) if res < atol or breakdown: break if not np.isfinite(R[j, j]): raise LinAlgError() y, _, _, _, = lstsq(R[:j + 1, :j + 1], Q[0, :j + 1].conj()) B = B[:, :j + 1] return Q, R, B, vs, zs, y
def quadprog_solve(quadr_coeff_G, linear_coeff_a, n_ineq, ineq_coef_C, ineq_vec_d, m_eq, eq_coef_A, eq_vec_b): DONE = False FULL_STEP = False if eq_coef_A is not None: ADDING_EQ_CONSTRAINTS = True else: ADDING_EQ_CONSTRAINTS = False first_pass = True ###-------------------------------------### ## Solution iterate sol = (-1) * np.linalg.inv(quadr_coeff_G) * linear_coeff_a ## We only need to keep ahold of L^{-1} for the implementation. ## L is the triangular result of L = np.linalg.cholesky(quadr_coeff_G) Linv = np.linalg.inv(L) ## Need to adopt the conventions used for directly computing ## the values of ``z`` and ``r``, we need to hold onto ## the values of the factorization QR = B = L^{-1} N. Q = None R = None J1 = None J2 = None ## indices of constraints being considered active_set = np.array([], dtype=(np.dtype(int))) ## normal vector of a given constraint n_p = None ## index of n_p in the choice of equality or inequality constraint p = None ## number of considered constraints in active set q = 0 u = None lagr = None # index out of all constraints which was dropped k_dropped = None # index of active constraint which was dropped j_dropped = None z = None ## Step direction in `primal' space r = 0 ## Step direction in `dual' space ###-------------------------------------### ###~~~~~~~~ Step 1 ~~~~~~~~### while not DONE: ineq = np.ravel((ineq_coef_C.T * sol) - ineq_vec_d) if ADDING_EQ_CONSTRAINTS: eq_prb = np.ravel((eq_coef_A.T * sol) - eq_vec_b) if (len(active_set) == m_eq): ADDING_EQ_CONSTRAINTS = False if (np.any(ineq < 0) or ADDING_EQ_CONSTRAINTS): if (ADDING_EQ_CONSTRAINTS): p = len(active_set) n_p = eq_coef_A[:, p] else: ## Choose a violated constraint not in active set. ## This is the most naive way, can be improved. #violated_constraints = np.ravel(np.where(ineq < 0)) #v = [x for x in violated_constraints if x not in active_set] ## Pick the first violated constraint. #p = v[0] ## normal vector for each constraint, vector normal to the plane. #n_p = ineq_coef_C[:,p] ## Instead, just loop through inequality constraints for blah in range(0, n_ineq): if ineq[blah] < 0: p = blah burn_flag = False for jack in range(m_eq + 1, q): if active_set[jack] == p: burn_flag = True break if burn_flag: continue else: break n_p = ineq_coef_C[:, p] if q == 0: u = 0 lagr = np.hstack((u, 0)) ###~~~~~~~~ Step 2 ~~~~~~~~### FULL_STEP = False while not FULL_STEP: # algo as writ will cycle back here after taking a step # in dual space, update inequality portion ineq = np.ravel((ineq_coef_C.T * sol) - ineq_vec_d) if ADDING_EQ_CONSTRAINTS: eq_prb = np.ravel((eq_coef_A.T * sol) - eq_vec_b) ###~~~~~~~~ Step 2(a) ~~~~~~~~### ## Calculate step directions if first_pass: z = np.linalg.inv(quadr_coeff_G) * n_p first_pass = False else: # step direction in the primal space z = J2 * J2.T * n_p if (q > 0): # negative of step direction in the dual space # r will have num_rows = len(active_set) r = np.linalg.inv(R[0:q, 0:q]) * J1.T * n_p ###~~~~~~~~ Step 2(b) ~~~~~~~~### # partial step length t1 - max step in dual space if ((q == 0) or (np.any(r <= 0)) or ADDING_EQ_CONSTRAINTS): t1 = np.inf else: t1 = np.inf k_dropped = None for j in range(m_eq, len(active_set)): k = active_set[j] if (r[j] > 0) and (lagr[j] / r[j]) < t1: t1 = lagr[j] / r[j] k_dropped = k j_dropped = j t1 = np.ravel(t1)[0] # full step length t2 - min step in primal space if (np.all(z == 0)): # If no step in primal space t2 = np.inf else: if ADDING_EQ_CONSTRAINTS: t2 = (-1) * eq_prb[p] / (z.T * n_p) else: t2 = (-1) * ineq[p] / (z.T * n_p) t2 = np.ravel(t2)[0] # current step length t = np.min([t1, t2]) ###~~~~~~~~ Step 2(c) ~~~~~~~~### if (t == np.inf): print("infeasible! Stop here!") FULL_STEP = True DONE = True return None #break # If t2 is infinite, then we took a partial step in the dual space. if (t2 == np.inf): #print("t2 infinite") # Update lagrangian lagr = lagr + t * np.hstack((np.ravel(-1 * r), 1)) # Drop the constraint which minimized the step we took at that # point. active_set = np.delete(active_set, j_dropped) q = q - 1 Q, R = qr_delete(Q, R, j_dropped, 1, 'col') J1 = Linv.T * Q[:, [x for x in range(0, q)]] J2 = Linv.T * Q[:, [x for x in range(q, Q.shape[1])]] # go back to step 2(a) continue # Update iterate for x, and the lagrangian sol = sol + t * z lagr = lagr + t * np.hstack((np.ravel(-1 * r), 1)) # if we took a full step if (t == t2): #print("full step") ## Add the constraint to the active set active_set = np.hstack((active_set, p)) q = q + 1 u = lagr[-q:] if Q is None: Q, R = np.linalg.qr(Linv * n_p, mode="complete") J1 = Linv.T * Q[:, [x for x in range(0, q)]] J2 = Linv.T * Q[:, [x for x in range(q, Q.shape[1])]] else: Q, R = qr_insert(Q, R, (Linv * n_p), len(active_set) - 1, 'col') J1 = Linv.T * Q[:, [x for x in range(0, q)]] J2 = Linv.T * Q[:, [x for x in range(q, Q.shape[1])]] # Exit current loop for Step 2, go back to Step 1 FULL_STEP = True break # if we took a partial step if (t == t1): #print("partial step") # Drop constraint k active_set = np.delete(active_set, j_dropped) q = q - 1 Q, R = qr_delete(Q, R, j_dropped, 1, 'col') J1 = Linv.T * Q[:, [x for x in range(0, q)]] J2 = Linv.T * Q[:, [x for x in range(q, Q.shape[1])]] # Go back to step 2(a) continue else: DONE = True #print(ineq) #print(sol) return sol