""" This contains a copy of the future version of scipy.sparse.linalg.eigen.arpack.eigsh It's an upgraded wrapper of the ARPACK library which allows the use of shift-invert mode for symmetric matrices. Find a few eigenvectors and eigenvalues of a matrix. Uses ARPACK: http://www.caam.rice.edu/software/ARPACK/ """ # Wrapper implementation notes # # ARPACK Entry Points # ------------------- # The entry points to ARPACK are # - (s,d)seupd : single and double precision symmetric matrix # - (s,d,c,z)neupd: single,double,complex,double complex general matrix # This wrapper puts the *neupd (general matrix) interfaces in eigs() # and the *seupd (symmetric matrix) in eigsh(). # There is no Hermetian complex/double complex interface. # To find eigenvalues of a Hermetian matrix you # must use eigs() and not eigsh() # It might be desirable to handle the Hermetian case differently # and, for example, return real eigenvalues. # Number of eigenvalues returned and complex eigenvalues # ------------------------------------------------------ # The ARPACK nonsymmetric real and double interface (s,d)naupd return # eigenvalues and eigenvectors in real (float,double) arrays. # Since the eigenvalues and eigenvectors are, in general, complex # ARPACK puts the real and imaginary parts in consecutive entries # in real-valued arrays. This wrapper puts the real entries # into complex data types and attempts to return the requested eigenvalues # and eigenvectors. # Solver modes # ------------ # ARPACK and handle shifted and shift-inverse computations # for eigenvalues by providing a shift (sigma) and a solver. __docformat__ = "restructuredtext en" __all__ = ['eigs', 'eigsh', 'svds', 'ArpackError', 'ArpackNoConvergence'] from scipy.sparse.linalg.eigen.arpack import _arpack import numpy as np from scipy.sparse.linalg.interface import aslinearoperator, LinearOperator from scipy.sparse import identity, isspmatrix, isspmatrix_csr from scipy.linalg import lu_factor, lu_solve from scipy.sparse.sputils import isdense from scipy.sparse.linalg import gmres, splu import scipy from distutils.version import LooseVersion _type_conv = {'f': 's', 'd': 'd', 'F': 'c', 'D': 'z'} _ndigits = {'f': 5, 'd': 12, 'F': 5, 'D': 12} DNAUPD_ERRORS = { 0: "Normal exit.", 1: "Maximum number of iterations taken. " "All possible eigenvalues of OP has been found. IPARAM(5) " "returns the number of wanted converged Ritz values.", 2: "No longer an informational error. Deprecated starting " "with release 2 of ARPACK.", 3: "No shifts could be applied during a cycle of the " "Implicitly restarted Arnoldi iteration. One possibility " "is to increase the size of NCV relative to NEV. ", -1: "N must be positive.", -2: "NEV must be positive.", -3: "NCV-NEV >= 2 and less than or equal to N.", -4: "The maximum number of Arnoldi update iterations allowed " "must be greater than zero.", -5: " WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'", -6: "BMAT must be one of 'I' or 'G'.", -7: "Length of private work array WORKL is not sufficient.", -8: "Error return from LAPACK eigenvalue calculation;", -9: "Starting vector is zero.", -10: "IPARAM(7) must be 1,2,3,4.", -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatable.", -12: "IPARAM(1) must be equal to 0 or 1.", -13: "NEV and WHICH = 'BE' are incompatable.", -9999: "Could not build an Arnoldi factorization. " "IPARAM(5) returns the size of the current Arnoldi " "factorization. The user is advised to check that " "enough workspace and array storage has been allocated." } SNAUPD_ERRORS = DNAUPD_ERRORS ZNAUPD_ERRORS = DNAUPD_ERRORS.copy() ZNAUPD_ERRORS[-10] = "IPARAM(7) must be 1,2,3." CNAUPD_ERRORS = ZNAUPD_ERRORS DSAUPD_ERRORS = { 0: "Normal exit.", 1: "Maximum number of iterations taken. " "All possible eigenvalues of OP has been found.", 2: "No longer an informational error. Deprecated starting with " "release 2 of ARPACK.", 3: "No shifts could be applied during a cycle of the Implicitly " "restarted Arnoldi iteration. One possibility is to increase " "the size of NCV relative to NEV. ", -1: "N must be positive.", -2: "NEV must be positive.", -3: "NCV must be greater than NEV and less than or equal to N.", -4: "The maximum number of Arnoldi update iterations allowed " "must be greater than zero.", -5: "WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.", -6: "BMAT must be one of 'I' or 'G'.", -7: "Length of private work array WORKL is not sufficient.", -8: "Error return from trid. eigenvalue calculation; " "Informational error from LAPACK routine dsteqr .", -9: "Starting vector is zero.", -10: "IPARAM(7) must be 1,2,3,4,5.", -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatable.", -12: "IPARAM(1) must be equal to 0 or 1.", -13: "NEV and WHICH = 'BE' are incompatable. ", -9999: "Could not build an Arnoldi factorization. " "IPARAM(5) returns the size of the current Arnoldi " "factorization. The user is advised to check that " "enough workspace and array storage has been allocated.", } SSAUPD_ERRORS = DSAUPD_ERRORS DNEUPD_ERRORS = { 0: "Normal exit.", 1: "The Schur form computed by LAPACK routine dlahqr " "could not be reordered by LAPACK routine dtrsen. " "Re-enter subroutine dneupd with IPARAM(5)NCV and " "increase the size of the arrays DR and DI to have " "dimension at least dimension NCV and allocate at least NCV " "columns for Z. NOTE: Not necessary if Z and V share " "the same space. Please notify the authors if this error" "occurs.", -1: "N must be positive.", -2: "NEV must be positive.", -3: "NCV-NEV >= 2 and less than or equal to N.", -5: "WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'", -6: "BMAT must be one of 'I' or 'G'.", -7: "Length of private work WORKL array is not sufficient.", -8: "Error return from calculation of a real Schur form. " "Informational error from LAPACK routine dlahqr .", -9: "Error return from calculation of eigenvectors. " "Informational error from LAPACK routine dtrevc.", -10: "IPARAM(7) must be 1,2,3,4.", -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.", -12: "HOWMNY = 'S' not yet implemented", -13: "HOWMNY must be one of 'A' or 'P' if RVEC = .true.", -14: "DNAUPD did not find any eigenvalues to sufficient " "accuracy.", -15: "DNEUPD got a different count of the number of converged " "Ritz values than DNAUPD got. This indicates the user " "probably made an error in passing data from DNAUPD to " "DNEUPD or that the data was modified before entering " "DNEUPD", } SNEUPD_ERRORS = DNEUPD_ERRORS.copy() SNEUPD_ERRORS[1] = ("The Schur form computed by LAPACK routine slahqr " "could not be reordered by LAPACK routine strsen . " "Re-enter subroutine dneupd with IPARAM(5)=NCV and " "increase the size of the arrays DR and DI to have " "dimension at least dimension NCV and allocate at least " "NCV columns for Z. NOTE: Not necessary if Z and V share " "the same space. Please notify the authors if this error " "occurs.") SNEUPD_ERRORS[-14] = ("SNAUPD did not find any eigenvalues to sufficient " "accuracy.") SNEUPD_ERRORS[-15] = ("SNEUPD got a different count of the number of " "converged Ritz values than SNAUPD got. This indicates " "the user probably made an error in passing data from " "SNAUPD to SNEUPD or that the data was modified before " "entering SNEUPD") ZNEUPD_ERRORS = {0: "Normal exit.", 1: "The Schur form computed by LAPACK routine csheqr " "could not be reordered by LAPACK routine ztrsen. " "Re-enter subroutine zneupd with IPARAM(5)=NCV and " "increase the size of the array D to have " "dimension at least dimension NCV and allocate at least " "NCV columns for Z. NOTE: Not necessary if Z and V share " "the same space. Please notify the authors if this error " "occurs.", -1: "N must be positive.", -2: "NEV must be positive.", -3: "NCV-NEV >= 1 and less than or equal to N.", -5: "WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'", -6: "BMAT must be one of 'I' or 'G'.", -7: "Length of private work WORKL array is not sufficient.", -8: "Error return from LAPACK eigenvalue calculation. " "This should never happened.", -9: "Error return from calculation of eigenvectors. " "Informational error from LAPACK routine ztrevc.", -10: "IPARAM(7) must be 1,2,3", -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.", -12: "HOWMNY = 'S' not yet implemented", -13: "HOWMNY must be one of 'A' or 'P' if RVEC = .true.", -14: "ZNAUPD did not find any eigenvalues to sufficient " "accuracy.", -15: "ZNEUPD got a different count of the number of " "converged Ritz values than ZNAUPD got. This " "indicates the user probably made an error in passing " "data from ZNAUPD to ZNEUPD or that the data was " "modified before entering ZNEUPD" } CNEUPD_ERRORS = ZNEUPD_ERRORS.copy() CNEUPD_ERRORS[-14] = ("CNAUPD did not find any eigenvalues to sufficient " "accuracy.") CNEUPD_ERRORS[-15] = ("CNEUPD got a different count of the number of " "converged Ritz values than CNAUPD got. This indicates " "the user probably made an error in passing data from " "CNAUPD to CNEUPD or that the data was modified before " "entering CNEUPD") DSEUPD_ERRORS = { 0: "Normal exit.", -1: "N must be positive.", -2: "NEV must be positive.", -3: "NCV must be greater than NEV and less than or equal to N.", -5: "WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.", -6: "BMAT must be one of 'I' or 'G'.", -7: "Length of private work WORKL array is not sufficient.", -8: ("Error return from trid. eigenvalue calculation; " "Information error from LAPACK routine dsteqr."), -9: "Starting vector is zero.", -10: "IPARAM(7) must be 1,2,3,4,5.", -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.", -12: "NEV and WHICH = 'BE' are incompatible.", -14: "DSAUPD did not find any eigenvalues to sufficient accuracy.", -15: "HOWMNY must be one of 'A' or 'S' if RVEC = .true.", -16: "HOWMNY = 'S' not yet implemented", -17: ("DSEUPD got a different count of the number of converged " "Ritz values than DSAUPD got. This indicates the user " "probably made an error in passing data from DSAUPD to " "DSEUPD or that the data was modified before entering " "DSEUPD.") } SSEUPD_ERRORS = DSEUPD_ERRORS.copy() SSEUPD_ERRORS[-14] = ("SSAUPD did not find any eigenvalues " "to sufficient accuracy.") SSEUPD_ERRORS[-17] = ("SSEUPD got a different count of the number of " "converged " "Ritz values than SSAUPD got. This indicates the user " "probably made an error in passing data from SSAUPD to " "SSEUPD or that the data was modified before entering " "SSEUPD.") _SAUPD_ERRORS = {'d': DSAUPD_ERRORS, 's': SSAUPD_ERRORS} _NAUPD_ERRORS = {'d': DNAUPD_ERRORS, 's': SNAUPD_ERRORS, 'z': ZNAUPD_ERRORS, 'c': CNAUPD_ERRORS} _SEUPD_ERRORS = {'d': DSEUPD_ERRORS, 's': SSEUPD_ERRORS} _NEUPD_ERRORS = {'d': DNEUPD_ERRORS, 's': SNEUPD_ERRORS, 'z': ZNEUPD_ERRORS, 'c': CNEUPD_ERRORS} # accepted values of parameter WHICH in _SEUPD _SEUPD_WHICH = ['LM', 'SM', 'LA', 'SA', 'BE'] # accepted values of parameter WHICH in _NAUPD _NEUPD_WHICH = ['LM', 'SM', 'LR', 'SR', 'LI', 'SI'] class ArpackError(RuntimeError): """ ARPACK error """ def __init__(self, info, infodict=_NAUPD_ERRORS): msg = infodict.get(info, "Unknown error") RuntimeError.__init__(self, "ARPACK error %d: %s" % (info, msg)) class ArpackNoConvergence(ArpackError): """ ARPACK iteration did not converge Attributes ---------- eigenvalues : ndarray Partial result. Converged eigenvalues. eigenvectors : ndarray Partial result. Converged eigenvectors. """ def __init__(self, msg, eigenvalues, eigenvectors): ArpackError.__init__(self, -1, {-1: msg}) self.eigenvalues = eigenvalues self.eigenvectors = eigenvectors class _ArpackParams(object): def __init__(self, n, k, tp, mode=1, sigma=None, ncv=None, v0=None, maxiter=None, which="LM", tol=0): if k <= 0: raise ValueError("k must be positive, k=%d" % k) if maxiter is None: maxiter = n * 10 if maxiter <= 0: raise ValueError("maxiter must be positive, maxiter=%d" % maxiter) if tp not in 'fdFD': raise ValueError("matrix type must be 'f', 'd', 'F', or 'D'") if v0 is not None: # ARPACK overwrites its initial resid, make a copy self.resid = np.array(v0, copy=True) info = 1 else: self.resid = np.zeros(n, tp) info = 0 if sigma is None: #sigma not used self.sigma = 0 else: self.sigma = sigma if ncv is None: ncv = 2 * k + 1 ncv = min(ncv, n) self.v = np.zeros((n, ncv), tp) # holds Ritz vectors self.iparam = np.zeros(11, "int") # set solver mode and parameters ishfts = 1 self.mode = mode self.iparam[0] = ishfts self.iparam[2] = maxiter self.iparam[3] = 1 self.iparam[6] = mode self.n = n self.tol = tol self.k = k self.maxiter = maxiter self.ncv = ncv self.which = which self.tp = tp self.info = info self.converged = False self.ido = 0 def _raise_no_convergence(self): msg = "No convergence (%d iterations, %d/%d eigenvectors converged)" k_ok = self.iparam[4] num_iter = self.iparam[2] try: ev, vec = self.extract(True) except ArpackError as err: msg = "%s [%s]" % (msg, err) ev = np.zeros((0,)) vec = np.zeros((self.n, 0)) k_ok = 0 raise ArpackNoConvergence(msg % (num_iter, k_ok, self.k), ev, vec) class _SymmetricArpackParams(_ArpackParams): def __init__(self, n, k, tp, matvec, mode=1, M_matvec=None, Minv_matvec=None, sigma=None, ncv=None, v0=None, maxiter=None, which="LM", tol=0): # The following modes are supported: # mode = 1: # Solve the standard eigenvalue problem: # A*x = lambda*x : # A - symmetric # Arguments should be # matvec = left multiplication by A # M_matvec = None [not used] # Minv_matvec = None [not used] # # mode = 2: # Solve the general eigenvalue problem: # A*x = lambda*M*x # A - symmetric # M - symmetric positive definite # Arguments should be # matvec = left multiplication by A # M_matvec = left multiplication by M # Minv_matvec = left multiplication by M^-1 # # mode = 3: # Solve the general eigenvalue problem in shift-invert mode: # A*x = lambda*M*x # A - symmetric # M - symmetric positive semi-definite # Arguments should be # matvec = None [not used] # M_matvec = left multiplication by M # or None, if M is the identity # Minv_matvec = left multiplication by [A-sigma*M]^-1 # # mode = 4: # Solve the general eigenvalue problem in Buckling mode: # A*x = lambda*AG*x # A - symmetric positive semi-definite # AG - symmetric indefinite # Arguments should be # matvec = left multiplication by A # M_matvec = None [not used] # Minv_matvec = left multiplication by [A-sigma*AG]^-1 # # mode = 5: # Solve the general eigenvalue problem in Cayley-transformed mode: # A*x = lambda*M*x # A - symmetric # M - symmetric positive semi-definite # Arguments should be # matvec = left multiplication by A # M_matvec = left multiplication by M # or None, if M is the identity # Minv_matvec = left multiplication by [A-sigma*M]^-1 if mode == 1: if matvec is None: raise ValueError("matvec must be specified for mode=1") if M_matvec is not None: raise ValueError("M_matvec cannot be specified for mode=1") if Minv_matvec is not None: raise ValueError("Minv_matvec cannot be specified for mode=1") self.OP = matvec self.B = lambda x: x self.bmat = 'I' elif mode == 2: if matvec is None: raise ValueError("matvec must be specified for mode=2") if M_matvec is None: raise ValueError("M_matvec must be specified for mode=2") if Minv_matvec is None: raise ValueError("Minv_matvec must be specified for mode=2") self.OP = lambda x: Minv_matvec(matvec(x)) self.OPa = Minv_matvec self.OPb = matvec self.B = M_matvec self.bmat = 'G' elif mode == 3: if matvec is not None: raise ValueError("matvec must not be specified for mode=3") if Minv_matvec is None: raise ValueError("Minv_matvec must be specified for mode=3") if M_matvec is None: self.OP = Minv_matvec self.OPa = Minv_matvec self.B = lambda x: x self.bmat = 'I' else: self.OP = lambda x: Minv_matvec(M_matvec(x)) self.OPa = Minv_matvec self.B = M_matvec self.bmat = 'G' elif mode == 4: if matvec is None: raise ValueError("matvec must be specified for mode=4") if M_matvec is not None: raise ValueError("M_matvec must not be specified for mode=4") if Minv_matvec is None: raise ValueError("Minv_matvec must be specified for mode=4") self.OPa = Minv_matvec self.OP = lambda x: self.OPa(matvec(x)) self.B = matvec self.bmat = 'G' elif mode == 5: if matvec is None: raise ValueError("matvec must be specified for mode=5") if Minv_matvec is None: raise ValueError("Minv_matvec must be specified for mode=5") self.OPa = Minv_matvec self.A_matvec = matvec if M_matvec is None: self.OP = lambda x: Minv_matvec(matvec(x) + sigma * x) self.B = lambda x: x self.bmat = 'I' else: self.OP = lambda x: Minv_matvec(matvec(x) + sigma * M_matvec(x)) self.B = M_matvec self.bmat = 'G' else: raise ValueError("mode=%i not implemented" % mode) if which not in _SEUPD_WHICH: raise ValueError("which must be one of %s" % ' '.join(_SEUPD_WHICH)) if k >= n: raise ValueError("k must be less than rank(A), k=%d" % k) _ArpackParams.__init__(self, n, k, tp, mode, sigma, ncv, v0, maxiter, which, tol) if self.ncv > n or self.ncv <= k: raise ValueError("ncv must be k= n - 1: raise ValueError("k must be less than rank(A)-1, k=%d" % k) _ArpackParams.__init__(self, n, k, tp, mode, sigma, ncv, v0, maxiter, which, tol) if self.ncv > n or self.ncv <= k + 1: raise ValueError("ncv must be k+1 k, so we'll # throw out this case. nreturned -= 1 i += 1 else: # real matrix, mode 3 or 4, imag(sigma) is nonzero: # see remark 3 in neupd.f # Build complex eigenvalues from real and imaginary parts i = 0 while i <= k: if abs(d[i].imag) == 0: d[i] = np.dot(zr[:, i], self.matvec(zr[:, i])) else: if i < k: z[:, i] = zr[:, i] + 1.0j * zr[:, i + 1] z[:, i + 1] = z[:, i].conjugate() d[i] = ((np.dot(zr[:, i], self.matvec(zr[:, i])) + np.dot(zr[:, i + 1], self.matvec(zr[:, i + 1]))) + 1j * (np.dot(zr[:, i], self.matvec(zr[:, i + 1])) - np.dot(zr[:, i + 1], self.matvec(zr[:, i])))) d[i + 1] = d[i].conj() i += 1 else: #last eigenvalue is complex: the imaginary part of # the eigenvector has not been returned #this can only happen if nreturned > k, so we'll # throw out this case. nreturned -= 1 i += 1 # Now we have k+1 possible eigenvalues and eigenvectors # Return the ones specified by the keyword "which" if nreturned <= k: # we got less or equal as many eigenvalues we wanted d = d[:nreturned] z = z[:, :nreturned] else: # we got one extra eigenvalue (likely a cc pair, but which?) # cut at approx precision for sorting rd = np.round(d, decimals=_ndigits[self.tp]) if self.which in ['LR', 'SR']: ind = np.argsort(rd.real) elif self.which in ['LI', 'SI']: # for LI,SI ARPACK returns largest,smallest # abs(imaginary) why? ind = np.argsort(abs(rd.imag)) else: ind = np.argsort(abs(rd)) if self.which in ['LR', 'LM', 'LI']: d = d[ind[-k:]] z = z[:, ind[-k:]] if self.which in ['SR', 'SM', 'SI']: d = d[ind[:k]] z = z[:, ind[:k]] else: # complex is so much simpler... d, z, ierr =\ self._arpack_extract(return_eigenvectors, howmny, sselect, self.sigma, workev, self.bmat, self.which, k, self.tol, self.resid, self.v, self.iparam, self.ipntr, self.workd, self.workl, self.rwork, ierr) if ierr != 0: raise ArpackError(ierr, infodict=self.extract_infodict) k_ok = self.iparam[4] d = d[:k_ok] z = z[:, :k_ok] if return_eigenvectors: return d, z else: return d def _aslinearoperator_with_dtype(m): m = aslinearoperator(m) if not hasattr(m, 'dtype'): x = np.zeros(m.shape[1]) m.dtype = (m * x).dtype return m class SpLuInv(LinearOperator): """ SpLuInv: helper class to repeatedly solve M*x=b using a sparse LU-decopposition of M """ def __init__(self, M): self.M_lu = splu(M) LinearOperator.__init__(self, M.shape, self._matvec, dtype=M.dtype) self.isreal = not np.issubdtype(self.dtype, np.complexfloating) def _matvec(self, x): # careful here: splu.solve will throw away imaginary # part of x if M is real if self.isreal and np.issubdtype(x.dtype, np.complexfloating): return (self.M_lu.solve(np.real(x)) + 1j * self.M_lu.solve(np.imag(x))) else: return self.M_lu.solve(x) class LuInv(LinearOperator): """ LuInv: helper class to repeatedly solve M*x=b using an LU-decomposition of M """ def __init__(self, M): self.M_lu = lu_factor(M) LinearOperator.__init__(self, M.shape, self._matvec, dtype=M.dtype) def _matvec(self, x): return lu_solve(self.M_lu, x) class IterInv(LinearOperator): """ IterInv: helper class to repeatedly solve M*x=b using an iterative method. """ def __init__(self, M, ifunc=gmres, tol=0): if tol <= 0: # when tol=0, ARPACK uses machine tolerance as calculated # by LAPACK's _LAMCH function. We should match this tol = np.finfo(M.dtype).eps self.M = M self.ifunc = ifunc self.tol = tol if hasattr(M, 'dtype'): dtype = M.dtype else: x = np.zeros(M.shape[1]) dtype = (M * x).dtype LinearOperator.__init__(self, M.shape, self._matvec, dtype=dtype) def _matvec(self, x): b, info = self.ifunc(self.M, x, tol=self.tol) if info != 0: raise ValueError("Error in inverting M: function " "%s did not converge (info = %i)." % (self.ifunc.__name__, info)) return b class IterOpInv(LinearOperator): """ IterOpInv: helper class to repeatedly solve [A-sigma*M]*x = b using an iterative method """ def __init__(self, A, M, sigma, ifunc=gmres, tol=0): if tol <= 0: # when tol=0, ARPACK uses machine tolerance as calculated # by LAPACK's _LAMCH function. We should match this tol = np.finfo(A.dtype).eps self.A = A self.M = M self.sigma = sigma self.ifunc = ifunc self.tol = tol x = np.zeros(A.shape[1]) if M is None: dtype = self.mult_func_M_None(x).dtype self.OP = LinearOperator(self.A.shape, self.mult_func_M_None, dtype=dtype) else: dtype = self.mult_func(x).dtype self.OP = LinearOperator(self.A.shape, self.mult_func, dtype=dtype) LinearOperator.__init__(self, A.shape, self._matvec, dtype=dtype) def mult_func(self, x): return self.A.matvec(x) - self.sigma * self.M.matvec(x) def mult_func_M_None(self, x): return self.A.matvec(x) - self.sigma * x def _matvec(self, x): b, info = self.ifunc(self.OP, x, tol=self.tol) if info != 0: raise ValueError("Error in inverting [A-sigma*M]: function " "%s did not converge (info = %i)." % (self.ifunc.__name__, info)) return b def get_inv_matvec(M, symmetric=False, tol=0): if isdense(M): return LuInv(M).matvec elif isspmatrix(M): if isspmatrix_csr(M) and symmetric: M = M.T return SpLuInv(M).matvec else: return IterInv(M, tol=tol).matvec def get_OPinv_matvec(A, M, sigma, symmetric=False, tol=0): if sigma == 0: return get_inv_matvec(A, symmetric=symmetric, tol=tol) if M is None: #M is the identity matrix if isdense(A): if (np.issubdtype(A.dtype, np.complexfloating) or np.imag(sigma) == 0): A = np.copy(A) else: A = A + 0j A.flat[::A.shape[1] + 1] -= sigma return LuInv(A).matvec elif isspmatrix(A): A = A - sigma * identity(A.shape[0]) if symmetric and isspmatrix_csr(A): A = A.T return SpLuInv(A.tocsc()).matvec else: return IterOpInv(_aslinearoperator_with_dtype(A), M, sigma, tol=tol).matvec else: if ((not isdense(A) and not isspmatrix(A)) or (not isdense(M) and not isspmatrix(M))): return IterOpInv(_aslinearoperator_with_dtype(A), _aslinearoperator_with_dtype(M), sigma, tol=tol).matvec elif isdense(A) or isdense(M): return LuInv(A - sigma * M).matvec else: OP = A - sigma * M if symmetric and isspmatrix_csr(OP): OP = OP.T return SpLuInv(OP.tocsc()).matvec def _eigs(A, k=6, M=None, sigma=None, which='LM', v0=None, ncv=None, maxiter=None, tol=0, return_eigenvectors=True, Minv=None, OPinv=None, OPpart=None): """ Find k eigenvalues and eigenvectors of the square matrix A. Solves ``A * x[i] = w[i] * x[i]``, the standard eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i]. If M is specified, solves ``A * x[i] = w[i] * M * x[i]``, the generalized eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i] Parameters ---------- A : An N x N matrix, array, sparse matrix, or LinearOperator representing the operation A * x, where A is a real or complex square matrix. k : integer The number of eigenvalues and eigenvectors desired. `k` must be smaller than N. It is not possible to compute all eigenvectors of a matrix. Returns ------- w : array Array of k eigenvalues. v : array An array of `k` eigenvectors. ``v[:, i]`` is the eigenvector corresponding to the eigenvalue w[i]. Other Parameters ---------------- M : An N x N matrix, array, sparse matrix, or LinearOperator representing the operation M*x for the generalized eigenvalue problem ``A * x = w * M * x`` M must represent a real symmetric matrix. For best results, M should be of the same type as A. Additionally: * If sigma==None, M is positive definite * If sigma is specified, M is positive semi-definite If sigma==None, eigs requires an operator to compute the solution of the linear equation `M * x = b`. This is done internally via a (sparse) LU decomposition for an explicit matrix M, or via an iterative solver for a general linear operator. Alternatively, the user can supply the matrix or operator Minv, which gives x = Minv * b = M^-1 * b sigma : real or complex Find eigenvalues near sigma using shift-invert mode. This requires an operator to compute the solution of the linear system `[A - sigma * M] * x = b`, where M is the identity matrix if unspecified. This is computed internally via a (sparse) LU decomposition for explicit matrices A & M, or via an iterative solver if either A or M is a general linear operator. Alternatively, the user can supply the matrix or operator OPinv, which gives x = OPinv * b = [A - sigma * M]^-1 * b. For a real matrix A, shift-invert can either be done in imaginary mode or real mode, specified by the parameter OPpart ('r' or 'i'). Note that when sigma is specified, the keyword 'which' (below) refers to the shifted eigenvalues w'[i] where: * If A is real and OPpart == 'r' (default), w'[i] = 1/2 * [ 1/(w[i]-sigma) + 1/(w[i]-conj(sigma)) ] * If A is real and OPpart == 'i', w'[i] = 1/2i * [ 1/(w[i]-sigma) - 1/(w[i]-conj(sigma)) ] * If A is complex, w'[i] = 1/(w[i]-sigma) v0 : array Starting vector for iteration. ncv : integer The number of Lanczos vectors generated `ncv` must be greater than `k`; it is recommended that ``ncv > 2*k``. which : string ['LM' | 'SM' | 'LR' | 'SR' | 'LI' | 'SI'] Which `k` eigenvectors and eigenvalues to find: - 'LM' : largest magnitude - 'SM' : smallest magnitude - 'LR' : largest real part - 'SR' : smallest real part - 'LI' : largest imaginary part - 'SI' : smallest imaginary part When sigma != None, 'which' refers to the shifted eigenvalues w'[i] (see discussion in 'sigma', above). ARPACK is generally better at finding large values than small values. If small eigenvalues are desired, consider using shift-invert mode for better performance. maxiter : integer Maximum number of Arnoldi update iterations allowed tol : float Relative accuracy for eigenvalues (stopping criterion) The default value of 0 implies machine precision. return_eigenvectors : boolean Return eigenvectors (True) in addition to eigenvalues Minv : N x N matrix, array, sparse matrix, or linear operator See notes in M, above. OPinv : N x N matrix, array, sparse matrix, or linear operator See notes in sigma, above. OPpart : 'r' or 'i'. See notes in sigma, above Raises ------ ArpackNoConvergence When the requested convergence is not obtained. The currently converged eigenvalues and eigenvectors can be found as ``eigenvalues`` and ``eigenvectors`` attributes of the exception object. See Also -------- eigsh : eigenvalues and eigenvectors for symmetric matrix A svds : singular value decomposition for a matrix A Examples -------- Find 6 eigenvectors of the identity matrix: >>> from sklearn.utils.arpack import eigs >>> id = np.identity(13) >>> vals, vecs = eigs(id, k=6) >>> vals array([ 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j]) >>> vecs.shape (13, 6) Notes ----- This function is a wrapper to the ARPACK [1]_ SNEUPD, DNEUPD, CNEUPD, ZNEUPD, functions which use the Implicitly Restarted Arnoldi Method to find the eigenvalues and eigenvectors [2]_. References ---------- .. [1] ARPACK Software, http://www.caam.rice.edu/software/ARPACK/ .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA, 1998. """ if A.shape[0] != A.shape[1]: raise ValueError('expected square matrix (shape=%s)' % (A.shape,)) if M is not None: if M.shape != A.shape: raise ValueError('wrong M dimensions %s, should be %s' % (M.shape, A.shape)) if np.dtype(M.dtype).char.lower() != np.dtype(A.dtype).char.lower(): import warnings warnings.warn('M does not have the same type precision as A. ' 'This may adversely affect ARPACK convergence') n = A.shape[0] if k <= 0 or k >= n: raise ValueError("k must be between 1 and rank(A)-1") if sigma is None: matvec = _aslinearoperator_with_dtype(A).matvec if OPinv is not None: raise ValueError("OPinv should not be specified " "with sigma = None.") if OPpart is not None: raise ValueError("OPpart should not be specified with " "sigma = None or complex A") if M is None: #standard eigenvalue problem mode = 1 M_matvec = None Minv_matvec = None if Minv is not None: raise ValueError("Minv should not be " "specified with M = None.") else: #general eigenvalue problem mode = 2 if Minv is None: Minv_matvec = get_inv_matvec(M, symmetric=True, tol=tol) else: Minv = _aslinearoperator_with_dtype(Minv) Minv_matvec = Minv.matvec M_matvec = _aslinearoperator_with_dtype(M).matvec else: #sigma is not None: shift-invert mode if np.issubdtype(A.dtype, np.complexfloating): if OPpart is not None: raise ValueError("OPpart should not be specified " "with sigma=None or complex A") mode = 3 elif OPpart is None or OPpart.lower() == 'r': mode = 3 elif OPpart.lower() == 'i': if np.imag(sigma) == 0: raise ValueError("OPpart cannot be 'i' if sigma is real") mode = 4 else: raise ValueError("OPpart must be one of ('r','i')") matvec = _aslinearoperator_with_dtype(A).matvec if Minv is not None: raise ValueError("Minv should not be specified when sigma is") if OPinv is None: Minv_matvec = get_OPinv_matvec(A, M, sigma, symmetric=False, tol=tol) else: OPinv = _aslinearoperator_with_dtype(OPinv) Minv_matvec = OPinv.matvec if M is None: M_matvec = None else: M_matvec = _aslinearoperator_with_dtype(M).matvec params = _UnsymmetricArpackParams(n, k, A.dtype.char, matvec, mode, M_matvec, Minv_matvec, sigma, ncv, v0, maxiter, which, tol) while not params.converged: params.iterate() return params.extract(return_eigenvectors) def _eigsh(A, k=6, M=None, sigma=None, which='LM', v0=None, ncv=None, maxiter=None, tol=0, return_eigenvectors=True, Minv=None, OPinv=None, mode='normal'): """ Find k eigenvalues and eigenvectors of the real symmetric square matrix or complex hermitian matrix A. Solves ``A * x[i] = w[i] * x[i]``, the standard eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i]. If M is specified, solves ``A * x[i] = w[i] * M * x[i]``, the generalized eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i] Parameters ---------- A : An N x N matrix, array, sparse matrix, or LinearOperator representing the operation A * x, where A is a real symmetric matrix For buckling mode (see below) A must additionally be positive-definite k : integer The number of eigenvalues and eigenvectors desired. `k` must be smaller than N. It is not possible to compute all eigenvectors of a matrix. Returns ------- w : array Array of k eigenvalues v : array An array of k eigenvectors The v[i] is the eigenvector corresponding to the eigenvector w[i] Other Parameters ---------------- M : An N x N matrix, array, sparse matrix, or linear operator representing the operation M * x for the generalized eigenvalue problem ``A * x = w * M * x``. M must represent a real, symmetric matrix. For best results, M should be of the same type as A. Additionally: * If sigma == None, M is symmetric positive definite * If sigma is specified, M is symmetric positive semi-definite * In buckling mode, M is symmetric indefinite. If sigma == None, eigsh requires an operator to compute the solution of the linear equation `M * x = b`. This is done internally via a (sparse) LU decomposition for an explicit matrix M, or via an iterative solver for a general linear operator. Alternatively, the user can supply the matrix or operator Minv, which gives x = Minv * b = M^-1 * b sigma : real Find eigenvalues near sigma using shift-invert mode. This requires an operator to compute the solution of the linear system `[A - sigma * M] x = b`, where M is the identity matrix if unspecified. This is computed internally via a (sparse) LU decomposition for explicit matrices A & M, or via an iterative solver if either A or M is a general linear operator. Alternatively, the user can supply the matrix or operator OPinv, which gives x = OPinv * b = [A - sigma * M]^-1 * b. Note that when sigma is specified, the keyword 'which' refers to the shifted eigenvalues w'[i] where: - if mode == 'normal', w'[i] = 1 / (w[i] - sigma) - if mode == 'cayley', w'[i] = (w[i] + sigma) / (w[i] - sigma) - if mode == 'buckling', w'[i] = w[i] / (w[i] - sigma) (see further discussion in 'mode' below) v0 : array Starting vector for iteration. ncv : integer The number of Lanczos vectors generated ncv must be greater than k and smaller than n; it is recommended that ncv > 2*k which : string ['LM' | 'SM' | 'LA' | 'SA' | 'BE'] If A is a complex hermitian matrix, 'BE' is invalid. Which `k` eigenvectors and eigenvalues to find: - 'LM' : Largest (in magnitude) eigenvalues - 'SM' : Smallest (in magnitude) eigenvalues - 'LA' : Largest (algebraic) eigenvalues - 'SA' : Smallest (algebraic) eigenvalues - 'BE' : Half (k/2) from each end of the spectrum When k is odd, return one more (k/2+1) from the high end When sigma != None, 'which' refers to the shifted eigenvalues w'[i] (see discussion in 'sigma', above). ARPACK is generally better at finding large values than small values. If small eigenvalues are desired, consider using shift-invert mode for better performance. maxiter : integer Maximum number of Arnoldi update iterations allowed tol : float Relative accuracy for eigenvalues (stopping criterion). The default value of 0 implies machine precision. Minv : N x N matrix, array, sparse matrix, or LinearOperator See notes in M, above OPinv : N x N matrix, array, sparse matrix, or LinearOperator See notes in sigma, above. return_eigenvectors : boolean Return eigenvectors (True) in addition to eigenvalues mode : string ['normal' | 'buckling' | 'cayley'] Specify strategy to use for shift-invert mode. This argument applies only for real-valued A and sigma != None. For shift-invert mode, ARPACK internally solves the eigenvalue problem ``OP * x'[i] = w'[i] * B * x'[i]`` and transforms the resulting Ritz vectors x'[i] and Ritz values w'[i] into the desired eigenvectors and eigenvalues of the problem ``A * x[i] = w[i] * M * x[i]``. The modes are as follows: - 'normal' : OP = [A - sigma * M]^-1 * M B = M w'[i] = 1 / (w[i] - sigma) - 'buckling' : OP = [A - sigma * M]^-1 * A B = A w'[i] = w[i] / (w[i] - sigma) - 'cayley' : OP = [A - sigma * M]^-1 * [A + sigma * M] B = M w'[i] = (w[i] + sigma) / (w[i] - sigma) The choice of mode will affect which eigenvalues are selected by the keyword 'which', and can also impact the stability of convergence (see [2] for a discussion) Raises ------ ArpackNoConvergence When the requested convergence is not obtained. The currently converged eigenvalues and eigenvectors can be found as ``eigenvalues`` and ``eigenvectors`` attributes of the exception object. See Also -------- eigs : eigenvalues and eigenvectors for a general (nonsymmetric) matrix A svds : singular value decomposition for a matrix A Notes ----- This function is a wrapper to the ARPACK [1]_ SSEUPD and DSEUPD functions which use the Implicitly Restarted Lanczos Method to find the eigenvalues and eigenvectors [2]_. Examples -------- >>> from sklearn.utils.arpack import eigsh >>> id = np.identity(13) >>> vals, vecs = eigsh(id, k=6) >>> vals # doctest: +SKIP array([ 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j]) >>> vecs.shape (13, 6) References ---------- .. [1] ARPACK Software, http://www.caam.rice.edu/software/ARPACK/ .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA, 1998. """ # complex hermitian matrices should be solved with eigs if np.issubdtype(A.dtype, np.complexfloating): if mode != 'normal': raise ValueError("mode=%s cannot be used with " "complex matrix A" % mode) if which == 'BE': raise ValueError("which='BE' cannot be used with complex matrix A") elif which == 'LA': which = 'LR' elif which == 'SA': which = 'SR' ret = eigs(A, k, M=M, sigma=sigma, which=which, v0=v0, ncv=ncv, maxiter=maxiter, tol=tol, return_eigenvectors=return_eigenvectors, Minv=Minv, OPinv=OPinv) if return_eigenvectors: return ret[0].real, ret[1] else: return ret.real if A.shape[0] != A.shape[1]: raise ValueError('expected square matrix (shape=%s)' % (A.shape,)) if M is not None: if M.shape != A.shape: raise ValueError('wrong M dimensions %s, should be %s' % (M.shape, A.shape)) if np.dtype(M.dtype).char.lower() != np.dtype(A.dtype).char.lower(): import warnings warnings.warn('M does not have the same type precision as A. ' 'This may adversely affect ARPACK convergence') n = A.shape[0] if k <= 0 or k >= n: raise ValueError("k must be between 1 and rank(A)-1") if sigma is None: A = _aslinearoperator_with_dtype(A) matvec = A.matvec if OPinv is not None: raise ValueError("OPinv should not be specified " "with sigma = None.") if M is None: #standard eigenvalue problem mode = 1 M_matvec = None Minv_matvec = None if Minv is not None: raise ValueError("Minv should not be " "specified with M = None.") else: #general eigenvalue problem mode = 2 if Minv is None: Minv_matvec = get_inv_matvec(M, symmetric=True, tol=tol) else: Minv = _aslinearoperator_with_dtype(Minv) Minv_matvec = Minv.matvec M_matvec = _aslinearoperator_with_dtype(M).matvec else: # sigma is not None: shift-invert mode if Minv is not None: raise ValueError("Minv should not be specified when sigma is") # normal mode if mode == 'normal': mode = 3 matvec = None if OPinv is None: Minv_matvec = get_OPinv_matvec(A, M, sigma, symmetric=True, tol=tol) else: OPinv = _aslinearoperator_with_dtype(OPinv) Minv_matvec = OPinv.matvec if M is None: M_matvec = None else: M = _aslinearoperator_with_dtype(M) M_matvec = M.matvec # buckling mode elif mode == 'buckling': mode = 4 if OPinv is None: Minv_matvec = get_OPinv_matvec(A, M, sigma, symmetric=True, tol=tol) else: Minv_matvec = _aslinearoperator_with_dtype(OPinv).matvec matvec = _aslinearoperator_with_dtype(A).matvec M_matvec = None # cayley-transform mode elif mode == 'cayley': mode = 5 matvec = _aslinearoperator_with_dtype(A).matvec if OPinv is None: Minv_matvec = get_OPinv_matvec(A, M, sigma, symmetric=True, tol=tol) else: Minv_matvec = _aslinearoperator_with_dtype(OPinv).matvec if M is None: M_matvec = None else: M_matvec = _aslinearoperator_with_dtype(M).matvec # unrecognized mode else: raise ValueError("unrecognized mode '%s'" % mode) params = _SymmetricArpackParams(n, k, A.dtype.char, matvec, mode, M_matvec, Minv_matvec, sigma, ncv, v0, maxiter, which, tol) while not params.converged: params.iterate() return params.extract(return_eigenvectors) def _svds(A, k=6, ncv=None, tol=0): """Compute k singular values/vectors for a sparse matrix using ARPACK. Parameters ---------- A : sparse matrix Array to compute the SVD on k : int, optional Number of singular values and vectors to compute. ncv : integer The number of Lanczos vectors generated ncv must be greater than k+1 and smaller than n; it is recommended that ncv > 2*k tol : float, optional Tolerance for singular values. Zero (default) means machine precision. Notes ----- This is a naive implementation using an eigensolver on A.H * A or A * A.H, depending on which one is more efficient. """ if not (isinstance(A, np.ndarray) or isspmatrix(A)): A = np.asarray(A) n, m = A.shape if np.issubdtype(A.dtype, np.complexfloating): herm = lambda x: x.T.conjugate() eigensolver = eigs else: herm = lambda x: x.T eigensolver = eigsh if n > m: X = A XH = herm(A) else: XH = A X = herm(A) def matvec_XH_X(x): return XH.dot(X.dot(x)) XH_X = LinearOperator(matvec=matvec_XH_X, dtype=X.dtype, shape=(X.shape[1], X.shape[1])) eigvals, eigvec = eigensolver(XH_X, k=k, tol=tol ** 2) s = np.sqrt(eigvals) if n > m: v = eigvec u = X.dot(v) / s vh = herm(v) else: u = eigvec vh = herm(X.dot(u) / s) return u, s, vh # check if backport is actually needed: if scipy.version.version >= LooseVersion('0.10'): from scipy.sparse.linalg import eigs, eigsh, svds else: eigs, eigsh, svds = _eigs, _eigsh, _svds