ó î&]\c @`sdZddlmZmZmZddddddgZdd lZdd lZd d l m Z dd l m Z d d l mZddlmZddlmZidd6dd6dd6dd6ZdZdZd„Zd„Zddd„Zeddƒeƒd d d d d d d!„ƒƒZed"d#ƒeƒd d d d d d d$„ƒƒZed%d&ƒeƒd d d d d d d'„ƒƒZed(d)ƒeƒd d d d d d d*„ƒƒZeƒd d d d d d d d d+„ƒZeƒd d d d d d d d,„ƒZd S(-s,Iterative methods for solving linear systemsi(tdivisiontprint_functiontabsolute_importtbicgtbicgstabtcgtcgstgmrestqmrNi(t _iterative(tLinearOperator(t make_system(t_aligned_zeros(t non_reentranttstftdtctFtztDsH Parameters ---------- A : {sparse matrix, dense matrix, LinearOperator}swb : {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, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior. .. warning:: The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. 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. cC`s6tjj|ƒ}||kr(|dfS|dfSdS(s; Successful termination condition for the solvers. iiN(tnptlinalgtnorm(tresidualtatoltresid((sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyt _stoptestEs  cC`s­|d kr:tjdjd|ƒdtddƒd}nt|ƒ}|dkrŒ|ƒ}||krkdS|dkr{|S|t|ƒSntt|ƒ|t|ƒƒSd S( sž Parse arguments for absolute tolerance in termination condition. Parameters ---------- tol, atol : object The arguments passed into the solver routine by user. bnrm2 : float 2-norm of the rhs vector. get_residual : callable Callable ``get_residual()`` that returns the initial value of the residual. routine_name : str Name of the routine. s scipy.sparse.linalg.{name} called without specifying `atol`. The default value will be changed in a future release. For compatibility, specify a value for `atol` explicitly, e.g., ``{name}(..., atol=0)``, or to retain the old behavior ``{name}(..., atol='legacy')``tnametcategoryt stacklevelitlegacytexitiN(tNonetwarningstwarntformattDeprecationWarningtfloattmax(ttolRtbnrm2t get_residualt routine_nameR((sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyt _get_atolPs         tt0c`s‡‡‡fd†}|S(Nc`s5djˆtdˆjddƒtˆfƒ|_|S(Ns s s (tjoint common_doc1treplacet common_doc2t__doc__(tfn(tAinfotfootertheader(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pytcombinezs ((R7R5R6t atol_defaultR8((R5R6R7sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyt set_docstringyss7Use BIConjugate Gradient iteration to solve ``Ax = b``.sThe real or complex N-by-N matrix of the linear system. It is required that the linear operator can produce ``Ax`` and ``A^T x``.gñh㈵øä>c `szt|||ˆƒ\}}‰‰}tˆƒ} |dkrI| d}n|j|j‰} |j|j} } tˆjj} tt | dƒ}‡‡‡fd†}t ||t j j ˆƒ|dƒ}|dkrç|ˆƒdfS|}d}d}td | d ˆjƒ}d}d}t}|}xtr;|}|ˆˆ|||||||ƒ \ ‰}}}}}}}}|dk r||kr|ˆƒnt|d|d| ƒ}t|d|d| ƒ}|dkrü|dk rø|ˆƒnPn6|dkr9||c|9<||c|ˆ||ƒ7“sRR iiiÿÿÿÿitdtypeiiii(R tlenR!R=trmatvect _type_convR@tchartgetattrR R,RRRR tTruetslicetFalseR(tAR<tx0R(tmaxitertMtcallbackRt postprocesstnRBtpsolvetrpsolvetltrtrevcomR*Rtndx1tndx2tworktijobtinfotftflagtiter_tolditertsclr1tsclr2tslice1tslice2((R<R=R>sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyR‚sj$   $  ?     ! !      % sBUse BIConjugate Gradient STABilized iteration to solve ``Ax = b``.s7The real or complex N-by-N matrix of the linear system.c `st|||ˆƒ\}}‰‰}tˆƒ} |dkrI| d}n|j‰|j} tˆjj} tt| dƒ} ‡‡‡fd†} t ||t j j ˆƒ| dƒ}|dkrÓ|ˆƒdfS|}d}d}t d | d ˆjƒ}d}d}t}|}x¯trÇ|}| ˆˆ|||||||ƒ \ ‰}}}}}}}}|dk r‰||kr‰|ˆƒnt|d|d| ƒ}t|d|d| ƒ}|dkrè|dk rä|ˆƒnPnÖ|dkr%||c|9<||c|ˆ||ƒ7(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyR?ØsRR iiiÿÿÿÿiR@iii(R RAR!R=RCR@RDRER R,RRRR RFRGRHR(RIR<RJR(RKRLRMRRNRORPRRRSR*RRTRURVRWRXRYRZR[R\R]R^R_((R<R=R>sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyRÈs`$     $  ?     !     % s5Use Conjugate Gradient iteration to solve ``Ax = b``.ssThe real or complex N-by-N matrix of the linear system. ``A`` must represent a hermitian, positive definite matrix.c `sNt|||ˆƒ\}}‰‰}tˆƒ} |dkrI| d}n|j‰|j} tˆjj} tt| dƒ} ‡‡‡fd†} t ||t j j ˆƒ| dƒ}|dkrÓ|ˆƒdfS|}d}d}t d | d ˆjƒ}d}d}t}|}x÷tr|}| ˆˆ|||||||ƒ \ ‰}}}}}}}}|dk r‰||kr‰|ˆƒnt|d|d| ƒ}t|d|d| ƒ}|dkrè|dk rä|ˆƒnPn|dkr%||c|9<||c|ˆ||ƒ7||kr>||k r>|}n|ˆƒ|fS( Ni tcgrevcomc`stjjˆˆƒˆƒS(N(RRR((R<R=R>(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyR?sRR iiiÿÿÿÿiR@ii(R RAR!R=RCR@RDRER R,RRRR RFRGRHR(RIR<RJR(RKRLRMRRNRORPRRRSR*RRTRURVRWRXRYRZR[R\R]R^R_((R<R=R>sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyRsf$     $  ?     !     % s=Use Conjugate Gradient Squared iteration to solve ``Ax = b``.s3The real-valued N-by-N matrix of the linear system.c `s‹t|||ˆƒ\}}‰‰}tˆƒ} |dkrI| d}n|j‰|j} tˆjj} tt| dƒ} ‡‡‡fd†} t ||t j j ˆƒ| dƒ}|dkrÓ|ˆƒdfS|}d}d}t d | d ˆjƒ}d}d}t}|}x÷tr|}| ˆˆ|||||||ƒ \ ‰}}}}}}}}|dk r‰||kr‰|ˆƒnt|d|d| ƒ}t|d|d| ƒ}|dkrè|dk rä|ˆƒnPn|dkr%||c|9<||c|ˆ||ƒ7(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyR?\sRR iiiÿÿÿÿiR@iiiiöÿÿÿ(R RAR!R=RCR@RDRER R,RRRR RFRGRHR(RIR<RJR(RKRLRMRRNRORPRRRSR*RRTRURVRWRXRYRZR[R\R]R^R_tok((R<R=R>sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyRMsn$     $  ?     !       % c & `s¼|dkr|}n|dk r0tdƒ‚nt|||ˆƒ\}}‰‰} tˆƒ} |dkry| d}n|dkrŽd}nt|| ƒ}|j‰|j} tˆjj} t t | dƒ}t j j ˆƒ}t j j | ˆƒƒ}‡‡‡fd†}t|| ||dƒ} | dkrE| ˆƒdfS|dkra| ˆƒdfSd }|t|| |ƒ}t j}t j}d }d }td || d ˆjƒ}t|d d|dd ˆjƒ}d }d}t}|}|}t}t} d }!xytr‰|ˆˆ||||||||||ƒ \ ‰}}}}}}"}#}t|d |d | ƒ}$t|d |d | ƒ}%|d krÎ| rÊ|dk rÊ|||ƒt} nPn–|d kr||%c|#9<||%c|"ˆˆƒ70 : convergence to tolerance not achieved, number of iterations * <0 : illegal input or breakdown Other parameters ---------------- x0 : {array, matrix} Starting guess for the solution (a vector of zeros by default). tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is ``'legacy'``, which emulates a different legacy behavior. .. warning:: The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. restart : int, optional Number of iterations between restarts. Larger values increase iteration cost, but may be necessary for convergence. Default is 20. maxiter : int, optional Maximum number of iterations (restart cycles). Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : {sparse matrix, dense matrix, LinearOperator} Inverse of the preconditioner of A. M should approximate the inverse of A and be easy to solve for (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. By default, no preconditioner is used. callback : function User-supplied function to call after each iteration. It is called as callback(rk), where rk is the current residual vector. restrt : int, optional DEPRECATED - use `restart` instead. See Also -------- LinearOperator Notes ----- A preconditioner, P, is chosen such that P is close to A but easy to solve for. The preconditioner parameter required by this routine is ``M = P^-1``. The inverse should preferably not be calculated explicitly. Rather, use the following template to produce M:: # Construct a linear operator that computes P^-1 * x. import scipy.sparse.linalg as spla M_x = lambda x: spla.spsolve(P, x) M = spla.LinearOperator((n, n), M_x) Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import gmres >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> b = np.array([2, 4, -1], dtype=float) >>> x, exitCode = gmres(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True sOCannot specify both restart and restrt keywords. Preferably use 'restart' only.i it gmresrevcomc`stjjˆˆƒˆƒS(N(RRR((R<R=R>(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyR?sRR igð?iiÿÿÿÿiR@iiigø?g¼‰Ø—²ÒœsClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyR—s¤V   $           % H             c ! `sˆ‰tˆd|ˆƒ\‰} ‰‰} |dkr|dkrtˆdƒrƇfd†} ‡fd†} ‡fd†} ‡fd†}tˆjd| d| ƒ}tˆjd| d|ƒ}qd„}tˆjd|d|ƒ}tˆjd|d|ƒ}ntˆƒ}|dkr-|d }ntˆjj}t t |d ƒ}‡‡‡fd †}t ||t j jˆƒ|d ƒ}|d kr¥| ˆƒdfS|}d}d}td|ˆjƒ}d}d}t}|}xjtrQ|}|ˆˆ|||||||ƒ \ ‰}}}}}}}}|dk rX||krX|ˆƒnt|d|d|ƒ}t|d|d|ƒ} |dkr·|dk r³|ˆƒnPn‘|dkr÷|| c|9<|| c|ˆj||ƒ70 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown Other Parameters ---------------- 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 ``'legacy'``, which emulates a different legacy behavior. .. warning:: The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M1 : {sparse matrix, dense matrix, LinearOperator} Left preconditioner for A. M2 : {sparse matrix, dense matrix, LinearOperator} Right preconditioner for A. Used together with the left preconditioner M1. The matrix M1*A*M2 should have better conditioned than A alone. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector. See Also -------- LinearOperator Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import qmr >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> b = np.array([2, 4, -1], dtype=float) >>> x, exitCode = qmr(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True RPc`sˆj|dƒS(Ntleft(RP(R<(tA_(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyt left_psolve¡sc`sˆj|dƒS(Ntright(RP(R<(Rt(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyt right_psolve¤sc`sˆj|dƒS(NRs(RQ(R<(Rt(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyt left_rpsolve§sc`sˆj|dƒS(NRv(RQ(R<(Rt(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyt right_rpsolveªsR=RBcS`s|S(N((R<((sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pytid¯si t qmrrevcomc`stjjˆjˆƒˆƒS(N(RRRR=((RIR<R>(sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyR?»sRR iiiÿÿÿÿi iiiiiiiN(R R!thasattrR tshapeRARCR@RDRER R,RRRR RFRGR=RBRHR(!RIR<RJR(RKtM1tM2RMRRLRNRuRwRxRyRzRORRRSR*RRTRURVRWRXRYRZR[R\R]R^R_((RIRtR<R>sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pyRZs†B$    $  ?     $ $         % ( R3t __future__RRRt__all__R"tnumpyRR-R tscipy.sparse.linalg.interfaceR tutilsR tscipy._lib._utilR tscipy._lib._threadsafetyR RCR0R2RR,R:R!RRRRRR(((sClib/python2.7/site-packages/scipy/sparse/linalg/isolve/iterative.pytsJ  ") ) !A!<!A!GÁ