ó ˽÷Xc@`sadZddlmZmZmZddlZddlmZm Z m Z m Z m Z ddl mZmZmZddlmZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZmZdddddgZ d„Z!d„Z"d„Z#d„Z$dd„Z&dde'd„Z(dde'd„Z)dd„Z*dS(sMatrix equation solver routinesi(tdivisiontprint_functiontabsolute_importN(tinvt LinAlgErrortnormtcondtsvdi(tsolvetsolve_triangulartmatrix_balance(tget_lapack_funcs(tschur(tlu(tqr(tordqz(t_asarray_validated(tkront block_diagtsolve_sylvestertsolve_lyapunovtsolve_discrete_lyapunovtsolve_continuous_aretsolve_discrete_arec C`st|ddƒ\}}t|jƒjƒddƒ\}}tjtj|jƒjƒ|ƒ|ƒ}td |||fƒ\}|d krœtdƒ‚n||||ddƒ\} } } | | } | dkrêtd| fƒ‚ntjtj|| ƒ|jƒjƒƒS( sâ Computes a solution (X) to the Sylvester equation :math:`AX + XB = Q`. Parameters ---------- a : (M, M) array_like Leading matrix of the Sylvester equation b : (N, N) array_like Trailing matrix of the Sylvester equation q : (M, N) array_like Right-hand side Returns ------- x : (M, N) ndarray The solution to the Sylvester equation. Raises ------ LinAlgError If solution was not found Notes ----- Computes a solution to the Sylvester matrix equation via the Bartels- Stewart algorithm. The A and B matrices first undergo Schur decompositions. The resulting matrices are used to construct an alternative Sylvester equation (``RY + YS^T = F``) where the R and S matrices are in quasi-triangular form (or, when R, S or F are complex, triangular form). The simplified equation is then solved using ``*TRSYL`` from LAPACK directly. .. versionadded:: 0.11.0 toutputtrealttrsylsQLAPACK implementation does not contain a proper Sylvester equation solver (TRSYL)ttranbtCis(Illegal value encountered in the %d term(strsylN( R tconjt transposetnptdotR tNonet RuntimeErrorR( tatbtqtrtutstvtfRtytscaletinfo((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/_solvers.pyRs&$* !  cC`st||jƒjƒ|ƒS(s£ Solves the continuous Lyapunov equation :math:`AX + XA^H = Q`. Uses the Bartels-Stewart algorithm to find :math:`X`. Parameters ---------- a : array_like A square matrix q : array_like Right-hand side square matrix Returns ------- x : array_like Solution to the continuous Lyapunov equation See Also -------- solve_sylvester : computes the solution to the Sylvester equation Notes ----- Because the continuous Lyapunov equation is just a special form of the Sylvester equation, this solver relies entirely on solve_sylvester for a solution. .. versionadded:: 0.11.0 (RRR(R#R%((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/_solvers.pyR[s!cC`sWt||jƒƒ}tj|jdƒ|}t||jƒƒ}tj||jƒS(sÄ Solves the discrete Lyapunov equation directly. This function is called by the `solve_discrete_lyapunov` function with `method=direct`. It is not supposed to be called directly. i(RRRteyetshapeRtflattentreshape(R#R%tlhstx((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/_solvers.pyt_solve_discrete_lyapunov_directscC`s”tj|jdƒ}|jƒjƒ}t||ƒ}tj|||ƒ}dtjtjt||ƒ|ƒ|ƒ}t|jƒjƒ| ƒS(sÝ Solves the discrete Lyapunov equation using a bilinear transformation. This function is called by the `solve_discrete_lyapunov` function with `method=bilinear`. It is not supposed to be called directly. ii(RR.R/RRRR R(R#R%R.taHtaHI_invR$tc((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/_solvers.pyt!_solve_discrete_lyapunov_bilinearŽs ,cC`s«tj|ƒ}tj|ƒ}|dkrO|jddkrFd}qOd}n|jƒ}|dkryt||ƒ}n.|dkr—t||ƒ}ntd|ƒ‚|S(sÉ Solves the discrete Lyapunov equation :math:`AXA^H - X + Q = 0`. Parameters ---------- a, q : (M, M) array_like Square matrices corresponding to A and Q in the equation above respectively. Must have the same shape. method : {'direct', 'bilinear'}, optional Type of solver. If not given, chosen to be ``direct`` if ``M`` is less than 10 and ``bilinear`` otherwise. Returns ------- x : ndarray Solution to the discrete Lyapunov equation See Also -------- solve_lyapunov : computes the solution to the continuous Lyapunov equation Notes ----- This section describes the available solvers that can be selected by the 'method' parameter. The default method is *direct* if ``M`` is less than 10 and ``bilinear`` otherwise. Method *direct* uses a direct analytical solution to the discrete Lyapunov equation. The algorithm is given in, for example, [1]_. However it requires the linear solution of a system with dimension :math:`M^2` so that performance degrades rapidly for even moderately sized matrices. Method *bilinear* uses a bilinear transformation to convert the discrete Lyapunov equation to a continuous Lyapunov equation :math:`(BX+XB'=-C)` where :math:`B=(A-I)(A+I)^{-1}` and :math:`C=2(A' + I)^{-1} Q (A + I)^{-1}`. The continuous equation can be efficiently solved since it is a special case of a Sylvester equation. The transformation algorithm is from Popov (1964) as described in [2]_. .. versionadded:: 0.11.0 References ---------- .. [1] Hamilton, James D. Time Series Analysis, Princeton: Princeton University Press, 1994. 265. Print. http://www.scribd.com/doc/20577138/Hamilton-1994-Time-Series-Analysis .. [2] Gajic, Z., and M.T.J. Qureshi. 2008. Lyapunov Matrix Equation in System Stability and Control. Dover Books on Engineering Series. Dover Publications. ii tbilineartdirectsUnknown solver %sN(RtasarrayR!R/tlowerR4R8t ValueError(R#R%tmethodtmethR3((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/_solvers.pyRs7      cC`sVt||||||dƒ\ }}}}}}}}} } tjd||d||fd| ƒ} || d|…d|…fs*  (   > $   L»¾