U Gv f/<@svdZddlZddlmZmZmZmZddlm Z m Z dddd gZ dd dZ d dZ d dZddZGdd d e ZdS)z2Nearly exact trust-region optimization subproblem.N)normget_lapack_funcssolve_triangular cho_solve)_minimize_trust_regionBaseQuadraticSubproblem_minimize_trustregion_exact estimate_smallest_singular_valuesingular_leading_submatrixIterativeSubproblemcKs<|dkrtdt|s tdt||f|||td|S)a Minimization of scalar function of one or more variables using a nearly exact trust-region algorithm. Options ------- initial_tr_radius : float Initial trust-region radius. max_tr_radius : float Maximum value of the trust-region radius. No steps that are longer than this value will be proposed. eta : float Trust region related acceptance stringency for proposed steps. gtol : float Gradient norm must be less than ``gtol`` before successful termination. Nz9Jacobian is required for trust region exact minimization.z?Hessian matrix is required for trust region exact minimization.)argsjachessZ subproblem) ValueErrorcallablerr )funZx0rrrZtrust_region_optionsr r E/tmp/pip-unpacked-wheel-96ln3f52/scipy/optimize/_trustregion_exact.pyr scCsXt|}|j\}}||kr$tdt|}t|}t|D]}d|||j||f}d|||j||f}||dd|j|dd|f|}||dd|j|dd|f|} t|t |dt|t | dkr |||<|||dd<q@|||<| ||dd<q@t ||} t | } t |} | | } | | }| |fS)aZGiven upper triangular matrix ``U`` estimate the smallest singular value and the correspondent right singular vector in O(n**2) operations. Parameters ---------- U : ndarray Square upper triangular matrix. Returns ------- s_min : float Estimated smallest singular value of the provided matrix. z_min : ndarray Estimatied right singular vector. Notes ----- The procedure is based on [1]_ and is done in two steps. First, it finds a vector ``e`` with components selected from {+1, -1} such that the solution ``w`` from the system ``U.T w = e`` is as large as possible. Next it estimate ``U v = w``. The smallest singular value is close to ``norm(w)/norm(v)`` and the right singular vector is close to ``v/norm(v)``. The estimation will be better more ill-conditioned is the matrix. References ---------- .. [1] Cline, A. K., Moler, C. B., Stewart, G. W., Wilkinson, J. H. An estimate for the condition number of a matrix. 1979. SIAM Journal on Numerical Analysis, 16(2), 368-375. z.A square triangular matrix should be provided.rN) npZ atleast_2dshaperzerosemptyrangeTabsrr)UmnpwkZwpZwmppZpmvv_normw_norms_minz_minr r rr ,s,"     **& cCsTt|}t|}tjt|dd}t|||}t|||}||fS)a Given a square matrix ``H`` compute upper and lower bounds for its eigenvalues (Gregoshgorin Bounds). Defined ref. [1]. References ---------- .. [1] Conn, A. R., Gould, N. I., & Toint, P. L. Trust region methods. 2000. Siam. pp. 19. r)Zaxis)rZdiagrsumminmax)HZH_diagZ H_diag_absZ H_row_sumsZlbZubr r rgershgorin_bounds{s  r-cCst|d|d|dfd||d|df}t|}t|}d||d<|dkrt|d|dd|df|d|d|df |d|d<||fS)a  Compute term that makes the leading ``k`` by ``k`` submatrix from ``A`` singular. Parameters ---------- A : ndarray Symmetric matrix that is not positive definite. U : ndarray Upper triangular matrix resulting of an incomplete Cholesky decomposition of matrix ``A``. k : int Positive integer such that the leading k by k submatrix from `A` is the first non-positive definite leading submatrix. Returns ------- delta : float Amount that should be added to the element (k, k) of the leading k by k submatrix of ``A`` to make it singular. v : ndarray A vector such that ``v.T B v = 0``. Where B is the matrix A after ``delta`` is added to its element (k, k). Nr)rr)lenrr)Arr"deltarr$r r rr s6  DcsBeZdZdZdZeejZ d fdd Z dd Z d d Z Z S) r aQuadratic subproblem solved by nearly exact iterative method. Notes ----- This subproblem solver was based on [1]_, [2]_ and [3]_, which implement similar algorithms. The algorithm is basically that of [1]_ but ideas from [2]_ and [3]_ were also used. References ---------- .. [1] A.R. Conn, N.I. Gould, and P.L. Toint, "Trust region methods", Siam, pp. 169-200, 2000. .. [2] J. Nocedal and S. Wright, "Numerical optimization", Springer Science & Business Media. pp. 83-91, 2006. .. [3] J.J. More and D.C. Sorensen, "Computing a trust region step", SIAM Journal on Scientific and Statistical Computing, vol. 4(3), pp. 553-572, 1983. g{Gz?N皙?皙?cst||||d|_d|_d|_||_||_td|jf\|_ t |j|_ t |j\|_ |_t|jtj|_t|jd|_|j |j|j|_dS)Nrr)ZpotrfZfro)super__init__previous_tr_radius lambda_lbniterk_easyk_hardrrcholeskyr/ dimensionr-hess_gershgorin_lbhess_gershgorin_ubrrZInfhess_infhess_froEPS CLOSE_TO_ZERO)selfxrrrZhesspr9r: __class__r rr5s zIterativeSubproblem.__init__cCstd|j|t|j |j|j}tdt|j |j|t|j|j|j}||j krjt|j |}|dkrxd}n"tt ||||j ||}|||fS)zGiven a trust radius, return a good initial guess for the damping factor, the lower bound and the upper bound. The values were chosen accordingly to the guidelines on section 7.3.8 (p. 192) from [1]_. r)r+jac_magr*r=r@r?rZdiagonalr>r6r7rsqrt UPDATE_COEFF)rC tr_radius lambda_ubr7Zlambda_initialr r r_initial_valuess$  z#IterativeSubproblem._initial_valuescCs4||\}}}|j}d}d}d|_|r.d}n*|j|t|}|j|dddd\} } |jd7_| dkr|j|jkrt | df|j } t | } | |kr|dkrd}qt | | dd} t | }| |d| ||}||}| |krt | \}}|| ||\}}t||gtd }t| t|| }|d|d|||d}||jkrn| ||7} q|}t|||d}|j|t|}|j|dddd\}} | dkr|}d}n,t||}tt||||j||}n(t| ||}||jkrq|}|}q$| dkr|j|jkr|dkrNt|} d}qt | \}}|}|d|d|j||dkr||} q|}t|||d}tt||||j||}q$t|| | \}}t |}t||||d}tt||||j||}q$||_||_||_| |fS) zSolve quadratic subproblemTFr)lowerZ overwrite_acleanrr)Ztransr.)key)rLr<r8rrZeyer;rGrBrrrrr Zget_boundaries_intersectionsr*rdotr:r+rHrIr9rr r7lambda_currentr6)rCrJrQr7rKrZ hits_boundaryZalready_factorizedr,rinfor Zp_normr!r&Z delta_lambdaZ lambda_newr'r(tatbZstep_lenZquadratic_termZrelative_errorcr1r$r%r r rsolves             $zIterativeSubproblem.solve)Nr2r3)__name__ __module__ __qualname____doc__rIrZfinfofloatZepsrAr5rLrV __classcell__r r rErr s ))r NN)rZZnumpyrZ scipy.linalgrrrrZ _trustregionrr__all__r r r-r r r r r rs O*