U /‰d~Pã@sFdZddlmZddlZddlmZddlmZm Z m Z ddl m Z ddl mZmZe e¡jZdd „Zd@d d „Zdd„Zdd„ZdAdd„ZdBdd„ZdCdd„Zdd„Zdd„ZdDdd„ZdEdd „Zd!d"„Zd#d$„Z d%d&„Z!d'd(„Z"d)d*„Z#d+d,„Z$d-d.„Z%dFd/d0„Z&d1d2„Z'd3d4„Z(d5d6„Z)dGd8d9„Z*dHd:d;„Z+dd?„Z-dS)Iz+Functions used by least-squares algorithms.é)ÚcopysignN)Únorm)Ú cho_factorÚ cho_solveÚ LinAlgError)Úissparse)ÚLinearOperatorÚaslinearoperatorc Csžt ||¡}|dkrtdƒ‚t ||¡}t ||¡|d}|dkrLtdƒ‚t ||||¡}|t||ƒ }||}||} || kr’|| fS| |fSdS)aqFind the intersection of a line with the boundary of a trust region. This function solves the quadratic equation with respect to t ||(x + s*t)||**2 = Delta**2. Returns ------- t_neg, t_pos : tuple of float Negative and positive roots. Raises ------ ValueError If `s` is zero or `x` is not within the trust region. rz `s` is zero.éz#`x` is not within the trust region.N)ÚnpÚdotÚ ValueErrorÚsqrtr) ÚxÚsÚDeltaÚaÚbÚcÚdÚqÚt1Út2©rú>/tmp/pip-unpacked-wheel-9gxwnfpp/scipy/optimize/_lsq/common.pyÚintersect_trust_regions  rç{®Gáz„?é c Cs’dd„} ||} ||kr6t||d} |d| k} nd} | rd| ||¡ } t| ƒ|krd| ddfSt| ƒ|}| r’| d| ||ƒ\}}| |}nd}|dksª| sÂ|dkrÂtd|||d ƒ}n|}t|ƒD]Œ}||ksâ||krøtd|||d ƒ}| || ||ƒ\}}|dkr|}||}t|||ƒ}|||||8}t |¡||krÎq\qÎ| | |d |¡ } | |t| ƒ9} | ||d fS) aÁSolve a trust-region problem arising in least-squares minimization. This function implements a method described by J. J. More [1]_ and used in MINPACK, but it relies on a single SVD of Jacobian instead of series of Cholesky decompositions. Before running this function, compute: ``U, s, VT = svd(J, full_matrices=False)``. Parameters ---------- n : int Number of variables. m : int Number of residuals. uf : ndarray Computed as U.T.dot(f). s : ndarray Singular values of J. V : ndarray Transpose of VT. Delta : float Radius of a trust region. initial_alpha : float, optional Initial guess for alpha, which might be available from a previous iteration. If None, determined automatically. rtol : float, optional Stopping tolerance for the root-finding procedure. Namely, the solution ``p`` will satisfy ``abs(norm(p) - Delta) < rtol * Delta``. max_iter : int, optional Maximum allowed number of iterations for the root-finding procedure. Returns ------- p : ndarray, shape (n,) Found solution of a trust-region problem. alpha : float Positive value such that (J.T*J + alpha*I)*p = -J.T*f. Sometimes called Levenberg-Marquardt parameter. n_iter : int Number of iterations made by root-finding procedure. Zero means that Gauss-Newton step was selected as the solution. References ---------- .. [1] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977. cSsD|d|}t||ƒ}||}t |d|d¡ |}||fS)z Function of which to find zero. It is defined as "norm of regularized (by alpha) least-squares solution minus `Delta`". Refer to [1]_. r é)rr Úsum)ÚalphaÚsufrrZdenomZp_normÚphiÚ phi_primerrrÚphi_and_derivativejs   z2solve_lsq_trust_region..phi_and_derivativeréÿÿÿÿFgNgü©ñÒMbP?çà?r é)ÚEPSr rÚmaxÚranger Úabs)ÚnÚmZufrÚVrZ initial_alphaÚrtolZmax_iterr$r!Ú thresholdZ full_rankÚpZ alpha_upperr"r#Z alpha_lowerr ÚitÚratiorrrÚsolve_lsq_trust_region9s@1       r4cCsxz>t|ƒ\}}t||f|ƒ }t ||¡|dkr<|dfWSWntk rRYnX|d|d}|d|d}|d|d}|d|} |d|} t | | d||| d|d| || | | g¡} t | ¡} t | t | ¡¡} |t  d| d| dd| dd| df¡}d tj || |¡dd t ||¡} t  | ¡}|d d …|f}|d fS) azSolve a general trust-region problem in 2 dimensions. The problem is reformulated as a 4th order algebraic equation, the solution of which is found by numpy.roots. Parameters ---------- B : ndarray, shape (2, 2) Symmetric matrix, defines a quadratic term of the function. g : ndarray, shape (2,) Defines a linear term of the function. Delta : float Radius of a trust region. Returns ------- p : ndarray, shape (2,) Found solution. newton_step : bool Whether the returned solution is the Newton step which lies within the trust region. r T)rr)rr')r'r'rr'ér&©ZaxisNF) rrr r rÚarrayÚrootsÚrealZisrealZvstackrÚargmin)ÚBÚgrÚRÚlowerr1rrrrÚfZcoeffsÚtÚvalueÚirrrÚsolve_trust_region_2d«s,   6ÿ 6( rCcCsb|dkr||}n"||kr&dkr0nnd}nd}|dkrFd|}n|dkrZ|rZ|d9}||fS)zÍUpdate the radius of a trust region based on the cost reduction. Returns ------- Delta : float New radius. ratio : float Ratio between actual and predicted reductions. rr'çÐ?gè?g@r)rZactual_reductionZpredicted_reductionÚ step_normZ bound_hitr3rrrÚupdate_tr_radiusÞs    rFc CsÎ| |¡}t ||¡}|dk r2|t |||¡7}|d9}t ||¡}|dk rÂ| |¡}|t ||¡7}dt ||¡t ||¡} |dk r¸|t |||¡7}| dt |||¡7} ||| fS||fSdS)a­Parameterize a multivariate quadratic function along a line. The resulting univariate quadratic function is given as follows:: f(t) = 0.5 * (s0 + s*t).T * (J.T*J + diag) * (s0 + s*t) + g.T * (s0 + s*t) Parameters ---------- J : ndarray, sparse matrix or LinearOperator shape (m, n) Jacobian matrix, affects the quadratic term. g : ndarray, shape (n,) Gradient, defines the linear term. s : ndarray, shape (n,) Direction vector of a line. diag : None or ndarray with shape (n,), optional Addition diagonal part, affects the quadratic term. If None, assumed to be 0. s0 : None or ndarray with shape (n,), optional Initial point. If None, assumed to be 0. Returns ------- a : float Coefficient for t**2. b : float Coefficient for t. c : float Free term. Returned only if `s0` is provided. Nr&)r r ) ÚJr<rÚdiagÚs0ÚvrrÚurrrrÚbuild_quadratic_1dûs     rLc Csv||g}|dkr>d||}||kr0|kr>nn | |¡t |¡}|||||}t |¡}||||fS)zÛMinimize a 1-D quadratic function subject to bounds. The free term `c` is 0 by default. Bounds must be finite. Returns ------- t : float Minimum point. y : float Minimum value. rgà¿)Úappendr Úasarrayr:) rrÚlbÚubrr@ZextremumÚyZ min_indexrrrÚminimize_quadratic_1d.s     rRcCs–|jdkr>| |¡}t ||¡}|dk r~|t |||¡7}n@| |j¡}tj|ddd}|dk r~|tj||ddd7}t ||¡}d||S)aìCompute values of a quadratic function arising in least squares. The function is 0.5 * s.T * (J.T * J + diag) * s + g.T * s. Parameters ---------- J : ndarray, sparse matrix or LinearOperator, shape (m, n) Jacobian matrix, affects the quadratic term. g : ndarray, shape (n,) Gradient, defines the linear term. s : ndarray, shape (k, n) or (n,) Array containing steps as rows. diag : ndarray, shape (n,), optional Addition diagonal part, affects the quadratic term. If None, assumed to be 0. Returns ------- values : ndarray with shape (k,) or float Values of the function. If `s` was 2-D, then ndarray is returned, otherwise, float is returned. r'Nr rr6r&)Úndimr r ÚTr)rGr<rrHZJsrÚlrrrÚevaluate_quadraticEs     rVcCst ||k||k@¡S)z$Check if a point lies within bounds.)r Úall)rrOrPrrrÚ in_boundsosrXc Cs’t |¡}||}t |¡}| tj¡tjdd.t ||||||||¡||<W5QRXt |¡}|t ||¡t  |¡  t ¡fS)aðCompute a min_step size required to reach a bound. The function computes a positive scalar t, such that x + s * t is on the bound. Returns ------- step : float Computed step. Non-negative value. hits : ndarray of int with shape of x Each element indicates whether a corresponding variable reaches the bound: * 0 - the bound was not hit. * -1 - the lower bound was hit. * 1 - the upper bound was hit. Úignore)Zover) r ZnonzeroZ empty_likeÚfillÚinfZerrstateÚmaximumÚminÚequalÚsignZastypeÚint)rrrOrPZnon_zeroZ s_non_zeroZstepsZmin_steprrrÚstep_size_to_boundts   ÿ ra绽×Ùß|Û=c Cs¶tj|td}|dkr2d|||k<d|||k<|S||}||}|t dt |¡¡}|t dt |¡¡}t |¡|t ||¡k@} d|| <t |¡|t ||¡k@} d|| <|S)a·Determine which constraints are active in a given point. The threshold is computed using `rtol` and the absolute value of the closest bound. Returns ------- active : ndarray of int with shape of x Each component shows whether the corresponding constraint is active: * 0 - a constraint is not active. * -1 - a lower bound is active. * 1 - a upper bound is active. ©Zdtyperr%r')r Ú zeros_liker`r\r+ÚisfiniteÚminimum) rrOrPr/ÚactiveZ lower_distZ upper_distZlower_thresholdZupper_thresholdZ lower_activeZ upper_activerrrÚfind_active_constraints‘s$  ÿÿrhc Csà| ¡}t||||ƒ}t |d¡}t |d¡}|dkrht ||||¡||<t ||||¡||<nL|||t dt ||¡¡||<|||t dt ||¡¡||<||k||kB}d||||||<|S)zÔShift a point to the interior of a feasible region. Each element of the returned vector is at least at a relative distance `rstep` from the closest bound. If ``rstep=0`` then `np.nextafter` is used. r%r'rr&)Úcopyrhr r^Z nextafterr\r+) rrOrPZrstepZx_newrgZ lower_maskZ upper_maskZ tight_boundsrrrÚmake_strictly_feasible¸s   ÿÿrjcCsxt |¡}t |¡}|dkt |¡@}||||||<d||<|dkt |¡@}||||||<d||<||fS)a4Compute Coleman-Li scaling vector and its derivatives. Components of a vector v are defined as follows:: | ub[i] - x[i], if g[i] < 0 and ub[i] < np.inf v[i] = | x[i] - lb[i], if g[i] > 0 and lb[i] > -np.inf | 1, otherwise According to this definition v[i] >= 0 for all i. It differs from the definition in paper [1]_ (eq. (2.2)), where the absolute value of v is used. Both definitions are equivalent down the line. Derivatives of v with respect to x take value 1, -1 or 0 depending on a case. Returns ------- v : ndarray with shape of x Scaling vector. dv : ndarray with shape of x Derivatives of v[i] with respect to x[i], diagonal elements of v's Jacobian. References ---------- .. [1] M.A. Branch, T.F. Coleman, and Y. Li, "A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems," SIAM Journal on Scientific Computing, Vol. 21, Number 1, pp 1-23, 1999. rr%r')r Ú ones_likerdre)rr<rOrPrJZdvÚmaskrrrÚCL_scaling_vectorÓs  rmc CsFt|||ƒr|t |¡fSt |¡}t |¡}| ¡}tj|td}||@}t ||d||||¡||<||||k||<||@}t ||d||||¡||<||||k||<||@}||}t  ||||d||¡} ||t | d||| ¡||<| ||k||<t |¡} d| |<|| fS)z3Compute reflective transformation and its gradient.rcr r%) rXr rkrerirdÚboolr\rfÚ remainder) rQrOrPZ lb_finiteZ ub_finiterZ g_negativerlrr@r<rrrÚreflective_transformationÿs(    $ $ $ rpc Cstd dddddd¡ƒdS)Nz*{0:^15}{1:^15}{2:^15}{3:^15}{4:^15}{5:^15}Ú Iterationz Total nfevÚCostúCost reductionú Step normÚ Optimality©ÚprintÚformatrrrrÚprint_header_nonlinear!sþryc CsL|dkrd}n d |¡}|dkr&d}n d |¡}td ||||||¡ƒdS)Nú ú {0:^15.2e}z({0:^15}{1:^15}{2:^15.4e}{3}{4}{5:^15.2e}©rxrw)Ú iterationZnfevÚcostÚcost_reductionrEÚ optimalityrrrÚprint_iteration_nonlinear's  þrcCstd ddddd¡ƒdS)Nz#{0:^15}{1:^15}{2:^15}{3:^15}{4:^15}rqrrrsrtrurvrrrrÚprint_header_linear8sþr‚cCsJ|dkrd}n d |¡}|dkr&d}n d |¡}td |||||¡ƒdS)Nrzr{z!{0:^15}{1:^15.4e}{2}{3}{4:^15.2e}r|)r}r~rrEr€rrrÚprint_iteration_linear>s  ÿrƒcCs$t|tƒr| |¡S|j |¡SdS)z4Compute gradient of the least-squares cost function.N)Ú isinstancerÚrmatvecrTr )rGr?rrrÚ compute_gradQs  r†cCsnt|ƒr*t | d¡jdd¡ ¡d}ntj|dddd}|dkrVd||dk<n t ||¡}d||fS)z5Compute variables scale based on the Jacobian matrix.r rr6r&Nr')rr rNÚpowerrÚravelr\)rGZ scale_inv_oldZ scale_invrrrÚcompute_jac_scaleYs" r‰csDtˆƒ‰‡‡fdd„}‡‡fdd„}‡‡fdd„}tˆj|||dS)z#Return diag(d) J as LinearOperator.csˆˆ |¡S©N)Úmatvec©r©rGrrrr‹lsz(left_multiplied_operator..matveccsˆdd…tjfˆ |¡SrŠ)r ÚnewaxisÚmatmat©ÚXrrrrosz(left_multiplied_operator..matmatcsˆ | ¡ˆ¡SrŠ)r…rˆrŒrrrr…rsz)left_multiplied_operator..rmatvec©r‹rr…©r rÚshape©rGrr‹rr…rrrÚleft_multiplied_operatorhs ÿr–csDtˆƒ‰‡‡fdd„}‡‡fdd„}‡‡fdd„}tˆj|||dS)z#Return J diag(d) as LinearOperator.csˆ t |¡ˆ¡SrŠ)r‹r rˆrŒrrrr‹}sz)right_multiplied_operator..matveccsˆ |ˆdd…tjf¡SrŠ)rr rŽrrrrr€sz)right_multiplied_operator..matmatcsˆˆ |¡SrŠ©r…rŒrrrr…ƒsz*right_multiplied_operator..rmatvecr’r“r•rrrÚright_multiplied_operatorys ÿr˜csFtˆƒ‰ˆj\‰}‡‡fdd„}‡‡‡fdd„}tˆ||f||dS)zµReturn a matrix arising in regularized least squares as LinearOperator. The matrix is [ J ] [ D ] where D is diagonal matrix with elements from `diag`. cst ˆ |¡ˆ|f¡SrŠ)r Zhstackr‹rŒ)rGrHrrr‹•sz(regularized_lsq_operator..matveccs*|dˆ…}|ˆd…}ˆ |¡ˆ|SrŠr—)rÚx1Zx2©rGrHr-rrr…˜s  z)regularized_lsq_operator..rmatvec)r‹r…)r r”r)rGrHr,r‹r…rršrÚregularized_lsq_operatorŠs  r›TcCs\|rt|tƒs| ¡}t|ƒr:|j|j|jdd9_nt|tƒrPt||ƒ}n||9}|S)zhCompute J diag(d). If `copy` is False, `J` is modified in place (unless being LinearOperator). Zclip)Úmode)r„rrirÚdataZtakeÚindicesr˜©rGrrirrrÚright_multiply s  r cCsn|rt|tƒs| ¡}t|ƒr>|jt |t |j¡¡9_n,t|tƒrTt ||ƒ}n||dd…tj f9}|S)zhCompute diag(d) J. If `copy` is False, `J` is modified in place (unless being LinearOperator). N) r„rrirrr ÚrepeatZdiffZindptrr–rŽrŸrrrÚ left_multiply²s   r¢c CsH|||ko|dk}||||k}|r0|r0dS|r8dS|r@dSdSdS)z8Check termination condition for nonlinear least squares.rDér rNr) ZdFÚFZdx_normZx_normr3ZftolZxtolZftol_satisfiedZxtol_satisfiedrrrÚcheck_terminationÄsr¥cCsR|dd|d|d}t||tk<|dC}||d|9}t||dd|fS)z`Scale Jacobian and residuals for a robust loss function. Arrays are modified in place. r'r r&F)ri)r(r¢)rGr?ÚrhoZJ_scalerrrÚscale_for_robust_loss_functionÓs  r§)Nrr)NN)r)N)rb)rb)N)T)T).Ú__doc__ÚmathrZnumpyr Z numpy.linalgrZ scipy.linalgrrrZ scipy.sparserZscipy.sparse.linalgrr ZfinfoÚfloatZepsr(rr4rCrFrLrRrVrXrarhrjrmrpryrr‚rƒr†r‰r–r˜r›r r¢r¥r§rrrrÚsH    'ÿ r3 3  * ' ,"