U -e~P@sFdZddlmZddlZddlmZddlmZm Z m Z ddl m Z ddl mZmZeejZdd Zd@d d ZddZddZdAddZdBddZdCddZddZddZdDddZdEdd Zd!d"Zd#d$Z d%d&Z!d'd(Z"d)d*Z#d+d,Z$d-d.Z%dFd/d0Z&d1d2Z'd3d4Z(d5d6Z)dGd8d9Z*dHd:d;Z+dd?Z-dS)Iz+Functions used by least-squares algorithms.)copysignN)norm) cho_factor cho_solve LinAlgError)issparse)LinearOperatoraslinearoperatorc Cst||}|dkrtdt||}t|||d}|dkrLtdt||||}|t|| }||}||} || 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.z#`x` is not within the trust region.N)npdot ValueErrorsqrtr) xsDeltaabcdqt1t2r[/var/www/html/Darija-Ai-Train/env/lib/python3.8/site-packages/scipy/optimize/_lsq/common.pyintersect_trust_regions  r{Gz? c Csdd} ||} ||kr6t||d} |d| k} nd} | rd||| } t| |krd| ddfSt| |}| r| d| ||\}}| |}nd}|dks| s|dkrtd|||d }n|}t|D]}||ks||krtd|||d }| || ||\}}|dkr|}||}t|||}|||||8}t|||krq\q|| |d | } | |t| 9} | ||d fS) aSolve 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||}||}t|d|d |}||fS)zFunction of which to find zero. It is defined as "norm of regularized (by alpha) least-squares solution minus `Delta`". Refer to [1]_. r )rr sum)alphasufrrdenomZp_normphi phi_primerrrphi_and_derivativejs   z2solve_lsq_trust_region..phi_and_derivativerFgNgMbP??r )EPSr rmaxranger abs)nmZufrVrZ initial_alphartolZmax_iterr%r! thresholdZ full_rankpZ alpha_upperr#r$Z alpha_lowerr itratiorrrsolve_lsq_trust_region9s@1       r5cCsxz>t|\}}t||f| }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(r'ZaxisNF) rrr r rarrayrootsrealZisrealZvstackrargmin)BgrRlowerr2rrrrfZcoeffstvalueirrrsolve_trust_region_2ds,   6 6( rDcCsb|dkr||}n"||kr&dkr0nnd}nd}|dkrFd|}n|dkrZ|rZ|d9}||fS)zUpdate 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(?g?g@r)rZactual_reductionZpredicted_reduction step_normZ bound_hitr4rrrupdate_tr_radiuss    rGc Cs||}t||}|dk r2|t|||7}|d9}t||}|dk r||}|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 ) Jr=rdiags0vrrurrrrbuild_quadratic_1ds     rMc Csv||g}|dkr>d||}||kr0|kr>nn ||t|}|||||}t|}||||fS)zMinimize 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. rg)appendr asarrayr;) rrlbubrrAZextremumy min_indexrrrminimize_quadratic_1d.s     rTcCs|jdkr>||}t||}|dk r~|t|||7}n@||j}tj|ddd}|dk r~|tj||ddd7}t||}d||S)aCompute 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 rr7r')ndimr r Tr)rHr=rrIZJsrlrrrevaluate_quadraticEs     rXcCst||k||k@S)z$Check if a point lies within bounds.)r all)rrPrQrrr in_boundsosrZc Cst|}||}t|}|tjtjdd.t||||||||||<W5QRXt|}|t||t | t fS)aCompute 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. ignore)Zover) r ZnonzeroZ empty_likefillinfZerrstatemaximumminequalsignZastypeint)rrrPrQZnon_zeroZ s_non_zeroZstepsZmin_steprrrstep_size_to_boundts    rc绽|=c Cstj|td}|dkr2d|||k<d|||k<|S||}||}|tdt|}|tdt|}t||t||k@} d|| <t||t||k@} d|| <|S)aDetermine 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 zeros_likerbr^r,isfiniteminimum) rrPrQr0activeZ lower_distZ upper_distZlower_thresholdZupper_thresholdZ lower_activeZ upper_activerrrfind_active_constraintss$  rjc Cs|}t||||}t|d}t|d}|dkrht||||||<t||||||<nL|||tdt||||<|||tdt||||<||k||kB}d||||||<|S)zShift 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')copyrjr r`Z nextafterr^r,) rrPrQZrstepZx_newriZ lower_maskZ upper_maskZ tight_boundsrrrmake_strictly_feasibles   rlcCsxt|}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 ones_likerfrg)rr=rPrQrKZdvmaskrrrCL_scaling_vectors  roc CsFt|||r|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.rer r&) rZr rmrgrkrfboolr^rh remainder) rRrPrQZ lb_finiteZ ub_finiterZ g_negativernrrAr=rrrreflective_transformations(    $ $ $ rrc CstddddddddS)Nz*{0:^15}{1:^15}{2:^15}{3:^15}{4:^15}{5:^15} Iterationz Total nfevCostCost reduction Step norm Optimalityprintformatrrrrprint_header_nonlinear!sr{c CsL|dkrd}n d|}|dkr&d}n d|}td||||||dS)N {0:^15.2e}z({0:^15}{1:^15}{2:^15.4e}{3}{4}{5:^15.2e}rzry) iterationZnfevcostcost_reductionrF optimalityrrrprint_iteration_nonlinear's  rcCstdddddddS)Nz#{0:^15}{1:^15}{2:^15}{3:^15}{4:^15}rsrtrurvrwrxrrrrprint_header_linear8srcCsJ|dkrd}n d|}|dkr&d}n d|}td|||||dS)Nr|r}z!{0:^15}{1:^15.4e}{2}{3}{4:^15.2e}r~)rrrrFrrrrprint_iteration_linear>s  rcCs$t|tr||S|j|SdS)z4Compute gradient of the least-squares cost function.N) isinstancerrmatvecrVr )rHr@rrr compute_gradQs  rcCsnt|r*t|djddd}ntj|dddd}|dkrVd||dk<n t||}d||fS)z5Compute variables scale based on the Jacobian matrix.r rr7r'Nr()rr rOpowerrravelr^)rHZ scale_inv_oldZ scale_invrrrcompute_jac_scaleYs" rcsDtfdd}fdd}fdd}tj|||dS)z#Return diag(d) J as LinearOperator.cs|SN)matvecrrHrrrrlsz(left_multiplied_operator..matveccsddtjf|Sr)r newaxismatmatXrrrrosz(left_multiplied_operator..matmatcs|Sr)rrrrrrrrsz)left_multiplied_operator..rmatvecrrrr rshaperHrrrrrrrleft_multiplied_operatorhs rcsDtfdd}fdd}fdd}tj|||dS)z#Return J diag(d) as LinearOperator.cst|Sr)rr rrrrrr}sz)right_multiplied_operator..matveccs|ddtjfSr)rr rrrrrrsz)right_multiplied_operator..matmatcs|Srrrrrrrsz*right_multiplied_operator..rmatvecrrrrrrright_multiplied_operatorys rcsFtj\}fdd}fdd}t||f||dS)zReturn a matrix arising in regularized least squares as LinearOperator. The matrix is [ J ] [ D ] where D is diagonal matrix with elements from `diag`. cst||fSr)r Zhstackrr)rHrIrrrsz(regularized_lsq_operator..matveccs*|d}|d}||Srr)rx1Zx2rHrIr.rrrs  z)regularized_lsq_operator..rmatvec)rr)r rr)rHrIr-rrrrrregularized_lsq_operators  rTcCs\|rt|ts|}t|r:|j|j|jdd9_nt|trPt||}n||9}|S)zhCompute J diag(d). If `copy` is False, `J` is modified in place (unless being LinearOperator). Zclip)mode)rrrkrdataZtakeindicesrrHrrkrrrright_multiplys  rcCsn|rt|ts|}t|r>|jt|t|j9_n,t|trTt ||}n||ddtj f9}|S)zhCompute diag(d) J. If `copy` is False, `J` is modified in place (unless being LinearOperator). N) rrrkrrr repeatdiffZindptrrrrrrr left_multiplys   rc CsH|||ko|dk}||||k}|r0|r0dS|r8dS|r@dSdSdS)z8Check termination condition for nonlinear least squares.rEr rNr) ZdFFZdx_normZx_normr4ZftolZxtolZftol_satisfiedZxtol_satisfiedrrrcheck_terminationsrcCsR|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)rk)r)r)rHr@rhoZJ_scalerrrscale_for_robust_loss_functions  r)Nrr)NN)r)N)rd)rd)N)T)T).__doc__mathrnumpyr Z numpy.linalgrZ scipy.linalgrrrZ scipy.sparserZscipy.sparse.linalgrr ZfinfofloatZepsr)rr5rDrGrMrTrXrZrcrjrlrorrr{rrrrrrrrrrrrrrrrsH    ' r3 3  * ' ,"