+ /iPRRt^RIHt^RIt^RIHt^RIHtH t H t ^RI H t H t ^RIHt]P !]4P$tRtR!RltR tR tR"R ltR#R ltR$R ltRtRtR%RltR%RltRtRt Rt!Rt"Rt#Rt$Rt%R$Rlt&Rt'Rt(Rt)R&Rlt*R&Rlt+Rt,R t-R#)'z+Functions used by least-squares algorithms.)copysignN)norm) cho_factor cho_solve LinAlgError)LinearOperatoraslinearoperator)issparsec\P!W4pV^8Xd \R4h\P!W4p\P!W4V^,, pV^8d \R4h\P!WD,W5,, 4pV\ Wd4,)pWs, pWW, p W8dW3#W3#)aEFind 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. z `s` is zero.z#`x` is not within the trust region.)npdot ValueErrorsqrtr) xsDeltaabcdqt1t2s &&& X/var/www/html/photoedit/myenv/lib/python3.14/site-packages/scipy/optimize/_lsq/common.pyintersect_trust_regionrs q AAv(( q A q uaxA1u>?? ac A hqn A B B wv v c Rp W2,p W8d)\V,V^,,p VR,V 8p MRp V 'd.VPW#, 4)p \V 4V8:dV R^3#\V 4V, pV 'dV !RWV4wppV)V, pMRpVeV 'g*V^8Xd#\RV,VV,R,4pMTp\ V4FpVV8gVV8d"\RV,VV,R,4pV !VWV4wppV^8dTpVV, p\VVV, 4pVW,V,V, ,p\ P !V4Wu,8gKM VPW^,V,, 4)p W\V 4, ,p V VX^,3#)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. cV^,V,p\W, 4pWS, p\P!V^,V^,, 4)V, pWg3#)zFunction of which to find zero. It is defined as "norm of regularized (by alpha) least-squares solution minus `Delta`". Refer to [1]_. )rr sum)alphasufrrdenomp_normphi phi_primes&&&& rphi_and_derivative2solve_lsq_trust_region..phi_and_derivativejsO 1u ck"nVVC1Huax/0069 ~rFggMbP??)EPSr rmaxranger abs)nmufrVr initial_alphartolmax_iterr%r threshold full_rankp alpha_upperr#r$ alpha_lowerritratios&&&&&&&&& rsolve_lsq_trust_regionr;9sb  &C v!GadN bEI%   UU26]N 7e c19 s)e#K+C?YdY&  I-12DEK'+ *Cc)IJHo ; %+"5 +kK.G#-MNE+E35AY 7Ki+uu}5  #+&.. 66#; %  sdUl# $$A aA eR!V rc`\V4wr4\W43V4)p\P!WU4V^,8:dVR3#VR,V^,,pVR,V^,,pVR,V^,,pV^,V,p V^,V,p \P !V)V ,^Wh, V ,,^V,^V)V,V ,,V)V , .4p \P !V 4p \P!V \P!V 4,4p V\P!^V ,^V ^,,, ^V ^,, ^V ^,,, 34,pR\P!WPPV4,^R7,\P!W4,p \P!V 4pVRV3,pVR3# \dELi;i) a2Solve 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. Tr'axisNNNF)r@)r@)rArA) rrr r rarrayrootsrealisrealvstackrargmin)BgrRlowerr6rrrrfcoeffstvalueis&&& rsolve_trust_region_2drQs. a= z1 % % 66!<5!8 #d7N $ $%(A $%(A $%(A !u A !u A XX aaeai!a%qb1fqj)9A26BDF A "))A, A  1q5A1H-AqDQAX/FGHHA "&&UU1XA. . =E %A !Q$A e8O)    sAH H-,H-cV^8d W, pMY!u;8Xd^8XdMM^pM^pVR8d RV,pW3#VR8dV'd VR,pW3#)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. ?g?g@)ractual_reductionpredicted_reduction step_norm bound_hitr:s&&&&& rupdate_tr_radiusrYs`Q 6  5A 5 t|y  < )   <rcfVPV4p\P!WU4pVe%V\P!W#,V4, pVR,p\P!W4pVeVPV4pV\P!W4, pR\P!W4,\P!W4,p VePV\P!WC,V4, pV R\P!WC,V4,, p WgV 3#Wg3#)aHParameterize 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 array 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. r')r r ) JrIrdiags0vrrurs &&&&& rbuild_quadratic_1dr`s> aA q A  RVVAHa  HA q A ~ EE"I RVVA\ "&&,  .    1% %A rvvbi,, ,AQwt rc,W#.pV^8wd3RV,V, pY&u;8dV8dMMVPV4\P!V4pWPV,V,,V,p\P!V4pWX,Wx,3#)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. g)appendr asarrayrG) rrlbubrrNextremumy min_indexs &&&&& rminimize_quadratic_1dri.ss AAv!8a< 2  HHX  1 A UQY!A ! I < %%rcVP^8XdQVPV4p\P!WD4pVe%V\P!W#,V4, pMjVPVP4p\P!V^,^R7pVe-V\P!W2^,,^R7, p\P!W!4pRV,V,#)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 array 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=r')ndimr r Tr)r[rIrr\Jsrls&&&& revaluate_quadraticroEs. vv{ UU1X FF2N   !$ $A UU133Z FF2q5q !   !t !, ,A q A 7Q;rcD\P!W8W8*,4#)z$Check if a point lies within bounds.)r all)rrdres&&&r in_boundsrros 6617qw' ((rcX\P!V4pW,p\P!V4pVP\P4\P !RR7;_uu_4\P !W , V,V, W0, V,V, 4Wd&RRR4\P!V4pV\P!Wg4\P!V4P\4,3# +'giLl;i)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)overN) r nonzero empty_likefillinferrstatemaximumminequalsignastypeint)rrrdrenon_zero s_non_zerostepsmin_steps&&&& rstep_size_to_boundrts$zz!}HJ MM! E JJrvv ( # #**bfh%7*%D&(fh%7*%DF $vve}H RXXe.1B1B31GG GG $ #s 3AD D) c4\P!V\R7pV^8XdRW@V8*&^W@V8&V#W, pW , pV\P!^\P!V44,pV\P!^\P!V44,p\P !V4V\P !Wg48*,p RWI&\P !V4V\P !WX48*,p ^WJ&V#)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. dtyper()r zeros_likerr{r,isfiniteminimum) rrdrer2active lower_dist upper_distlower_thresholdupper_threshold lower_active upper_actives &&&& rfind_active_constraintsrs]]1C (F qyBwBw JJRZZ266":66ORZZ266":66OKKO2::j#JJLLFKKO2::j#JJLLF Mrc VP4p\WW#4p\P!VR4p\P!V^4pV^8XdL\P!W,W&,4WF&\P!W',W,4WG&MW,V\P !^\P !W,44,,WF&W',V\P !^\P !W',44,, WG&WA8WB8,pRW,W(,,,WH&V#)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()copyrr r} nextafterr{r,) rrdrerstepx_newr lower_mask upper_mask tight_boundss &&&& rmake_strictly_feasiblers FFHE $QB 6F&"%J&!$J zLLHLLH^"RZZ266".3I%JJK^"RZZ266".3I%JJKJ5:.L!1B4D!DEE LrcL\P!V4p\P!V4pV^8\P!V4,pW6,W,, WF&RWV&V^8\P!V4,pW,W&,, WF&^WV&WE3#)aCompute 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. r()r ones_likerr)rrIrdrer^dvmasks&&&& rCL_scaling_vectorrs< QA q B ER[[_ $Dh AGBH ER[[_ $Dg AGBH 5Lrc\WV4'dV\P!V43#\P!V4p\P!V4pVP 4p\P !V\ R7pW4(,p\P!W,^W,,W,, 4WW&W,W,8Wg&V(V,p\P!W,^W',,W,, 4WW&W,W',8Wg&W4,pW!, p\P!W,W,, ^W,,4p W,\P!V ^W,,V , 4,WW&WV,8Wg&\P!V4p RW&WZ3#)z3Compute reflective transformation and its gradient.rr() rrr rrrrboolr{r remainder) rgrdre lb_finite ub_finiter g_negativerrrNrIs &&& rreflective_transformationrsK",,q/!! BI BI Aq-J z !Djj!bh,"89AGw)J : !Djj!bh,"89AGw)J  D A QWrx'QW5AhAq17{Q77AGT7{J QAAM 4Krc D\RPRRRRRR44R#)z${:^15}{:^15}{:^15}{:^15}{:^15}{:^15} Iterationz Total nfevCostCost reduction Step norm OptimalityNprintformatrTrrprint_header_nonlinearr!s% 0 6+|V5E| -.rc hVfRpMVR pVfRpMVR p\VR VR VR V V VR 24R#Nz^15.2ez^15z^15.4ez r) iterationnfevcostcost_reductionrW optimalitys&&&&&&rprint_iteration_nonlinearr'sX!*62  (  YsOD:d6]>2B9+jY_M` abrc B\RPRRRRR44R#)z{:^15}{:^15}{:^15}{:^15}{:^15}rrrrrNrrTrrprint_header_linearr6s# * 6+v'7 !rcbVfRpMVR pVfRpMVR p\VR VR V V VR 24R#rr)rrrrWrs&&&&&rprint_iteration_linearr<sQ!*62  (  YsOD=(8 JvCV WXrc\V\4'dVPV4#VPP V4#)z4Compute gradient of the least-squares cost function.) isinstancerrmatvecrlr )r[rLs&&r compute_gradrNs/!^$$yy|sswwqzrc`\V4'dL\P!VP^4P ^R74P 4R,pM&\P!V^,^R7R,pVf ^W"^8H&M\P !W!4p^V, V3#)z5Compute variables scale based on the Jacobian matrix.r=r')r r rcpowerrravelr{)r[ scale_inv_old scale_invs&& rcompute_jac_scalerVs{{JJqwwqz~~1~56<<>C FF1a4a(#- $% q.!JJy8 y=) ##rcvaa\S4oVV3RlpVV3RlpVV3Rlp\SPW#VR7#)z#Return diag(d) J as LinearOperator.c4<SSPV4,#N)matvecrr[rs&rr(left_multiplied_operator..matvecis188A;rcb<SR\P3,SPV4,#r?)r newaxismatmatXr[rs&rr(left_multiplied_operator..matmatls#BJJ!((1+--rcP<SPVP4S,4#r)rrrs&rr)left_multiplied_operator..rmatvecosyyQ''rrrrrrshaper[rrrrsff rleft_multiplied_operatorres6A.( !''&") ++rcvaa\S4oVV3RlpVV3RlpVV3Rlp\SPW#VR7#)z#Return J diag(d) as LinearOperator.c\<SP\P!V4S,4#r)rr rrs&rr)right_multiplied_operator..matveczsxx a((rcb<SPVSR\P3,,4#r)rr rrs&rr)right_multiplied_operator..matmat}s$xxAam,,--rc4<SSPV4,#rrrs&rr*right_multiplied_operator..rmatvecs199Q<rrrrsff rright_multiplied_operatorrvs6A).  !''&") ++rcaaa\S4oSPwopVV3RlpVVV3Rlp\SV,V3W4R7#)zReturn a matrix arising in regularized least squares as LinearOperator. The matrix is [ J ] [ D ] where D is diagonal matrix with elements from `diag`. c`<\P!SPV4SV,34#r)r hstackr)rr[r\s&rr(regularized_lsq_operator..matvecs#yy!((1+tax011rcV<VRSpVSRpSPV4SV,,#rr)rx1x2r[r\r.s& rr)regularized_lsq_operator..rmatvecs0 rU qrUyy}tby((r)rr)rrr)r[r\r-rrr.sff @rregularized_lsq_operatorrs= A 77DAq2) 1q5!*V EErcFV'd'\V\4'gVP4p\V4'd7V;PVP VP RR7,unV#\V\4'd\W4pV#W,pV#)z`Compute J diag(d). If `copy` is False, `J` is modified in place (unless being LinearOperator). clip)mode)rrrr datatakeindicesrr[rrs&&&rright_multiplyrs{  Jq.11 FFH{{ !&&&00 H A~ & & %a + H  HrcV'd'\V\4'gVP4p\V4'dOV;P\ P !V\ P!VP44,unV#\V\4'd\W4pV#WR\ P3,,pV#)z`Compute diag(d) J. If `copy` is False, `J` is modified in place (unless being LinearOperator). r?) rrrr rr repeatdiffindptrrrrs&&&r left_multiplyrs  Jq.11 FFH{{ "))Arwwqxx011 H A~ & & $Q * H q"**}  HrcWV,8;'dVR8pW&Wc,,8pV'd V'd^#V'd^#V'd^#R#)z8Check termination condition for nonlinear least squares.rSNrT) dFFdx_normx_normr:ftolxtolftol_satisfiedxtol_satisfieds &&&&&&& rcheck_terminationrs?(]33ut|Nt}55N.  rcV^,^V^,,V^,,,p\W3\8&VR,pW^,V, ,p\WRR7V3#)zXScale Jacobian and residuals for a robust loss function. Arrays are modified in place. r'F)r)r)r)r[rLrhoJ_scales&&& rscale_for_robust_loss_functionr s[ !fq3q6zAqD((G GcM OGQ' A % 0! 33r)Ng{Gz? )NN)r@r)g|=)T).__doc__mathrnumpyr numpy.linalgr scipy.linalgrrrscipy.sparse.linalgrrscipy._lib._sparser finfofloatepsr)rr;rQrYr`rirorrrrrrrrrrrrrrrrrrrr rTrrrs1;;@' hhuo $Nod0f:0f&.$T) H:$N6)XD. c! Y$ $+"+"F, $ $  4r