(phOTSrSSKJr SSKrSSKJr SSKJrJ r J r SSK J r SSK JrJr \R "\5R$rSrS"S jrS rS rS#S jrS$S jrS%SjrSrSrS&SjrS&SjrSrSr Sr!Sr"Sr#Sr$Sr%S%Sjr&Sr'Sr(Sr)S'Sjr*S'Sjr+S r,S!r-g)(z+Functions used by least-squares algorithms.)copysignN)norm) cho_factor cho_solve LinAlgError)issparse)LinearOperatoraslinearoperatorcT[R"X5nUS:Xa [S5e[R"X5n[R"X5US-- nUS:a [S5e[R"XD-X5-- 5nU[ Xd5-*nXs- nXW- n X:aX4$X4$)aEFind the intersection of a line with the boundary of a trust region. This function solves the quadratic equation with respect to t ||(x + s*t)||**2 = Delta**2. Returns ------- t_neg, t_pos : tuple of float Negative and positive roots. Raises ------ ValueError If `s` is zero or `x` is not within the trust region. rz `s` is zero.z#`x` is not within the trust region.)npdot ValueErrorsqrtr) xsDeltaabcdqt1t2s M/var/www/html/venv/lib/python3.13/site-packages/scipy/optimize/_lsq/common.pyintersect_trust_regionrs q AAv(( q A q uaxA1u>?? ac A hqn A B B wv v c Sn X2-n X:a[U-US-n USU :n OSn U (a(URX#- 5*n [U 5U::aU SS4$[U 5U- nU (aU "SXU5unnU*U- nOSnUb U (dUS:Xa[SU-UU-S-5nOUn[ U5HxnUU:dUU:a[SU-UU-S-5nU "UXU5unnUS:aUnUU- n[UUU- 5nUX-U-U- -n[ R "U5Xu-:dMx O URXS-U-- 5*n X[U 5- -n U UWS -4$) aSolve a trust-region problem arising in least-squares minimization. This function implements a method described by J. J. More [1]_ and used in MINPACK, but it relies on a single SVD of Jacobian instead of series of Cholesky decompositions. Before running this function, compute: ``U, s, VT = svd(J, full_matrices=False)``. Parameters ---------- n : int Number of variables. m : int Number of residuals. uf : ndarray Computed as U.T.dot(f). s : ndarray Singular values of J. V : ndarray Transpose of VT. Delta : float Radius of a trust region. initial_alpha : float, optional Initial guess for alpha, which might be available from a previous iteration. If None, determined automatically. rtol : float, optional Stopping tolerance for the root-finding procedure. Namely, the solution ``p`` will satisfy ``abs(norm(p) - Delta) < rtol * Delta``. max_iter : int, optional Maximum allowed number of iterations for the root-finding procedure. Returns ------- p : ndarray, shape (n,) Found solution of a trust-region problem. alpha : float Positive value such that (J.T*J + alpha*I)*p = -J.T*f. Sometimes called Levenberg-Marquardt parameter. n_iter : int Number of iterations made by root-finding procedure. Zero means that Gauss-Newton step was selected as the solution. References ---------- .. [1] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977. cUS-U-n[X- 5nXS- n[R"US-US-- 5*U- nXg4$)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)alphasufrrdenomp_normphi phi_primes rphi_and_derivative2solve_lsq_trust_region..phi_and_derivativejsO 1u ck"nVVC1Huax/0069 ~rrFggMbP??r )EPSrrmaxranger abs)nmufrVr initial_alphartolmax_iterr(r# threshold full_rankp alpha_upperr&r' alpha_lowerr"itratios 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*[U5up4[X44U5*n[R"XU5US-::aUS4$USUS--nUSUS--nUSUS--nUSU-n USU-n [R "U*U -SXh- U --SU-SU*U-U --U*U - /5n [R "U 5n [R"U [R"U 55n U[R"SU -SU S--- SU S-- SU S--- 45-nS [R"XPRU5-SS 9-[R"X5-n [R"U 5nUS S 2U4nUS 4$![a GNRf=f) 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. r T)rr)rr,)r,r,rr,r+axisNF) rrr rrarrayrootsrealisrealvstackr!argmin)BgrRlowerr:rrrrfcoeffstvalueis rsolve_trust_region_2drSs. 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)    s;F FFcUS:aX- nOX!s=:XaS:XaO OSnOSnUS:aSU-nX4$US:a U(aUS-nX4$)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@)ractual_reductionpredicted_reduction step_norm bound_hitr>s rupdate_tr_radiusr[s`Q 6  5A 5 t|y  < )   <rcURU5n[R"XU5nUbU[R"X#-U5- nUS-n[R"X5nUbURU5nU[R"X5- nS[R"X5-[R"X5-n Ub;U[R"XC-U5- nU S[R"XC-U5-- n XgU 4$Xg4$)aIParameterize 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. r+)rr ) JrKrdiags0vrrurs rbuild_quadratic_1drbs> aA q A  RVVAHa  HA q A ~ EE"I RVVA\ "&&,  .    1% %A rvvbi,, ,AQwt rcX#/nUS:wa(SU-U- nX&s=:aU:aO OURU5 [R"U5nXPU-U--U-n[R"U5nXXXx4$)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 asarrayrI) rrlbubrrPextremumy min_indexs rminimize_quadratic_1drk.ss AAv!8a< 2  HHX  1 A UQY!A ! I < %%rcURS:XaGURU5n[R"XD5nUbU[R"X#-U5- nOSURUR5n[R"US-SS9nUbU[R"X2S--SS9- n[R"X!5nSU-U-$)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,r rrBr+)ndimrr Tr!)r]rKrr^Jsrls revaluate_quadraticrqEs. vv{ UU1X FF2N   !$ $A UU133Z FF2q5q !   !t !, ,A q A 7Q;rc<[R"X:X:*-5$)z$Check if a point lies within bounds.)r all)rrfrgs r in_boundsrtos 6617qw' ((rc[R"U5nXn[R"U5nUR[R5 [R "SS9 [R "X - UU- X0- UU- 5Xd'SSS5 [R"U5nU[R"Xg5[R"U5R[5-4$!,(df  Nf=f)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)rrrfrgnon_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 $*C.. C<c[R"U[S9nUS:XaSX@U:*'SX@U:'U$X- nX - nU[R"S[R"U55-nU[R"S[R"U55-n[R "U5U[R "Xg5:*-n SXI'[R "U5U[R "XX5:*-n SXJ'U$)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. dtyperr*r,)r zeros_likerr}r0isfiniteminimum) rrfrgr6active lower_dist upper_distlower_thresholdupper_threshold lower_active upper_actives rfind_active_constraintsrs]]1C (F qyBwBw JJRZZ266":66ORZZ266":66OKKO2::j#JJLLFKKO2::j#JJLLF Mrc &UR5n[XX#5n[R"US5n[R"US5nUS:Xa;[R"XX&5XF'[R"X'X5XG'OnXU[R "S[R "X55--XF'X'U[R "S[R "X'55-- XG'XA:XB:-nSXX(--XH'U$)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+)copyrr r nextafterr}r0) rrfrgrstepx_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 Lrc [R"U5n[R"U5nUS:[R"U5-nX6X- XF'SXV'US:[R"U5-nXX&- XF'SXV'XE4$)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. rr*r,)r ones_likerr)rrKrfrgr`dvmasks rCL_scaling_vectorrs< QA q B ER[[_ $Dh AGBH ER[[_ $Dg AGBH 5Lrc[XU5(aU[R"U54$[R"U5n[R"U5nUR 5n[R "U[ S9nX4)-n[R"XSX-X- 5XW'XX:Xg'U)U-n[R"XSX'-X- 5XW'XX':Xg'X4-nX!- n[R"XX- SX-5n X[R"U SX-U - 5-XW'XU:Xg'[R"U5n SX'XZ4$)z3Compute reflective transformation and its gradient.rr r*) rtr rrrrboolr}r remainder) rirfrg lb_finite ub_finiter g_negativerrrPrKs rreflective_transformationrsK",,q/!! BI BI Aq-J z !Djj!bh,"89AGw)J : !Djj!bh,"89AGw)J  D A QWrx'QW5AhAq17{Q77AGT7{J QAAM 4Krc B[SRSSSSSS55 g)Nz${:^15}{:^15}{:^15}{:^15}{:^15}{:^15} Iterationz Total nfevCostCost reduction Step norm OptimalityprintformatrVrrprint_header_nonlinearr!s% 0 6+|V5E| -.rc bUcSnOUSnUcSnOUSn[USUSUSUUUS35 gNz z^15.2ez^15z^15.4er) iterationnfevcostcost_reductionrY optimalitys rprint_iteration_nonlinearr'sX!*62  (  YsOD:d6]>2B9+jY_M` abrc @[SRSSSSS55 g)Nz{:^15}{:^15}{:^15}{:^15}{:^15}rrrrrrrVrrprint_header_linearr6s# * 6+v'7 !rc\UcSnOUSnUcSnOUSn[USUSUUUS35 grr)rrrrYrs rprint_iteration_linearr<sQ!*62  (  YsOD=(8 JvCV WXrc[U[5(aURU5$URR U5$)z4Compute gradient of the least-squares cost function.) isinstancer rmatvecrnr)r]rNs r compute_gradrNs/!^$$yy|sswwqzrc0[U5(aD[R"URS5R SS95R 5S-nO[R"US-SS9S-nUcSX"S:H'O[R "X!5nSU- U4$)z5Compute variables scale based on the Jacobian matrix.r rrBr+r,)rr repowerr!ravelr})r] scale_inv_old scale_invs rcompute_jac_scalerVs{{JJqwwqz~~1~56<<>C FF1a4a(#- $% q.!JJy8 y=) ##rcp^^[T5mUU4SjnUU4SjnUU4Sjn[TRX#US9$)z#Return diag(d) J as LinearOperator.c,>TTRU5-$N)matvecrr]rs rr(left_multiplied_operator..matvecis188A;rcV>TSS2[R4TRU5-$r)r newaxismatmatXr]rs rr(left_multiplied_operator..matmatls#BJJ!((1+--rcH>TRUR5T-5$r)rrrs rr)left_multiplied_operator..rmatvecosyyQ''rrrrr r shaper]rrrrs`` rleft_multiplied_operatorres6A.( !''&") ++rcp^^[T5mUU4SjnUU4SjnUU4Sjn[TRX#US9$)z#Return J diag(d) as LinearOperator.cT>TR[R"U5T-5$r)rr rrs rr)right_multiplied_operator..matveczsxx a((rcV>TRUTSS2[R4-5$r)rr rrs rr)right_multiplied_operator..matmat}s$xxAam,,--rc,>TTRU5-$rrrs rr*right_multiplied_operator..rmatvecs199Q<rrrrs`` rright_multiplied_operatorrvs6A).  !''&") ++rcx^^^[T5mTRumnUU4SjnUUU4Sjn[TU-U4X4S9$)zReturn a matrix arising in regularized least squares as LinearOperator. The matrix is [ J ] [ D ] where D is diagonal matrix with elements from `diag`. cX>[R"TRU5TU-45$r)r hstackr)rr]r^s rr(regularized_lsq_operator..matvecs#yy!((1+tax011rcF>USTnUTSnTRU5TU--$rr)rx1x2r]r^r2s rr)regularized_lsq_operator..rmatvecs0 rU qrUyy}tby((r)rr)r rr )r]r^r1rrr2s`` @rregularized_lsq_operatorrs= A 77DAq2) 1q5!*V EErc(U(a%[U[5(dUR5n[U5(a/U=RUR UR SS9-slU$[U[5(a [X5nU$X-nU$)z`Compute J diag(d). If `copy` is False, `J` is modified in place (unless being LinearOperator). clip)mode)rr rrdatatakeindicesrr]rrs rright_multiplyrs{  Jq.11 FFH{{ !&&&00 H A~ & & %a + H  HrcU(a%[U[5(dUR5n[U5(aJU=R[ R "U[ R"UR55-slU$[U[5(a [X5nU$XSS2[ R4-nU$)z`Compute diag(d) J. If `copy` is False, `J` is modified in place (unless being LinearOperator). N) rr rrrr repeatdiffindptrrrrs r left_multiplyrs  Jq.11 FFH{{ "))Arwwqxx011 H A~ & & $Q * H q"**}  HrczXU-:=(a US:nX&Xc--:nU(aU(agU(agU(agg)z8Check termination condition for nonlinear least squares.rUr r NrV) dFFdx_normx_normr>ftolxtolftol_satisfiedxtol_satisfieds rcheck_terminationrs<(]3ut|Nt}55N.  rc~USSUS-US---n[X3[:'US-nXSU- -n[XSS9U4$)zXScale Jacobian and residuals for a robust loss function. Arrays are modified in place. r,r r+F)r)r-r)r]rNrhoJ_scales rscale_for_robust_loss_functionr s[ !fq3q6zAqD((G GcM OGQ' A % 0! 33r)Ng{Gz? )NN)rr)g|=)T).__doc__mathrnumpyr numpy.linalgr scipy.linalgrrr scipy.sparserscipy.sparse.linalgr r finfofloatepsr-rr?rSr[rbrkrqrtrrrrrrrrrrrrrrrrrr rVrrrs1;;!@ hhuo $NAE/1od0f:0f&.$T) H:$N6)XD. c! Y$ $+"+"F, $ $  4r