9i vSrSSKrSSKrSSKJr SSKJr SSKJ r J r J r J r J r Jr SSKJrJr SS KJr SS KJr SS KJr S r\R4S 5rSrSrSSjrSSSS\R>*\R>4SSSSSS4 Sjr Sr!Sr"Sr#"SS5r$\R>*\R>4SS4Sjr%g)z'Routines for numerical differentiation.N)norm)LinearOperator)issparse isspmatrixfind csc_array csr_array csr_matrix) group_dense group_sparse)array_namespace) MapWrapper)array_api_extracUS:Xa[R"U[S9nOAUS:Xa0[R"U5n[R"U[S9nO [ S5e[R "U[R*:HU[R:H-5(aX4$X-nUR5nX- n XP- n US:XaoX-n X:X:-n [R"U5[R"X5:*n XU -==S-ss'X:U )-nXU- X'X:U )-nX*U- X'X4$US:XaX:X:-nX:U)-n[R"XSX-U- 5X'SXn'X:U)-n[R"XSX-U- 5*X'SXo'[R"X5U- nU)[R"U5U:*-nUUUU'SUU'X4$) a<Adjust final difference scheme to the presence of bounds. Parameters ---------- x0 : ndarray, shape (n,) Point at which we wish to estimate derivative. h : ndarray, shape (n,) Desired absolute finite difference steps. num_steps : int Number of `h` steps in one direction required to implement finite difference scheme. For example, 2 means that we need to evaluate f(x0 + 2 * h) or f(x0 - 2 * h) scheme : {'1-sided', '2-sided'} Whether steps in one or both directions are required. In other words '1-sided' applies to forward and backward schemes, '2-sided' applies to center schemes. lb : ndarray, shape (n,) Lower bounds on independent variables. ub : ndarray, shape (n,) Upper bounds on independent variables. Returns ------- h_adjusted : ndarray, shape (n,) Adjusted absolute step sizes. Step size decreases only if a sign flip or switching to one-sided scheme doesn't allow to take a full step. use_one_sided : ndarray of bool, shape (n,) Whether to switch to one-sided scheme. Informative only for ``scheme='2-sided'``. 1-sideddtype2-sidedz(`scheme` must be '1-sided' or '2-sided'.?TF) np ones_likeboolabs zeros_like ValueErrorallinfcopymaximumminimum)x0h num_stepsschemelbub use_one_sidedh_total h_adjusted lower_dist upper_distxviolatedfittingforwardbackwardcentralmin_distadjusted_centrals W/var/www/html/land-doc-ocr/venv/lib/python3.13/site-packages/scipy/optimize/_numdiff.py_adjust_scheme_to_boundsr8s> Qd3 9  FF1I at4 CDD vvrbffW}rvv.//mGJJJ  LFqv&&&/RZZ %GGg%&",&+x7(1I= +x7 * 44y@ &  $$% 9 (Z-BC+x7 jj Jj11I=? !% +x7 " Kz33i?!A A "& ::j5 A$Hz(:h(FG'/0@'A #$*/ &'  $$cj[R"[R5RnSn[R"U[R 5(aB[R"U5Rn[R "U5RnSn[R"U[R 5(aM[R "U5RnU(a&UW:a [R"U5RnUS;aUS-$US;aUS-$[S5e)aX Calculates relative EPS step to use for a given data type and numdiff step method. Progressively smaller steps are used for larger floating point types. Parameters ---------- f0_dtype: np.dtype dtype of function evaluation x0_dtype: np.dtype dtype of parameter vector method: {'2-point', '3-point', 'cs'} Returns ------- EPS: float relative step size. May be np.float16, np.float32, np.float64 Notes ----- The default relative step will be np.float64. However, if x0 or f0 are smaller floating point types (np.float16, np.float32), then the smallest floating point type is chosen. FT)2-pointcsr)3-pointgUUUUUU?zBUnknown step method, should be one of {'2-point', '3-point', 'cs'}) rfinfofloat64eps issubdtypeinexactritemsize RuntimeError)x0_dtypef0_dtypemethodEPSx0_is_fp x0_itemsize f0_itemsizes r7_eps_for_methodrL]s< ((2::  " "CH }}Xrzz**hhx $$hhx(11  }}Xrzz**hhx(11  k1((8$((C ""Cx ; Sz:; ;r9c US:R[5S-S- n[URURU5nUc2XT-[R "S[R "U55-nU$X-[R "U5-nX-U- n[R"US:HXT-[R "S[R "U55-U5nU$)a* Computes an absolute step from a relative step for finite difference calculation. Parameters ---------- rel_step: None or array-like Relative step for the finite difference calculation x0 : np.ndarray Parameter vector f0 : np.ndarray or scalar method : {'2-point', '3-point', 'cs'} Returns ------- h : float The absolute step size Notes ----- `h` will always be np.float64. However, if `x0` or `f0` are smaller floating point dtypes (e.g. np.float32), then the absolute step size will be calculated from the smallest floating point size. rrr ?)astypefloatrLrrr"rwhere)rel_stepr$f0rGsign_x0rstepabs_stepdxs r7_compute_absolute_steprXs6Qwu%)A-G BHHbhh 7E?RZZRVVBZ%@@ O%r 2}"88B!G!ObjjbffRj.II$& Or9cSU5up#URS:Xa [R"X!R5nURS:Xa [R"X1R5nX#4$)aA Prepares new-style bounds from a two-tuple specifying the lower and upper limits for values in x0. If a value is not bound then the lower/upper bound will be expected to be -np.inf/np.inf. Examples -------- >>> _prepare_bounds([(0, 1, 2), (1, 2, np.inf)], [0.5, 1.5, 2.5]) (array([0., 1., 2.]), array([ 1., 2., inf])) c3T# UHn[R"U[S9v M g7f)rN)rasarrayrP).0bs r7 "_prepare_bounds..s 9&Qbjj%(&s&(r)ndimrresizeshape)boundsr$r(r)s r7_prepare_boundsrdsQ:& 9FB ww!| YYr88 $ ww!| YYr88 $ 6Mr9c[U5(a [U5nO8[R"U5nUS:gR [R 5nUR S:wa [S5eURup#Ub[R"U5(a1[RRU5nURU5nO2[R"U5nURU4:wa [S5eUSS2U4n[U5(a"[X#URUR 5nO [#X#U5nUR%5XQ'U$)aiGroup columns of a 2-D matrix for sparse finite differencing [1]_. Two columns are in the same group if in each row at least one of them has zero. A greedy sequential algorithm is used to construct groups. Parameters ---------- A : array_like or sparse array, shape (m, n) Matrix of which to group columns. order : int, iterable of int with shape (n,) or None Permutation array which defines the order of columns enumeration. If int or None, a random permutation is used with `order` used as a random seed. Default is 0, that is use a random permutation but guarantee repeatability. Returns ------- groups : ndarray of int, shape (n,) Contains values from 0 to n_groups-1, where n_groups is the number of found groups. Each value ``groups[i]`` is an index of a group to which ith column assigned. The procedure was helpful only if n_groups is significantly less than n. References ---------- .. [1] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of sparse Jacobian matrices", Journal of the Institute of Mathematics and its Applications, 13 (1974), pp. 117-120. rrz`A` must be 2-dimensional.Nz`order` has incorrect shape.)rr r atleast_2drOint32r`rrbisscalarrandom RandomState permutationr[rindicesindptrr r!)Aordermnrnggroupss r7 group_columnsrts<{{ aL MM!  !VOOBHH %vv{566 77DA } E**ii##E*" 5! ;;1$ ;< < !U( A{{aAIIqxx8Q1%KKMFM Mr9r=Fc US;a[SUS35eSS0n [U5n[R"UR U5SUS9nUR nUR URS5(a URnURUU5nURS:a [S 5e[Xa5unnURUR:wdURUR:wa [S 5eU(ai[R"[R"U55(a/[R"[R"U55(d [S 5eU c0n [XX5nS =nnUc U"U5nSnO1[R "U5nURS:a [S 5e[R""UU:UU:-5(a [S5eU(a7Uc![%URURU5n['UUXSU5unnGOUc [)X1XR5nOUS :R[*5S-S- nUnUU-U- n[R,"US :H[%URURU5U-[R."S[R0"U55-U5nUS:Xa[3UUSSUU5unnO"US:Xa[3UUSSUU5unnOUS:XaSnU =(d [4n [7U 5nUc[9UXUWUU5unnO[;U5(d[=U5S:XaUunnO Un[?U5n[;U5(aURA5nO[RB"U5n[R "U5n[EUXUWUUUU5 unnSSS5 U (aUU- nUU S'WU 4$W$!,(df  N%=f)uJ'Compute finite difference approximation of the derivatives of a vector-valued function. If a function maps from R^n to R^m, its derivatives form m-by-n matrix called the Jacobian, where an element (i, j) is a partial derivative of f[i] with respect to x[j]. Parameters ---------- fun : callable Function of which to estimate the derivatives. The argument x passed to this function is ndarray of shape (n,) (never a scalar even if n=1). It must return 1-D array_like of shape (m,) or a scalar. x0 : array_like of shape (n,) or float Point at which to estimate the derivatives. Float will be converted to a 1-D array. method : {'3-point', '2-point', 'cs'}, optional Finite difference method to use: - '2-point' - use the first order accuracy forward or backward difference. - '3-point' - use central difference in interior points and the second order accuracy forward or backward difference near the boundary. - 'cs' - use a complex-step finite difference scheme. This assumes that the user function is real-valued and can be analytically continued to the complex plane. Otherwise, produces bogus results. rel_step : None or array_like, optional Relative step size to use. If None (default) the absolute step size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``, with `rel_step` being selected automatically, see Notes. Otherwise ``h = rel_step * sign(x0) * abs(x0)``. For ``method='3-point'`` the sign of `h` is ignored. The calculated step size is possibly adjusted to fit into the bounds. abs_step : array_like, optional Absolute step size to use, possibly adjusted to fit into the bounds. For ``method='3-point'`` the sign of `abs_step` is ignored. By default relative steps are used, only if ``abs_step is not None`` are absolute steps used. f0 : None or array_like, optional If not None it is assumed to be equal to ``fun(x0)``, in this case the ``fun(x0)`` is not called. Default is None. bounds : tuple of array_like, optional Lower and upper bounds on independent variables. Defaults to no bounds. Each bound must match the size of `x0` or be a scalar, in the latter case the bound will be the same for all variables. Use it to limit the range of function evaluation. Bounds checking is not implemented when `as_linear_operator` is True. sparsity : {None, array_like, sparse array, 2-tuple}, optional Defines a sparsity structure of the Jacobian matrix. If the Jacobian matrix is known to have only few non-zero elements in each row, then it's possible to estimate its several columns by a single function evaluation [3]_. To perform such economic computations two ingredients are required: * structure : array_like or sparse array of shape (m, n). A zero element means that a corresponding element of the Jacobian identically equals to zero. * groups : array_like of shape (n,). A column grouping for a given sparsity structure, use `group_columns` to obtain it. A single array or a sparse array is interpreted as a sparsity structure, and groups are computed inside the function. A tuple is interpreted as (structure, groups). If None (default), a standard dense differencing will be used. Note, that sparse differencing makes sense only for large Jacobian matrices where each row contains few non-zero elements. as_linear_operator : bool, optional When True the function returns an `scipy.sparse.linalg.LinearOperator`. Otherwise it returns a dense array or a sparse array depending on `sparsity`. The linear operator provides an efficient way of computing ``J.dot(p)`` for any vector ``p`` of shape (n,), but does not allow direct access to individual elements of the matrix. By default `as_linear_operator` is False. args, kwargs : tuple and dict, optional Additional arguments passed to `fun`. Both empty by default. The calling signature is ``fun(x, *args, **kwargs)``. full_output : bool, optional If True then the function also returns a dictionary with extra information about the calculation. workers : int or map-like callable, optional Supply a map-like callable, such as `multiprocessing.Pool.map` for evaluating the population in parallel. This evaluation is carried out as ``workers(fun, iterable)``. Alternatively, if `workers` is an int the task is subdivided into `workers` sections and the fun evaluated in parallel (uses `multiprocessing.Pool `). Supply -1 to use all available CPU cores. It is recommended that a map-like be used instead of int, as repeated calls to `approx_derivative` will incur large overhead from setting up new processes. Returns ------- J : {ndarray, sparse array, LinearOperator} Finite difference approximation of the Jacobian matrix. If `as_linear_operator` is True returns a LinearOperator with shape (m, n). Otherwise it returns a dense array or sparse array depending on how `sparsity` is defined. If `sparsity` is None then a ndarray with shape (m, n) is returned. If `sparsity` is not None returns a csr_array or csr_matrix with shape (m, n) following the array/matrix type of the incoming structure. For sparse arrays and linear operators it is always returned as a 2-D structure. For ndarrays, if m=1 it is returned as a 1-D gradient array with shape (n,). info_dict : dict Dictionary containing extra information about the calculation. The keys include: - `nfev`, number of function evaluations. If `as_linear_operator` is True then `fun` is expected to track the number of evaluations itself. This is because multiple calls may be made to the linear operator which are not trackable here. See Also -------- check_derivative : Check correctness of a function computing derivatives. Notes ----- If `rel_step` is not provided, it assigned as ``EPS**(1/s)``, where EPS is determined from the smallest floating point dtype of `x0` or `fun(x0)`, ``np.finfo(x0.dtype).eps``, s=2 for '2-point' method and s=3 for '3-point' method. Such relative step approximately minimizes a sum of truncation and round-off errors, see [1]_. Relative steps are used by default. However, absolute steps are used when ``abs_step is not None``. If any of the absolute or relative steps produces an indistinguishable difference from the original `x0`, ``(x0 + dx) - x0 == 0``, then a automatic step size is substituted for that particular entry. A finite difference scheme for '3-point' method is selected automatically. The well-known central difference scheme is used for points sufficiently far from the boundary, and 3-point forward or backward scheme is used for points near the boundary. Both schemes have the second-order accuracy in terms of Taylor expansion. Refer to [2]_ for the formulas of 3-point forward and backward difference schemes. For dense differencing when m=1 Jacobian is returned with a shape (n,), on the other hand when n=1 Jacobian is returned with a shape (m, 1). Our motivation is the following: a) It handles a case of gradient computation (m=1) in a conventional way. b) It clearly separates these two different cases. b) In all cases np.atleast_2d can be called to get 2-D Jacobian with correct dimensions. References ---------- .. [1] W. H. Press et. al. "Numerical Recipes. The Art of Scientific Computing. 3rd edition", sec. 5.7. .. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of sparse Jacobian matrices", Journal of the Institute of Mathematics and its Applications, 13 (1974), pp. 117-120. .. [3] B. Fornberg, "Generation of Finite Difference Formulas on Arbitrarily Spaced Grids", Mathematics of Computation 51, 1988. Examples -------- >>> import numpy as np >>> from scipy.optimize._numdiff import approx_derivative >>> >>> def f(x, c1, c2): ... return np.array([x[0] * np.sin(c1 * x[1]), ... x[0] * np.cos(c2 * x[1])]) ... >>> x0 = np.array([1.0, 0.5 * np.pi]) >>> approx_derivative(f, x0, args=(1, 2)) array([[ 1., 0.], [-1., 0.]]) Bounds can be used to limit the region of function evaluation. In the example below we compute left and right derivative at point 1.0. >>> def g(x): ... return x**2 if x >= 1 else x ... >>> x0 = 1.0 >>> approx_derivative(g, x0, bounds=(-np.inf, 1.0)) array([ 1.]) >>> approx_derivative(g, x0, bounds=(1.0, np.inf)) array([ 2.]) We can also parallelize the derivative calculation using the workers keyword. >>> from multiprocessing import Pool >>> import time >>> def fun2(x): # import from an external file for use with multiprocessing ... time.sleep(0.002) ... return rosen(x) >>> rng = np.random.default_rng() >>> x0 = rng.uniform(high=10, size=(2000,)) >>> f0 = rosen(x0) >>> %timeit approx_derivative(fun2, x0, f0=f0) # may vary 10.5 s ± 5.91 ms per loop (mean ± std. dev. of 7 runs, 1 loop each) >>> elapsed = [] >>> with Pool() as workers: ... for i in range(10): ... t = time.perf_counter() ... approx_derivative(fun2, x0, workers=workers.map, f0=f0) ... et = time.perf_counter() ... elapsed.append(et - t) >>> np.mean(elapsed) # may vary np.float64(1.442545195999901) Create a map-like vectorized version. `x` is a generator, so first of all a 2-D array, `xx`, is reconstituted. Here `xx` has shape `(Y, N)` where `Y` is the number of function evaluations to perform and `N` is the dimensionality of the objective function. The underlying objective function is `rosen`, which requires `xx` to have shape `(N, Y)`, so a transpose is required. >>> def fun(f, x, *args, **kwds): ... xx = np.r_[[xs for xs in x]] ... return f(xx.T) >>> %timeit approx_derivative(fun2, x0, workers=fun, f0=f0) # may vary 91.8 ms ± 755 μs per loop (mean ± std. dev. of 7 runs, 10 loops each) )r;r=r<zUnknown method 'z'. nfevNr )r`xp real floatingz#`x0` must have at most 1 dimension.z,Inconsistent shapes between bounds and `x0`.z7Bounds not supported when `as_linear_operator` is True.rz&`f0` passed has more than 1 dimension.z `x0` violates bound constraints.rrNr;rr=rr<F)#rrxpx atleast_ndr[r?isdtyperrOr`rdrbrrisinf _Fun_Wrapper atleast_1danyrL_linear_operator_differencerXrPrQr"rr8mapr_dense_differencerlenrttocscrf_sparse_difference)funr$rGrRrVrSrcsparsityas_linear_operatorargskwargs full_outputworkers info_dictrx_x_dtyper(r) fun_wrappedrw_nfevJ_r%rTrWr*mf structurerss r7approx_derivativersF11+F83788I  B  2Q2 6B ZZF zz"((O,, 2v B ww{>?? V (FB xx288rxx2883GHH266"((2,#7#7')vvbhhrl';';9: :~s5KD5 z _ ]]2  77Q;EF F vvrBw27#$$;<<  &rxx6BH*;+-A1  &xR@AQw&&u-1A5GA6R-Bq(288VD !#%::c266":#>?A Y 7Aq)R - A} y 7Aq)R - A} t^!M.S  B,["!)6)+-5 ))c(mq.@(0%Iv (I*84FI&& ) 1I " i 8Iv.-k21-:I-3VRA5#!*   &)|5! s %B,O00 O>c^^^^^TRmTRnUS:Xa UUUUU4SjnO+US:Xa UUUU4SjnOUS:Xa UUUU4SjnO [S5e[TU4U5S4$) Nr;c>[R"U[R"U55(a[R"T5$T[ U5- nTX--nT"U5T- nX1- $Nr array_equalrzerosr) prWr/dfrSrr%rpr$s r7matvec+_linear_operator_difference..matvecrsW~~aq!122xx{"T!WBRT AQ"B7Nr9r=c>[R"U[R"U55(a[R"T 5$ST-[ U5- nT US- U-- nT US- U--nT"U5nT"U5nXT- nXa- $Nrr) rrWx1x2f1f2rrr%rpr$s r7rr|s~~~aq!122xx{"1tAwBr!tQhBr!tQhBRBRBB7Nr9r<c>[R"U[R"U55(a[R"T5$T[ U5- nTX-S--nT"U5nUR nXA- $N?)rrrrrimag) rrWr/rrrr%rpr$s r7rrs`~~aq!122xx{"T!WBRT#X AQBB7Nr9Never be here.r)sizerDr)rr$rSr%rGrqrrps```` @r7rrlsr A A    9    4  +,, 1a&& )1 ,,r9c R^URnURm[R"TU45nSn US:XaU4Sjn U"X "X55n [T5V s/sHoU X<-X- PM n n U Vs/sHoU- PM nn[ X5VVs/sH unnUU- PM nnnU [ U5- n GOUS:XGaSn[ U"UU"XU555n U"XU5n[5n [5n[U5Hun n[U5n[U5n[U 5n[U 5nU(a8U RUU X- 5 URSU-SU--U- 5 MqU RUU UU - 5 URUU- 5 M [ X5VVs/sH unnUU- PM nnnU S[ U5-- n OkUS :XaZU4S jn[ U"UU"X555n [ X5VVs/sHunnURU- PM nnnU [ U5- n O [S 5e[U5H un nUX'M US :Xa[R"U5nURU 4$s sn fs snfs snnfs snnfs snnf) Nrr;c3|># [T5H(n[R"U5nXX-X2'Uv M* g7fr)rangerr!)r$r%irrqs r7 x_generator2'_dense_difference..x_generator2s71XWWR[ s9<r=c3 # [U5Hpup4[R"U5n[R"U5nU(aXX-XS'XSX--Xc'OXX- XS'XX-Xc'Uv Uv Mr g7fr) enumeraterr!)r$r%r*r one_sidedrrs r7 x_generator3'_dense_difference..x_generator3s} )- 8 WWR[WWR[EADLBEEAadFNBEEADLBEEADLBE!9sBBgrr<c3># [T5H,nUR[SS9nX2==XS-- ss'Uv M. g7f)NTr!r)rrOcomplex)r$r%rxcrqs r7x_generator_cs)_dense_difference..x_generator_css=1XYYwTY2#s=Arr )rremptyrzipriterlistrnextappendrrDravelT)rr$rSr%r*rGrrp J_transposedrwrf_evalsrrWf_evalrdelfdelxdf_dxrgenrlurrrhivrqs @r7rrs{ A A88QF#L D  #|B23.3Ah 7h!uqt|ru$h 7(/ 0frk 0/22{;{t{; E  9  wsL $FGH2-0 V V%m4LAyS AS AgBgB !A$,' $)a"f,r12 !A$1+& "r'"5032{;{t{; CJ 4  wsN2$9:;,/O>4 u8 0;F<=sJ 9JJ8JJ#c ^^^^^#^$URn TRn /n /n /n [R"T5S-m$SnUU$4Sjm#U#UU4SjnU#UUU4SjnU#UU4SjnUS:Xa[U"X"555nU"5nOBUS:Xa[U"UU"555nU"5nOUS :Xa[U"UU"555nT#"5GHn[R"U5un[ USS2U45unnnUUnUS:Xa"[ W5T- n[ W5U- nUS- nOUS:Xa[ W5n[ U5nTU-nT)U-n[R"U 5nUUTU- UU'UUUU- UU'[ W5n[ U5n US - nTUn![R"U 5nUU!n"S UU"-S UU"--U U"- UU"'UU!)n"U U"UU"- UU"'O3US :Xa"[ W5nUS- nURnTU-nO [S 5eU RU5 U RU5 U RUUUU- 5 GM [R"U 5n [R"U 5n [R"U 5n [U5(a[XU 44X4S9U4$[XU 44X4S9U4$)Nr rc3d># [T5Hn[R"UT5v M g7fr)rrequal)grouprsn_groupss r7 e_generator'_sparse_difference..e_generators%8_E((5&) )%s-0c3J># T"5nUHnTU-nTU-nUv M g7frru)e_geneh_vecr/rr%r$s r7r(_sparse_difference..x_generator2s. AEEU AGs #c3># T"5nUHvnTU-nT R5nT R5nT U-nX5==X%- ss'XE==SX%-- ss'T )U-nX6==X&-ss'XF==X&- ss'Uv Uv Mx g7frr) rrrrrmask_1mask_2rr%r*r$s r7r(_sparse_difference..x_generator3s AEEBB"Q&F J%- 'J J!em+ +J#^a'F J%- 'J J%- 'JHHsBBc3L># T"5nUHnTU-nTUS--v M g7frru)rrrrr%r$s r7r*_sparse_difference..x_generator_css/ AEEus{" "s!$r;r=r<rrr)rb)rrmaxrnonzerorrrrrrrhstackrr r )%rr$rSr%r*rrsrGrrprq row_indices col_indices fractionsrwrrrrxsrcolsrjrrWrrrrrrrmaskrowsrrs% ` `` ` @@r7rrs A AKKIvvf~!H D* "#wsLN34 ^ 9 wsLN34 ^ 4wsN$456 ] 1 yD)*1a G Y bBBg#B AID y bBbB"Q&F#^a'F!BFbj0BvJFbj0BvJgBgB AID #D!BT7DBtH}q2d8|3bh>BtHdU8D$x"T(*BtH t^gB AIDBQB-. . 11AA'eh))K(K))K(K )$I)9K&@A!PRVVV i{!;>> import numpy as np >>> from scipy.optimize._numdiff import check_derivative >>> >>> >>> def f(x, c1, c2): ... return np.array([x[0] * np.sin(c1 * x[1]), ... x[0] * np.cos(c2 * x[1])]) ... >>> def jac(x, c1, c2): ... return np.array([ ... [np.sin(c1 * x[1]), c1 * x[0] * np.cos(c1 * x[1])], ... [np.cos(c2 * x[1]), -c2 * x[0] * np.sin(c2 * x[1])] ... ]) ... >>> >>> x0 = np.array([1.0, 0.5 * np.pi]) >>> check_derivative(f, jac, x0, args=(1, 2)) 2.4492935982947064e-16 )rcrrrr )rcrr) rrr rrr[rrrr") rjacr$rcrr J_to_testJ_diffabs_errrr abs_err_data J_diff_datas r7check_derivativervsz~B(((I "36(,=i( $!']ljj1.446 vvbff\*jjBFF;$789: :#36(,=&&+,vvg 1bffVn ==>>r9)r)&__doc__ functoolsnumpyr numpy.linalgrscipy.sparse.linalgrsparserrrr r r _group_columnsr rscipy._lib._array_apirscipy._lib._utilr scipy._librrzr8 lru_cacherLrXrdrtr rrrrr~rrur9r7rs-.QQ51'-L%^ 2;2;j.b*:z'0$w&7$).R"'Sl (-VP frRj.-/FF7BFF*;" M?r9