doi SrSSKJr SSKJr SrSrSrSrS r S r S r S r S r Sr\\\ \ \ \ \ \S.rSrSrSrSrSrg)aj 'Generic' Array API backend for RBF interpolation. The general logic is this: `_rbfinterp.py` implements the user API and calls into either `_rbfinterp_np` (the "numpy backend"), or `_rbfinterp_xp` (the "generic backend". The numpy backend offloads performance-critical computations to the pythran-compiled `_rbfinterp_pythran` extension. This way, the call chain is _rbfinterp.py <-- _rbfinterp_np.py <-- _rbfinterp_pythran.py The "generic" backend here is a drop-in replacement of the API of `_rbfinterp_np.py` for use in `_rbfinterp.py` with non-numpy arrays. The implementation closely follows `_rbfinterp_np + _rbfinterp_pythran`, with the following differences: - We used vectorized code not explicit loops in `_build_system` and `_build_evaluation_coefficients`; this is more torch/jax friendly; - RBF kernels are also "vectorized" and not scalar: they receive an array of norms not a single norm; - RBF kernels accept an extra xp= argument; In general, we would prefer less code duplication. The main blocker ATM is that pythran cannot compile functions with an xp= argument where xp is numpy. ) LinAlgError)_monomial_powers_implc[X5nURU5nURSS:XaURUSU45nU$)Nr)rasarrayshapereshape)ndimdegreexpouts [/var/www/html/land-ocr/venv/lib/python3.13/site-packages/scipy/interpolate/_rbfinterp_xp.py_monomial_powersr sB  -C **S/C yy|qjjq$i( Jc 4[XX#XEU5upxpURRXx5n XU 4$![a[ Sn URSn U S:a8[ X - U - XVS9nURR U5nX:a SUSU S3n [U 5ef=f)aBuild and solve the RBF interpolation system of equations. Parameters ---------- y : (P, N) float ndarray Data point coordinates. d : (P, S) float ndarray Data values at `y`. smoothing : (P,) float ndarray Smoothing parameter for each data point. kernel : str Name of the RBF. epsilon : float Shape parameter. powers : (R, N) int ndarray The exponents for each monomial in the polynomial. Returns ------- coeffs : (P + R, S) float ndarray Coefficients for each RBF and monomial. shift : (N,) float ndarray Domain shift used to create the polynomial matrix. scale : (N,) float ndarray Domain scaling used to create the polynomial matrix. zSingular matrixr)r zqSingular matrix. The matrix of monomials evaluated at the data point coordinates does not have full column rank (/z).) _build_systemlinalgsolve Exceptionrpolynomial_matrix matrix_rankr)yd smoothingkernelepsilonpowersr lhsrhsshiftscalecoeffsmsgnmonospmatranks r_build_and_solve_systemr((s8+ i" Ce*&  %   a A:$ai%6FD99((.D}!F!F82/ #!s 2A%BcU*$Nrr s rlinearr.^s 2IrcXURUS:HSUS-URU5-5$)Nr)wherelogr,s rthin_plate_spliner3bs* 88AFAq!tbffQi/ 00rc US-$)Nr+r,s rcubicr6gs a4KrcUS-*$)Nr+r,s rquinticr9ks qD5Lrc2URUS-S-5*$)Nr0rsqrtr,s r multiquadricr=os GGAqD1H  rc6SURUS-S-5- $N?r0r;r,s rinverse_multiquadricrAss A$ $$rcSUS-S-- $r?r+r,s rinverse_quadraticrCws !Q$* rc,URUS-*5$)Nr0)expr,s rgaussianrF{s 661a4%=r)r.r3r6r9r=rArCrFc rU"URRUSSS2SS24USS2SSS24- SS9U5$)z+Evaluate RBFs, with centers at `x`, at `x`.Naxis)r vector_norm)x kernel_funcr s r kernel_matrixrNsA  aa ma4 m;"Er rc<URUSS2SSS24U-SS9$)z9Evaluate monomials, with exponents from `powers`, at `x`.NrHrI)prod)rLrr s rrrs$ 771QaZ=F*7 44rc dURSnURSn[Un URUSS9n URUSS9n X-S- n X- S- n UR U S:HSU 5n X-nX - U - n[ XU5n[ XU5nURURUU4SS9URURURX454SS9/SS9URURX&RU5/55-nURXRX45/SS9nUUX4$)aBuild the system used to solve for the RBF interpolant coefficients. Parameters ---------- y : (P, N) float ndarray Data point coordinates. d : (P, S) float ndarray Data values at `y`. smoothing : (P,) float ndarray Smoothing parameter for each data point. kernel : str Name of the RBF. epsilon : float Shape parameter. powers : (R, N) int ndarray The exponents for each monomial in the polynomial. Returns ------- lhs : (P + R, P + R) float ndarray Left-hand side matrix. rhs : (P + R, S) float ndarray Right-hand side matrix. shift : (N,) float ndarray Domain shift used to create the polynomial matrix. scale : (N,) float ndarray Domain scaling used to create the polynomial matrix. rrrIr0gr@) r NAME_TO_FUNCminmaxr1rNrconcatTzerosdiag)rrrrrrr sr-rMminsmaxsr!r"yepsyhat out_kernelsout_polyrr s rrrsS<  A QAv&K 66!!6 D 66!!6 D [!OE [!OE HHUc\3 .E 9D Iu D B7K r2H )) K* 3 HJJ! 01 :     9hhqk":;<  =C ))Q!()) 2C U !!rc [UnX-n X-n X- U- n URU"URRU SS2SSS24U SSS2SS24- SS9U5UR U SS2SSS24U-SS9/SS9n U $)a-Construct the coefficients needed to evaluate the RBF. Parameters ---------- x : (Q, N) float ndarray Evaluation point coordinates. y : (P, N) float ndarray Data point coordinates. kernel : str Name of the RBF. epsilon : float Shape parameter. powers : (R, N) int ndarray The exponents for each monomial in the polynomial. shift : (N,) float ndarray Shifts the polynomial domain for numerical stability. scale : (N,) float ndarray Scales the polynomial domain for numerical stability. Returns ------- (Q, P + R) float ndarray NrHrI)rRrUrrKrP) rLrrrrr!r"r rMr\xepsxhatvecs r_build_evaluation_coefficientsrds8v&K 9D 9D Iu D ))  %%D!$tD!QJ'77b&  GGDD!$.RG 8    C Jrc &[XX#XEXh5n X-$r*)rd) rLrrrrr!r"r#r rcs rcompute_interpolationrfs ( fve C <rN)__doc__ numpy.linalgr_rbfinterp_commonrrr(r.r3r6r9r=rArCrFrRrNrrrdrfr+rrrjs6%43 l1 % ) /)  5 ;"|.br