+ iL'^RIHtHt^RIHtHt^RIHtHt^RI H t ^RI H t ^RI HtRtRt]!^R 7R 4tR tR tR #))Ssympify)Dummysymbols) Piecewisepiecewise_fold)And)Interval) lru_cachec\V\4'dT\VP4^8Xd:VPwr#VPV8XdY2r2VPVP 3#\ RV,4h)zreturn the interval corresponding to the condition Conditions in spline's Piecewise give the range over which an expression is valid like (lo <= x) & (x <= hi). This function returns (lo, hi). zunexpected cond type: %s) isinstancer lenargsltsgts TypeError)condxabs&& ڀ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/functions/special/bsplines.py_ivlr s^$TYY1!4yy 55A:quuaee| .5 66c\\PW39d"\W#,4pVP4#\PW239d"\W,4pVP4#.p\W,4p\W#,4p\VPRR4p VPRRFp V P p V P p \W4^,p \V 4FewrVP pVP p\VV4wppVV 8XdV V, p WM)VV 8gKIVV 8:gKRVPV4WM VPW34K VPV 4VPR4\VRR/pVP4#)zConstruct c*b1 + d*b2.NevaluateFrT) rZerorlistrexprrr enumerateappendextendrexpand)cb1db2rrvnew_argsp1p2p2argsargr rloweriarg2expr2cond2lower_2upper_2s&&&&& r _add_splinesr6s} vv" AF #h 99;g B7  AF #d 99;a AF # AF #bggcrl#773B D=EMD u_E)9OOD) %-* OOTL )9 >   "  15 1 99;r)maxsizec Tp\4p\;QJd.RV4FNK 5M !RV44p\V4p\V4p\V4pV^, pW ,^,V8d \ R4hV^8XdK\ \ P\W,W^,,4PV43R4pEM%V^8Ed WV,^,,W^,,, pV\ P8wdAWV,^,,V, V, p \V^, W^,V4p M\ P;rWV,,W,, pV\ P8wd,W1V,, V, p \V^, WV4p M\ P;r\WWV4pM\ RV,4hVPW4/4#)aL The $n$-th B-spline at $x$ of degree $d$ with knots. Explanation =========== B-Splines are piecewise polynomials of degree $d$. They are defined on a set of knots, which is a sequence of integers or floats. Examples ======== The 0th degree splines have a value of 1 on a single interval: >>> from sympy import bspline_basis >>> from sympy.abc import x >>> d = 0 >>> knots = tuple(range(5)) >>> bspline_basis(d, knots, 0, x) Piecewise((1, (x >= 0) & (x <= 1)), (0, True)) For a given ``(d, knots)`` there are ``len(knots)-d-1`` B-splines defined, that are indexed by ``n`` (starting at 0). Here is an example of a cubic B-spline: >>> bspline_basis(3, tuple(range(5)), 0, x) Piecewise((x**3/6, (x >= 0) & (x <= 1)), (-x**3/2 + 2*x**2 - 2*x + 2/3, (x >= 1) & (x <= 2)), (x**3/2 - 4*x**2 + 10*x - 22/3, (x >= 2) & (x <= 3)), (-x**3/6 + 2*x**2 - 8*x + 32/3, (x >= 3) & (x <= 4)), (0, True)) By repeating knot points, you can introduce discontinuities in the B-splines and their derivatives: >>> d = 1 >>> knots = (0, 0, 2, 3, 4) >>> bspline_basis(d, knots, 0, x) Piecewise((1 - x/2, (x >= 0) & (x <= 2)), (0, True)) It is quite time consuming to construct and evaluate B-splines. If you need to evaluate a B-spline many times, it is best to lambdify them first: >>> from sympy import lambdify >>> d = 3 >>> knots = tuple(range(10)) >>> b0 = bspline_basis(d, knots, 0, x) >>> f = lambdify(x, b0) >>> y = f(0.5) Parameters ========== d : integer degree of bspline knots : list of integer values list of knots points of bspline n : integer $n$-th B-spline x : symbol See Also ======== bspline_basis_set References ========== .. [1] https://en.wikipedia.org/wiki/B-spline c38"TFp\V4xK R#5iN)r).0ks& r bspline_basis..s,e'!**esz(n + d + 1 must not exceed len(knots) - 1zdegree must be non-negative: %rr)rtupleintr ValueErrorrrOner containsr bspline_basisr6xreplace) r'knotsnrxvarn_knots n_intervalsresultdenomBr(Ar&s &&&& rrDrDTsf D A E,e,EE,e, ,E AA AA%jGA+Kuqy;CDDAv UUHUXuU|4==a@ A9  Q!eai 5Q</ AFF?1uqy!A%.Aq1ueUA6BVVOB!e ux' AFF?1X&Aq1ue2BVVOBaQA.:Q>?? ??A9 %%rc \V4V, ^, p\V4Uu.uFp\V\V4WB4NK up#uupi)a Return the ``len(knots)-d-1`` B-splines at *x* of degree *d* with *knots*. Explanation =========== This function returns a list of piecewise polynomials that are the ``len(knots)-d-1`` B-splines of degree *d* for the given knots. This function calls ``bspline_basis(d, knots, n, x)`` for different values of *n*. Examples ======== >>> from sympy import bspline_basis_set >>> from sympy.abc import x >>> d = 2 >>> knots = range(5) >>> splines = bspline_basis_set(d, knots, x) >>> splines [Piecewise((x**2/2, (x >= 0) & (x <= 1)), (-x**2 + 3*x - 3/2, (x >= 1) & (x <= 2)), (x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)), (0, True)), Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x <= 2)), (-x**2 + 5*x - 11/2, (x >= 2) & (x <= 3)), (x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)), (0, True))] Parameters ========== d : integer degree of bspline knots : list of integers list of knots points of bspline x : symbol See Also ======== bspline_basis )rrangerDr?)r'rFr n_splinesr0s&&& rbspline_basis_setrRsC`E Q"I:? :J K:JQM!U5\1 0:J KK KsA c a^RIHp^RIHp\ V4pVP 'dVP 'g\RV,4h\V4\V48wd \R4h\V4V^,8d \R4h\;QJd,R\W"R,44F 'dK RM! R M!R\W"R,444'g \R 4hVUu.uFp\ V4NK ppVP'dV^,^,pW'V)pMIV^,p\W'V)^, W'^,V)4U U u.uFwrW,^, NK pp p V^,.V^,,\V4,VR,.V^,,,p \W S4p VU U u.uF#qU u.uFqPSV 4NK up NK% pp p V!V!V4V!V43\R P!\V44\"R 74p\V4^,pV U UUu0uF#qP$FwppVR 8wgKVkK K% ppp p\'VV3R lR7pV U UUu.uF$qP$UUu/uF wppVVbK uppNK& ppp p.pVFtp\)\VV4UUu.uF-wppVVP+V\,P.4,NK/ upp\,P.4pVP1VV34Kv \3V!#uupiuup p iuup iuup p iuuppp iuuppiuuppp iuuppi)a Return spline of degree *d*, passing through the given *X* and *Y* values. Explanation =========== This function returns a piecewise function such that each part is a polynomial of degree not greater than *d*. The value of *d* must be 1 or greater and the values of *X* must be strictly increasing. Examples ======== >>> from sympy import interpolating_spline >>> from sympy.abc import x >>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7]) Piecewise((3*x, (x >= 1) & (x <= 2)), (7 - x/2, (x >= 2) & (x <= 4)), (2*x/3 + 7/3, (x >= 4) & (x <= 7))) >>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3]) Piecewise((7*x**3/117 + 7*x**2/117 - 131*x/117 + 2, (x >= -2) & (x <= 1)), (10*x**3/117 - 2*x**2/117 - 122*x/117 + 77/39, (x >= 1) & (x <= 4))) Parameters ========== d : integer Degree of Bspline strictly greater than equal to one x : symbol X : list of strictly increasing real values list of X coordinates through which the spline passes Y : list of real values list of corresponding Y coordinates through which the spline passes See Also ======== bspline_basis_set, interpolating_poly )linsolve)Matrixz1Spline degree must be a positive integer, not %s.z/Number of X and Y coordinates must be the same.z6Degree must be less than the number of control points.c3."TF wrW8xK R#5ir:)r;rrs& rr='interpolating_spline..9s/qus:NNFTz.The x-coordinates must be strictly increasing.zc0:{})clsc<\VS4#r:)r)r%rs&r&interpolating_spline..Ss Q r)keyr)sympy.solvers.solvesetrTsympy.matrices.denserUr is_Integer is_positiverArallzipis_oddrrRsubsrformatrrsortedsumgetrrr"r)r'rXYrTrUr0jinterior_knotsrrrFbasisvrNcoeffer% intervals basis_dictssplinepieces&f&& rinterpolating_splinerws\0+  A LLLQ]]]LqPQQ 1vQJKK 1vA~QRR 3/Q"/333/Q"/ / /IJJQQA xxx UqLaR F"%aQBFmQ1ur]"C "C$!QUAII"C  qTFa!e tN3 3qugQ6G GE a *E0121 &1&&A, &A2 fQi+WW^^CF5KQV-W XE KNE!DEqfq!!t)EIDy&:;I8=>1vv.vVaAqDv.K> F 03E;0G H0Gfq!Qq!&&! ! !0G H!&&   uaj!  f C   '2E /> IsN>M(MM$#M =M$M*< M*!M74M1M7*3M> M$1M7N) sympy.corerrsympy.core.symbolrrsympy.functionsrrsympy.logic.boolalgr sympy.sets.setsr functoolsr rr6rDrRrwrWrrr~sK!.5#$ 78v 3t&t&n1Lh\r