\hL'SSKJrJr SSKJrJr SSKJrJr SSK J r SSK J r SSK Jr SrSr\"S S 9S 5rS rS rg))Ssympify)Dummysymbols) Piecewisepiecewise_fold)And)Interval) lru_cachec[U[5(aQ[UR5S:Xa8URup#URU:XaX2p2URUR 4$[ SU-5e)zreturn the interval corresponding to the condition Conditions in spline's Piecewise give the range over which an expression is valid like (lo <= x) & (x <= hi). This function returns (lo, hi). zunexpected cond type: %s) isinstancer lenargsltsgts TypeError)condxabs X/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/special/bsplines.py_ivlr s^$TYY1!4yy 55A:quuaee| .5 66c"[RX4;a[X#-5nUR5$[RX24;a[X-5nUR5$/n[X-5n[X#-5n[URSS5n URSSHn U R n U R n [X5Sn [U 5H^upUR nUR n[UU5unnUU :Xa U U- n X O'UU :dMCUU ::dMKURU5 X O URX45 M URU 5 URS5 [USS06nUR5$)zConstruct c*b1 + d*b2.NrrTevaluateF) rZerorlistrexprrr enumerateappendextendrexpand)cb1db2rrvnew_argsp1p2p2argsargr!rloweriarg2expr2cond2lower_2upper_2s r _add_splinesr7s} vv" AF #h 99;g B7  AF #d 99;a AF # AF #bggcrl#773B D=EMD u_E)9OOD) %-* OOTL )9 >   "  15 1 99;r)maxsizec Un[5n[SU55n[U5n[U5n[U5nUS- nX -S-U:a [ S5eUS:Xa=[ [ R[XXS-5RU54S5nOUS:aXU-S-XS-- nU[ R:wa$XU-S-U- U- n [US- XS-U5n O[ R=pXU-X- nU[ R:waX1U- U- n [US- XU5n O[ R=p[XXU5nO[ SU-5eURX405$)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# UHn[U5v M g7fN)r).0ks r bspline_basis..s,e'!**esz(n + d + 1 must not exceed len(knots) - 1rrzdegree must be non-negative: %r)rtupleintr ValueErrorrrOner containsr bspline_basisr7xreplace) r(knotsnrxvarn_knots n_intervalsresultdenomBr)Ar's rrGrGTs~f D A ,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 [U5U- S- n[U5Vs/sHn[U[U5XB5PM sn$s snf)a Return the ``len(knots)-d-1`` B-splines at *x* of degree *d* with *knots*. Explanation =========== This function returns a list of piecewise polynomials that are the ``len(knots)-d-1`` B-splines of degree *d* for the given knots. This function calls ``bspline_basis(d, knots, n, x)`` for different values of *n*. Examples ======== >>> from sympy import bspline_basis_set >>> from sympy.abc import x >>> d = 2 >>> knots = range(5) >>> splines = bspline_basis_set(d, knots, x) >>> splines [Piecewise((x**2/2, (x >= 0) & (x <= 1)), (-x**2 + 3*x - 3/2, (x >= 1) & (x <= 2)), (x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)), (0, True)), Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x <= 2)), (-x**2 + 5*x - 11/2, (x >= 2) & (x <= 3)), (x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)), (0, True))] Parameters ========== d : integer degree of bspline knots : list of integers list of knots points of bspline x : symbol See Also ======== bspline_basis rA)rrangerGrB)r(rIr n_splinesr1s rbspline_basis_setrUsC`E Q"I:? :J K:JQM!U5\1 0:J KK Ks Ac ^SSKJn SSKJn [ U5nUR (aUR (d[SU-5e[U5[U5:wa [S5e[U5US-:a [S5e[S[X"SS 555(d [S 5eUVs/sHn[ U5PM nnUR(aUS-S -nX'U*nO6US -n[X'U*S- X'S-U*5V V s/sH upX-S - PM nn n US/US--[U5-US /US---n [X T5n UV V s/sH$oV s/sHoRTU 5PM sn PM& nn n U"U"U5U"U54[S R![U55["S95n[U5SnU V VVs1sH#oR$HunnUS:wdMUiM M% nnn n['UU4SjS9nU V VVs/sH%oR$VVs0sH unnUU_M snnPM' nnn n/nUHqn[)[UU5VVs/sH)unnUUR+U[,R.5-PM+ snn[,R.5nUR1UU45 Ms [3U6$s snfs sn n fs sn fs sn n fs snnn fs snnfs snnn fs snnf)a Return spline of degree *d*, passing through the given *X* and *Y* values. Explanation =========== This function returns a piecewise function such that each part is a polynomial of degree not greater than *d*. The value of *d* must be 1 or greater and the values of *X* must be strictly increasing. Examples ======== >>> from sympy import interpolating_spline >>> from sympy.abc import x >>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7]) Piecewise((3*x, (x >= 1) & (x <= 2)), (7 - x/2, (x >= 2) & (x <= 4)), (2*x/3 + 7/3, (x >= 4) & (x <= 7))) >>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3]) Piecewise((7*x**3/117 + 7*x**2/117 - 131*x/117 + 2, (x >= -2) & (x <= 1)), (10*x**3/117 - 2*x**2/117 - 122*x/117 + 77/39, (x >= 1) & (x <= 4))) Parameters ========== d : integer Degree of Bspline strictly greater than equal to one x : symbol X : list of strictly increasing real values list of X coordinates through which the spline passes Y : list of real values list of corresponding Y coordinates through which the spline passes See Also ======== bspline_basis_set, interpolating_poly r)linsolve)Matrixz1Spline degree must be a positive integer, not %s.z/Number of X and Y coordinates must be the same.rAz6Degree must be less than the number of control points.c3.# UH upX:v M g7fr<)r=rrs rr?'interpolating_spline..9s/qusNz.The x-coordinates must be strictly increasing.r rzc0:{})clsTc>[UT5$r<)r)r&rs r&interpolating_spline..Ss Q r)key)sympy.solvers.solvesetrWsympy.matrices.denserXr is_Integer is_positiverDrallzipis_oddr rUsubsrformatrrsortedsumgetrrr#r)r(rXYrWrXr1jinterior_knotsrrrIbasisvrQcoeffer& intervals basis_dictssplinepieces ` rinterpolating_splinerys\0+  A LLQ]]LqPQQ 1vQJKK 1vA~QRR /Q!"/ / /IJJQQA xx UqLaR F"%aQBFmQ1ur]"C "C$!QUAI"C  qTFa!e tN3 3qugQ6G GE a *E0121 &1&&A, &A2 fQi+WW^^CF5KQV-W XE KNE!DEqfq!!t)EIDy&:;I8=>1vv.vVaAqDv.K> F 03E;0G H0Gfq!Qq!&&! !0G H!&&   uaj!  f C   '2E /> IsN;K K K&K KK#: K#K00K*K0'0K7 K*K0N) sympy.corerrsympy.core.symbolrrsympy.functionsrrsympy.logic.boolalgr sympy.sets.setsr functoolsr rr7rGrUryrZrrrsK!.5#$ 78v 3t&t&n1Lh\r