Ed'ddlmZmZddlmZmZddlmZmZddl m Z ddl m Z ddl mZdZdZed d Zd Zd ZdS))Ssympify)Dummysymbols) Piecewisepiecewise_fold)And)Interval) lru_cachect|tr?t|jdkr'|j\}}|j|kr||}}|j|jfSt d|z)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 @lib/python3.11/site-packages/sympy/functions/special/bsplines.py_ivlr sn$TY1!4y1 5A: aqAuae| .5 6 66ctj||fvrt||z}nUtj||fvrt||z}n1g}t||z}t||z}t|jdd} |jddD]} | j} | j} t| |d} t| D]\\}}|j}|j}t||\}}|| kr | |z } | |=n'|| kr || kr| || |=n]| | | f| | | dt|ddi}| S)zConstruct c*b1 + d*b2.NrrTevaluateF) rZerorlistrexprrr enumerateappendextendrexpand)cb1db2rrvnew_argsp1p2p2argsargr!rloweriarg2expr2cond2lower_2upper_2s r _add_splinesr7s v"a32 AF # # B7 12 AF # # AF # # AF # #bgcrcl##73B3< * *C8D8DqMM!$E%V,,  4  #'q>> D= EMDq Eu_E)9OOD)))q E OOT4L ) ) ) )   """  15 1 1 99;;r)maxsizec |}t}td|D}t|}t|}t|}|dz }||zdz|krt d|dkrMt t jt||||dz |fd}n|dkr|||zdz||dzz }|t j kr-|||zdz|z |z } t|dz ||dz|} nt j x} } |||z||z }|t j kr$|||z |z } t|dz |||} nt j x} } t| | | | |}nt d|z| ||iS)a0 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 c34K|]}t|VdSNr).0ks r z bspline_basis..s(,,'!**,,,,,,rz(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 rrGrGTsf D A ,,e,,, , ,E AA AA%jjGA+K1uqy;ECDDDAv@ UHU1XuQU|44==a@@ A9   Q@a!eai 5Q</ AF? q1uqy!A%.Aq1ueQUA66BBVOBa!e uQx' AF? U1X&Aq1ueQ22BBVOBaQA..:Q>??? ??At9 % %%rcltz dz }fdt|DS)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 rAcNg|]!}tt|"S)rGrB)r>r1r(rIrs r z%bspline_basis_set..s- K K KQM!U5\\1a 0 0 K K Kr)rrange)r(rIr n_spliness``` rbspline_basis_setrXsD`E Q"I K K K K K K% :J:J K K KKrc ,ddlm}ddlm}t |}|jr|jstd|zt|t|krtdt||dzkrtdtdt||dd Dstd d |D}|j r|dzd z}||| }n7|d z}d t||| dz ||dz| D}|dg|dzzt|z|dg|dzzz}t||fd|D} ||| ||ftdt|t } t| d} dD} fd| D} t| | } t#| d} d| D} dD}g}| D]Mt%fdt| |Dt&j}||fNt-|S)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 integer values list of X coordinates through which the spline passes Y : list of strictly increasing integer values list of 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(K|] \}}||kVdSr<rTr>rrs rr@z'interpolating_spline..9s*//Aq1u//////rNz.The x-coordinates must be strictly increasing.c,g|]}t|SrTr=)r>r1s rrUz(interpolating_spline..;srr c$g|] \}}||zdz S)r rTr]s rrUz(interpolating_spline..Cs1   !QQUAI   rrc0g|]fdDS)c<g|]}|SrT)subs)r>rvrs rrUz3interpolating_spline...Ks% & & &1!&&A,, & & &rrT)r>rcbasisrs @rrUz(interpolating_spline..Ks22221 & & & & & & & &222rzc0:{})clsc8h|]}|jD] \}}|dk |S)Tr)r>rer&s r z'interpolating_spline..Os4DDDqDDfq!!t)DDDDDrc0g|]}t|SrT)r)r>r&rs rrUz(interpolating_spline..Ss! * * *1DAJJ * * *rc|dS)NrrT)rs rz&interpolating_spline..Us AaDr)keycg|]\}}|SrTrT)r>rys rrUz(interpolating_spline..Vs###tq!###rc0g|]}d|jDS)ci|]\}}|| SrTrT)r>rhr&s r z3interpolating_spline...Xs...VaAq...rrg)r>rs rrUz(interpolating_spline..Xs)>>>1..qv...>>>rc\g|](\}}||tjz)SrT)getrr)r>r&r(r1s rrUz(interpolating_spline..\s2 H H Hfq!Qq!&!! ! 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