U Zʹj'@sddlmZmZddlmZmZddlmZmZddl m Z ddl m Z ddl mZddZd d Zed d d dZddZddZdS))Ssympify)Dummysymbols) Piecewisepiecewise_fold)And)Interval) lru_cachecCsRt|trBt|jdkrB|j\}}|j|kr6||}}|j|jfStd|dS)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: %sN) isinstancerlenargsZltsZgts TypeError)condxabr_/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/functions/special/bsplines.py_ivl s     rcCs@tj||fkrt||}ntj||fkr>> 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 css|]}t|VqdSNr).0krrr sz bspline_basis..z(n + d + 1 must not exceed len(knots) - 1rrzdegree must be non-negative: %r)rtupleintr ValueErrorrrZOner containsr bspline_basisr+Zxreplace) r#knotsnrZxvarZn_knotsZ n_intervalsresultdenomBr$Ar"rrrr8Ts8S       r8cs*td}fddt|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 r3csg|]}tt|qSr)r8r4r0r*r#r9rrr sz%bspline_basis_set..)r range)r#r9rZ n_splinesrr@rbspline_basis_sets0rCcsddlm}ddlm}t|}|jr,|js8td|t|t|krPtdt||dkrhtdt dd t ||dd Dstd d d |D}|j r|dd}||| }n8|d}dd t ||| d||d| D}|dg|dt ||dg|d}t ||fdd |D} ||| ||ftdt|td} t | d} ddD} fdd | D} t | | } t| ddd} dd | D} dd D}g}| D]4tfdd t | |Dtj}||fqt|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 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.r3z6Degree must be less than the number of control points.css|]\}}||kVqdSr.rr0rrrrrr29sz'interpolating_spline..Nz.The x-coordinates must be strictly increasing.cSsg|] }t|qSrr/r?rrrrA;sz(interpolating_spline..r cSsg|]\}}||dqS)r rrFrrrrACsrcs g|]fddDqS)csg|]}|qSr)subsr0r)vrrrrAKsz3interpolating_spline...r)r0)basisr)rIrrAKszc0:{})clscSs(h|] }|jD]\}}|dkr|qqS)Tr)r0rer!rrr Os z'interpolating_spline..csg|]}t|qSr)r)r0r!rrrrASscSs|dS)NrrrOrrrUz&interpolating_spline..)keycSsg|] \}}|qSrr)r0ryrrrrAVscSsg|]}dd|jDqS)cSsi|]\}}||qSrr)r0rMr!rrr Xsz3interpolating_spline...rLrHrrrrAXscs"g|]\}}||tjqSr)getrr)r0r!r#)r*rrrA\s)Zsympy.solvers.solvesetrDZsympy.matrices.denserErZ is_IntegerZ is_positiver6r allzipZis_oddrrCrformatrsortedsumrrrr)r#rXYrDrEjZinterior_knotsr9r>ZcoeffZ intervalsZivalcomZ basis_dictsZsplineZpiecer)rJr*rrinterpolating_splinesL.      $, (  r_N)Z sympy.corerrZsympy.core.symbolrrZsympy.functionsrrZsympy.logic.boolalgrZsympy.sets.setsr functoolsr rr+r8rCr_rrrrs   ; w4