KhXSrSSKrSSKrSSKrSSKrSSKJrJr SSK J r /SQr Sr SSjr SSjrS rS rS rS rS rSrSrSrSrSrSrSrSSjrSrSrSSjrg)an Utility classes and functions for the polynomial modules. This module provides: error and warning objects; a polynomial base class; and some routines used in both the `polynomial` and `chebyshev` modules. Functions --------- .. autosummary:: :toctree: generated/ as_series convert list of array_likes into 1-D arrays of common type. trimseq remove trailing zeros. trimcoef remove small trailing coefficients. getdomain return the domain appropriate for a given set of abscissae. mapdomain maps points between domains. mapparms parameters of the linear map between domains. N)dragon4_positionaldragon4_scientific) RankWarning) as_seriestrimseqtrimcoef getdomain mapdomainmapparms format_floatc[U5S:Xd USS:waU$[[U5S- SS5H nXS:wdM O USWS-$)aRemove small Poly series coefficients. Parameters ---------- seq : sequence Sequence of Poly series coefficients. Returns ------- series : sequence Subsequence with trailing zeros removed. If the resulting sequence would be empty, return the first element. The returned sequence may or may not be a view. Notes ----- Do not lose the type info if the sequence contains unknown objects. rN)lenrange)seqis L/var/www/html/env/lib/python3.13/site-packages/numpy/polynomial/polyutils.pyrr%sW( 3x1}B1  s3x!|R,Av{-4AaCyc hUVs/sHn[R"USSS9PM nnUHnURS:XdM[S5e [ SU55(a [S5eU(aUVs/sHn[ U5PM nn[ SU55(a/nUHnUR [R "[5:waQ[R"[U5[R "[5S 9nUSSUSS&URU5 M|URUR55 M U$[R"U6nUVs/sHn[R"US US 9PM nnU$s snfs snf![an[S 5UeSnAff=fs snf) a Return argument as a list of 1-d arrays. The returned list contains array(s) of dtype double, complex double, or object. A 1-d argument of shape ``(N,)`` is parsed into ``N`` arrays of size one; a 2-d argument of shape ``(M,N)`` is parsed into ``M`` arrays of size ``N`` (i.e., is "parsed by row"); and a higher dimensional array raises a Value Error if it is not first reshaped into either a 1-d or 2-d array. Parameters ---------- alist : array_like A 1- or 2-d array_like trim : boolean, optional When True, trailing zeros are removed from the inputs. When False, the inputs are passed through intact. Returns ------- [a1, a2,...] : list of 1-D arrays A copy of the input data as a list of 1-d arrays. Raises ------ ValueError Raised when `as_series` cannot convert its input to 1-d arrays, or at least one of the resulting arrays is empty. Examples -------- >>> import numpy as np >>> from numpy.polynomial import polyutils as pu >>> a = np.arange(4) >>> pu.as_series(a) [array([0.]), array([1.]), array([2.]), array([3.])] >>> b = np.arange(6).reshape((2,3)) >>> pu.as_series(b) [array([0., 1., 2.]), array([3., 4., 5.])] >>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16))) [array([1.]), array([0., 1., 2.]), array([0., 1.])] >>> pu.as_series([2, [1.1, 0.]]) [array([2.]), array([1.1])] >>> pu.as_series([2, [1.1, 0.]], trim=False) [array([2.]), array([1.1, 0. ])] rN)ndmincopyrzCoefficient array is emptyc3># UHoRS:gv M g7f)rN)ndim.0as r as_series..ys '166Q;szCoefficient array is not 1-dc3n# UH+oR[R"[5:Hv M- g7fN)dtypenpobjectrs rrr~s 7177bhhv& &s35r"z&Coefficient arrays have no common typeT)rr")r#arraysize ValueErroranyrr"r$emptyrappendr common_type Exception)alisttrimrarraysrettmpr"es rrrBsof8= =u!bhhq-uF =  66Q;9: : ' '''788 &,-f'!*f- 7 777Aww"((6**hhs1vRXXf-=>1A 3 1668$  J  NNNF+E=CCFqrxxE2FC J1>. NEFA M NCs) F8F  F# F/ F, F''F,cUS:a [S5e[U/5un[R"[R"U5U:5un[ U5S:XaUSSS-$USUSS-R 5$)a Remove "small" "trailing" coefficients from a polynomial. "Small" means "small in absolute value" and is controlled by the parameter `tol`; "trailing" means highest order coefficient(s), e.g., in ``[0, 1, 1, 0, 0]`` (which represents ``0 + x + x**2 + 0*x**3 + 0*x**4``) both the 3-rd and 4-th order coefficients would be "trimmed." Parameters ---------- c : array_like 1-d array of coefficients, ordered from lowest order to highest. tol : number, optional Trailing (i.e., highest order) elements with absolute value less than or equal to `tol` (default value is zero) are removed. Returns ------- trimmed : ndarray 1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned. Raises ------ ValueError If `tol` < 0 Examples -------- >>> from numpy.polynomial import polyutils as pu >>> pu.trimcoef((0,0,3,0,5,0,0)) array([0., 0., 3., 0., 5.]) >>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed array([0.]) >>> i = complex(0,1) # works for complex >>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3) array([0.0003+0.j , 0.001 -0.001j]) rztol must be non-negativeNrr)r(rr#nonzeroabsrr)ctolinds rrrsxP Qw344 QC.CQ JJrvvay3 'ES 3x1}!uQw#b'A+##%%rc[U/SS9unURR[RS;aUR R 5UR R5p!URR 5URR5pC[R"[X5[X$545$[R"UR 5UR545$)a Return a domain suitable for given abscissae. Find a domain suitable for a polynomial or Chebyshev series defined at the values supplied. Parameters ---------- x : array_like 1-d array of abscissae whose domain will be determined. Returns ------- domain : ndarray 1-d array containing two values. If the inputs are complex, then the two returned points are the lower left and upper right corners of the smallest rectangle (aligned with the axes) in the complex plane containing the points `x`. If the inputs are real, then the two points are the ends of the smallest interval containing the points `x`. See Also -------- mapparms, mapdomain Examples -------- >>> import numpy as np >>> from numpy.polynomial import polyutils as pu >>> points = np.arange(4)**2 - 5; points array([-5, -4, -1, 4]) >>> pu.getdomain(points) array([-5., 4.]) >>> c = np.exp(complex(0,1)*np.pi*np.arange(12)/6) # unit circle >>> pu.getdomain(c) array([-1.-1.j, 1.+1.j]) Fr/Complex) rr"charr# typecodesrealminmaximagr&complex)xrminrmaximinimaxs rr r sN QCe $CQww||r||I..VVZZ\166:: new[i]``, ``i = 0, 1``. Parameters ---------- old, new : array_like Domains. Each domain must (successfully) convert to a 1-d array containing precisely two values. Returns ------- offset, scale : scalars The map ``L(x) = offset + scale*x`` maps the first domain to the second. See Also -------- getdomain, mapdomain Notes ----- Also works for complex numbers, and thus can be used to calculate the parameters required to map any line in the complex plane to any other line therein. Examples -------- >>> from numpy.polynomial import polyutils as pu >>> pu.mapparms((-1,1),(-1,1)) (0.0, 1.0) >>> pu.mapparms((1,-1),(-1,1)) (-0.0, -1.0) >>> i = complex(0,1) >>> pu.mapparms((-i,-1),(1,i)) ((1+1j), (1-0j)) rr)oldnewoldlennewlenoffscls rr r s\RVc!f_F Vc!f_F q6#a&=3q6#a&= (& 0C -C 8Orc[U5[[[4;a5[ U[ R 5(d[ R"U5n[X5up4X4U--$)a Apply linear map to input points. The linear map ``offset + scale*x`` that maps the domain `old` to the domain `new` is applied to the points `x`. Parameters ---------- x : array_like Points to be mapped. If `x` is a subtype of ndarray the subtype will be preserved. old, new : array_like The two domains that determine the map. Each must (successfully) convert to 1-d arrays containing precisely two values. Returns ------- x_out : ndarray Array of points of the same shape as `x`, after application of the linear map between the two domains. See Also -------- getdomain, mapparms Notes ----- Effectively, this implements: .. math:: x\_out = new[0] + m(x - old[0]) where .. math:: m = \frac{new[1]-new[0]}{old[1]-old[0]} Examples -------- >>> import numpy as np >>> from numpy.polynomial import polyutils as pu >>> old_domain = (-1,1) >>> new_domain = (0,2*np.pi) >>> x = np.linspace(-1,1,6); x array([-1. , -0.6, -0.2, 0.2, 0.6, 1. ]) >>> x_out = pu.mapdomain(x, old_domain, new_domain); x_out array([ 0. , 1.25663706, 2.51327412, 3.76991118, 5.02654825, # may vary 6.28318531]) >>> x - pu.mapdomain(x_out, new_domain, old_domain) array([0., 0., 0., 0., 0., 0.]) Also works for complex numbers (and thus can be used to map any line in the complex plane to any other line therein). >>> i = complex(0,1) >>> old = (-1 - i, 1 + i) >>> new = (-1 + i, 1 - i) >>> z = np.linspace(old[0], old[1], 6); z array([-1. -1.j , -0.6-0.6j, -0.2-0.2j, 0.2+0.2j, 0.6+0.6j, 1. +1.j ]) >>> new_z = pu.mapdomain(z, old, new); new_z array([-1.0+1.j , -0.6+0.6j, -0.2+0.2j, 0.2-0.2j, 0.6-0.6j, 1.0-1.j ]) # may vary ) typeintfloatrC isinstancer#generic asanyarrayr )rDrKrLrOrPs rr r sN@ AwsE7++Jq"**4M4M MM! !HC Q;rcZ[R/U-n[S5X '[U5$r!)r#newaxisslicetuple)rrsls r _nth_slicer]fs' ** B $KBE 9rc^^^^[T5mT[T5:wa[STS[T535eT[T5:wa[STS[T535eTS:Xa [S5e[[R"[T55S-5mUUUU4Sj[ T55n[ R"[RU5$)a A generalization of the Vandermonde matrix for N dimensions The result is built by combining the results of 1d Vandermonde matrices, .. math:: W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{V_k(x_k)[i_0, \ldots, i_M, j_k]} where .. math:: N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\ M &= \texttt{points[k].ndim} \\ V_k &= \texttt{vander\_fs[k]} \\ x_k &= \texttt{points[k]} \\ 0 \le j_k &\le \texttt{degrees[k]} Expanding the one-dimensional :math:`V_k` functions gives: .. math:: W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{B_{k, j_k}(x_k[i_0, \ldots, i_M])} where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`. Parameters ---------- vander_fs : Sequence[function(array_like, int) -> ndarray] The 1d vander function to use for each axis, such as ``polyvander`` points : Sequence[array_like] Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. This must be the same length as `vander_fs`. degrees : Sequence[int] The maximum degree (inclusive) to use for each axis. This must be the same length as `vander_fs`. Returns ------- vander_nd : ndarray An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``. z Expected z" dimensions of sample points, got z dimensions of degrees, got rz9Unable to guess a dtype or shape when no points are givenc3h># UH'nTU"TUTU5S[UT5-v M) g7f)).N)r])rrdegreesn_dimspoints vander_fss rr_vander_nd..s<A ! VAY +FZ65J,JKs/2) rr(r[r#asarrayr functoolsreduceoperatormul)rdrcra vander_arraysrbs``` @r _vander_ndrllsZ^F VxA#f+ OQ Q Wx;CL> JL L {TUU2::eFm,s2 3FvM   HLL- 88rcp[XU5nURURS[U5*S-5$)z Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis Used to implement the public ``vanderd`` functions. N)r)rlreshapeshaper)rdrcravs r_vander_nd_flatrqs5 9g.A 99QWW^s7|m,u4 55rc [U5S:Xa[R"S5$[U/SS9unUR 5 UVs/sH o0"U*S5PM nn[U5nUS:a[[ US5upc[ U5Vs/sHoq"XGXGU-5PM nnU(aU"USUS5US'UnUnUS:aM[US$s snfs snf)af Helper function used to implement the ``fromroots`` functions. Parameters ---------- line_f : function(float, float) -> ndarray The ``line`` function, such as ``polyline`` mul_f : function(array_like, array_like) -> ndarray The ``mul`` function, such as ``polymul`` roots See the ``fromroots`` functions for more detail rrFr;r)rr#onesrsortdivmodr) line_fmul_frootsrpnmrr2s r _fromrootsr~s 5zQwwqzUG%0 $) *EqVQB]E * F!e!Qvald`` functions. Parameters ---------- val_f : function(array_like, array_like, tensor: bool) -> array_like The ``val`` function, such as ``polyval`` c, args See the ``vald`` functions for more detail rc3@># UHoRT:Hv M g7fr!)ro)rrshape0s rr_valnd..s3(Qww& (srNzx, y, z are incompatiblerszx, y are incompatiblezordinates are incompatibleF)tensor)r#rWroallrr(iternext)val_fr7argsritx0xirs @r_valndrs'+ +dBMM! dD + !W]]F 3$qr(3 3 3 t9>78 8 Y!^45 59: : dB bB b A " & H! ,s Cc(UH nU"X15nM U$)a Helper function used to implement the ``gridd`` functions. Parameters ---------- val_f : function(array_like, array_like, tensor: bool) -> array_like The ``val`` function, such as ``polyval`` c, args See the ``gridd`` functions for more detail rJ)rr7rrs r_gridndrs "L Hrc[X/5upUSS:Xa[e[U5n[U5nX4:a USSS-U4$US:XaXS- USSS-4$[R"X4- S-UR S9nUn[ X4- SS5H0nU"S/U-S/-U5nUSUS- n USSXSS-- nXU'M2 U[U54$)a Helper function used to implement the ``div`` functions. Implementation uses repeated subtraction of c2 multiplied by the nth basis. For some polynomial types, a more efficient approach may be possible. Parameters ---------- mul_f : function(array_like, array_like) -> array_like The ``mul`` function, such as ``polymul`` c1, c2 See the ``div`` functions for more detail rrNrr%)rZeroDivisionErrorrr#r*r"rr) rxc1c2lc1lc2quoremrr{qs r_divrs""HR "v{ b'C b'C y"1vax| R&y"Ra&(""hhsy1}BHH5sy#r*Aqc!eqck2&AB" Acr(Q"vX%CF + GCL  rc[X/5up[U5[U5:aUSUR===U- sss&UnOUSUR===U- sss&Un[U5$)z?Helper function used to implement the ``add`` functions. Nrrr'rrrr1s r_addr+s^""HR 2wR 8BGG   8BGG   3<rc[X/5up[U5[U5:aUSUR===U-sss&UnOU*nUSUR===U- sss&Un[U5$)z?Helper function used to implement the ``sub`` functions. Nrrs r_subr8se""HR 2wR 8BGG  S 8BGG   3<rc2[R"U5S-n[R"U5S-n[R"U5nURS:d*URRS;dUR S:Xa [ S5eUR5S:a [S5eURS:wa [ S5eUR S:Xa [ S5eURS:dURS :a [ S 5e[U5[U5:wa [ S 5eURS:XaUnUS-nU"X5n O5[R"U5nUS n[U5nU"X5S S 2U4n U Rn URn Ub_[R"U5S-nURS:wa [ S5e[U5[U5:wa [ S5eX-n X-n Uc6[U5[R"UR5R-n[U RR[R 5(ae[R""[R$"U R&5[R$"U R(5-R+S55n O9[R""[R$"U 5R+S55n SXS:H'[R,R/U RU - U RU5upnnU RU - Rn URS:akU RS :Xa2[R0"US-U R2S4U RS9nO"[R0"US-U RS9nU UU'Un X:wa#U(dSn[4R6"U[8S S9 U(aXUUU/4$U $)z Helper function used to implement the ``fit`` functions. Parameters ---------- vander_f : function(array_like, int) -> ndarray The 1d vander function, such as ``polyvander`` c1, c2 See the ``fit`` functions for more detail r_riurz0deg must be an int or non-empty 1-D array of intzexpected deg >= 0zexpected 1D vector for xzexpected non-empty vector for xrszexpected 1D or 2D array for yz$expected x and y to have same lengthrNzexpected 1D vector for wz$expected x and w to have same lengthr%z!The fit may be poorly conditioned) stacklevel)r#rfrr"kindr' TypeErrorr@r(rruTfinfoeps issubclassrRcomplexfloatingsqrtsquarer?rBsumlinalglstsqzerosrowarningswarnr)vander_frDydegrcondfullwlmaxordervanlhsrhsrPr7residsranksccmsgs r_fitrFs. 1 A 1 A **S/C xx!|syy~~T1SXX]JKK wwy1},--vv{233vv{9::vvzQVVaZ788 1vQ>?? xx1}qqggcl2wCq3' %%C ##C} JJqMC  66Q;67 7 q6SV BC Cgg }Arxx(,,,#))..""4"455ggryy*RYYsxx-@@EEaHIggbiin((+,CqMsCEE5AAtQ S A xx!| 66Q;461771:.agg>B$q&0B3  }T1 c;15 4E***rc&[U/5un[U5nXB:wdUS:a [S5eUbXC:a [S5eUS:Xa [R"S/UR S9$US:XaU$Un[ SUS-5H nU"XQ5nM U$)a> Helper function used to implement the ``pow`` functions. Parameters ---------- mul_f : function(array_like, array_like) -> ndarray The ``mul`` function, such as ``polymul`` c : array_like 1-D array of array of series coefficients pow, maxpower See the ``pow`` functions for more detail rz%Power must be a non-negative integer.zPower is too largerr%rs)rrSr(r#r&r"r)rxr7powmaxpowerpowerprdrs r_powrs QC.CQ HE |uqy@AA  %"2-.. !xx177++ !q%!)$A-C% rcv[R"U5$![an[USU35UeSnAff=f)a Like `operator.index`, but emits a custom exception when passed an incorrect type Parameters ---------- x : int-like Value to interpret as an integer desc : str description to include in any error message Raises ------ TypeError : if x is a float or non-numeric z must be an integer, received N)riindexr)rDdescr3s r_as_intrsB K~~a  K4& >qcBCJKs 838c V[R"[U5[R5(d [ U5$[R "5n[R "U5(aUS$[R"U5(aUS$SnUS:wa;[R"U5nUS:dUS[SUSS- *S -5-:aS nS upVUS S :XaSupVU(a%[XSXeUSS:HS9nU(aSU-S-nU$[XSS XeUSS:HS9nU$)NnanstrinfstrFrgחA precisionrrsT)0T floatmodefixed)kFsign+)runiquer/r())r fractionalrr/r) r# issubdtyperRfloatingstrget_printoptionsisnanisinfr6r@rr)rDparensopts exp_formatrr/rrs rr r s- ==a"++ . .1v   D xx{{H~ !H~JAv FF1I 9BAk):1)<'=q'@ AAAJLD KG#!  q,=&,$(LC$7 9 a# A H q,=*.&,$(LC$7 9 Hr)T)r)NFN)F)__doc__rirgrnumpyr#numpy._core.multiarrayrrnumpy.exceptionsr__all__rrrr r r r]rlrqr~rrrrrrrrr rJrrrs(I( :K\0&d-,^-^CL B9J6< <  !!H  UpBK,  r