o 6a[V@sdZddlZddlZddlZddlZgdZGdddeZddZ d,d d Z d-d d Z d dZ ddZ ddZddZddZddZddZddZddZdd Zd!d"Zd#d$Zd.d&d'Zd(d)Zd*d+ZdS)/a Utility classes and functions for the polynomial modules. This module provides: error and warning objects; a polynomial base class; and some routines used in both the `polynomial` and `chebyshev` modules. Warning objects --------------- .. autosummary:: :toctree: generated/ RankWarning raised in least-squares fit for rank-deficient matrix. Functions --------- .. autosummary:: :toctree: generated/ as_series convert list of array_likes into 1-D arrays of common type. trimseq remove trailing zeros. trimcoef remove small trailing coefficients. getdomain return the domain appropriate for a given set of abscissae. mapdomain maps points between domains. mapparms parameters of the linear map between domains. N) RankWarning as_seriestrimseqtrimcoef getdomain mapdomainmapparmsc@seZdZdZdS)rz;Issued by chebfit when the design matrix is rank deficient.N)__name__ __module__ __qualname____doc__r r >> from numpy.polynomial import polyutils as pu >>> a = np.arange(4) >>> pu.as_series(a) [array([0.]), array([1.]), array([2.]), array([3.])] >>> b = np.arange(6).reshape((2,3)) >>> pu.as_series(b) [array([0., 1., 2.]), array([3., 4., 5.])] >>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16))) [array([1.]), array([0., 1., 2.]), array([0., 1.])] >>> pu.as_series([2, [1.1, 0.]]) [array([2.]), array([1.1])] >>> pu.as_series([2, [1.1, 0.]], trim=False) [array([2.]), array([1.1, 0. ])] cSsg|] }tj|dddqS)rF)ndmincopynparray.0ar r r zas_series..cSsg|]}|jqSr )sizerr r rrsrzCoefficient array is emptycss|]}|jdkVqdS)rN)ndimrr r r zas_series..zCoefficient array is not 1-dcSsg|]}t|qSr )rrr r rrscss |] }|jttkVqdSN)dtyperobjectrr r rr!sr$Nz&Coefficient arrays have no common typecsg|] }tj|ddqS)T)rr$rrr&r rrr) min ValueErroranyr$rr%emptyrappendr common_type Exception)alisttrimarraysretrtmper r&rrPs02  rcCsf|dkrtdt|g\}tt||k\}t|dkr'|dddS|d|ddS)a0 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 See Also -------- trimseq 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(rrnonzeroabsrr)ctolindr r rrs,  rcCs~t|gdd\}|jjtjdvr4|j|j}}|j|j}}t t ||t ||fSt ||fS)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 -------- >>> 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$charr typecodesrealr'maximagrcomplex)xrminrmaximinimaxr r rrs &rcCsT|d|d}|d|d}|d|d|d|d|}||}||fS)a Linear map parameters between domains. Return the parameters of the linear map ``offset + scale*x`` that maps `old` to `new` such that ``old[i] -> new[i]``, ``i = 0, 1``. Parameters ---------- old, new : array_like Domains. Each domain must (successfully) convert to a 1-d array containing precisely two values. Returns ------- offset, scale : scalars The map ``L(x) = offset + scale*x`` maps the first domain to the second. See Also -------- getdomain, mapdomain Notes ----- Also works for complex numbers, and thus can be used to calculate the parameters required to map any line in the complex plane to any other line therein. Examples -------- >>> from numpy.polynomial import polyutils as pu >>> pu.mapparms((-1,1),(-1,1)) (0.0, 1.0) >>> pu.mapparms((1,-1),(-1,1)) (-0.0, -1.0) >>> i = complex(0,1) >>> pu.mapparms((-i,-1),(1,i)) ((1+1j), (1-0j)) rrr )oldnewoldlennewlenoffsclr r rrs )$rcCs$t|}t||\}}|||S)a5 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 -------- >>> 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 )r asanyarrayr)rArFrGrJrKr r rr/s ? rcCs tjg|}td||<t|Sr#)rnewaxisslicetuple)rr slr r r _nth_slicess  rQcsttkrtddttkr(tddtdkr0tdttjtdddfd d tD}ttj |S) am 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 givenF)rc3s4|]}|||dt|VqdS)).N)rQrrdegreesn_dimspoints vander_fsr rr!s " z_vander_nd..) rr(rOrrr functoolsreduceoperatormul)rXrWrU vander_arraysr rTr _vander_ndys -  r^cCs*t|||}||jdt| dS)z Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis Used to implement the public ``vanderd`` functions. N)r)r^reshapeshaper)rXrWrUvr r r_vander_nd_flats rbcst|dkr tdSt|gdd\}|fdd|Dt}|dkrQt|d\}fddtD}|rI|dd |d<|}|dks(dS) a 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 rrFr9csg|]}| dqS)rr )rr)line_fr rrsz_fromroots..cs"g|] }||qSr r rS)mmul_fpr rrs"r)rronesrsortdivmodr)rdrgrootsnrcr2r )rdrfrgrhr _fromrootss rncsdd|D}|djtfdd|ddDs3t|dkr%td t|d kr/td td t|}t|}|||}|D] }|||d d}qB|S)a4 Helper function used to implement the ``vald`` 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 cSsg|]}t|qSr )rrLrr r rrsz_valnd..rc3s|]}|jkVqdSr#)r`rshape0r rr!r"z_valnd..rNzx, y, z are incompatiblerezx, y are incompatiblezordinates are incompatibleF)tensor)r`allrr(iternext)val_fr6argsitx0xir ror_valnds    r{cGs|D]}|||}q|S)a6 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 r )rvr6rwrzr r r_gridnds r|c Cst||g\}}|ddkrtt|}t|}||kr'|ddd|fS|dkr9||d|dddfStj||d|jd}|}t||ddD]'}|dg|dg|}|d|d} |dd| |dd}| ||<qO|t|fS)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&)rZeroDivisionErrorrrr*r$rr) rgc1c2lc1lc2quoremrrhqr r r_divs"   rcCsdt||g\}}t|t|kr!|d|j|7<|}t|S|d|j|7<|}t|S)z@ Helper function used to implement the ``add`` functions. Nrrrrr~rr1r r r_add8srcCsjt||g\}}t|t|kr!|d|j|8<|}t|S| }|d|j|7<|}t|S)z@ Helper function used to implement the ``sub`` functions. Nrrr r r_subEsrFcCst|d}t|d}t|}|jdks#|jjdvs#|jdkr'td|dkr1td|jdkr:td|jdkrCtd|jdksM|jd krQtd t |t |kr]td |jdkrn|}|d}|||} nt |}|d }t |}|||d d |f} | j } |j } |d urt|d}|jdkrtdt |t |krtd| |} | |} |d urt |t |jj }t| jjtjrtt| jt| jd} n tt| d} d| | dk<tj| j | | j |\} }}}| j | j } |jdkr1| jd kr!tj|d| jdf| jd}n tj|d| jd}| ||<|} ||krC|sCd}tj|td d|rN| ||||gfS| S)a 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 rRriurz0deg must be an int or non-empty 1-D array of intzexpected deg >= 0zexpected 1D vector for xzexpected non-empty vector for xrezexpected 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)rasarrayr r$kindr TypeErrorr'r(rrjTfinfoeps issubclasstypecomplexfloatingsqrtsquarer=r?sumlinalglstsqzerosr`warningswarnr)vander_frAydegrcondfullwlmaxordervanlhsrhsrKr6residsranksccmsgr r r_fitSsj          &      rcCst|g\}t|}||ks|dkrtd|dur"||kr"td|dkr/tjdg|jdS|dkr5|S|}td|dD]}|||}q>|S)af Helper function used to implement the ``pow`` functions. Parameters ---------- mul_f : function(array_like, array_like) -> ndarray The ``mul`` function, such as ``polymul`` c : array_like 1-D array of array of series coefficients pow, maxpower See the ``pow`` functions for more detail rz%Power must be a non-negative integer.NzPower is too largerr&re)rintr(rrr$r)rgr6powmaxpowerpowerprdrr r r_pows  rc Cszt|WStyB}z0zt|}Wn tyYnw||kr6tjd|dtdd|WYd}~St|d|d}~ww)a Like `operator.index`, but emits a deprecation warning when passed a float Parameters ---------- x : int-like, or float with integral value Value to interpret as an integer desc : str description to include in any error message Raises ------ TypeError : if x is a non-integral float or non-numeric DeprecationWarning : if x is an integral float z)In future, this will raise TypeError, as z7 will need to be an integer not just an integral float.rqrNz must be an integer)r[indexrrrrDeprecationWarning)rAdescr3ixr r r_deprecate_as_ints$    r)T)r)NFN)r r[rYrnumpyr__all__ UserWarningrrrrrrrrQr^rbrnr{r|rrrrrrr r r rs2  L6./DE $ X !