KhSrSSKrSSKJr SSKJr SSKJ r SSK J r /SQr \ RrSrS r\R$"S S /5r\R$"S/5r\R$"S/5r\R$"SS /5rS rSrSrSrSrSrSrS'SjrS(SjrS/SSS4Sjr S)Sjr!Sr"Sr#Sr$Sr%Sr&Sr'Sr(S*Sjr)S r*S!r+S"r,S#r-S$r."S%S&\ 5r/g)+a ============================================================== Hermite Series, "Physicists" (:mod:`numpy.polynomial.hermite`) ============================================================== This module provides a number of objects (mostly functions) useful for dealing with Hermite series, including a `Hermite` class that encapsulates the usual arithmetic operations. (General information on how this module represents and works with such polynomials is in the docstring for its "parent" sub-package, `numpy.polynomial`). Classes ------- .. autosummary:: :toctree: generated/ Hermite Constants --------- .. autosummary:: :toctree: generated/ hermdomain hermzero hermone hermx Arithmetic ---------- .. autosummary:: :toctree: generated/ hermadd hermsub hermmulx hermmul hermdiv hermpow hermval hermval2d hermval3d hermgrid2d hermgrid3d Calculus -------- .. autosummary:: :toctree: generated/ hermder hermint Misc Functions -------------- .. autosummary:: :toctree: generated/ hermfromroots hermroots hermvander hermvander2d hermvander3d hermgauss hermweight hermcompanion hermfit hermtrim hermline herm2poly poly2herm See also -------- `numpy.polynomial` N)normalize_axis_index) polyutils) ABCPolyBase)hermzerohermonehermx hermdomainhermlinehermaddhermsubhermmulxhermmulhermdivhermpowhermvalhermderhermint herm2poly poly2herm hermfromroots hermvanderhermfithermtrim hermrootsHermite hermval2d hermval3d hermgrid2d hermgrid3d hermvander2d hermvander3d hermcompanion hermgauss hermweightc[R"U/5un[U5S- nSn[USS5Hn[ [ U5X5nM U$)a poly2herm(pol) Convert a polynomial to a Hermite series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Hermite series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Hermite series. See Also -------- herm2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.hermite import poly2herm >>> poly2herm(np.arange(4)) array([1. , 2.75 , 0.5 , 0.375]) rr)pu as_serieslenranger r)poldegresis J/var/www/html/env/lib/python3.13/site-packages/numpy/polynomial/hermite.pyrr`sSL LL# ES c(Q,C C 3B hsmSV, JcPSSKJnJnJn [R "U/5un[ U5nUS:XaU$US:XaUS==S-ss'U$USnUSn[US- SS5H-nUnU"XS- USUS- --5nU"X"U5S-5nM/ U"XS"U5S-5$)a Convert a Hermite series to a polynomial. Convert an array representing the coefficients of a Hermite series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Hermite series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2herm Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.hermite import herm2poly >>> herm2poly([ 1. , 2.75 , 0.5 , 0.375]) array([0., 1., 2., 3.]) r)polyaddpolysubpolymulxr') polynomialr3r4r5r(r)r*r+) cr3r4r5nc0c1r/tmps r0rrsL76 ,,s CQ AAAvAv !  rU rUq1ua$ACq52q!a%y>2Bhrl1n-B%r8B<>**r1g??cpUS:wa[R"XS- /5$[R"U/5$)aU Hermite series whose graph is a straight line. Parameters ---------- off, scl : scalars The specified line is given by ``off + scl*x``. Returns ------- y : ndarray This module's representation of the Hermite series for ``off + scl*x``. See Also -------- numpy.polynomial.polynomial.polyline numpy.polynomial.chebyshev.chebline numpy.polynomial.legendre.legline numpy.polynomial.laguerre.lagline numpy.polynomial.hermite_e.hermeline Examples -------- >>> from numpy.polynomial.hermite import hermline, hermval >>> hermval(0,hermline(3, 2)) 3.0 >>> hermval(1,hermline(3, 2)) 5.0 rr6)nparray)offscls r0r r s1D axxx!e %%xxr1cB[R"[[U5$)a Generate a Hermite series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Hermite form, where the :math:`r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Hermite form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> from numpy.polynomial.hermite import hermfromroots, hermval >>> coef = hermfromroots((-1, 0, 1)) >>> hermval((-1, 0, 1), coef) array([0., 0., 0.]) >>> coef = hermfromroots((-1j, 1j)) >>> hermval((-1j, 1j), coef) array([0.+0.j, 0.+0.j]) )r( _fromrootsr r)rootss r0rrsj ==7E 22r1c.[R"X5$)at Add one Hermite series to another. Returns the sum of two Hermite series `c1` + `c2`. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the Hermite series of their sum. See Also -------- hermsub, hermmulx, hermmul, hermdiv, hermpow Notes ----- Unlike multiplication, division, etc., the sum of two Hermite series is a Hermite series (without having to "reproject" the result onto the basis set) so addition, just like that of "standard" polynomials, is simply "component-wise." Examples -------- >>> from numpy.polynomial.hermite import hermadd >>> hermadd([1, 2, 3], [1, 2, 3, 4]) array([2., 4., 6., 4.]) )r(_addr<c2s r0r r :J 772?r1c.[R"X5$)a Subtract one Hermite series from another. Returns the difference of two Hermite series `c1` - `c2`. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- out : ndarray Of Hermite series coefficients representing their difference. See Also -------- hermadd, hermmulx, hermmul, hermdiv, hermpow Notes ----- Unlike multiplication, division, etc., the difference of two Hermite series is a Hermite series (without having to "reproject" the result onto the basis set) so subtraction, just like that of "standard" polynomials, is simply "component-wise." Examples -------- >>> from numpy.polynomial.hermite import hermsub >>> hermsub([1, 2, 3, 4], [1, 2, 3]) array([0., 0., 0., 4.]) )r(_subrJs r0r r brLr1cf[R"U/5un[U5S:Xa USS:XaU$[R"[U5S-UR S9nUSS-US'USS- US'[ S[U55H#nXS- XS-'XS- ==XU-- ss'M% U$)aMultiply a Hermite series by x. Multiply the Hermite series `c` by x, where x is the independent variable. Parameters ---------- c : array_like 1-D array of Hermite series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the result of the multiplication. See Also -------- hermadd, hermsub, hermmul, hermdiv, hermpow Notes ----- The multiplication uses the recursion relationship for Hermite polynomials in the form .. math:: xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x)) Examples -------- >>> from numpy.polynomial.hermite import hermmulx >>> hermmulx([1, 2, 3]) array([2. , 6.5, 1. , 1.5]) rrdtyper6)r(r)r*rAemptyrQr+)r9prdr/s r0rrsN ,,s CQ 1v{qtqy ((3q6A:QWW -C qT!VCF qT!VCF 1c!f T!VE  E ad1f  Jr1c[R"X/5up[U5[U5:aUnUnOUnUn[U5S:Xa USU-nSnO[U5S:XaUSU-nUSU-nOt[U5nUSU-nUSU-n[S[U5S-5H=nUnUS- n[ X&*U-USUS- --5n[ U[ U5S-5nM? [ U[ U5S-5$)a Multiply one Hermite series by another. Returns the product of two Hermite series `c1` * `c2`. The arguments are sequences of coefficients, from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- out : ndarray Of Hermite series coefficients representing their product. See Also -------- hermadd, hermsub, hermmulx, hermdiv, hermpow Notes ----- In general, the (polynomial) product of two C-series results in terms that are not in the Hermite polynomial basis set. Thus, to express the product as a Hermite series, it is necessary to "reproject" the product onto said basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermmul >>> hermmul([1, 2, 3], [0, 1, 2]) array([52., 29., 52., 7., 6.]) rrr6r7r')r(r)r*r+r r r)r<rKr9xsr;ndr/r=s r0rrsN||RH%HR 2wR     1v{ qT"W  Q1 qT"W qT"W V rU2X rU2Xq#a&1*%ACaB2r2q"q&z?3Bhrl1n-B & 2x|A~ &&r1c8[R"[X5$)a Divide one Hermite series by another. Returns the quotient-with-remainder of two Hermite series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Hermite series coefficients representing the quotient and remainder. See Also -------- hermadd, hermsub, hermmulx, hermmul, hermpow Notes ----- In general, the (polynomial) division of one Hermite series by another results in quotient and remainder terms that are not in the Hermite polynomial basis set. Thus, to express these results as a Hermite series, it is necessary to "reproject" the results onto the Hermite basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermdiv >>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([0.])) >>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([2., 2.])) >>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([1., 1.])) )r(_divrrJs r0rrsZ 777B ##r1c:[R"[XU5$)aRaise a Hermite series to a power. Returns the Hermite series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` Parameters ---------- c : array_like 1-D array of Hermite series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Hermite series of power. See Also -------- hermadd, hermsub, hermmulx, hermmul, hermdiv Examples -------- >>> from numpy.polynomial.hermite import hermpow >>> hermpow([1, 2, 3], 2) array([81., 52., 82., 12., 9.]) )r(_powr)r9powmaxpowers r0rr1sD 777AH --r1c[R"USSS9nURRS;aUR [R 5n[ R"US5n[ R"US5nUS:a [S5e[XPR5nUS:XaU$[R"XS5n[U5nXF:a US SS-nOp[U5HanUS- nX-n[R"U4URSS -URS 9n[USS 5Hn S U -X -XS- 'M UnMc [R"USU5nU$) al Differentiate a Hermite series. Returns the Hermite series coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Hermite series coefficients. If `c` is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). Returns ------- der : ndarray Hermite series of the derivative. See Also -------- hermint Notes ----- In general, the result of differentiating a Hermite series does not resemble the same operation on a power series. Thus the result of this function may be "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermder >>> hermder([ 1. , 0.5, 0.5, 0.5]) array([1., 2., 3.]) >>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2) array([1., 2., 3.]) rTndmincopy ?bBhHiIlLqQpPzthe order of derivationthe axisrz,The order of derivation must be non-negativeNrPr'r6)rArBrQcharastypedoubler(_as_int ValueErrorrndimmoveaxisr*r+rRshape) r9mrDaxiscntiaxisr:r/derjs r0rrVs<j !$'Aww||& HHRYY  **Q1 2C JJtZ (E QwGHH  /E ax Aa A AA x bqE!GsAAA HA((A4!''!"+-QWW=C1a_c14ZE %A  Aq% A Hr1c [R"USSS9nURRS;aUR [R 5n[R "U5(dU/n[R"US5n[R"US5nUS:a [S5e[U5U:a [S 5e[R"U5S:wa [S 5e[R"U5S:wa [S 5e[XpR5nUS:XaU$[R"XS5n[U5S/U[U5- --n[U5Hn[U5n X-nU S:Xa2[R "USS:H5(aUS==X(- ss'MJ[R""U S-4UR$SS -URS 9n USS-U S'USS- U S'[SU 5Hn X SU S--- XS-'M U S==X(['X:5- - ss'U nM [R"USU5nU$)a' Integrate a Hermite series. Returns the Hermite series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Hermite series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). Returns ------- S : ndarray Hermite series coefficients of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- hermder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermint >>> hermint([1,2,3]) # integrate once, value 0 at 0. array([1. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0 array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary >>> hermint([1,2,3], k=1) # integrate once, value 1 at 0. array([2. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1 array([-2. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1) array([ 1.66666667, -0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary rTr_rbzthe order of integrationrcrz-The order of integration must be non-negativezToo many integration constantszlbnd must be a scalar.zscl must be a scalar.NrPr6)rArBrQrdrerfiterabler(rgrhr*rirrjlistr+allrRrkr) r9rlklbndrDrmrnror/r:r=rqs r0rrsb !$'Aww||& HHRYY  ;;q>> C **Q2 3C JJtZ (E QwHII 1v|9:: wwt}122 wws|q011  /E ax Aa A Q1#sSV|$$A 3Z F  6bffQqTQY'' aDADLD((AE8aggabk1ACqT!VCFqT!VCF1a[T1a!e9-E ! FadWT// /FA Aq% A Hr1c[R"USSS9nURRS;aUR [R 5n[ U[[45(a[R"U5n[ U[R5(a2U(a+URURSUR--5nUS-n[U5S:XaUSnSnOm[U5S:Xa USnUSnOS[U5nUSnUS n[S [U5S-5H"nUnUS- nX*USUS- --- nXU--nM$ XEU--$) a9 Evaluate an Hermite series at points x. If `c` is of length ``n + 1``, this function returns the value: .. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x) The parameter `x` is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either `x` or its elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array, then ``p(x)`` will have the same shape as `x`. If `c` is multidimensional, then the shape of the result depends on the value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that scalars have shape (,). Trailing zeros in the coefficients will be used in the evaluation, so they should be avoided if efficiency is a concern. Parameters ---------- x : array_like, compatible object If `x` is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, `x` or its elements must support addition and multiplication with themselves and with the elements of `c`. c : array_like Array of coefficients ordered so that the coefficients for terms of degree n are contained in c[n]. If `c` is multidimensional the remaining indices enumerate multiple polynomials. In the two dimensional case the coefficients may be thought of as stored in the columns of `c`. tensor : boolean, optional If True, the shape of the coefficient array is extended with ones on the right, one for each dimension of `x`. Scalars have dimension 0 for this action. The result is that every column of coefficients in `c` is evaluated for every element of `x`. If False, `x` is broadcast over the columns of `c` for the evaluation. This keyword is useful when `c` is multidimensional. The default value is True. Returns ------- values : ndarray, algebra_like The shape of the return value is described above. See Also -------- hermval2d, hermgrid2d, hermval3d, hermgrid3d Notes ----- The evaluation uses Clenshaw recursion, aka synthetic division. Examples -------- >>> from numpy.polynomial.hermite import hermval >>> coef = [1,2,3] >>> hermval(1, coef) 11.0 >>> hermval([[1,2],[3,4]], coef) array([[ 11., 51.], [115., 203.]]) rNr_rb)rr6rr7r'rU)rArBrQrdrerf isinstancetuplertasarrayndarrayreshaperkrir*r+) xr9tensorx2r;r<rWr/r=s r0rrs=F !$'Aww||& HHRYY !eT]## JJqM!RZZ  V IIaggQVV + , 1B 1v{ qT  Q1 qT qT V rU rUq#a&1*%ACaB2QQZ(B"uB & 2:r1c:[R"[X U5$)aF Evaluate a 2-D Hermite series at points (x, y). This function returns the values: .. math:: p(x,y) = \sum_{i,j} c_{i,j} * H_i(x) * H_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points ``(x, y)``, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points formed with pairs of corresponding values from `x` and `y`. See Also -------- hermval, hermgrid2d, hermval3d, hermgrid3d Examples -------- >>> from numpy.polynomial.hermite import hermval2d >>> x = [1, 2] >>> y = [4, 5] >>> c = [[1, 2, 3], [4, 5, 6]] >>> hermval2d(x, y, c) array([1035., 2883.]) r(_valndrr~yr9s r0rr}sd 99WaA &&r1c:[R"[X U5$)a Evaluate a 2-D Hermite series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \sum_{i,j} c_{i,j} * H_i(a) * H_j(b) where the points ``(a, b)`` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- hermval, hermval2d, hermval3d, hermgrid3d Examples -------- >>> from numpy.polynomial.hermite import hermgrid2d >>> x = [1, 2, 3] >>> y = [4, 5] >>> c = [[1, 2, 3], [4, 5, 6]] >>> hermgrid2d(x, y, c) array([[1035., 1599.], [1867., 2883.], [2699., 4167.]]) r(_gridndrrs r0rrsp ::gqQ ''r1c:[R"[X0X5$)a Evaluate a 3-D Hermite series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points ``(x, y, z)``, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- hermval, hermval2d, hermgrid2d, hermgrid3d Examples -------- >>> from numpy.polynomial.hermite import hermval3d >>> x = [1, 2] >>> y = [4, 5] >>> z = [6, 7] >>> c = [[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]] >>> hermval3d(x, y, z, c) array([ 40077., 120131.]) rr~rzr9s r0rrsj 99WaA ))r1c:[R"[X0X5$)a Evaluate a 3-D Hermite series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \sum_{i,j,k} c_{i,j,k} * H_i(a) * H_j(b) * H_k(c) where the points ``(a, b, c)`` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- hermval, hermval2d, hermgrid2d, hermval3d Examples -------- >>> from numpy.polynomial.hermite import hermgrid3d >>> x = [1, 2] >>> y = [4, 5] >>> z = [6, 7] >>> c = [[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]] >>> hermgrid3d(x, y, z, c) array([[[ 40077., 54117.], [ 49293., 66561.]], [[ 72375., 97719.], [ 88975., 120131.]]]) rrs r0r r %sz ::gqQ **r1c[R"US5nUS:a [S5e[R"USSS9S-nUS-4UR -nUR n[R"X4S9nUS-S-US'US:a<US -nXeS'[S US-5H nXWS- U-XWS - S US- --- XW'M" [R"USS 5$) aPseudo-Vandermonde matrix of given degree. Returns the pseudo-Vandermonde matrix of degree `deg` and sample points `x`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., i] = H_i(x), where ``0 <= i <= deg``. The leading indices of `V` index the elements of `x` and the last index is the degree of the Hermite polynomial. If `c` is a 1-D array of coefficients of length ``n + 1`` and `V` is the array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and ``hermval(x, c)`` are the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of Hermite series of the same degree and sample points. Parameters ---------- x : array_like Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If `x` is scalar it is converted to a 1-D array. deg : int Degree of the resulting matrix. Returns ------- vander : ndarray The pseudo-Vandermonde matrix. The shape of the returned matrix is ``x.shape + (deg + 1,)``, where The last index is the degree of the corresponding Hermite polynomial. The dtype will be the same as the converted `x`. Examples -------- >>> import numpy as np >>> from numpy.polynomial.hermite import hermvander >>> x = np.array([-1, 0, 1]) >>> hermvander(x, 3) array([[ 1., -2., 2., 4.], [ 1., 0., -2., -0.], [ 1., 2., 2., -4.]]) r-rzdeg must be non-negativeNr)rar`rPr6r') r(rgrhrArBrkrQrRr+rj)r~r-idegdimsdtypvrr/s r0rresZ ::c5 !D ax344 Q'#-A 1H; D 77D "A Q37AaD ax qS!q$(#AcF2IA#1q5 22AD$ ;;q!R  r1cH[R"[[4X4U5$)a>Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points ``(x, y)``. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = H_i(x) * H_j(y), where ``0 <= i <= deg[0]`` and ``0 <= j <= deg[1]``. The leading indices of `V` index the points ``(x, y)`` and the last index encodes the degrees of the Hermite polynomials. If ``V = hermvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``hermval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Hermite series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- hermvander, hermvander3d, hermval2d, hermval3d Examples -------- >>> import numpy as np >>> from numpy.polynomial.hermite import hermvander2d >>> x = np.array([-1, 0, 1]) >>> y = np.array([-1, 0, 1]) >>> hermvander2d(x, y, [2, 2]) array([[ 1., -2., 2., -2., 4., -4., 2., -4., 4.], [ 1., 0., -2., 0., 0., -0., -2., -0., 4.], [ 1., 2., 2., 2., 4., 4., 2., 4., 4.]]) r(_vander_nd_flatr)r~rr-s r0r!r!s!p   z:6 DDr1cT[R"[[[4XU4U5$)aPseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points ``(x, y, z)``. If `l`, `m`, `n` are the given degrees in `x`, `y`, `z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = H_i(x)*H_j(y)*H_k(z), where ``0 <= i <= l``, ``0 <= j <= m``, and ``0 <= j <= n``. The leading indices of `V` index the points ``(x, y, z)`` and the last index encodes the degrees of the Hermite polynomials. If ``V = hermvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``hermval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Hermite series of the same degrees and sample points. Parameters ---------- x, y, z : 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. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- hermvander, hermvander3d, hermval2d, hermval3d Examples -------- >>> from numpy.polynomial.hermite import hermvander3d >>> x = np.array([-1, 0, 1]) >>> y = np.array([-1, 0, 1]) >>> z = np.array([-1, 0, 1]) >>> hermvander3d(x, y, z, [0, 1, 2]) array([[ 1., -2., 2., -2., 4., -4.], [ 1., 0., -2., 0., 0., -0.], [ 1., 2., 2., 2., 4., 4.]]) r)r~rrr-s r0r"r"s%r   z:zBQ1Is SSr1c <[R"[XX#XE5$)a Least squares fit of Hermite series to data. Return the coefficients of a Hermite series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. Returns ------- coef : ndarray, shape (M,) or (M, K) Hermite coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.exceptions.RankWarning) See Also -------- numpy.polynomial.chebyshev.chebfit numpy.polynomial.legendre.legfit numpy.polynomial.laguerre.lagfit numpy.polynomial.polynomial.polyfit numpy.polynomial.hermite_e.hermefit hermval : Evaluates a Hermite series. hermvander : Vandermonde matrix of Hermite series. hermweight : Hermite weight function numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Hermite series `p` that minimizes the sum of the weighted squared errors .. math:: E = \sum_j w_j^2 * |y_j - p(x_j)|^2, where the :math:`w_j` are the weights. This problem is solved by setting up the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `~exceptions.RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Hermite series are probably most useful when the data can be approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Hermite weight. In that case the weight ``sqrt(w(x[i]))`` should be used together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is available as `hermweight`. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- >>> import numpy as np >>> from numpy.polynomial.hermite import hermfit, hermval >>> x = np.linspace(-10, 10) >>> rng = np.random.default_rng() >>> err = rng.normal(scale=1./10, size=len(x)) >>> y = hermval(x, [1, 2, 3]) + err >>> hermfit(x, y, 2) array([1.02294967, 2.00016403, 2.99994614]) # may vary )r(_fitr)r~rr-rcondfullws r0rrsF 77:qS 99r1c [R"U/5un[U5S:a [S5e[U5S:Xa$[R "SUS-US- //5$[U5S- n[R "X4URS9n[R"SS[R"S[R"US- SS 5-5- 45n[RRU5S S S 2nURS 5SS US-2nURS 5US US-2n[R"S [R"SU5-5US 'XES 'US S 2S 4==X0S S -SUS -- -ss'U$) aReturn the scaled companion matrix of c. The basis polynomials are scaled so that the companion matrix is symmetric when `c` is an Hermite basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be real if `numpy.linalg.eigvalsh` is used to obtain them. Parameters ---------- c : array_like 1-D array of Hermite series coefficients ordered from low to high degree. Returns ------- mat : ndarray Scaled companion matrix of dimensions (deg, deg). Examples -------- >>> from numpy.polynomial.hermite import hermcompanion >>> hermcompanion([1, 0, 1]) array([[0. , 0.35355339], [0.70710678, 0. ]]) r6z.Series must have maximum degree of at least 1.rrrPr>@r'Nr?.)r(r)r*rhrArBzerosrQhstacksqrtarangemultiply accumulater})r9r:matrDtopbots r0r#r#s_: ,,s CQ 1vzIJJ 1v{xx#ad(1Q4-)** A A ((A6 )C ))RBGGBryyQ2'>$>??@ AC ++  %dd +C ++b/!&QqS& !C ++b/!&QqS& !Cwwr"))Aq/)*CHH2J#f*c!B%i((J Jr1cp[R"U/5un[U5S::a[R"/UR S9$[U5S:Xa#[R"SUS-US- /5$[ U5SSS2SSS24n[R"U5nUR5 U$)ap Compute the roots of a Hermite series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \sum_i c[i] * H_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.legendre.legroots numpy.polynomial.laguerre.lagroots numpy.polynomial.chebyshev.chebroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Hermite series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive. Examples -------- >>> from numpy.polynomial.hermite import hermroots, hermfromroots >>> coef = hermfromroots([-1, 0, 1]) >>> coef array([0. , 0.25 , 0. , 0.125]) >>> hermroots(coef) array([-1.00000000e+00, -1.38777878e-17, 1.00000000e+00]) rrPr6rrNr') r(r)r*rArBrQr#laeigvalssort)r9rlrs r0rrsf ,,s CQ 1v{xx!''** 1v{xxQqT!A$(( a2dd#A 1 AFFH Hr1c HUS:XaZ[R"URS[R"[R"[R55- 5$SnS[R"[R"[R55- n[ U5n[ US- 5HKnUnU*[R"US- U- 5-nXcU-[R"SU- 5--nUS- nMM X#U-[R"S5--$)a5 Evaluate a normalized Hermite polynomial. Compute the value of the normalized Hermite polynomial of degree ``n`` at the points ``x``. Parameters ---------- x : ndarray of double. Points at which to evaluate the function n : int Degree of the normalized Hermite function to be evaluated. Returns ------- values : ndarray The shape of the return value is described above. Notes ----- This function is needed for finding the Gauss points and integration weights for high degrees. The values of the standard Hermite functions overflow when n >= 207. rrrr>rr6)rArrkrpifloatr+)r~r:r;r<rWr/r=s r0_normed_hermite_nrs6 Avwwqww"''"''"%%."9 9:: B BGGBGGBEEN # #B qB 1q5\S"r'2& & a42& & #X  1RWWQZ r1c[R"US5nUS::a [S5e[R"S/U-S/-[R S9n[ U5n[R"U5n[XA5n[XAS- 5[R"SU-5-nXEU- -n[XAS- 5nU[R"U5R5-nSXw-- nXSSS2-S- nXDSSS2- S- nU[R"[R5UR5- -nXH4$) a Gauss-Hermite quadrature. Computes the sample points and weights for Gauss-Hermite quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[-\inf, \inf]` with the weight function :math:`f(x) = \exp(-x^2)`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that .. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k)) where :math:`c` is a constant independent of :math:`k` and :math:`x_k` is the k'th root of :math:`H_n`, and then scaling the results to get the right value when integrating 1. Examples -------- >>> from numpy.polynomial.hermite import hermgauss >>> hermgauss(2) (array([-0.70710678, 0.70710678]), array([0.88622693, 0.88622693])) r-rzdeg must be a positive integerrrPr6Nr')r(rgrhrArBfloat64r#reigvalshrrabsmaxrsum) r-rr9rlr~dydffmrs r0r$r$8sN ::c5 !D qy9:: !SA3bjj1AaA AA 1 #B 1Qh '"''!D&/ 9BBJA 1Qh 'B"&&*.. B 27 A ttWaA ttWaA!%%' !!A 4Kr1c:[R"US-*5nU$)aQ Weight function of the Hermite polynomials. The weight function is :math:`\exp(-x^2)` and the interval of integration is :math:`[-\inf, \inf]`. the Hermite polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Examples -------- >>> import numpy as np >>> from numpy.polynomial.hermite import hermweight >>> x = np.arange(-2, 2) >>> hermweight(x) array([0.01831564, 0.36787944, 1. , 0.36787944]) r6)rAexp)r~rs r0r%r%~s6 1u A Hr1c$\rSrSrSr\"\5r\"\5r \"\ 5r \"\ 5r \"\5r\"\5r\"\5r\"\5r\"\5r\"\5r\"\5r\"\5r\R>"\ 5r!\R>"\ 5r"Sr#Sr$g)riaAn Hermite series class. The Hermite class provides the standard Python numerical methods '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the attributes and methods listed below. Parameters ---------- coef : array_like Hermite coefficients in order of increasing degree, i.e, ``(1, 2, 3)`` gives ``1*H_0(x) + 2*H_1(x) + 3*H_2(x)``. domain : (2,) array_like, optional Domain to use. The interval ``[domain[0], domain[1]]`` is mapped to the interval ``[window[0], window[1]]`` by shifting and scaling. The default value is [-1., 1.]. window : (2,) array_like, optional Window, see `domain` for its use. The default value is [-1., 1.]. symbol : str, optional Symbol used to represent the independent variable in string representations of the polynomial expression, e.g. for printing. The symbol must be a valid Python identifier. Default value is 'x'. .. versionadded:: 1.24 HN)%__name__ __module__ __qualname____firstlineno____doc__ staticmethodr rIr rNr_mulrrYrr[r_valr_intr_derrrr _liner_rootsrrFrArBr domainwindow basis_name__static_attributes__rr1r0rrs4  D  D  D  D  D  D  D  D  D  "E ) $Fm,JXXj !F XXj !FJr1r))rrr)T)NFN)0rnumpyrA numpy.linalglinalgrnumpy.lib.array_utilsrrr( _polybaser__all__trimcoefrrrrBr rrr r rr r rrrrrrrrrrr rr!r"rr#rrr$r%rrr1r0rs6LZ6" @ ;;+\7+@XXsBi  88QC= ((A3- !S%P53p%P%P2j?'D-$`".JN bbqaau p[|2'j8(v5*p=+@;!|8Ev9TxC:L,^= @& RCL F+k+r1