KhSr/SQrSSKrSSKrSSKrSSKJr SSKJs J r SSK J r J r JrJrJrJrJrJr SSK Jr SSKJr SS KJrJr SS KJr SS KJrJrJrJ r SS K!J"r"J#r#J$r$ \RJ"\RLS S9r&Sr'\&"\'5S5r(Sr)\&"\)5S5r*S&Sjr+\&"\+5S'Sj5r,S(Sjr-\&"\-5S)Sj5r.S*Sjr/\&"\/5S+Sj5r0Sr1\&"\15S5r2Sr3\&"\35S5r4\&"\35S5r5\&"\35S5r6Sr7\&"\75S 5r8\Rr"S!5r:S,S"jr;\"S 5"S#S$55r<\Rz"S%\5 g)-z' Functions to operate on polynomials. ) polyrootspolyintpolyderpolyaddpolysubpolymulpolydivpolyvalpoly1dpolyfitN) set_module)isscalarabsfinfo atleast_1dhstackdotarrayones) overrides) RankWarning)diagvander) trim_zeros) iscomplexrealimag mintypecode)eigvalslstsqinvnumpy)modulecU$N) seq_of_zeross L/var/www/html/env/lib/python3.13/site-packages/numpy/lib/_polynomial_impl.py_poly_dispatcherr+s c V[U5nURn[U5S:Xa!USUS:XaUSS:wa [U5nOU[U5S:Xa;URnU[ :wa$UR [UR55nO [S5e[U5S:XagURn[SUS9nUH$n[R"U[SU*/US9SS 9nM& [URR[R 5(a[R""U[$5n[R&"[R("U5[R("UR+55:H5(aUR,R/5nU$) a( Find the coefficients of a polynomial with the given sequence of roots. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. Returns the coefficients of the polynomial whose leading coefficient is one for the given sequence of zeros (multiple roots must be included in the sequence as many times as their multiplicity; see Examples). A square matrix (or array, which will be treated as a matrix) can also be given, in which case the coefficients of the characteristic polynomial of the matrix are returned. Parameters ---------- seq_of_zeros : array_like, shape (N,) or (N, N) A sequence of polynomial roots, or a square array or matrix object. Returns ------- c : ndarray 1D array of polynomial coefficients from highest to lowest degree: ``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]`` where c[0] always equals 1. Raises ------ ValueError If input is the wrong shape (the input must be a 1-D or square 2-D array). See Also -------- polyval : Compute polynomial values. roots : Return the roots of a polynomial. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class. Notes ----- Specifying the roots of a polynomial still leaves one degree of freedom, typically represented by an undetermined leading coefficient. [1]_ In the case of this function, that coefficient - the first one in the returned array - is always taken as one. (If for some reason you have one other point, the only automatic way presently to leverage that information is to use ``polyfit``.) The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n` matrix **A** is given by :math:`p_a(t) = \mathrm{det}(t\, \mathbf{I} - \mathbf{A})`, where **I** is the `n`-by-`n` identity matrix. [2]_ References ---------- .. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trigonometry, Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996. .. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition," Academic Press, pg. 182, 1980. Examples -------- Given a sequence of a polynomial's zeros: >>> import numpy as np >>> np.poly((0, 0, 0)) # Multiple root example array([1., 0., 0., 0.]) The line above represents z**3 + 0*z**2 + 0*z + 0. >>> np.poly((-1./2, 0, 1./2)) array([ 1. , 0. , -0.25, 0. ]) The line above represents z**3 - z/4 >>> np.poly((np.random.random(1)[0], 0, np.random.random(1)[0])) array([ 1. , -0.77086955, 0.08618131, 0. ]) # random Given a square array object: >>> P = np.array([[0, 1./3], [-1./2, 0]]) >>> np.poly(P) array([1. , 0. , 0.16666667]) Note how in all cases the leading coefficient is always 1. rr z.input must be 1d or non-empty square 2d array.?r.dtypefull)mode)rshapelenr!r2objectastyper char ValueErrorrNXconvolver issubclasstypecomplexfloatingasarraycomplexallsort conjugatercopy)r)shdtazerors r*rr"sG@l+L   B 2w!|1A2a5A:|, RA    <'..{277/CDLIJJ <A   B TA KK5!dU26V D!'',, 2 233 <1 66"''%.BGGEOO,=$>> ? ? A Hr,cU$r'r()ps r*_roots_dispatcherrLs Hr,cN[U5nURS:wa [S5e[R"[R "U55Sn[ U5S:Xa[R"/5$[ U5US- S- nU[US5[US5S-n[URR[R[R45(dUR[5n[ U5nUS:aN[![R""US- 4UR5S5nUSS*US- USSS24'[%U5nO[R"/5n['U[R("X%R545nU$)aO Return the roots of a polynomial with coefficients given in p. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. The values in the rank-1 array `p` are coefficients of a polynomial. If the length of `p` is n+1 then the polynomial is described by:: p[0] * x**n + p[1] * x**(n-1) + ... + p[n-1]*x + p[n] Parameters ---------- p : array_like Rank-1 array of polynomial coefficients. Returns ------- out : ndarray An array containing the roots of the polynomial. Raises ------ ValueError When `p` cannot be converted to a rank-1 array. See also -------- poly : Find the coefficients of a polynomial with a given sequence of roots. polyval : Compute polynomial values. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class. Notes ----- The algorithm relies on computing the eigenvalues of the companion matrix [1]_. References ---------- .. [1] R. A. Horn & C. R. Johnson, *Matrix Analysis*. Cambridge, UK: Cambridge University Press, 1999, pp. 146-7. Examples -------- >>> import numpy as np >>> coeff = [3.2, 2, 1] >>> np.roots(coeff) array([-0.3125+0.46351241j, -0.3125-0.46351241j]) r.zInput must be a rank-1 array.r rN)rndimr:r;nonzeroravelr6rintr=r2r>floatingr?r8floatrrr!rzeros)rKnon_zerotrailing_zerosNArs r*rrsVt 1 Avv{899zz"((1+&q)H 8}xx|Vhrl*Q.N #hqk 3x|,Q./A aggllR[["2D2D$E F F HHUO AA1u !A#)2 .AB%!A$!A#   E288NKK@A BE Lr,cU4$r'r()rKmks r*_polyint_dispatcherr] 4Kr,c [U5nUS:a [S5eUc[R"U[5n[ U5n[ U5S:Xa'US:a!US[R"U[5-n[ U5U:a [S5e[U[5n[R"U5nUS:XaU(a [U5$U$[R"UR[R"[ U5SS55US/45n[XAS- USSS9nU(a [U5$U$)a Return an antiderivative (indefinite integral) of a polynomial. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. The returned order `m` antiderivative `P` of polynomial `p` satisfies :math:`\frac{d^m}{dx^m}P(x) = p(x)` and is defined up to `m - 1` integration constants `k`. The constants determine the low-order polynomial part .. math:: \frac{k_{m-1}}{0!} x^0 + \ldots + \frac{k_0}{(m-1)!}x^{m-1} of `P` so that :math:`P^{(j)}(0) = k_{m-j-1}`. Parameters ---------- p : array_like or poly1d Polynomial to integrate. A sequence is interpreted as polynomial coefficients, see `poly1d`. m : int, optional Order of the antiderivative. (Default: 1) k : list of `m` scalars or scalar, optional Integration constants. They are given in the order of integration: those corresponding to highest-order terms come first. If ``None`` (default), all constants are assumed to be zero. If `m = 1`, a single scalar can be given instead of a list. See Also -------- polyder : derivative of a polynomial poly1d.integ : equivalent method Examples -------- The defining property of the antiderivative: >>> import numpy as np >>> p = np.poly1d([1,1,1]) >>> P = np.polyint(p) >>> P poly1d([ 0.33333333, 0.5 , 1. , 0. ]) # may vary >>> np.polyder(P) == p True The integration constants default to zero, but can be specified: >>> P = np.polyint(p, 3) >>> P(0) 0.0 >>> np.polyder(P)(0) 0.0 >>> np.polyder(P, 2)(0) 0.0 >>> P = np.polyint(p, 3, k=[6,5,3]) >>> P poly1d([ 0.01666667, 0.04166667, 0.16666667, 3. , 5. , 3. ]) # may vary Note that 3 = 6 / 2!, and that the constants are given in the order of integrations. Constant of the highest-order polynomial term comes first: >>> np.polyder(P, 2)(0) 6.0 >>> np.polyder(P, 1)(0) 5.0 >>> P(0) 3.0 r z0Order of integral must be positive (see polyder)Nr.z7k must be a scalar or a rank-1 array of length 1 or >m.rN)r\)rRr:r;rUrTrr6r isinstancer r@ concatenate __truediv__aranger)rKr[r\truepolyyvals r*rrsX AA1uKLLy HHQ 1 A 1v{q1u aDE" " 1vzGI I!V$H 1 AAv !9  NNAMM"))CFAr*BCadVL MaQ!AB%( #;  r,cU4$r'r()rKr[s r*_polyder_dispatcherrhkr^r,c4[U5nUS:a [S5e[U[5n[R "U5n[ U5S- nUSS[R"USS5-nUS:XaUnO[XAS- 5nU(a [U5nU$)a Return the derivative of the specified order of a polynomial. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. Parameters ---------- p : poly1d or sequence Polynomial to differentiate. A sequence is interpreted as polynomial coefficients, see `poly1d`. m : int, optional Order of differentiation (default: 1) Returns ------- der : poly1d A new polynomial representing the derivative. See Also -------- polyint : Anti-derivative of a polynomial. poly1d : Class for one-dimensional polynomials. Examples -------- The derivative of the polynomial :math:`x^3 + x^2 + x^1 + 1` is: >>> import numpy as np >>> p = np.poly1d([1,1,1,1]) >>> p2 = np.polyder(p) >>> p2 poly1d([3, 2, 1]) which evaluates to: >>> p2(2.) 17.0 We can verify this, approximating the derivative with ``(f(x + h) - f(x))/h``: >>> (p(2. + 0.001) - p(2.)) / 0.001 17.007000999997857 The fourth-order derivative of a 3rd-order polynomial is zero: >>> np.polyder(p, 2) poly1d([6, 2]) >>> np.polyder(p, 3) poly1d([6]) >>> np.polyder(p, 4) poly1d([0]) r z2Order of derivative must be positive (see polyint)r.NrN) rRr:r`r r;r@r6rcr)rKr[rdnrerfs r*rrosz AA1uMNN!V$H 1 A A A #21a$$AAvaQSk Jr,c XU4$r'r()xredegrcondr3wcovs r*_polyfit_dispatcherrqs !9r,c[U5S-n[R"U5S-n[R"U5S-nUS:a [S5eURS:wa [ S5eUR S:Xa [ S5eURS:dURS:a [ S5eURSURS:wa [ S 5eUc+[U5[UR5R-n[X5nUn Ub[R"U5S-nURS:wa [ S 5eURSURS:wa [ S 5eXS S 2[R4-nU RS:XaXS S 2[R4-n OX-n [R"X-RSS 95n X-n[!XU5uppU R"U - R"n X:wa#U(dSn[$R&"U[(SS9 U(aXXU4$U(a[+[-UR"U55nU[R."X5-nUS:XaSnO+[U5U::a [S5eU [U5U- - nURS:XaU UU-4$U US S 2S S 2[R4U-4$U $)a Least squares polynomial fit. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. Fit a polynomial ``p(x) = p[0] * x**deg + ... + p[deg]`` of degree `deg` to points `(x, y)`. Returns a vector of coefficients `p` that minimises the squared error in the order `deg`, `deg-1`, ... `0`. The `Polynomial.fit ` class method is recommended for new code as it is more stable numerically. See the documentation of the method for more information. 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 Degree of the fitting polynomial 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. cov : bool or str, optional If given and not `False`, return not just the estimate but also its covariance matrix. By default, the covariance are scaled by chi2/dof, where dof = M - (deg + 1), i.e., the weights are presumed to be unreliable except in a relative sense and everything is scaled such that the reduced chi2 is unity. This scaling is omitted if ``cov='unscaled'``, as is relevant for the case that the weights are w = 1/sigma, with sigma known to be a reliable estimate of the uncertainty. Returns ------- p : ndarray, shape (deg + 1,) or (deg + 1, K) Polynomial coefficients, highest power first. If `y` was 2-D, the coefficients for `k`-th data set are in ``p[:,k]``. residuals, rank, singular_values, rcond These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the effective rank of the scaled Vandermonde coefficient matrix - singular_values -- singular values of the scaled Vandermonde coefficient matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. V : ndarray, shape (deg + 1, deg + 1) or (deg + 1, deg + 1, K) Present only if ``full == False`` and ``cov == True``. The covariance matrix of the polynomial coefficient estimates. The diagonal of this matrix are the variance estimates for each coefficient. If y is a 2-D array, then the covariance matrix for the `k`-th data set are in ``V[:,:,k]`` 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 -------- polyval : Compute polynomial values. linalg.lstsq : Computes a least-squares fit. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution minimizes the squared error .. math:: E = \sum_{j=0}^k |p(x_j) - y_j|^2 in the equations:: x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0] x[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1] ... x[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k] The coefficient matrix of the coefficients `p` is a Vandermonde matrix. `polyfit` issues a `~exceptions.RankWarning` when the least-squares fit is badly conditioned. This implies that the best fit is not well-defined due to numerical error. The results may be improved by lowering the polynomial degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious: including contributions from the small singular values can add numerical noise to the result. Note that fitting polynomial coefficients is inherently badly conditioned when the degree of the polynomial is large or the interval of sample points is badly centered. The quality of the fit should always be checked in these cases. When polynomial fits are not satisfactory, splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting .. [2] Wikipedia, "Polynomial interpolation", https://en.wikipedia.org/wiki/Polynomial_interpolation Examples -------- >>> import numpy as np >>> import warnings >>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0]) >>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0]) >>> z = np.polyfit(x, y, 3) >>> z array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254]) # may vary It is convenient to use `poly1d` objects for dealing with polynomials: >>> p = np.poly1d(z) >>> p(0.5) 0.6143849206349179 # may vary >>> p(3.5) -0.34732142857143039 # may vary >>> p(10) 22.579365079365115 # may vary High-order polynomials may oscillate wildly: >>> with warnings.catch_warnings(): ... warnings.simplefilter('ignore', np.exceptions.RankWarning) ... p30 = np.poly1d(np.polyfit(x, y, 30)) ... >>> p30(4) -0.80000000000000204 # may vary >>> p30(5) -0.99999999999999445 # may vary >>> p30(4.5) -0.10547061179440398 # may vary Illustration: >>> import matplotlib.pyplot as plt >>> xp = np.linspace(-2, 6, 100) >>> _ = plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--') >>> plt.ylim(-2,2) (-2, 2) >>> plt.show() r.r zexpected deg >= 0zexpected 1D vector for xzexpected non-empty vector for xrzexpected 1D or 2D array for yz$expected x and y to have same lengthNz expected a 1-d array for weightsz(expected w and y to have the same length)axisz!Polyfit may be poorly conditioned stacklevelunscaledzJthe number of data points must exceed order to scale the covariance matrix)rRr;r@r:rO TypeErrorsizer5r6rr2epsrnewaxissqrtsumr"Twarningswarnrr#router)rlrermrnr3rorporderlhsrhsscalecresidsranksmsgVbasefacs r*r r sb HqLE 1 A 1 A Qw,--vv{233vv{9::vvzQVVaZ788wwqzQWWQZ>?? }AuQWW~)))  C C } JJqMC  66Q;>? ? 771: #FG G BJJ 88q= Q ]# #C HC GGSWMMqM) *ELCs/At U A }T1 c;15 $5(( CsO$ %'' * C1v "BCC CFUN+C 66Q;eck> !eAaO,s22 2r,cX4$r'r()rKrls r*_polyval_dispatcherr 6Mr,c[R"U5n[U[5(aSnO,[R"U5n[R "U5nUH nX!-U-nM U$)a Evaluate a polynomial at specific values. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. If `p` is of length N, this function returns the value:: p[0]*x**(N-1) + p[1]*x**(N-2) + ... + p[N-2]*x + p[N-1] If `x` is a sequence, then ``p(x)`` is returned for each element of ``x``. If `x` is another polynomial then the composite polynomial ``p(x(t))`` is returned. Parameters ---------- p : array_like or poly1d object 1D array of polynomial coefficients (including coefficients equal to zero) from highest degree to the constant term, or an instance of poly1d. x : array_like or poly1d object A number, an array of numbers, or an instance of poly1d, at which to evaluate `p`. Returns ------- values : ndarray or poly1d If `x` is a poly1d instance, the result is the composition of the two polynomials, i.e., `x` is "substituted" in `p` and the simplified result is returned. In addition, the type of `x` - array_like or poly1d - governs the type of the output: `x` array_like => `values` array_like, `x` a poly1d object => `values` is also. See Also -------- poly1d: A polynomial class. Notes ----- Horner's scheme [1]_ is used to evaluate the polynomial. Even so, for polynomials of high degree the values may be inaccurate due to rounding errors. Use carefully. If `x` is a subtype of `ndarray` the return value will be of the same type. References ---------- .. [1] I. N. Bronshtein, K. A. Semendyayev, and K. A. Hirsch (Eng. trans. Ed.), *Handbook of Mathematics*, New York, Van Nostrand Reinhold Co., 1985, pg. 720. Examples -------- >>> import numpy as np >>> np.polyval([3,0,1], 5) # 3 * 5**2 + 0 * 5**1 + 1 76 >>> np.polyval([3,0,1], np.poly1d(5)) poly1d([76]) >>> np.polyval(np.poly1d([3,0,1]), 5) 76 >>> np.polyval(np.poly1d([3,0,1]), np.poly1d(5)) poly1d([76]) r )r;r@r`r asanyarray zeros_like)rKrlrepvs r*r r sZJ 1 A!V  MM!  MM!  EBJ Hr,cX4$r'r()a1a2s r*_binary_op_dispatcherrs 8Or,c[U[5=(d [U[5n[U5n[U5n[U5[U5- nUS:XaX-nOUS:a;[R "X0R 5n[R"XP45U-nOD[R "[U5UR 5nU[R"XQ45-nU(a [U5nU$)a Find the sum of two polynomials. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. Returns the polynomial resulting from the sum of two input polynomials. Each input must be either a poly1d object or a 1D sequence of polynomial coefficients, from highest to lowest degree. Parameters ---------- a1, a2 : array_like or poly1d object Input polynomials. Returns ------- out : ndarray or poly1d object The sum of the inputs. If either input is a poly1d object, then the output is also a poly1d object. Otherwise, it is a 1D array of polynomial coefficients from highest to lowest degree. See Also -------- poly1d : A one-dimensional polynomial class. poly, polyadd, polyder, polydiv, polyfit, polyint, polysub, polyval Examples -------- >>> import numpy as np >>> np.polyadd([1, 2], [9, 5, 4]) array([9, 6, 6]) Using poly1d objects: >>> p1 = np.poly1d([1, 2]) >>> p2 = np.poly1d([9, 5, 4]) >>> print(p1) 1 x + 2 >>> print(p2) 2 9 x + 5 x + 4 >>> print(np.polyadd(p1, p2)) 2 9 x + 6 x + 6 r r`r rr6r;rUr2rarrrrddiffrfzrs r*rrsh2v&@*R*@H BB BB r7SW D qyg  XXdHH %nnbX&+ XXc$i *2>>2(++Sk Jr,c[U[5=(d [U[5n[U5n[U5n[U5[U5- nUS:XaX- nOUS:a;[R "X0R 5n[R"XP45U- nOD[R "[U5UR 5nU[R"XQ45- nU(a [U5nU$)a Difference (subtraction) of two polynomials. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. Given two polynomials `a1` and `a2`, returns ``a1 - a2``. `a1` and `a2` can be either array_like sequences of the polynomials' coefficients (including coefficients equal to zero), or `poly1d` objects. Parameters ---------- a1, a2 : array_like or poly1d Minuend and subtrahend polynomials, respectively. Returns ------- out : ndarray or poly1d Array or `poly1d` object of the difference polynomial's coefficients. See Also -------- polyval, polydiv, polymul, polyadd Examples -------- .. math:: (2 x^2 + 10 x - 2) - (3 x^2 + 10 x -4) = (-x^2 + 2) >>> import numpy as np >>> np.polysub([2, 10, -2], [3, 10, -4]) array([-1, 0, 2]) r rrs r*rrWsN2v&@*R*@H BB BB r7SW D qyg  XXdHH %nnbX&+ XXc$i *2>>2(++Sk Jr,c[U[5=(d [U[5n[U5[U5p[R"X5nU(a [U5nU$)a Find the product of two polynomials. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. Finds the polynomial resulting from the multiplication of the two input polynomials. Each input must be either a poly1d object or a 1D sequence of polynomial coefficients, from highest to lowest degree. Parameters ---------- a1, a2 : array_like or poly1d object Input polynomials. Returns ------- out : ndarray or poly1d object The polynomial resulting from the multiplication of the inputs. If either inputs is a poly1d object, then the output is also a poly1d object. Otherwise, it is a 1D array of polynomial coefficients from highest to lowest degree. See Also -------- poly1d : A one-dimensional polynomial class. poly, polyadd, polyder, polydiv, polyfit, polyint, polysub, polyval convolve : Array convolution. Same output as polymul, but has parameter for overlap mode. Examples -------- >>> import numpy as np >>> np.polymul([1, 2, 3], [9, 5, 1]) array([ 9, 23, 38, 17, 3]) Using poly1d objects: >>> p1 = np.poly1d([1, 2, 3]) >>> p2 = np.poly1d([9, 5, 1]) >>> print(p1) 2 1 x + 2 x + 3 >>> print(p2) 2 9 x + 5 x + 1 >>> print(np.polymul(p1, p2)) 4 3 2 9 x + 23 x + 38 x + 17 x + 3 )r`r r;r<)rrrdrfs r*rrsKp2v&@*R*@H BZ ++b CSk Jr,cX4$r'r()uvs r*_polydiv_dispatcherrrr,c[U[5=(d [U[5n[U5S-n[U5S-nUSUS-n[U5S- n[U5S- nSUS- n[R "[ XE- S-S54UR5nURUR5n[SXE- S-5H"n XhU -n XU 'XX-S-===X--sss&M$ [R"USSSS9(aKURSS:a8USSn[R"USSSS9(aURSS:aM8U(a[U5[U54$Xx4$) a Returns the quotient and remainder of polynomial division. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. The input arrays are the coefficients (including any coefficients equal to zero) of the "numerator" (dividend) and "denominator" (divisor) polynomials, respectively. Parameters ---------- u : array_like or poly1d Dividend polynomial's coefficients. v : array_like or poly1d Divisor polynomial's coefficients. Returns ------- q : ndarray Coefficients, including those equal to zero, of the quotient. r : ndarray Coefficients, including those equal to zero, of the remainder. See Also -------- poly, polyadd, polyder, polydiv, polyfit, polyint, polymul, polysub polyval Notes ----- Both `u` and `v` must be 0-d or 1-d (ndim = 0 or 1), but `u.ndim` need not equal `v.ndim`. In other words, all four possible combinations - ``u.ndim = v.ndim = 0``, ``u.ndim = v.ndim = 1``, ``u.ndim = 1, v.ndim = 0``, and ``u.ndim = 0, v.ndim = 1`` - work. Examples -------- .. math:: \frac{3x^2 + 5x + 2}{2x + 1} = 1.5x + 1.75, remainder 0.25 >>> import numpy as np >>> x = np.array([3.0, 5.0, 2.0]) >>> y = np.array([2.0, 1.0]) >>> np.polydiv(x, y) (array([1.5 , 1.75]), array([0.25])) rsr r.r/g+=)rtolrNN) r`r rr6r;rUmaxr2r8rangeallcloser5) rrrdror[rjrqrr\ds r*r r sUj1f%>Av)>H1 A1 A !qt A A A A A 1IE #aeai#%qww/A A 1ac!e_ aDL! ACE ac  ++adAE * a abE ++adAE * aay&)## 4Kr,z \*\*([0-9]*)cSnSnSnSn[RX5nUcOUR5nUR5SnXUSn USnU S[ U5S- --n S[ U 5S- -U-n [ U5[ U 5-U:d[ U5[ U 5-U:aXSS-U-S-- nU nU nO-XIS[ U5S- --- nUS[ U 5S- -U-- nMXSS-U-- nXPUS-$)Nr  r. z ) _poly_matsearchspangroupsr6) astrwraprjline1line2outputmatrpowerpartstrtoadd2toadd1s r* _raise_powerrs6 A E E F t' ; xxz Qa/ G3E 1 --c'l1n%- Z#f+ % ,Uc&k)D0 dlU*U2 2FEE sCJqL11 1E S#g,q.)E1 1E# $ dlU""F H r,cp\rSrSrSrSr\S5r\RS5r\S5r \S5r \S5r \S 5r \ RS 5r \ r \=r=rr\ rS$S jrS%S jrS rSrSrSrSrSrSrSrSrSrSrSrSr Sr!\!r"Sr#\#r$Sr%Sr&Sr'Sr(S r)S&S!jr*S'S"jr+S#r,g)(r i8aD A one-dimensional polynomial class. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide `. A convenience class, used to encapsulate "natural" operations on polynomials so that said operations may take on their customary form in code (see Examples). Parameters ---------- c_or_r : array_like The polynomial's coefficients, in decreasing powers, or if the value of the second parameter is True, the polynomial's roots (values where the polynomial evaluates to 0). For example, ``poly1d([1, 2, 3])`` returns an object that represents :math:`x^2 + 2x + 3`, whereas ``poly1d([1, 2, 3], True)`` returns one that represents :math:`(x-1)(x-2)(x-3) = x^3 - 6x^2 + 11x -6`. r : bool, optional If True, `c_or_r` specifies the polynomial's roots; the default is False. variable : str, optional Changes the variable used when printing `p` from `x` to `variable` (see Examples). Examples -------- Construct the polynomial :math:`x^2 + 2x + 3`: >>> import numpy as np >>> p = np.poly1d([1, 2, 3]) >>> print(np.poly1d(p)) 2 1 x + 2 x + 3 Evaluate the polynomial at :math:`x = 0.5`: >>> p(0.5) 4.25 Find the roots: >>> p.r array([-1.+1.41421356j, -1.-1.41421356j]) >>> p(p.r) array([ -4.44089210e-16+0.j, -4.44089210e-16+0.j]) # may vary These numbers in the previous line represent (0, 0) to machine precision Show the coefficients: >>> p.c array([1, 2, 3]) Display the order (the leading zero-coefficients are removed): >>> p.order 2 Show the coefficient of the k-th power in the polynomial (which is equivalent to ``p.c[-(i+1)]``): >>> p[1] 2 Polynomials can be added, subtracted, multiplied, and divided (returns quotient and remainder): >>> p * p poly1d([ 1, 4, 10, 12, 9]) >>> (p**3 + 4) / p (poly1d([ 1., 4., 10., 12., 9.]), poly1d([4.])) ``asarray(p)`` gives the coefficient array, so polynomials can be used in all functions that accept arrays: >>> p**2 # square of polynomial poly1d([ 1, 4, 10, 12, 9]) >>> np.square(p) # square of individual coefficients array([1, 4, 9]) The variable used in the string representation of `p` can be modified, using the `variable` parameter: >>> p = np.poly1d([1,2,3], variable='z') >>> print(p) 2 1 z + 2 z + 3 Construct a polynomial from its roots: >>> np.poly1d([1, 2], True) poly1d([ 1., -3., 2.]) This is the same polynomial as obtained by: >>> np.poly1d([1, -1]) * np.poly1d([1, -2]) poly1d([ 1, -3, 2]) NcUR$)zThe polynomial coefficients )_coeffsselfs r*coeffs poly1d.coeffss||r,c6XRLa [S5eg)NzCannot set attribute)rAttributeError)rvalues r*rrs  $ !78 8 %r,cUR$)z$The name of the polynomial variable ) _variablers r*variablepoly1d.variables~~r,c2[UR5S- $)z&The order or degree of the polynomial r.)r6rrs r*r poly1d.orders4<< 1$$r,c,[UR5$)z0The roots of the polynomial, where self(x) == 0 )rrrs r*r poly1d.rootssT\\""r,c URS$Nr__dict__rs r*rpoly1d._coeffss}}X&&r,c XRS'grr)rrs r*rrs"( hr,cl[U[5(aURUlURUl[ UR 5[ UR 5- (aASn[ R"U[SS9 UR RUR 5 UbX0lgU(a [U5n[U5nURS:a [S5e[USS9n[U5S:Xa [ R""S/UR$S 9nXlUcS nX0lg) NzbIn the future extra properties will not be copied across when constructing one poly1d from anotherrrur.zPolynomial must be 1d only.f)trimr r1rl)r`r rrsetrrr FutureWarningupdaterrrOr:rr6r;rr2)rc_or_rrrrs r*__init__poly1d.__init__s ff % %#--DN!>>DL6??#c$--&88J c=Q? $$V__5#!)  &\FF# ;;?:; ;F- v;! 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