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diff DEPENDENCIES/mingw32/Python27/Lib/site-packages/numpy/polynomial/polynomial.py @ 87:2a2c65a20a8b
Add Python libs and headers
| author | Chris Cannam |
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| date | Wed, 25 Feb 2015 14:05:22 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/DEPENDENCIES/mingw32/Python27/Lib/site-packages/numpy/polynomial/polynomial.py Wed Feb 25 14:05:22 2015 +0000 @@ -0,0 +1,1532 @@ +""" +Objects for dealing with polynomials. + +This module provides a number of objects (mostly functions) useful for +dealing with polynomials, including a `Polynomial` class that +encapsulates the usual arithmetic operations. (General information +on how this module represents and works with polynomial objects is in +the docstring for its "parent" sub-package, `numpy.polynomial`). + +Constants +--------- +- `polydomain` -- Polynomial default domain, [-1,1]. +- `polyzero` -- (Coefficients of the) "zero polynomial." +- `polyone` -- (Coefficients of the) constant polynomial 1. +- `polyx` -- (Coefficients of the) identity map polynomial, ``f(x) = x``. + +Arithmetic +---------- +- `polyadd` -- add two polynomials. +- `polysub` -- subtract one polynomial from another. +- `polymul` -- multiply two polynomials. +- `polydiv` -- divide one polynomial by another. +- `polypow` -- raise a polynomial to an positive integer power +- `polyval` -- evaluate a polynomial at given points. +- `polyval2d` -- evaluate a 2D polynomial at given points. +- `polyval3d` -- evaluate a 3D polynomial at given points. +- `polygrid2d` -- evaluate a 2D polynomial on a Cartesian product. +- `polygrid3d` -- evaluate a 3D polynomial on a Cartesian product. + +Calculus +-------- +- `polyder` -- differentiate a polynomial. +- `polyint` -- integrate a polynomial. + +Misc Functions +-------------- +- `polyfromroots` -- create a polynomial with specified roots. +- `polyroots` -- find the roots of a polynomial. +- `polyvander` -- Vandermonde-like matrix for powers. +- `polyvander2d` -- Vandermonde-like matrix for 2D power series. +- `polyvander3d` -- Vandermonde-like matrix for 3D power series. +- `polycompanion` -- companion matrix in power series form. +- `polyfit` -- least-squares fit returning a polynomial. +- `polytrim` -- trim leading coefficients from a polynomial. +- `polyline` -- polynomial representing given straight line. + +Classes +------- +- `Polynomial` -- polynomial class. + +See Also +-------- +`numpy.polynomial` + +""" +from __future__ import division, absolute_import, print_function + +__all__ = [ + 'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd', + 'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval', + 'polyder', 'polyint', 'polyfromroots', 'polyvander', 'polyfit', + 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d', + 'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d'] + +import warnings +import numpy as np +import numpy.linalg as la + +from . import polyutils as pu +from ._polybase import ABCPolyBase + +polytrim = pu.trimcoef + +# +# These are constant arrays are of integer type so as to be compatible +# with the widest range of other types, such as Decimal. +# + +# Polynomial default domain. +polydomain = np.array([-1, 1]) + +# Polynomial coefficients representing zero. +polyzero = np.array([0]) + +# Polynomial coefficients representing one. +polyone = np.array([1]) + +# Polynomial coefficients representing the identity x. +polyx = np.array([0, 1]) + +# +# Polynomial series functions +# + + +def polyline(off, scl): + """ + Returns an array representing a linear polynomial. + + Parameters + ---------- + off, scl : scalars + The "y-intercept" and "slope" of the line, respectively. + + Returns + ------- + y : ndarray + This module's representation of the linear polynomial ``off + + scl*x``. + + See Also + -------- + chebline + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> P.polyline(1,-1) + array([ 1, -1]) + >>> P.polyval(1, P.polyline(1,-1)) # should be 0 + 0.0 + + """ + if scl != 0: + return np.array([off, scl]) + else: + return np.array([off]) + + +def polyfromroots(roots): + """ + Generate a monic polynomial with given roots. + + Return the coefficients of the polynomial + + .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), + + where the `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 * x + ... + x^n + + The coefficient of the last term is 1 for monic polynomials in this + form. + + Parameters + ---------- + roots : array_like + Sequence containing the roots. + + Returns + ------- + out : ndarray + 1-D array of the polynomial's coefficients If all the roots are + real, then `out` is also real, otherwise it is complex. (see + Examples below). + + See Also + -------- + chebfromroots, legfromroots, lagfromroots, hermfromroots + hermefromroots + + Notes + ----- + The coefficients are determined by multiplying together linear factors + of the form `(x - r_i)`, i.e. + + .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n) + + where ``n == len(roots) - 1``; note that this implies that `1` is always + returned for :math:`a_n`. + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x + array([ 0., -1., 0., 1.]) + >>> j = complex(0,1) + >>> P.polyfromroots((-j,j)) # complex returned, though values are real + array([ 1.+0.j, 0.+0.j, 1.+0.j]) + + """ + if len(roots) == 0: + return np.ones(1) + else: + [roots] = pu.as_series([roots], trim=False) + roots.sort() + p = [polyline(-r, 1) for r in roots] + n = len(p) + while n > 1: + m, r = divmod(n, 2) + tmp = [polymul(p[i], p[i+m]) for i in range(m)] + if r: + tmp[0] = polymul(tmp[0], p[-1]) + p = tmp + n = m + return p[0] + + +def polyadd(c1, c2): + """ + Add one polynomial to another. + + Returns the sum of two polynomials `c1` + `c2`. The arguments are + sequences of coefficients from lowest order term to highest, i.e., + [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of polynomial coefficients ordered from low to high. + + Returns + ------- + out : ndarray + The coefficient array representing their sum. + + See Also + -------- + polysub, polymul, polydiv, polypow + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> sum = P.polyadd(c1,c2); sum + array([ 4., 4., 4.]) + >>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2) + 28.0 + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + if len(c1) > len(c2): + c1[:c2.size] += c2 + ret = c1 + else: + c2[:c1.size] += c1 + ret = c2 + return pu.trimseq(ret) + + +def polysub(c1, c2): + """ + Subtract one polynomial from another. + + Returns the difference of two polynomials `c1` - `c2`. The arguments + are sequences of coefficients from lowest order term to highest, i.e., + [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of polynomial coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of coefficients representing their difference. + + See Also + -------- + polyadd, polymul, polydiv, polypow + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> P.polysub(c1,c2) + array([-2., 0., 2.]) + >>> P.polysub(c2,c1) # -P.polysub(c1,c2) + array([ 2., 0., -2.]) + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + if len(c1) > len(c2): + c1[:c2.size] -= c2 + ret = c1 + else: + c2 = -c2 + c2[:c1.size] += c1 + ret = c2 + return pu.trimseq(ret) + + +def polymulx(c): + """Multiply a polynomial by x. + + Multiply the polynomial `c` by x, where x is the independent + variable. + + + Parameters + ---------- + c : array_like + 1-D array of polynomial coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the result of the multiplication. + + Notes + ----- + + .. versionadded:: 1.5.0 + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + # The zero series needs special treatment + if len(c) == 1 and c[0] == 0: + return c + + prd = np.empty(len(c) + 1, dtype=c.dtype) + prd[0] = c[0]*0 + prd[1:] = c + return prd + + +def polymul(c1, c2): + """ + Multiply one polynomial by another. + + Returns the product of two polynomials `c1` * `c2`. The arguments are + sequences of coefficients, from lowest order term to highest, e.g., + [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.`` + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of coefficients representing a polynomial, relative to the + "standard" basis, and ordered from lowest order term to highest. + + Returns + ------- + out : ndarray + Of the coefficients of their product. + + See Also + -------- + polyadd, polysub, polydiv, polypow + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> P.polymul(c1,c2) + array([ 3., 8., 14., 8., 3.]) + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + ret = np.convolve(c1, c2) + return pu.trimseq(ret) + + +def polydiv(c1, c2): + """ + Divide one polynomial by another. + + Returns the quotient-with-remainder of two polynomials `c1` / `c2`. + The arguments are sequences of coefficients, from lowest order term + to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of polynomial coefficients ordered from low to high. + + Returns + ------- + [quo, rem] : ndarrays + Of coefficient series representing the quotient and remainder. + + See Also + -------- + polyadd, polysub, polymul, polypow + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> P.polydiv(c1,c2) + (array([ 3.]), array([-8., -4.])) + >>> P.polydiv(c2,c1) + (array([ 0.33333333]), array([ 2.66666667, 1.33333333])) + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + if c2[-1] == 0: + raise ZeroDivisionError() + + len1 = len(c1) + len2 = len(c2) + if len2 == 1: + return c1/c2[-1], c1[:1]*0 + elif len1 < len2: + return c1[:1]*0, c1 + else: + dlen = len1 - len2 + scl = c2[-1] + c2 = c2[:-1]/scl + i = dlen + j = len1 - 1 + while i >= 0: + c1[i:j] -= c2*c1[j] + i -= 1 + j -= 1 + return c1[j+1:]/scl, pu.trimseq(c1[:j+1]) + + +def polypow(c, pow, maxpower=None): + """Raise a polynomial to a power. + + Returns the polynomial `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 ``1 + 2*x + 3*x**2.`` + + Parameters + ---------- + c : array_like + 1-D array of array of series coefficients ordered from low to + high degree. + 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 + Power series of power. + + See Also + -------- + polyadd, polysub, polymul, polydiv + + Examples + -------- + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + power = int(pow) + if power != pow or power < 0: + raise ValueError("Power must be a non-negative integer.") + elif maxpower is not None and power > maxpower: + raise ValueError("Power is too large") + elif power == 0: + return np.array([1], dtype=c.dtype) + elif power == 1: + return c + else: + # This can be made more efficient by using powers of two + # in the usual way. + prd = c + for i in range(2, power + 1): + prd = np.convolve(prd, c) + return prd + + +def polyder(c, m=1, scl=1, axis=0): + """ + Differentiate a polynomial. + + Returns the polynomial 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 polynomial ``1 + 2*x + 3*x**2`` + while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is + ``x`` and axis=1 is ``y``. + + Parameters + ---------- + c : array_like + Array of polynomial 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). + + .. versionadded:: 1.7.0 + + Returns + ------- + der : ndarray + Polynomial coefficients of the derivative. + + See Also + -------- + polyint + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3 + >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2 + array([ 2., 6., 12.]) + >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24 + array([ 24.]) + >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2 + array([ -2., -6., -12.]) + >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x + array([ 6., 24.]) + + """ + c = np.array(c, ndmin=1, copy=1) + if c.dtype.char in '?bBhHiIlLqQpP': + # astype fails with NA + c = c + 0.0 + cdt = c.dtype + cnt, iaxis = [int(t) for t in [m, axis]] + + if cnt != m: + raise ValueError("The order of derivation must be integer") + if cnt < 0: + raise ValueError("The order of derivation must be non-negative") + if iaxis != axis: + raise ValueError("The axis must be integer") + if not -c.ndim <= iaxis < c.ndim: + raise ValueError("The axis is out of range") + if iaxis < 0: + iaxis += c.ndim + + if cnt == 0: + return c + + c = np.rollaxis(c, iaxis) + n = len(c) + if cnt >= n: + c = c[:1]*0 + else: + for i in range(cnt): + n = n - 1 + c *= scl + der = np.empty((n,) + c.shape[1:], dtype=cdt) + for j in range(n, 0, -1): + der[j - 1] = j*c[j] + c = der + c = np.rollaxis(c, 0, iaxis + 1) + return c + + +def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0): + """ + Integrate a polynomial. + + Returns the polynomial 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 polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]] + represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is + ``y``. + + Parameters + ---------- + c : array_like + 1-D array of polynomial coefficients, ordered from low to high. + 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 zero + is the first value in the list, the value of the second integral + at zero 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). + + .. versionadded:: 1.7.0 + + Returns + ------- + S : ndarray + Coefficient array of the integral. + + Raises + ------ + ValueError + If ``m < 1``, ``len(k) > m``. + + See Also + -------- + polyder + + 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. + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c = (1,2,3) + >>> P.polyint(c) # should return array([0, 1, 1, 1]) + array([ 0., 1., 1., 1.]) + >>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20]) + array([ 0. , 0. , 0. , 0.16666667, 0.08333333, + 0.05 ]) + >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1]) + array([ 3., 1., 1., 1.]) + >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1]) + array([ 6., 1., 1., 1.]) + >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2]) + array([ 0., -2., -2., -2.]) + + """ + c = np.array(c, ndmin=1, copy=1) + if c.dtype.char in '?bBhHiIlLqQpP': + # astype doesn't preserve mask attribute. + c = c + 0.0 + cdt = c.dtype + if not np.iterable(k): + k = [k] + cnt, iaxis = [int(t) for t in [m, axis]] + + if cnt != m: + raise ValueError("The order of integration must be integer") + if cnt < 0: + raise ValueError("The order of integration must be non-negative") + if len(k) > cnt: + raise ValueError("Too many integration constants") + if iaxis != axis: + raise ValueError("The axis must be integer") + if not -c.ndim <= iaxis < c.ndim: + raise ValueError("The axis is out of range") + if iaxis < 0: + iaxis += c.ndim + + if cnt == 0: + return c + + k = list(k) + [0]*(cnt - len(k)) + c = np.rollaxis(c, iaxis) + for i in range(cnt): + n = len(c) + c *= scl + if n == 1 and np.all(c[0] == 0): + c[0] += k[i] + else: + tmp = np.empty((n + 1,) + c.shape[1:], dtype=cdt) + tmp[0] = c[0]*0 + tmp[1] = c[0] + for j in range(1, n): + tmp[j + 1] = c[j]/(j + 1) + tmp[0] += k[i] - polyval(lbnd, tmp) + c = tmp + c = np.rollaxis(c, 0, iaxis + 1) + return c + + +def polyval(x, c, tensor=True): + """ + Evaluate a polynomial at points x. + + If `c` is of length `n + 1`, this function returns the value + + .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n + + 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 + 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. + + .. versionadded:: 1.7.0 + + Returns + ------- + values : ndarray, compatible object + The shape of the returned array is described above. + + See Also + -------- + polyval2d, polygrid2d, polyval3d, polygrid3d + + Notes + ----- + The evaluation uses Horner's method. + + Examples + -------- + >>> from numpy.polynomial.polynomial import polyval + >>> polyval(1, [1,2,3]) + 6.0 + >>> a = np.arange(4).reshape(2,2) + >>> a + array([[0, 1], + [2, 3]]) + >>> polyval(a, [1,2,3]) + array([[ 1., 6.], + [ 17., 34.]]) + >>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients + >>> coef + array([[0, 1], + [2, 3]]) + >>> polyval([1,2], coef, tensor=True) + array([[ 2., 4.], + [ 4., 7.]]) + >>> polyval([1,2], coef, tensor=False) + array([ 2., 7.]) + + """ + c = np.array(c, ndmin=1, copy=0) + if c.dtype.char in '?bBhHiIlLqQpP': + # astype fails with NA + c = c + 0.0 + if isinstance(x, (tuple, list)): + x = np.asarray(x) + if isinstance(x, np.ndarray) and tensor: + c = c.reshape(c.shape + (1,)*x.ndim) + + c0 = c[-1] + x*0 + for i in range(2, len(c) + 1): + c0 = c[-i] + c0*x + return c0 + + +def polyval2d(x, y, c): + """ + Evaluate a 2-D polynomial at points (x, y). + + This function returns the value + + .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j + + 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` 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 `(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 + -------- + polyval, polygrid2d, polyval3d, polygrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + try: + x, y = np.array((x, y), copy=0) + except: + raise ValueError('x, y are incompatible') + + c = polyval(x, c) + c = polyval(y, c, tensor=False) + return c + + +def polygrid2d(x, y, c): + """ + Evaluate a 2-D polynomial on the Cartesian product of x and y. + + This function returns the values: + + .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j + + 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 + y.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 + -------- + polyval, polyval2d, polyval3d, polygrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + c = polyval(x, c) + c = polyval(y, c) + return c + + +def polyval3d(x, y, z, c): + """ + Evaluate a 3-D polynomial at points (x, y, z). + + This function returns the values: + + .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k + + 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 + -------- + polyval, polyval2d, polygrid2d, polygrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + try: + x, y, z = np.array((x, y, z), copy=0) + except: + raise ValueError('x, y, z are incompatible') + + c = polyval(x, c) + c = polyval(y, c, tensor=False) + c = polyval(z, c, tensor=False) + return c + + +def polygrid3d(x, y, z, c): + """ + Evaluate a 3-D polynomial 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} * a^i * b^j * c^k + + 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 + -------- + polyval, polyval2d, polygrid2d, polyval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + c = polyval(x, c) + c = polyval(y, c) + c = polyval(z, c) + return c + + +def polyvander(x, deg): + """Vandermonde matrix of given degree. + + Returns the Vandermonde matrix of degree `deg` and sample points + `x`. The Vandermonde matrix is defined by + + .. math:: V[..., i] = x^i, + + where `0 <= i <= deg`. The leading indices of `V` index the elements of + `x` and the last index is the power of `x`. + + If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the + matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and + ``polyval(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 polynomials 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 Vandermonde matrix. The shape of the returned matrix is + ``x.shape + (deg + 1,)``, where the last index is the power of `x`. + The dtype will be the same as the converted `x`. + + See Also + -------- + polyvander2d, polyvander3d + + """ + ideg = int(deg) + if ideg != deg: + raise ValueError("deg must be integer") + if ideg < 0: + raise ValueError("deg must be non-negative") + + x = np.array(x, copy=0, ndmin=1) + 0.0 + dims = (ideg + 1,) + x.shape + dtyp = x.dtype + v = np.empty(dims, dtype=dtyp) + v[0] = x*0 + 1 + if ideg > 0: + v[1] = x + for i in range(2, ideg + 1): + v[i] = v[i-1]*x + return np.rollaxis(v, 0, v.ndim) + + +def polyvander2d(x, y, deg): + """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]*i + j] = x^i * y^j, + + 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 powers of + `x` and `y`. + + If ``V = polyvander2d(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 ``polyval2d(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 polynomials + 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 + -------- + polyvander, polyvander3d. polyval2d, polyval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy = ideg + x, y = np.array((x, y), copy=0) + 0.0 + + vx = polyvander(x, degx) + vy = polyvander(y, degy) + v = vx[..., None]*vy[..., None,:] + # einsum bug + #v = np.einsum("...i,...j->...ij", vx, vy) + return v.reshape(v.shape[:-2] + (-1,)) + + +def polyvander3d(x, y, z, deg): + """Pseudo-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] = x^i * y^j * z^k, + + 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 powers of `x`, `y`, and `z`. + + If ``V = polyvander3d(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 ``polyval3d(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 polynomials + 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 + -------- + polyvander, polyvander3d. polyval2d, polyval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy, degz = ideg + x, y, z = np.array((x, y, z), copy=0) + 0.0 + + vx = polyvander(x, degx) + vy = polyvander(y, degy) + vz = polyvander(z, degz) + v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:] + # einsum bug + #v = np.einsum("...i, ...j, ...k->...ijk", vx, vy, vz) + return v.reshape(v.shape[:-3] + (-1,)) + + +def polyfit(x, y, deg, rcond=None, full=False, w=None): + """ + Least-squares fit of a polynomial to data. + + Return the coefficients of a polynomial 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 * x + ... + c_n * x^n, + + where `n` is `deg`. + + Parameters + ---------- + x : array_like, shape (`M`,) + x-coordinates of the `M` sample (data) points ``(x[i], y[i])``. + y : array_like, shape (`M`,) or (`M`, `K`) + y-coordinates of the sample points. Several sets of sample points + sharing the same x-coordinates can be (independently) fit with one + call to `polyfit` by passing in for `y` a 2-D array that contains + one data set per column. + deg : int + Degree of the polynomial(s) to be fit. + rcond : float, optional + Relative condition number of the fit. Singular values smaller + than `rcond`, relative to the largest singular value, will be + ignored. The default value is ``len(x)*eps``, where `eps` is the + relative precision of the platform's float type, about 2e-16 in + most cases. + full : bool, optional + Switch determining the nature of the return value. When ``False`` + (the default) just the coefficients are returned; when ``True``, + diagnostic information from the singular value decomposition (used + to solve the fit's matrix equation) is also returned. + w : array_like, shape (`M`,), optional + Weights. If not None, the contribution of each point + ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the + weights are chosen so that the errors of the products ``w[i]*y[i]`` + all have the same variance. The default value is None. + + .. versionadded:: 1.5.0 + + Returns + ------- + coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`) + Polynomial coefficients ordered from low to high. If `y` was 2-D, + the coefficients in column `k` of `coef` represent the polynomial + fit to the data in `y`'s `k`-th column. + + [residuals, rank, singular_values, rcond] : list + These values are only returned if `full` = True + + resid -- sum of squared residuals of the least squares fit + rank -- the numerical rank of the scaled Vandermonde matrix + sv -- singular values of the scaled Vandermonde matrix + rcond -- value of `rcond`. + + For more details, see `linalg.lstsq`. + + Raises + ------ + RankWarning + Raised if the matrix in the least-squares fit is rank deficient. + The warning is only raised if `full` == False. The warnings can + be turned off by: + + >>> import warnings + >>> warnings.simplefilter('ignore', RankWarning) + + See Also + -------- + chebfit, legfit, lagfit, hermfit, hermefit + polyval : Evaluates a polynomial. + polyvander : Vandermonde matrix for powers. + linalg.lstsq : Computes a least-squares fit from the matrix. + scipy.interpolate.UnivariateSpline : Computes spline fits. + + Notes + ----- + The solution is the coefficients of the polynomial `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) over-determined 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, and `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 (and `full` == ``False``), a `RankWarning` will be raised. + This means that the coefficient values may be poorly determined. + Fitting to a lower order polynomial will usually get rid of the warning + (but may not be what you want, of course; if you have independent + reason(s) for choosing the degree which isn't working, you may have to: + a) reconsider those reasons, and/or b) reconsider the quality of your + data). 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. + + Polynomial fits using double precision tend to "fail" at about + (polynomial) degree 20. Fits using Chebyshev or Legendre series are + generally better conditioned, but much can still depend on the + distribution of the sample points and the smoothness of the data. If + the quality of the fit is inadequate, splines may be a good + alternative. + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1] + >>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + N(0,1) "noise" + >>> c, stats = P.polyfit(x,y,3,full=True) + >>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1 + array([ 0.01909725, -1.30598256, -0.00577963, 1.02644286]) + >>> stats # note the large SSR, explaining the rather poor results + [array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, + 0.28853036]), 1.1324274851176597e-014] + + Same thing without the added noise + + >>> y = x**3 - x + >>> c, stats = P.polyfit(x,y,3,full=True) + >>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1 + array([ -1.73362882e-17, -1.00000000e+00, -2.67471909e-16, + 1.00000000e+00]) + >>> stats # note the minuscule SSR + [array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, + 0.50443316, 0.28853036]), 1.1324274851176597e-014] + + """ + order = int(deg) + 1 + x = np.asarray(x) + 0.0 + y = np.asarray(y) + 0.0 + + # check arguments. + if deg < 0: + raise ValueError("expected deg >= 0") + if x.ndim != 1: + raise TypeError("expected 1D vector for x") + if x.size == 0: + raise TypeError("expected non-empty vector for x") + if y.ndim < 1 or y.ndim > 2: + raise TypeError("expected 1D or 2D array for y") + if len(x) != len(y): + raise TypeError("expected x and y to have same length") + + # set up the least squares matrices in transposed form + lhs = polyvander(x, deg).T + rhs = y.T + if w is not None: + w = np.asarray(w) + 0.0 + if w.ndim != 1: + raise TypeError("expected 1D vector for w") + if len(x) != len(w): + raise TypeError("expected x and w to have same length") + # apply weights. Don't use inplace operations as they + # can cause problems with NA. + lhs = lhs * w + rhs = rhs * w + + # set rcond + if rcond is None: + rcond = len(x)*np.finfo(x.dtype).eps + + # Determine the norms of the design matrix columns. + if issubclass(lhs.dtype.type, np.complexfloating): + scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1)) + else: + scl = np.sqrt(np.square(lhs).sum(1)) + scl[scl == 0] = 1 + + # Solve the least squares problem. + c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond) + c = (c.T/scl).T + + # warn on rank reduction + if rank != order and not full: + msg = "The fit may be poorly conditioned" + warnings.warn(msg, pu.RankWarning) + + if full: + return c, [resids, rank, s, rcond] + else: + return c + + +def polycompanion(c): + """ + Return the companion matrix of c. + + The companion matrix for power series cannot be made symmetric by + scaling the basis, so this function differs from those for the + orthogonal polynomials. + + Parameters + ---------- + c : array_like + 1-D array of polynomial coefficients ordered from low to high + degree. + + Returns + ------- + mat : ndarray + Companion matrix of dimensions (deg, deg). + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + raise ValueError('Series must have maximum degree of at least 1.') + if len(c) == 2: + return np.array([[-c[0]/c[1]]]) + + n = len(c) - 1 + mat = np.zeros((n, n), dtype=c.dtype) + bot = mat.reshape(-1)[n::n+1] + bot[...] = 1 + mat[:, -1] -= c[:-1]/c[-1] + return mat + + +def polyroots(c): + """ + Compute the roots of a polynomial. + + Return the roots (a.k.a. "zeros") of the polynomial + + .. math:: p(x) = \\sum_i c[i] * x^i. + + Parameters + ---------- + c : 1-D array_like + 1-D array of polynomial coefficients. + + Returns + ------- + out : ndarray + Array of the roots of the polynomial. If all the roots are real, + then `out` is also real, otherwise it is complex. + + See Also + -------- + chebroots + + 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 power 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. + + Examples + -------- + >>> import numpy.polynomial.polynomial as poly + >>> poly.polyroots(poly.polyfromroots((-1,0,1))) + array([-1., 0., 1.]) + >>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype + dtype('float64') + >>> j = complex(0,1) + >>> poly.polyroots(poly.polyfromroots((-j,0,j))) + array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j]) + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + return np.array([], dtype=c.dtype) + if len(c) == 2: + return np.array([-c[0]/c[1]]) + + m = polycompanion(c) + r = la.eigvals(m) + r.sort() + return r + + +# +# polynomial class +# + +class Polynomial(ABCPolyBase): + """A power series class. + + The Polynomial class provides the standard Python numerical methods + '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the + attributes and methods listed in the `ABCPolyBase` documentation. + + Parameters + ---------- + coef : array_like + Polynomial coefficients in order of increasing degree, i.e., + ``(1, 2, 3)`` give ``1 + 2*x + 3*x**2``. + 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]. + + .. versionadded:: 1.6.0 + + """ + # Virtual Functions + _add = staticmethod(polyadd) + _sub = staticmethod(polysub) + _mul = staticmethod(polymul) + _div = staticmethod(polydiv) + _pow = staticmethod(polypow) + _val = staticmethod(polyval) + _int = staticmethod(polyint) + _der = staticmethod(polyder) + _fit = staticmethod(polyfit) + _line = staticmethod(polyline) + _roots = staticmethod(polyroots) + _fromroots = staticmethod(polyfromroots) + + # Virtual properties + nickname = 'poly' + domain = np.array(polydomain) + window = np.array(polydomain)