diff options
Diffstat (limited to 'numpy/polynomial/legendre.py')
| -rw-r--r-- | numpy/polynomial/legendre.py | 121 |
1 files changed, 61 insertions, 60 deletions
diff --git a/numpy/polynomial/legendre.py b/numpy/polynomial/legendre.py index 58c130b7e..d2de28269 100644 --- a/numpy/polynomial/legendre.py +++ b/numpy/polynomial/legendre.py @@ -90,17 +90,18 @@ import numpy.linalg as la from . import polyutils as pu from ._polybase import ABCPolyBase -__all__ = ['legzero', 'legone', 'legx', 'legdomain', 'legline', - 'legadd', 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', - 'legder', 'legint', 'leg2poly', 'poly2leg', 'legfromroots', - 'legvander', 'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', - 'legval3d', 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', - 'legcompanion', 'leggauss', 'legweight'] +__all__ = [ + 'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd', + 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder', + 'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander', + 'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d', + 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion', + 'leggauss', 'legweight'] legtrim = pu.trimcoef -def poly2leg(pol) : +def poly2leg(pol): """ Convert a polynomial to a Legendre series. @@ -143,12 +144,12 @@ def poly2leg(pol) : [pol] = pu.as_series([pol]) deg = len(pol) - 1 res = 0 - for i in range(deg, -1, -1) : + for i in range(deg, -1, -1): res = legadd(legmulx(res), pol[i]) return res -def leg2poly(c) : +def leg2poly(c): """ Convert a Legendre series to a polynomial. @@ -202,7 +203,7 @@ def leg2poly(c) : c0 = c[-2] c1 = c[-1] # i is the current degree of c1 - for i in range(n - 1, 1, -1) : + for i in range(n - 1, 1, -1): tmp = c0 c0 = polysub(c[i - 2], (c1*(i - 1))/i) c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i) @@ -226,7 +227,7 @@ legone = np.array([1]) legx = np.array([0, 1]) -def legline(off, scl) : +def legline(off, scl): """ Legendre series whose graph is a straight line. @@ -256,13 +257,13 @@ def legline(off, scl) : -3.0 """ - if scl != 0 : + if scl != 0: return np.array([off, scl]) - else : + else: return np.array([off]) -def legfromroots(roots) : +def legfromroots(roots): """ Generate a Legendre series with given roots. @@ -311,9 +312,9 @@ def legfromroots(roots) : array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) """ - if len(roots) == 0 : + if len(roots) == 0: return np.ones(1) - else : + else: [roots] = pu.as_series([roots], trim=False) roots.sort() p = [legline(-r, 1) for r in roots] @@ -369,10 +370,10 @@ def legadd(c1, c2): """ # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) - if len(c1) > len(c2) : + if len(c1) > len(c2): c1[:c2.size] += c2 ret = c1 - else : + else: c2[:c1.size] += c1 ret = c2 return pu.trimseq(ret) @@ -421,10 +422,10 @@ def legsub(c1, c2): """ # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) - if len(c1) > len(c2) : + if len(c1) > len(c2): c1[:c2.size] -= c2 ret = c1 - else : + else: c2 = -c2 c2[:c1.size] += c1 ret = c2 @@ -533,13 +534,13 @@ def legmul(c1, c2): elif len(c) == 2: c0 = c[0]*xs c1 = c[1]*xs - else : + else: nd = len(c) c0 = c[-2]*xs c1 = c[-1]*xs - for i in range(3, len(c) + 1) : + for i in range(3, len(c) + 1): tmp = c0 - nd = nd - 1 + nd = nd - 1 c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd) c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd) return legadd(c0, legmulx(c1)) @@ -593,16 +594,16 @@ def legdiv(c1, c2): """ # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) - if c2[-1] == 0 : + if c2[-1] == 0: raise ZeroDivisionError() lc1 = len(c1) lc2 = len(c2) - if lc1 < lc2 : + if lc1 < lc2: return c1[:1]*0, c1 - elif lc2 == 1 : + elif lc2 == 1: return c1/c2[-1], c1[:1]*0 - else : + else: quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype) rem = c1 for i in range(lc1 - lc2, - 1, -1): @@ -613,7 +614,7 @@ def legdiv(c1, c2): return quo, pu.trimseq(rem) -def legpow(c, pow, maxpower=16) : +def legpow(c, pow, maxpower=16): """Raise a Legendre series to a power. Returns the Legendre series `c` raised to the power `pow`. The @@ -647,24 +648,24 @@ def legpow(c, pow, maxpower=16) : # c is a trimmed copy [c] = pu.as_series([c]) power = int(pow) - if power != pow or power < 0 : + if power != pow or power < 0: raise ValueError("Power must be a non-negative integer.") - elif maxpower is not None and power > maxpower : + elif maxpower is not None and power > maxpower: raise ValueError("Power is too large") - elif power == 0 : + elif power == 0: return np.array([1], dtype=c.dtype) - elif power == 1 : + elif power == 1: return c - else : + 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) : + for i in range(2, power + 1): prd = legmul(prd, c) return prd -def legder(c, m=1, scl=1, axis=0) : +def legder(c, m=1, scl=1, axis=0): """ Differentiate a Legendre series. @@ -747,7 +748,7 @@ def legder(c, m=1, scl=1, axis=0) : n = len(c) if cnt >= n: c = c[:1]*0 - else : + else: for i in range(cnt): n = n - 1 c *= scl @@ -857,9 +858,9 @@ def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0): if cnt != m: raise ValueError("The order of integration must be integer") - if cnt < 0 : + if cnt < 0: raise ValueError("The order of integration must be non-negative") - if len(k) > cnt : + if len(k) > cnt: raise ValueError("Too many integration constants") if iaxis != axis: raise ValueError("The axis must be integer") @@ -873,7 +874,7 @@ def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0): c = np.rollaxis(c, iaxis) k = list(k) + [0]*(cnt - len(k)) - for i in range(cnt) : + for i in range(cnt): n = len(c) c *= scl if n == 1 and np.all(c[0] == 0): @@ -964,19 +965,19 @@ def legval(x, c, tensor=True): if isinstance(x, np.ndarray) and tensor: c = c.reshape(c.shape + (1,)*x.ndim) - if len(c) == 1 : + if len(c) == 1: c0 = c[0] c1 = 0 - elif len(c) == 2 : + elif len(c) == 2: c0 = c[0] c1 = c[1] - else : + else: nd = len(c) c0 = c[-2] c1 = c[-1] - for i in range(3, len(c) + 1) : + for i in range(3, len(c) + 1): tmp = c0 - nd = nd - 1 + nd = nd - 1 c0 = c[-i] - (c1*(nd - 1))/nd c1 = tmp + (c1*x*(2*nd - 1))/nd return c0 + c1*x @@ -1211,7 +1212,7 @@ def leggrid3d(x, y, z, c): return c -def legvander(x, deg) : +def legvander(x, deg): """Pseudo-Vandermonde matrix of given degree. Returns the pseudo-Vandermonde matrix of degree `deg` and sample points @@ -1259,14 +1260,14 @@ def legvander(x, deg) : # Use forward recursion to generate the entries. This is not as accurate # as reverse recursion in this application but it is more efficient. v[0] = x*0 + 1 - if ideg > 0 : + if ideg > 0: v[1] = x - for i in range(2, ideg + 1) : + for i in range(2, ideg + 1): v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i return np.rollaxis(v, 0, v.ndim) -def legvander2d(x, y, deg) : +def legvander2d(x, y, deg): """Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample @@ -1329,7 +1330,7 @@ def legvander2d(x, y, deg) : return v.reshape(v.shape[:-2] + (-1,)) -def legvander3d(x, y, z, deg) : +def legvander3d(x, y, z, deg): """Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample @@ -1515,13 +1516,13 @@ def legfit(x, y, deg, rcond=None, full=False, w=None): y = np.asarray(y) + 0.0 # check arguments. - if deg < 0 : + 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 : + 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") @@ -1541,7 +1542,7 @@ def legfit(x, y, deg, rcond=None, full=False, w=None): rhs = rhs * w # set rcond - if rcond is None : + if rcond is None: rcond = len(x)*np.finfo(x.dtype).eps # Determine the norms of the design matrix columns. @@ -1560,9 +1561,9 @@ def legfit(x, y, deg, rcond=None, full=False, w=None): msg = "The fit may be poorly conditioned" warnings.warn(msg, pu.RankWarning) - if full : + if full: return c, [resids, rank, s, rcond] - else : + else: return c @@ -1637,11 +1638,11 @@ def legroots(c): ----- 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. + 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 Legendre series basis polynomials aren't powers of ``x`` so the results of this function may seem unintuitive. @@ -1649,7 +1650,7 @@ def legroots(c): Examples -------- >>> import numpy.polynomial.legendre as leg - >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0 has only real roots + >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots array([-0.85099543, -0.11407192, 0.51506735]) """ |