diff options
Diffstat (limited to 'numpy/polynomial/hermite.py')
| -rw-r--r-- | numpy/polynomial/hermite.py | 170 |
1 files changed, 88 insertions, 82 deletions
diff --git a/numpy/polynomial/hermite.py b/numpy/polynomial/hermite.py index 550e316de..d266a6453 100644 --- a/numpy/polynomial/hermite.py +++ b/numpy/polynomial/hermite.py @@ -96,13 +96,9 @@ def poly2herm(pol) : Examples -------- - >>> from numpy import polynomial as P - >>> p = P.Polynomial(np.arange(4)) - >>> p - Polynomial([ 0., 1., 2., 3.], [-1., 1.]) - >>> c = P.Hermite(P.poly2herm(p.coef)) - >>> c - Hermite([ 1. , 3.25, 1. , 0.75], [-1., 1.]) + >>> from numpy.polynomial.hermite_e import poly2herme + >>> poly2herm(np.arange(4)) + array([ 1. , 2.75 , 0.5 , 0.375]) """ [pol] = pu.as_series([pol]) @@ -146,15 +142,9 @@ def herm2poly(cs) : Examples -------- - >>> c = P.Hermite(range(4)) - >>> c - Hermite([ 0., 1., 2., 3.], [-1., 1.]) - >>> p = c.convert(kind=P.Polynomial) - >>> p - Polynomial([-1. , -3.5, 3. , 7.5], [-1., 1.]) - >>> P.herm2poly(range(4)) - array([-1. , -3.5, 3. , 7.5]) - + >>> from numpy.polynomial.hermite import herm2poly + >>> herm2poly([ 1. , 2.75 , 0.5 , 0.375]) + array([ 0., 1., 2., 3.]) """ from polynomial import polyadd, polysub, polymulx @@ -217,11 +207,11 @@ def hermline(off, scl) : Examples -------- - >>> import numpy.polynomial.legendre as L - >>> L.hermline(3,2) - array([3, 2]) - >>> L.hermval(-3, L.hermline(3,2)) # should be -3 - -3.0 + >>> from numpy.polynomial.hermite import hermline, hermval + >>> hermval(0,hermline(3, 2)) + 3.0 + >>> hermval(1,hermline(3, 2)) + 5.0 """ if scl != 0 : @@ -274,12 +264,13 @@ def hermfromroots(roots) : Examples -------- - >>> import numpy.polynomial.legendre as L - >>> L.hermfromroots((-1,0,1)) # x^3 - x relative to the standard basis - array([ 0. , -0.4, 0. , 0.4]) - >>> j = complex(0,1) - >>> L.hermfromroots((-j,j)) # x^2 + 1 relative to the standard basis - array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) + >>> 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]) """ if len(roots) == 0 : @@ -324,11 +315,9 @@ def hermadd(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.hermadd(c1,c2) - array([ 4., 4., 4.]) + >>> from numpy.polynomial.hermite import hermadd + >>> hermadd([1, 2, 3], [1, 2, 3, 4]) + array([ 2., 4., 6., 4.]) """ # c1, c2 are trimmed copies @@ -374,13 +363,9 @@ def hermsub(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.hermsub(c1,c2) - array([-2., 0., 2.]) - >>> L.hermsub(c2,c1) # -C.hermsub(c1,c2) - array([ 2., 0., -2.]) + >>> from numpy.polynomial.hermite import hermsub + >>> hermsub([1, 2, 3, 4], [1, 2, 3]) + array([ 0., 0., 0., 4.]) """ # c1, c2 are trimmed copies @@ -420,7 +405,13 @@ def hermmulx(cs): .. math:: - xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1) + 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]) """ # cs is a trimmed copy @@ -471,11 +462,9 @@ def hermmul(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2) - >>> P.hermmul(c1,c2) # multiplication requires "reprojection" - array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) + >>> from numpy.polynomial.hermite import hermmul + >>> hermmul([1, 2, 3], [0, 1, 2]) + array([ 52., 29., 52., 7., 6.]) """ # s1, s2 are trimmed copies @@ -542,14 +531,13 @@ def hermdiv(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.hermdiv(c1,c2) # quotient "intuitive," remainder not - (array([ 3.]), array([-8., -4.])) - >>> c2 = (0,1,2,3) - >>> L.hermdiv(c2,c1) # neither "intuitive" - (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) + >>> 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.])) """ # c1, c2 are trimmed copies @@ -603,6 +591,9 @@ def hermpow(cs, pow, maxpower=16) : Examples -------- + >>> from numpy.polynomial.hermite import hermpow + >>> hermpow([1, 2, 3], 2) + array([ 81., 52., 82., 12., 9.]) """ # cs is a trimmed copy @@ -664,16 +655,11 @@ def hermder(cs, m=1, scl=1) : Examples -------- - >>> from numpy.polynomial import legendre as L - >>> cs = (1,2,3,4) - >>> L.hermder(cs) - array([ 6., 9., 20.]) - >>> L.hermder(cs,3) - array([ 60.]) - >>> L.hermder(cs,scl=-1) - array([ -6., -9., -20.]) - >>> L.hermder(cs,2,-1) - array([ 9., 60.]) + >>> 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.]) """ cnt = int(m) @@ -762,19 +748,17 @@ def hermint(cs, m=1, k=[], lbnd=0, scl=1): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> cs = (1,2,3) - >>> L.hermint(cs) - array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.hermint(cs,3) - array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, - -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) - >>> L.hermint(cs, k=3) - array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.hermint(cs, lbnd=-2) - array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.hermint(cs, scl=2) - array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) + >>> 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 ]) + >>> 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 ]) """ cnt = int(m) @@ -847,6 +831,13 @@ def hermval(x, cs): 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.]]) """ # cs is a trimmed copy @@ -897,6 +888,15 @@ def hermvander(x, deg) : The shape of the returned matrix is ``x.shape + (deg+1,)``. The last index is the degree. + Examples + -------- + >>> 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.]]) + """ ideg = int(deg) if ideg != deg: @@ -1015,6 +1015,12 @@ def hermfit(x, y, deg, rcond=None, full=False, w=None): Examples -------- + >>> from numpy.polynomial.hermite import hermfit, hermval + >>> x = np.linspace(-10, 10) + >>> err = np.random.randn(len(x))/10 + >>> y = hermval(x, [1, 2, 3]) + err + >>> hermfit(x, y, 2) + array([ 0.97902637, 1.99849131, 3.00006 ]) """ order = int(deg) + 1 @@ -1104,12 +1110,12 @@ def hermroots(cs): Examples -------- - >>> import numpy.polynomial as P - >>> P.polyroots((1, 2, 3, 4)) # 4x^3 + 3x^2 + 2x + 1 has two complex roots - array([-0.60582959+0.j , -0.07208521-0.63832674j, - -0.07208521+0.63832674j]) - >>> P.hermroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0 has only real roots - array([-0.85099543, -0.11407192, 0.51506735]) + >>> 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]) """ # cs is a trimmed copy |