diff options
Diffstat (limited to 'numpy/polynomial')
| -rw-r--r-- | numpy/polynomial/hermite.py | 170 | ||||
| -rw-r--r-- | numpy/polynomial/hermite_e.py | 171 | ||||
| -rw-r--r-- | numpy/polynomial/laguerre.py | 169 |
3 files changed, 263 insertions, 247 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 diff --git a/numpy/polynomial/hermite_e.py b/numpy/polynomial/hermite_e.py index df0ff4e5e..ecb4bc3c3 100644 --- a/numpy/polynomial/hermite_e.py +++ b/numpy/polynomial/hermite_e.py @@ -96,13 +96,9 @@ def poly2herme(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.poly2herme(p.coef)) - >>> c - Hermite([ 1. , 3.25, 1. , 0.75], [-1., 1.]) + >>> from numpy.polynomial.hermite_e import poly2herme + >>> poly2herme(np.arange(4)) + array([ 2., 10., 2., 3.]) """ [pol] = pu.as_series([pol]) @@ -146,15 +142,9 @@ def herme2poly(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.herme2poly(range(4)) - array([-1. , -3.5, 3. , 7.5]) - + >>> from numpy.polynomial.hermite_e import herme2poly + >>> herme2poly([ 2., 10., 2., 3.]) + array([ 0., 1., 2., 3.]) """ from polynomial import polyadd, polysub, polymulx @@ -216,11 +206,12 @@ def hermeline(off, scl) : Examples -------- - >>> import numpy.polynomial.legendre as L - >>> L.hermeline(3,2) - array([3, 2]) - >>> L.hermeval(-3, L.hermeline(3,2)) # should be -3 - -3.0 + >>> from numpy.polynomial.hermite_e import hermeline + >>> from numpy.polynomial.hermite_e import hermeline, hermeval + >>> hermeval(0,hermeline(3, 2)) + 3.0 + >>> hermeval(1,hermeline(3, 2)) + 5.0 """ if scl != 0 : @@ -273,12 +264,13 @@ def hermefromroots(roots) : Examples -------- - >>> import numpy.polynomial.legendre as L - >>> L.hermefromroots((-1,0,1)) # x^3 - x relative to the standard basis - array([ 0. , -0.4, 0. , 0.4]) - >>> j = complex(0,1) - >>> L.hermefromroots((-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_e import hermefromroots, hermeval + >>> coef = hermefromroots((-1, 0, 1)) + >>> hermeval((-1, 0, 1), coef) + array([ 0., 0., 0.]) + >>> coef = hermefromroots((-1j, 1j)) + >>> hermeval((-1j, 1j), coef) + array([ 0.+0.j, 0.+0.j]) """ if len(roots) == 0 : @@ -323,11 +315,9 @@ def hermeadd(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.hermeadd(c1,c2) - array([ 4., 4., 4.]) + >>> from numpy.polynomial.hermite_e import hermeadd + >>> hermeadd([1, 2, 3], [1, 2, 3, 4]) + array([ 2., 4., 6., 4.]) """ # c1, c2 are trimmed copies @@ -373,13 +363,9 @@ def hermesub(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.hermesub(c1,c2) - array([-2., 0., 2.]) - >>> L.hermesub(c2,c1) # -C.hermesub(c1,c2) - array([ 2., 0., -2.]) + >>> from numpy.polynomial.hermite_e import hermesub + >>> hermesub([1, 2, 3, 4], [1, 2, 3]) + array([ 0., 0., 0., 4.]) """ # c1, c2 are trimmed copies @@ -419,7 +405,13 @@ def hermemulx(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) + iP_{i - 1}(x))) + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermemulx + >>> hermemulx([1, 2, 3]) + array([ 2., 7., 2., 3.]) """ # cs is a trimmed copy @@ -470,11 +462,9 @@ def hermemul(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2) - >>> P.hermemul(c1,c2) # multiplication requires "reprojection" - array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) + >>> from numpy.polynomial.hermite_e import hermemul + >>> hermemul([1, 2, 3], [0, 1, 2]) + array([ 14., 15., 28., 7., 6.]) """ # s1, s2 are trimmed copies @@ -541,14 +531,11 @@ def hermediv(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.hermediv(c1,c2) # quotient "intuitive," remainder not - (array([ 3.]), array([-8., -4.])) - >>> c2 = (0,1,2,3) - >>> L.hermediv(c2,c1) # neither "intuitive" - (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) + >>> from numpy.polynomial.hermite_e import hermediv + >>> hermediv([ 14., 15., 28., 7., 6.], [0, 1, 2]) + (array([ 1., 2., 3.]), array([ 0.])) + >>> hermediv([ 15., 17., 28., 7., 6.], [0, 1, 2]) + (array([ 1., 2., 3.]), array([ 1., 2.])) """ # c1, c2 are trimmed copies @@ -602,6 +589,9 @@ def hermepow(cs, pow, maxpower=16) : Examples -------- + >>> from numpy.polynomial.hermite_e import hermepow + >>> hermepow([1, 2, 3], 2) + array([ 23., 28., 46., 12., 9.]) """ # cs is a trimmed copy @@ -663,16 +653,11 @@ def hermeder(cs, m=1, scl=1) : Examples -------- - >>> from numpy.polynomial import legendre as L - >>> cs = (1,2,3,4) - >>> L.hermeder(cs) - array([ 6., 9., 20.]) - >>> L.hermeder(cs,3) - array([ 60.]) - >>> L.hermeder(cs,scl=-1) - array([ -6., -9., -20.]) - >>> L.hermeder(cs,2,-1) - array([ 9., 60.]) + >>> from numpy.polynomial.hermite_e import hermeder + >>> hermeder([ 1., 1., 1., 1.]) + array([ 1., 2., 3.]) + >>> hermeder([-0.25, 1., 1./2., 1./3., 1./4 ], m=2) + array([ 1., 2., 3.]) """ cnt = int(m) @@ -761,19 +746,17 @@ def hermeint(cs, m=1, k=[], lbnd=0, scl=1): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> cs = (1,2,3) - >>> L.hermeint(cs) - array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.hermeint(cs,3) - array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, - -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) - >>> L.hermeint(cs, k=3) - array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.hermeint(cs, lbnd=-2) - array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.hermeint(cs, scl=2) - array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) + >>> from numpy.polynomial.hermite_e import hermeint + >>> hermeint([1, 2, 3]) # integrate once, value 0 at 0. + array([ 1., 1., 1., 1.]) + >>> hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0 + array([-0.25 , 1. , 0.5 , 0.33333333, 0.25 ]) + >>> hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0. + array([ 2., 1., 1., 1.]) + >>> hermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1 + array([-1., 1., 1., 1.]) + >>> hermeint([1, 2, 3], m=2, k=[1,2], lbnd=-1) + array([ 1.83333333, 0. , 0.5 , 0.33333333, 0.25 ]) """ cnt = int(m) @@ -846,6 +829,13 @@ def hermeval(x, cs): Examples -------- + >>> from numpy.polynomial.hermite_e import hermeval + >>> coef = [1,2,3] + >>> hermeval(1, coef) + 3.0 + >>> hermeval([[1,2],[3,4]], coef) + array([[ 3., 14.], + [ 31., 54.]]) """ # cs is a trimmed copy @@ -895,6 +885,15 @@ def hermevander(x, deg) : The shape of the returned matrix is ``x.shape + (deg+1,)``. The last index is the degree. + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermevander + >>> x = np.array([-1, 0, 1]) + >>> hermevander(x, 3) + array([[ 1., -1., 0., 2.], + [ 1., 0., -1., -0.], + [ 1., 1., 0., -2.]]) + """ ideg = int(deg) if ideg != deg: @@ -1012,6 +1011,12 @@ def hermefit(x, y, deg, rcond=None, full=False, w=None): Examples -------- + >>> from numpy.polynomial.hermite_e import hermefit, hermeval + >>> x = np.linspace(-10, 10) + >>> err = np.random.randn(len(x))/10 + >>> y = hermeval(x, [1, 2, 3]) + err + >>> hermefit(x, y, 2) + array([ 1.01690445, 1.99951418, 2.99948696]) """ order = int(deg) + 1 @@ -1020,7 +1025,7 @@ def hermefit(x, y, deg, rcond=None, full=False, w=None): # check arguments. if deg < 0 : - raise ValueError, "expected deg >= 0" + raise valueerror, "expected deg >= 0" if x.ndim != 1: raise TypeError, "expected 1D vector for x" if x.size == 0: @@ -1101,12 +1106,12 @@ def hermeroots(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.hermeroots((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_e import hermeroots, hermefromroots + >>> coef = hermefromroots([-1, 0, 1]) + >>> coef + array([ 0., 2., 0., 1.]) + >>> hermeroots(coef) + array([-1., 0., 1.]) """ # cs is a trimmed copy diff --git a/numpy/polynomial/laguerre.py b/numpy/polynomial/laguerre.py index 94c495deb..b6389bf63 100644 --- a/numpy/polynomial/laguerre.py +++ b/numpy/polynomial/laguerre.py @@ -96,13 +96,9 @@ def poly2lag(pol) : Examples -------- - >>> from numpy import polynomial as P - >>> p = P.Polynomial(np.arange(4)) - >>> p - Polynomial([ 0., 1., 2., 3.], [-1., 1.]) - >>> c = P.Laguerre(P.poly2lag(p.coef)) - >>> c - Laguerre([ 1. , 3.25, 1. , 0.75], [-1., 1.]) + >>> from numpy.polynomial.laguerre import poly2lag + >>> poly2lag(np.arange(4)) + array([ 23., -63., 58., -18.]) """ [pol] = pu.as_series([pol]) @@ -146,15 +142,9 @@ def lag2poly(cs) : Examples -------- - >>> c = P.Laguerre(range(4)) - >>> c - Laguerre([ 0., 1., 2., 3.], [-1., 1.]) - >>> p = c.convert(kind=P.Polynomial) - >>> p - Polynomial([-1. , -3.5, 3. , 7.5], [-1., 1.]) - >>> P.lag2poly(range(4)) - array([-1. , -3.5, 3. , 7.5]) - + >>> from numpy.polynomial.laguerre import lag2poly + >>> lag2poly([ 23., -63., 58., -18.]) + array([ 0., 1., 2., 3.]) """ from polynomial import polyadd, polysub, polymulx @@ -214,11 +204,11 @@ def lagline(off, scl) : Examples -------- - >>> import numpy.polynomial.legendre as L - >>> L.lagline(3,2) - array([3, 2]) - >>> L.lagval(-3, L.lagline(3,2)) # should be -3 - -3.0 + >>> from numpy.polynomial.laguerre import lagline, lagval + >>> lagval(0,lagline(3, 2)) + 3.0 + >>> lagval(1,lagline(3, 2)) + 5.0 """ if scl != 0 : @@ -271,12 +261,13 @@ def lagfromroots(roots) : Examples -------- - >>> import numpy.polynomial.legendre as L - >>> L.lagfromroots((-1,0,1)) # x^3 - x relative to the standard basis - array([ 0. , -0.4, 0. , 0.4]) - >>> j = complex(0,1) - >>> L.lagfromroots((-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.laguerre import lagfromroots, lagval + >>> coef = lagfromroots((-1, 0, 1)) + >>> lagval((-1, 0, 1), coef) + array([ 0., 0., 0.]) + >>> coef = lagfromroots((-1j, 1j)) + >>> lagval((-1j, 1j), coef) + array([ 0.+0.j, 0.+0.j]) """ if len(roots) == 0 : @@ -321,11 +312,10 @@ def lagadd(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.lagadd(c1,c2) - array([ 4., 4., 4.]) + >>> from numpy.polynomial.laguerre import lagadd + >>> lagadd([1, 2, 3], [1, 2, 3, 4]) + array([ 2., 4., 6., 4.]) + """ # c1, c2 are trimmed copies @@ -371,13 +361,9 @@ def lagsub(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.lagsub(c1,c2) - array([-2., 0., 2.]) - >>> L.lagsub(c2,c1) # -C.lagsub(c1,c2) - array([ 2., 0., -2.]) + >>> from numpy.polynomial.laguerre import lagsub + >>> lagsub([1, 2, 3, 4], [1, 2, 3]) + array([ 0., 0., 0., 4.]) """ # c1, c2 are trimmed copies @@ -417,7 +403,13 @@ def lagmulx(cs): .. math:: - xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1) + xP_i(x) = (-(i + 1)*P_{i + 1}(x) + (2i + 1)P_{i}(x) - iP_{i - 1}(x)) + + Examples + -------- + >>> from numpy.polynomial.laguerre import lagmulx + >>> lagmulx([1, 2, 3]) + array([ -1., -1., 11., -9.]) """ # cs is a trimmed copy @@ -469,11 +461,9 @@ def lagmul(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2) - >>> P.lagmul(c1,c2) # multiplication requires "reprojection" - array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) + >>> from numpy.polynomial.laguerre import lagmul + >>> lagmul([1, 2, 3], [0, 1, 2]) + array([ 8., -13., 38., -51., 36.]) """ # s1, s2 are trimmed copies @@ -540,14 +530,11 @@ def lagdiv(c1, c2): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> c1 = (1,2,3) - >>> c2 = (3,2,1) - >>> L.lagdiv(c1,c2) # quotient "intuitive," remainder not - (array([ 3.]), array([-8., -4.])) - >>> c2 = (0,1,2,3) - >>> L.lagdiv(c2,c1) # neither "intuitive" - (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) + >>> from numpy.polynomial.laguerre import lagdiv + >>> lagdiv([ 8., -13., 38., -51., 36.], [0, 1, 2]) + (array([ 1., 2., 3.]), array([ 0.])) + >>> lagdiv([ 9., -12., 38., -51., 36.], [0, 1, 2]) + (array([ 1., 2., 3.]), array([ 1., 1.])) """ # c1, c2 are trimmed copies @@ -601,6 +588,9 @@ def lagpow(cs, pow, maxpower=16) : Examples -------- + >>> from numpy.polynomial.laguerre import lagpow + >>> lagpow([1, 2, 3], 2) + array([ 14., -16., 56., -72., 54.]) """ # cs is a trimmed copy @@ -662,16 +652,11 @@ def lagder(cs, m=1, scl=1) : Examples -------- - >>> from numpy.polynomial import legendre as L - >>> cs = (1,2,3,4) - >>> L.lagder(cs) - array([ 6., 9., 20.]) - >>> L.lagder(cs,3) - array([ 60.]) - >>> L.lagder(cs,scl=-1) - array([ -6., -9., -20.]) - >>> L.lagder(cs,2,-1) - array([ 9., 60.]) + >>> from numpy.polynomial.laguerre import lagder + >>> lagder([ 1., 1., 1., -3.]) + array([ 1., 2., 3.]) + >>> lagder([ 1., 0., 0., -4., 3.], m=2) + array([ 1., 2., 3.]) """ cnt = int(m) @@ -761,19 +746,17 @@ def lagint(cs, m=1, k=[], lbnd=0, scl=1): Examples -------- - >>> from numpy.polynomial import legendre as L - >>> cs = (1,2,3) - >>> L.lagint(cs) - array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.lagint(cs,3) - array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, - -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) - >>> L.lagint(cs, k=3) - array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.lagint(cs, lbnd=-2) - array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) - >>> L.lagint(cs, scl=2) - array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) + >>> from numpy.polynomial.laguerre import lagint + >>> lagint([1,2,3]) + array([ 1., 1., 1., -3.]) + >>> lagint([1,2,3], m=2) + array([ 1., 0., 0., -4., 3.]) + >>> lagint([1,2,3], k=1) + array([ 2., 1., 1., -3.]) + >>> lagint([1,2,3], lbnd=-1) + array([ 11.5, 1. , 1. , -3. ]) + >>> lagint([1,2], m=2, k=[1,2], lbnd=-1) + array([ 11.16666667, -5. , -3. , 2. ]) """ cnt = int(m) @@ -847,6 +830,13 @@ def lagval(x, cs): Examples -------- + >>> from numpy.polynomial.laguerre import lagval + >>> coef = [1,2,3] + >>> lagval(1, coef) + -0.5 + >>> lagval([[1,2],[3,4]], coef) + array([[-0.5, -4. ], + [-4.5, -2. ]]) """ # cs is a trimmed copy @@ -896,6 +886,15 @@ def lagvander(x, deg) : The shape of the returned matrix is ``x.shape + (deg+1,)``. The last index is the degree. + Examples + -------- + >>> from numpy.polynomial.laguerre import lagvander + >>> x = np.array([0, 1, 2]) + >>> lagvander(x, 3) + array([[ 1. , 1. , 1. , 1. ], + [ 1. , 0. , -0.5 , -0.66666667], + [ 1. , -1. , -1. , -0.33333333]]) + """ ideg = int(deg) if ideg != deg: @@ -1013,6 +1012,12 @@ def lagfit(x, y, deg, rcond=None, full=False, w=None): Examples -------- + >>> from numpy.polynomial.laguerre import lagfit, lagval + >>> x = np.linspace(0, 10) + >>> err = np.random.randn(len(x))/10 + >>> y = lagval(x, [1, 2, 3]) + err + >>> lagfit(x, y, 2) + array([ 0.96971004, 2.00193749, 3.00288744]) """ order = int(deg) + 1 @@ -1102,12 +1107,12 @@ def lagroots(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.lagroots((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.laguerre import lagroots, lagfromroots + >>> coef = lagfromroots([0, 1, 2]) + >>> coef + array([ 2., -8., 12., -6.]) + >>> lagroots(coef) + array([ -4.44089210e-16, 1.00000000e+00, 2.00000000e+00]) """ # cs is a trimmed copy |