The product of all xs[j] with j != k.
"""
- return product( xs[k] - xs[j] for j in xrange(len(xs)) if j != k )
+ return product( xs[k] - xs[j] for j in range(len(xs)) if j != k )
def lagrange_coefficient(k, x, xs):
TESTS::
sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
- sage: ys = map(sin, xs)
+ sage: ys = list(map(sin, xs))
sage: L = lagrange_polynomial(x, xs, ys)
sage: expected = 27/16*(pi - 6*x)*(pi - 2*x)*(pi + 2*x)*x/pi^4
sage: expected -= 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
True
"""
- ls = [ lagrange_coefficient(k, x, xs) for k in xrange(len(xs)) ]
- return sum( ys[k] * ls[k] for k in xrange(len(xs)) )
+ ls = [ lagrange_coefficient(k, x, xs) for k in range(len(xs)) ]
+ return sum( ys[k] * ls[k] for k in range(len(xs)) )
[1/2/pi^2, -1/pi^2, 1/2/pi^2]
"""
- return [ ~lagrange_denominator(k, xs) for k in xrange(len(xs)) ]
+ return [ ~lagrange_denominator(k, xs) for k in range(len(xs)) ]
def divided_difference(xs, ys):
TESTS::
sage: xs = [0]
- sage: ys = map(sin, xs)
+ sage: ys = list(map(sin, xs))
sage: divided_difference(xs, ys)
0
sage: xs = [0, pi]
- sage: ys = map(sin, xs)
+ sage: ys = list(map(sin, xs))
sage: divided_difference(xs, ys)
0
sage: xs = [0, pi, 2*pi]
- sage: ys = map(sin, xs)
+ sage: ys = list(map(sin, xs))
sage: divided_difference(xs, ys)
0
TESTS::
sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
- sage: ys = map(sin, xs)
+ sage: ys = list(map(sin, xs))
sage: L = lagrange_polynomial(x, xs, ys)
sage: N = newton_polynomial(x, xs, ys)
sage: bool(N == L)
"""
return sum( divided_difference(xs[:k+1], ys[:k+1])*lagrange_psi(x, xs[:k])
- for k in xrange(len(xs)) )
+ for k in range(len(xs)) )
def hermite_coefficient(k, x, xs):
TESTS::
sage: xs = [ 0, pi/6, pi/2 ]
- sage: ys = map(sin, xs)
- sage: y_primes = map(cos, xs)
+ sage: ys = list(map(sin, xs))
+ sage: y_primes = list(map(cos, xs))
sage: H = hermite_interpolant(x, xs, ys, y_primes)
sage: expected = -27/4*(pi - 6*x)*(pi - 2*x)^2*sqrt(3)*x^2/pi^4
sage: expected += (5*(pi - 2*x)/pi + 1)*(pi - 6*x)^2*x^2/pi^4
"""
s1 = sum( ys[k] * hermite_coefficient(k, x, xs)
- for k in xrange(len(xs)) )
+ for k in range(len(xs)) )
s2 = sum( y_primes[k] * hermite_deriv_coefficient(k, x, xs)
- for k in xrange(len(xs)) )
+ for k in range(len(xs)) )
return (s1 + s2)