from sage.all import *
-from misc import product
+product = prod
+
+
+def lagrange_denominator(k, xs):
+ """
+ Return the denominator of the kth Lagrange coefficient.
+
+ INPUT:
+
+ - ``k`` -- The index of the coefficient.
+
+ - ``xs`` -- The list of points at which the function values are
+ known.
+
+ OUTPUT:
+
+ The product of all xs[j] with j != k.
+
+ """
+ return product( xs[k] - xs[j] for j in range(len(xs)) if j != k )
+
def lagrange_coefficient(k, x, xs):
"""
INPUT:
- - ``k`` -- the index of the coefficient.
+ - ``k`` -- The index of the coefficient.
- - ``x`` -- the symbolic variable to use for the first argument
+ - ``x`` -- The symbolic variable to use for the first argument
of l_{k}.
- ``xs`` -- The list of points at which the function values are
A symbolic expression of one variable.
+ SETUP::
+
+ sage: from mjo.interpolation import lagrange_coefficient
+
TESTS::
sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
sage: lagrange_coefficient(0, x, xs)
- 1/8*(pi - 6*x)*(pi - 2*x)*(pi + 6*x)*x/pi^4
+ 1/8*(pi + 6*x)*(pi - 2*x)*(pi - 6*x)*x/pi^4
"""
- numerator = product([x - xs[j] for j in range(0, len(xs)) if j != k])
- denominator = product([xs[k] - xs[j] for j in range(0, len(xs)) if j != k])
+ numerator = lagrange_psi(x, xs)/(x - xs[k])
+ denominator = lagrange_denominator(k, xs)
return (numerator / denominator)
def lagrange_polynomial(x, xs, ys):
"""
- Return the Lagrange form of the interpolation polynomial in `x` of
+ Return the Lagrange form of the interpolating polynomial in `x`
at the points (xs[k], ys[k]).
INPUT:
A symbolic expression (polynomial) interpolating each (xs[k], ys[k]).
+ SETUP::
+
+ sage: from mjo.interpolation import lagrange_polynomial
+
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 range(0, len(xs)) ]
- sigma = sum([ ys[k] * ls[k] for k in range(0, len(xs)) ])
- return sigma
+ ls = [ lagrange_coefficient(k, x, xs) for k in range(len(xs)) ]
+ return sum( ys[k] * ls[k] for k in range(len(xs)) )
+
+
+
+def lagrange_interpolate(f, x, xs):
+ """
+ Interpolate the function ``f`` at the points ``xs`` using the
+ Lagrange form of the interpolating polynomial.
+
+ INPUT:
+
+ - ``f`` -- The function to interpolate.
+
+ - ``x`` -- The independent variable of the resulting polynomial.
+
+ - ``xs`` -- A list of points at which to interpolate ``f``.
+
+ OUTPUT:
+
+ A polynomial in ``x`` which interpolates ``f`` at ``xs``.
+
+ SETUP::
+
+ sage: from mjo.interpolation import lagrange_interpolate
+
+ EXAMPLES:
+
+ We're exact on polynomials of degree `n` if we use `n+1` points::
+
+ sage: t = SR.symbol('t', domain='real')
+ sage: lagrange_interpolate(x^2, t, [-1,0,1]).simplify_rational()
+ t^2
+
+ """
+ # f should be a function of one variable.
+ z = f.variables()[0]
+ # We're really just doing map(f, xs) here; the additional
+ # gymnastics are to avoid a warning when calling `f` with an
+ # unnamed argument.
+ ys = [ f({z: xk}) for xk in xs ]
+ return lagrange_polynomial(x, xs, ys)
Assuming some function `f`, compute the coefficients of the
divided difference f[xs[0], ..., xs[n]].
- TESTS:
+ SETUP::
+
+ sage: from mjo.interpolation import divided_difference_coefficients
+
+ TESTS::
sage: divided_difference_coefficients([0])
[1]
[1/2/pi^2, -1/pi^2, 1/2/pi^2]
"""
- coeffs = [ product([ (QQ(1) / (xj - xi)) for xi in xs if xi != xj ])
- for xj in xs ]
- return coeffs
+ return [ ~lagrange_denominator(k, xs) for k in range(len(xs)) ]
def divided_difference(xs, ys):
The (possibly symbolic) divided difference function.
+ SETUP::
+
+ sage: from mjo.interpolation import divided_difference
+
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
We try something entirely symbolic::
- sage: f = function('f', x)
+ sage: f = function('f')(x)
sage: divided_difference([x], [f(x=x)])
f(x)
- sage: x1,x2 = var('x1,x2')
+ sage: x1,x2 = SR.var('x1,x2')
sage: divided_difference([x1,x2], [f(x=x1),f(x=x2)])
f(x1)/(x1 - x2) - f(x2)/(x1 - x2)
A symbolic expression.
- TESTS:
+ SETUP::
+
+ sage: from mjo.interpolation import lagrange_polynomial, newton_polynomial
+
+ 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)
True
"""
- degree = len(xs) - 1
-
- N = SR(0)
-
- for k in range(0, degree+1):
- term = divided_difference(xs[:k+1], ys[:k+1])
- term *= product([ x - xk for xk in xs[:k]])
- N += term
-
- return N
+ return sum( divided_difference(xs[:k+1], ys[:k+1])*lagrange_psi(x, xs[:k])
+ for k in range(len(xs)) )
def hermite_coefficient(k, x, xs):
A symbolic expression.
- TESTS:
+ SETUP::
+
+ sage: from mjo.interpolation import hermite_interpolant
+
+ 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
True
"""
- s1 = sum([ ys[k] * hermite_coefficient(k, x, xs)
- for k in range(0, len(xs)) ])
+ s1 = sum( ys[k] * hermite_coefficient(k, x, xs)
+ for k in range(len(xs)) )
- s2 = sum([ y_primes[k] * hermite_deriv_coefficient(k, x, xs)
- for k in range(0, len(xs)) ])
+ s2 = sum( y_primes[k] * hermite_deriv_coefficient(k, x, xs)
+ for k in range(len(xs)) )
return (s1 + s2)
OUTPUT:
A symbolic expression in one variable, `x`.
+
"""
- return product([ (x - xj) for xj in xs ])
+ return product( (x - xj) for xj in xs )
projects / sage.d.git / blobdiff