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
OUTPUT:
- A symbolic function of one variable.
+ 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:
OUTPUT:
- A symbolic function (polynomial) interpolating each (xs[k], ys[k]).
+ 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 divided_difference(f, xs):
+def lagrange_interpolate(f, x, xs):
"""
- Return the Newton divided difference of `f` at the points
- `xs`. Reference:
+ 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)
+
+
+
+def divided_difference_coefficients(xs):
+ """
+ Assuming some function `f`, compute the coefficients of the
+ divided difference f[xs[0], ..., xs[n]].
+
+ SETUP::
+
+ sage: from mjo.interpolation import divided_difference_coefficients
+
+ TESTS::
+
+ sage: divided_difference_coefficients([0])
+ [1]
+ sage: divided_difference_coefficients([0, pi])
+ [-1/pi, 1/pi]
+ sage: divided_difference_coefficients([0, pi, 2*pi])
+ [1/2/pi^2, -1/pi^2, 1/2/pi^2]
+
+ """
+ return [ ~lagrange_denominator(k, xs) for k in range(len(xs)) ]
+
+
+def divided_difference(xs, ys):
+ """
+ Return the Newton divided difference of the points (xs[k],
+ ys[k]). Reference:
http://en.wikipedia.org/wiki/Divided_differences
INPUT:
- - ``f`` -- The function whose divided difference we seek.
+ - ``xs`` -- The list of x-values.
- - ``xs`` -- The list of points at which to compute `f`.
+ - ``ys`` -- The function values at `xs`.
OUTPUT:
- The divided difference of `f` at ``xs``.
+ The (possibly symbolic) divided difference function.
+
+ SETUP::
+
+ sage: from mjo.interpolation import divided_difference
TESTS::
- sage: divided_difference(sin, [0])
+ sage: xs = [0]
+ sage: ys = list(map(sin, xs))
+ sage: divided_difference(xs, ys)
0
- sage: divided_difference(sin, [0, pi])
+ sage: xs = [0, pi]
+ sage: ys = list(map(sin, xs))
+ sage: divided_difference(xs, ys)
0
- sage: divided_difference(sin, [0, pi, 2*pi])
+ sage: xs = [0, pi, 2*pi]
+ sage: ys = list(map(sin, xs))
+ sage: divided_difference(xs, ys)
0
We try something entirely symbolic::
- sage: f = function('f', x)
- sage: divided_difference(f, [x])
+ sage: f = function('f')(x)
+ sage: divided_difference([x], [f(x=x)])
f(x)
- sage: x1,x2 = var('x1,x2')
- sage: divided_difference(f, [x1,x2])
- (f(x1) - f(x2))/(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)
"""
- if (len(xs) == 1):
- # Avoid that goddamned DeprecationWarning when we have a named
- # argument but don't know what it is.
- if len(f.variables()) == 0:
- return f(xs[0])
- else:
- v = f.variables()[0]
- return f({ v: xs[0] })
+ coeffs = divided_difference_coefficients(xs)
+ v_cs = vector(coeffs)
+ v_ys = vector(ys)
+ return v_cs.dot_product(v_ys)
- # Use the recursive definition.
- numerator = divided_difference(f, xs[1:])
- numerator -= divided_difference(f, xs[:-1])
- return numerator / (xs[-1] - xs[0])
-
-def newton_polynomial(f, x, xs):
+def newton_polynomial(x, xs, ys):
"""
- Return the Newton form of the interpolating polynomial of `f` at
- the points `xs` in the variable `x`.
+ Return the Newton form of the interpolating polynomial of the
+ points (xs[k], ys[k]) in the variable `x`.
INPUT:
- - ``f`` -- The function to interpolate.
-
- ``x`` -- The independent variable to use for the interpolating
polynomial.
- - ``xs`` -- The list of points at which to interpolate `f`.
+ - ``xs`` -- The list of x-values.
+
+ - ``ys`` -- The function values at `xs`.
OUTPUT:
- A symbolic function.
+ 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(sin, x, xs)
+ sage: N = newton_polynomial(x, xs, ys)
sage: bool(N == L)
True
"""
- degree = len(xs) - 1
+ 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):
+ """
+ Return the Hermite coefficient h_{k}(x). See Atkinson, p. 160.
+
+ INPUT:
+
+ - ``k`` -- The index of the coefficient.
+
+ - ``x`` -- The symbolic variable to use as the argument of h_{k}.
+
+ - ``xs`` -- The list of points at which the function values are
+ known.
+
+ OUTPUT:
+
+ A symbolic expression.
+
+ """
+ lk = lagrange_coefficient(k, x, xs)
+ return (1 - 2*lk.diff(x)(x=xs[k])*(x - xs[k]))*(lk**2)
+
+
+def hermite_deriv_coefficient(k, x, xs):
+ """
+ Return the Hermite derivative coefficient, \tilde{h}_{k}(x). See
+ Atkinson, p. 160.
+
+ INPUT:
+
+ - ``k`` -- The index of the coefficient.
+
+ - ``x`` -- The symbolic variable to use as the argument of h_{k}.
+
+ - ``xs`` -- The list of points at which the function values are
+ known.
+
+ OUTPUT:
+
+ A symbolic expression.
+
+ """
+ lk = lagrange_coefficient(k, x, xs)
+ return (x - xs[k])*(lk**2)
+
+
+def hermite_interpolant(x, xs, ys, y_primes):
+ """
+ Return the Hermite interpolant `H(x)` such that H(xs[k]) = ys[k]
+ and H'(xs[k]) = y_primes[k] for each k.
+
+ Reference: Atkinson, p. 160.
+
+ INPUT:
+
+ - ``x`` -- The symbolic variable to use as the argument of H(x).
+
+ - ``xs`` -- The list of points at which the function values are
+ known.
+
+ - ``ys`` -- The function values at the `xs`.
+
+ - ``y_primes`` -- The derivatives at the `xs`.
+
+ OUTPUT:
+
+ A symbolic expression.
+
+ SETUP::
+
+ sage: from mjo.interpolation import hermite_interpolant
+
+ TESTS::
+
+ sage: xs = [ 0, pi/6, pi/2 ]
+ 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
+ sage: expected += 81/2*((pi - 6*x)/pi + 1)*(pi - 2*x)^2*x^2/pi^4
+ sage: expected += (pi - 6*x)^2*(pi - 2*x)^2*x/pi^4
+ sage: bool(H == expected)
+ True
+
+ """
+ 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(len(xs)) )
+
+ return (s1 + s2)
+
+
+def lagrange_psi(x, xs):
+ """
+ The function,
- N = SR(0)
+ Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])
- for k in range(0, degree+1):
- term = divided_difference(f, xs[:k+1])
- term *= product([ x - xk for xk in xs[:k]])
- N += term
+ used in Lagrange and Hermite interpolation.
+
+ INPUT:
+
+ - ``x`` -- The independent variable of the resulting expression.
+
+ - ``xs`` -- A list of points.
+
+ OUTPUT:
+
+ A symbolic expression in one variable, `x`.
+
+ """
- return N
+ return product( (x - xj) for xj in xs )
projects / sage.d.git / blobdiff