OUTPUT:
- A symbolic function of one variable.
+ A symbolic expression of one variable.
TESTS::
OUTPUT:
- A symbolic function (polynomial) interpolating each (xs[k], ys[k]).
+ A symbolic expression (polynomial) interpolating each (xs[k], ys[k]).
TESTS::
for xj in xs ]
return coeffs
-def divided_difference(f, xs):
+
+def divided_difference(xs, ys):
"""
- Return the Newton divided difference of `f` at the points
- `xs`. Reference:
+ 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.
TESTS::
- sage: divided_difference(sin, [0])
+ sage: xs = [0]
+ sage: ys = map(sin, xs)
+ sage: divided_difference(xs, ys)
0
- sage: divided_difference(sin, [0, pi])
+ sage: xs = [0, pi]
+ sage: ys = 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 = 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: 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: 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] })
-
- # Use the recursive definition.
- numerator = divided_difference(f, xs[1:])
- numerator -= divided_difference(f, xs[:-1])
- return numerator / (xs[-1] - xs[0])
+ coeffs = divided_difference_coefficients(xs)
+ v_cs = vector(coeffs)
+ v_ys = vector(ys)
+ return v_cs.dot_product(v_ys)
-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:
sage: xs = [ -pi/2, -pi/6, 0, pi/6, pi/2 ]
sage: ys = 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
N = SR(0)
for k in range(0, degree+1):
- term = divided_difference(f, xs[:k+1])
+ term = divided_difference(xs[:k+1], ys[:k+1])
term *= product([ x - xk for xk in xs[:k]])
N += term
return N
+
+
+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.
+
+ TESTS:
+
+ sage: xs = [ 0, pi/6, pi/2 ]
+ sage: ys = map(sin, xs)
+ sage: y_primes = 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(0, len(xs)) ])
+
+ s2 = sum([ y_primes[k] * hermite_deriv_coefficient(k, x, xs)
+ for k in range(0, len(xs)) ])
+
+ return (s1 + s2)
+
+
+def lagrange_psi(x, xs):
+ """
+ The function,
+
+ Psi(x) = (x - xs[0])*(x - xs[1])* ... *(x - xs[-1])
+
+ 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 product([ (x - xj) for xj in xs ])
projects / sage.d.git / blobdiff