def Nystrom(k):
r"""
Construct the k-step explicit Nystrom linear multistep method.
The methods are explicit and have order k.
They have the form:
`y_{n+1} = y_{n-1} + h \sum_{j=0}^{k-1} \beta_j f(y_n-k+j+1)`
They are generated using equations (1.13) and (1.7) from
[hairer1993]_ III.1, along with the binomial expansion
and the relation in exercise 4 on p. 367.
Note that the term "Nystrom method" is also commonly used to refer
to a class of methods for second-order ODEs; those are NOT
the methods generated by this function.
**Examples**::
>>> import nodepy.linear_multistep_method as lm
>>> nys3=lm.Nystrom(6)
>>> nys3.order()
6
References:
#. [hairer1993]_
"""
import sympy
from sympy import Rational
one = Rational(1, 1)
alpha = snp.zeros(k + 1)
alpha[k] = one
alpha[k - 2] = -one
beta = snp.zeros(k + 1)
kappa = snp.zeros(k)
gamma = snp.zeros(k)
gamma[0] = one
kappa[0] = 2 * one
beta[k - 1] = 2 * one
betaj = snp.zeros(k + 1)
for j in range(1, k):
gamma[j] = one - sum(gamma[:j] / snp.arange(j + 1, 1, -1))
kappa[j] = 2 * gamma[j] - gamma[j - 1]
for i in range(0, j + 1):
betaj[k - i -
1] = (-one)**i * sympy.combinatorial.factorials.binomial(
j, i) * kappa[j]
beta = beta + betaj
name = str(k) + '-step Nystrom'
return LinearMultistepMethod(alpha,
beta,
name=name,
shortname='Nys' + str(k))
def Milne_Simpson(k):
r"""
Construct the k-step, Milne-Simpson method.
The methods are implicit and (for k>=3) have order k+1.
They have the form:
`y_{n+1} = y_{n-1} + h \sum_{j=0}^{k} \beta_j f(y_n-k+j+1)`
They are generated using equation (1.15), the equation in
Exercise 3, and the relation in exercise 4, all from Hairer & Wanner
III.1, along with the binomial expansion.
**Examples**::
>>> import nodepy.linear_multistep_method as lm
>>> ms3=lm.Milne_Simpson(3)
>>> ms3.order()
4
References:
[hairer1993]_
"""
import sympy
alpha = snp.zeros(k + 1)
beta = snp.zeros(k + 1)
alpha[k] = 1
alpha[k - 2] = -1
gamma = snp.zeros(k + 1)
kappa = snp.zeros(k + 1)
gamma[0] = 1
kappa[0] = 2
beta[k] = 2
betaj = snp.zeros(k + 1)
for j in range(1, k + 1):
gamma[j] = -sum(gamma[:j] / snp.arange(j + 1, 1, -1))
kappa[j] = 2 * gamma[j] - gamma[j - 1]
for i in range(0, j + 1):
betaj[k - i] = (-1)**i * sympy.combinatorial.factorials.binomial(
j, i) * kappa[j]
beta = beta + betaj
name = str(k) + '-step Milne-Simpson'
return LinearMultistepMethod(alpha,
beta,
name=name,
shortname='MS' + str(k))
def Adams_Bashforth(k):
r"""
Construct the k-step, Adams-Bashforth method.
The methods are explicit and have order k.
They have the form:
`y_{n+1} = y_n + h \sum_{j=0}^{k-1} \beta_j f(y_n-k+j+1)`
They are generated using equations (1.5) and (1.7) from
[hairer1993]_ III.1, along with the binomial expansion.
**Examples**::
>>> import nodepy.linear_multistep_method as lm
>>> ab3=lm.Adams_Bashforth(3)
>>> ab3.order()
3
References:
#. [hairer1993]_
"""
import sympy
from sympy import Rational
one = Rational(1, 1)
alpha = snp.zeros(k + 1)
beta = snp.zeros(k + 1)
alpha[k] = one
alpha[k - 1] = -one
gamma = snp.zeros(k)
gamma[0] = one
beta[k - 1] = one
betaj = snp.zeros(k + 1)
for j in range(1, k):
gamma[j] = one - sum(gamma[:j] / snp.arange(j + 1, 1, -1))
for i in range(0, j + 1):
betaj[k - i -
1] = (-one)**i * sympy.combinatorial.factorials.binomial(
j, i) * gamma[j]
beta = beta + betaj
name = str(k) + '-step Adams-Bashforth'
return LinearMultistepMethod(alpha,
beta,
name=name,
shortname='AB' + str(k))
def Adams_Moulton(k):
r"""
Construct the k-step, Adams-Moulton method.
The methods are implicit and have order k+1.
They have the form:
`y_{n+1} = y_n + h \sum_{j=0}^{k} \beta_j f(y_n-k+j+1)`
They are generated using equation (1.9) and the equation in
Exercise 3 from Hairer & Wanner III.1, along with the binomial
expansion.
**Examples**::
>>> import nodepy.linear_multistep_method as lm
>>> am3=lm.Adams_Moulton(3)
>>> am3.order()
4
References:
[hairer1993]_
"""
import sympy
alpha = snp.zeros(k + 1)
beta = snp.zeros(k + 1)
alpha[k] = 1
alpha[k - 1] = -1
gamma = snp.zeros(k + 1)
gamma[0] = 1
beta[k] = 1
betaj = snp.zeros(k + 1)
for j in range(1, k + 1):
gamma[j] = -sum(gamma[:j] / snp.arange(j + 1, 1, -1))
for i in range(0, j + 1):
betaj[k - i] = (-1)**i * sympy.combinatorial.factorials.binomial(
j, i) * gamma[j]
beta = beta + betaj
name = str(k) + '-step Adams-Moulton'
return LinearMultistepMethod(alpha,
beta,
name=name,
shortname='AM' + str(k))
def _satisfies_order_conditions(self, p, tol):
""" Return True if the linear multistep method satisfies
the conditions of order p (only) """
ii = snp.arange(len(self.alpha))
return abs(sum(ii**p * self.alpha - p * self.beta * ii**(p - 1))) < tol