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 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 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 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 method'
return LinearMultistepMethod(alpha,beta,name=name)
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 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]
#betaj[k-i]=(-1.)**i*comb(j,i)*gamma[j]
beta=beta+betaj
name=str(k)+'-step Adams-Moulton method'
return LinearMultistepMethod(alpha,beta,name=name)
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
