def main():
# 一般項f(n)
fn = sym.simplify("f(n)")
# 左辺が0になるように漸化式を書く
y = sym.simplify("f(n)-f(n-1)-f(n-2)")
# 初期条件
init = sym.simplify({"f(0)": 0, "f(1)": 1})
# 初期条件iniの漸化式sをfについて解く
fib_n = sym.rsolve(y, fn, init)
print('Fibonacci sequenceの一般項の式は')
print(fib_n)
print('------------------------------')
# -sqrt(5)*(1/2 - sqrt(5)/2)**n/5 + sqrt(5)*(1/2 + sqrt(5)/2)**n/5
def fib_g(n):
x = sym.symbols('x', nonnegative=True, integer=True)
fib = -sym.sqrt(5) * (1 / 2 - sym.sqrt(5) / 2)**n / 5 + sym.sqrt(5) * (
1 / 2 + sym.sqrt(5) / 2)**n / 5
# Fibの式のxにnを代入
formula = fib.subs(x, n)
ans = int(sym.simplify(formula))
print(formula, '=', ans)
return ans
print('0以上の整数を標準入力するとFibonacci sequenceの一般項の式と検算結果を出力する')
n = int(input())
print('検算結果')
print('第',
n,
'項までの',
'Fibonacci sequence', [fib_g(n) for n in range(0, n)],
sep='')
def _selector(k, n):
'''Compute the closed form for sequence defined by
f(0) = 1,
f(1) = ... = f(k-1) = 0,
f(n+k) = f(n)'''
f = sp.Function('f')
inits = {f(0): 1}
inits.update({f(i): 0 for i in range(1, k)})
return sp.rsolve(f(n + k) - f(n), f(n), inits)
def apply(self, eqns, a, n, evaluation):
'RSolve[eqns_, a_, n_]'
# TODO: Do this with rules?
if not eqns.has_form('List', None):
eqns = Expression('List', eqns)
if len(eqns.leaves) == 0:
return
for eqn in eqns.leaves:
if eqn.get_head_name() != 'System`Equal':
evaluation.message('RSolve', 'deqn', eqn)
return
if (n.is_atom() and not n.is_symbol()) or \
n.get_head_name() in ('System`Plus', 'System`Times',
'System`Power') or \
'System`Constant' in n.get_attributes(evaluation.definitions):
# TODO: Factor out this check for dsvar into a separate
# function. DSolve uses this too.
evaluation.message('RSolve', 'dsvar')
return
try:
a.leaves
function_form = None
func = a
except AttributeError:
func = Expression(a, n)
function_form = Expression('List', n)
if func.is_atom() or len(func.leaves) != 1:
evaluation.message('RSolve', 'dsfun', a)
if n not in func.leaves:
evaluation.message('DSolve', 'deqx')
# Seperate relations from conditions
conditions = {}
def is_relation(eqn):
left, right = eqn.leaves
for l, r in [(left, right), (right, left)]:
if (left.get_head_name() == func.get_head_name() and # noqa
len(left.leaves) == 1 and isinstance(
l.leaves[0].to_python(), int) and r.is_numeric()):
r_sympy = r.to_sympy()
if r_sympy is None:
raise ValueError
conditions[l.leaves[0].to_python()] = r_sympy
return False
return True
# evaluate is_relation on all leaves to store conditions
try:
relations = [leaf for leaf in eqns.leaves if is_relation(leaf)]
except ValueError:
return
relation = relations[0]
left, right = relation.leaves
relation = Expression('Plus', left,
Expression('Times', -1,
right)).evaluate(evaluation)
sym_eq = relation.to_sympy(
converted_functions=set([func.get_head_name()]))
if sym_eq is None:
return
sym_n = sympy.core.symbols(str(sympy_symbol_prefix + n.name))
sym_func = sympy.Function(
str(sympy_symbol_prefix + func.get_head_name()))(sym_n)
sym_conds = {}
for cond in conditions:
sym_conds[sympy.Function(str(
sympy_symbol_prefix + func.get_head_name()))(cond)] = \
conditions[cond]
try:
# Sympy raises error when given empty conditions. Fixed in
# upcomming sympy release.
if sym_conds != {}:
sym_result = sympy.rsolve(sym_eq, sym_func, sym_conds)
else:
sym_result = sympy.rsolve(sym_eq, sym_func)
if not isinstance(sym_result, list):
sym_result = [sym_result]
except ValueError:
return
if function_form is None:
return Expression(
'List', *[
Expression('List', Expression('Rule', a, from_sympy(soln)))
for soln in sym_result
])
else:
return Expression(
'List', *[
Expression(
'List',
Expression(
'Rule', a,
Expression('Function', function_form,
from_sympy(soln))))
for soln in sym_result
])
def apply(self, eqns, a, n, evaluation):
'RSolve[eqns_, a_, n_]'
# TODO: Do this with rules?
if not eqns.has_form('List', None):
eqns = Expression('List', eqns)
if len(eqns.leaves) == 0:
return
for eqn in eqns.leaves:
if eqn.get_head_name() != 'System`Equal':
evaluation.message('RSolve', 'deqn', eqn)
return
if (n.is_atom() and not n.is_symbol()) or \
n.get_head_name() in ('System`Plus', 'System`Times',
'System`Power') or \
'System`Constant' in n.get_attributes(evaluation.definitions):
# TODO: Factor out this check for dsvar into a separate
# function. DSolve uses this too.
evaluation.message('RSolve', 'dsvar')
return
try:
a.leaves
function_form = None
func = a
except AttributeError:
func = Expression(a, n)
function_form = Expression('List', n)
if func.is_atom() or len(func.leaves) != 1:
evaluation.message('RSolve', 'dsfun', a)
if n not in func.leaves:
evaluation.message('DSolve', 'deqx')
# Seperate relations from conditions
conditions = {}
def is_relation(eqn):
left, right = eqn.leaves
for l, r in [(left, right), (right, left)]:
if (left.get_head_name() == func.get_head_name() and # noqa
len(left.leaves) == 1 and
isinstance(l.leaves[0].to_python(), int) and
r.is_numeric()):
conditions[l.leaves[0].to_python()] = r.to_sympy()
return False
return True
relation = filter(is_relation, eqns.leaves)[0]
left, right = relation.leaves
relation = Expression('Plus', left, Expression(
'Times', -1, right)).evaluate(evaluation)
sym_eq = relation.to_sympy(
converted_functions=set([func.get_head_name()]))
sym_n = sympy.symbols(str(sympy_symbol_prefix + n.name))
sym_func = sympy.Function(str(
sympy_symbol_prefix + func.get_head_name()))(sym_n)
sym_conds = {}
for cond in conditions:
sym_conds[sympy.Function(str(
sympy_symbol_prefix + func.get_head_name()))(cond)] = \
conditions[cond]
try:
# Sympy raises error when given empty conditions. Fixed in
# upcomming sympy release.
if sym_conds != {}:
sym_result = sympy.rsolve(sym_eq, sym_func, sym_conds)
else:
sym_result = sympy.rsolve(sym_eq, sym_func)
if not isinstance(sym_result, list):
sym_result = [sym_result]
except ValueError:
return
if function_form is None:
return Expression('List', *[
Expression('List', Expression('Rule', a, from_sympy(soln)))
for soln in sym_result])
else:
return Expression('List', *[
Expression('List', Expression(
'Rule', a, Expression('Function', function_form,
from_sympy(soln)))) for soln in sym_result])
#程序文件Pex13_2.py
import sympy as sp
sp.var('k')
sp.var('y', cls=sp.Function)
f = y(k + 1) - y(k) - 3 - 2 * k
f1 = sp.rsolve(f, y(k))
f2 = sp.simplify(f1)
print(f2)
#程序文件Pex13_1.py
from sympy import Function, rsolve
from sympy.abc import n
y = Function('y')
f = y(n + 2) - y(n + 1) - y(n)
ff = rsolve(f, y(n), {y(1): 1, y(2): 1})
print(ff)
# Program 13a: Computing bank interest.
# See Example 1.
from sympy import Function, rsolve
from sympy.abc import n
x = Function('x')
f = x(n + 1) - (1 + 3 / 100) * x(n)
sol = rsolve(f, x(n), {x(0): 10000})
print('x_n = {}'.format(sol))
x_5 = round(sol.subs(n, 5), 2)
print('x(5) = ${:,}'.format(x_5))
#程序文件ex3_1_3.py
import sympy as sp
sp.var('k')
y = sp.Function('y')
f = y(k + 2) - y(k + 1) - y(k)
s = sp.rsolve(f, y(k), {y(0): 1, y(1): 1})
print(s)
# Programs_13b: A second order recurrence relation.
# See Example 2.
from sympy import Function, rsolve
from sympy.abc import n
x = Function('x')
f = x(n + 2) - x(n + 1) - 6 * x(n)
sol = rsolve(f, x(n), {
x(0): 1,
x(1): 2
})
print('x_n=', sol)
#程序文件Pex13_2.py
import sympy as sp
sp.var('k'); sp.var('y',cls=sp.Function)
f = y(k+1)-y(k)-3-2*k
f1 = sp.rsolve(f, y(k)); f2 = sp.simplify(f1)
print(f2)
# #### The `glue:math` directive
#
# The `glue:math` directive is specific to LaTeX math outputs
# (glued variables that contain a `text/latex` MIME type),
# and works similarly to the [Sphinx math directive](https://www.sphinx-doc.org/en/1.8/usage/restructuredtext/directives.html#math).
# For example, with this code we glue an equation:
# In[13]:
import sympy as sym
f = sym.Function('f')
y = sym.Function('y')
n = sym.symbols(r'\alpha')
f = y(n) - 2 * y(n - 1 / sym.pi) - 5 * y(n - 2)
glue("sym_eq", sym.rsolve(f, y(n), [1, 4]))
# and now we can use the following code:
#
# ````md
# ```{glue:math} sym_eq
# :label: eq-sym
# ```
# ````
#
# to insert the equation here:
#
# ```{glue:math} sym_eq
# :label: eq-sym
# ```
#
# This file is aimed at being run as an interactive session via
# scitools file2interactive:
#
# Terminal> scitools file2interactive decay_analysis.py
#
import sys
import sympy as sym
p = sym.Symbol('p')
A_e = sym.exp(-p)
# Demo on Taylor polynomials
A_e.series(p, 0, 6)
"""
# NOTE: rsolve can solve recurrence relations:
a, dt, I, n = sym.symbols('a dt I n')
u = sym.Function('u')
f = u(n+1) - u(n) + dt*a*u(n+1)
sym.rsolve(f, u(n), {u(0): I})
# However, 0 is the answer!
# Experimentation shows that we cannot have symbols dt, a in the
# recurrence equation, just n or numbers.
# Even if we worked with scaled equations, dt is in there,
# rsolve cannot be applied.
"""
# Numerical amplification factor
theta = sym.Symbol('theta')
A = (1-(1-theta)*p)/(1+theta*p)
