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# Synthesize a procedure to numerically integrate the 3rd order poly spiral
tex = False
if tex:
mulsym = ' '
else:
mulsym = ' * '
class Poly:
def __init__(self, p0, coeffs):
self.p0 = p0
self.coeffs = coeffs
def eval(self, x):
y = x ** self.p0
z = 0
for c in coeffs:
z += y * c
y *= x
return z
def add(poly0, poly1, nmax):
lp0 = len(poly0.coeffs)
lp1 = len(poly1.coeffs)
p0 = min(poly0.p0, poly1.p0)
n = min(max(poly0.p0 + lp0, poly1.p1 + lp1), nmax) - p0
if n <= 0: return Poly(0, [])
coeffs = []
for i in range(n):
c = 0
if i >= poly0.p0 - p0 and i < lp0 + poly0.p0 - p0:
c += poly0.coeffs[i + p0 - poly0.p0]
if i >= poly1.p0 - p0 and i < lp1 + poly1.p0 - p0:
c += poly1.coeffs[i + p0 - poly1.p0]
coeffs.append(c)
return Poly(p0, coeffs)
def pr(str):
if tex:
print str, '\\\\'
else:
print '\t' + str + ';'
def prd(str):
if tex:
print str, '\\\\'
else:
print '\tdouble ' + str + ';'
def polymul(p0, p1, degree, basename, suppress_odd = False):
result = []
for i in range(min(degree, len(p0) + len(p1) - 1)):
terms = []
for j in range(i + 1):
if j < len(p0) and i - j < len(p1):
t0 = p0[j]
t1 = p1[i - j]
if t0 != None and t1 != None:
terms.append(t0 + mulsym + t1)
if terms == []:
result.append(None)
else:
var = basename % i
if (j % 2 == 0) or not suppress_odd:
prd(var + ' = ' + ' + '.join(terms))
result.append(var)
return result
def polysquare(p0, degree, basename):
result = []
for i in range(min(degree, 2 * len(p0) - 1)):
terms = []
for j in range((i + 1)/ 2):
if i - j < len(p0):
t0 = p0[j]
t1 = p0[i - j]
if t0 != None and t1 != None:
terms.append(t0 + mulsym + t1)
if len(terms) >= 1:
if tex and len(terms) == 1:
terms = ['2 ' + terms[0]]
else:
terms = ['2' + mulsym + '(' + ' + '.join(terms) + ')']
if (i % 2) == 0:
t = p0[i / 2]
if t != None:
if tex:
terms.append(t + '^2')
else:
terms.append(t + mulsym + t)
if terms == []:
result.append(None)
else:
var = basename % i
prd(var + ' = ' + ' + '.join(terms))
result.append(var)
return result
def mkspiro(degree):
if tex:
us = ['u = 1']
vs = ['v =']
else:
us = ['u = 1']
vs = ['v = 0']
if tex:
tp = [None, 't_{11}', 't_{12}', 't_{13}', 't_{14}']
else:
tp = [None, 't1_1', 't1_2', 't1_3', 't1_4']
if tex:
prd(tp[1] + ' = k_0')
prd(tp[2] + ' = \\frac{k_1}{2}')
prd(tp[3] + ' = \\frac{k_2}{6}')
prd(tp[4] + ' = \\frac{k_3}{24}')
else:
prd(tp[1] + ' = km0')
prd(tp[2] + ' = .5 * km1')
prd(tp[3] + ' = (1./6) * km2')
prd(tp[4] + ' = (1./24) * km3')
tlast = tp
coef = 1.
for i in range(1, degree - 1):
tmp = []
tcoef = coef
#print tlast
for j in range(len(tlast)):
c = tcoef / (j + 1)
if (j % 2) == 0 and tlast[j] != None:
if tex:
tmp.append('\\frac{%s}{%.0f}' % (tlast[j], 1./c))
else:
if c < 1e-9:
cstr = '%.16e' % c
else:
cstr = '(1./%d)' % int(.5 + (1./c))
tmp.append(cstr + ' * ' + tlast[j])
tcoef *= .5
if tmp != []:
sign = ('+', '-')[(i / 2) % 2]
var = ('u', 'v')[i % 2]
if tex:
if i == 1: pref = ''
else: pref = sign + ' '
str = pref + (' ' + sign + ' ').join(tmp)
else:
str = var + ' ' + sign + '= ' + ' + '.join(tmp)
if var == 'u': us.append(str)
else: vs.append(str)
if i < degree - 1:
if tex:
basename = 't_{%d%%d}' % (i + 1)
else:
basename = 't%d_%%d' % (i + 1)
if i == 1:
tnext = polysquare(tp, degree - 1, basename)
t2 = tnext
elif i == 3:
tnext = polysquare(t2l, degree - 1, basename)
elif (i % 2) == 0:
tnext = polymul(tlast, tp, degree - 1, basename, True)
else:
tnext = polymul(t2l, t2, degree - 1, basename)
t2l = tlast
tlast = tnext
coef /= (i + 1)
if tex:
pr(' '.join(us))
pr(' '.join(vs))
else:
for u in us:
pr(u)
for v in vs:
pr(v)
if __name__ == '__main__':
mkspiro(12)
|
