def sum_next(self, f, nodes, degree, prec, previous, verbose=False):
r"""
Evaluates the step sum `\sum w_k f(x_k)` where the *nodes* list
contains the `(w_k, x_k)` pairs.
:func:`summation` will supply the list *results* of
values computed by :func:`sum_next` at previous degrees, in
case the quadrature rule is able to reuse them.
"""
return fdot((w, f(x)) for (x,w) in nodes)
def sum_next(self, f, nodes, degree, prec, previous, verbose=False):
r"""
Evaluates the step sum `\sum w_k f(x_k)` where the *nodes* list
contains the `(w_k, x_k)` pairs.
:func:`summation` will supply the list *results* of
values computed by :func:`sum_next` at previous degrees, in
case the quadrature rule is able to reuse them.
"""
return fdot((w, f(x)) for (x,w) in nodes)
def sum_next(self, f, nodes, degree, prec, previous, verbose=False):
"""
Step sum for tanh-sinh quadrature of degree `m`. We exploit the
fact that half of the abscissas at degree `m` are precisely the
abscissas from degree `m-1`. Thus reusing the result from
the previous level allows a 2x speedup.
"""
h = mpf(2)**(-degree)
# Abscissas overlap, so reusing saves half of the time
if previous:
S = previous[-1]/(h*2)
else:
S = mpf(0)
S += fdot((w,f(x)) for (x,w) in nodes)
return h*S
def sum_next(self, f, nodes, degree, prec, previous, verbose=False):
"""
Step sum for tanh-sinh quadrature of degree `m`. We exploit the
fact that half of the abscissas at degree `m` are precisely the
abscissas from degree `m-1`. Thus reusing the result from
the previous level allows a 2x speedup.
"""
h = mpf(2)**(-degree)
# Abscissas overlap, so reusing saves half of the time
if previous:
S = previous[-1]/(h*2)
else:
S = mpf(0)
S += fdot((w,f(x)) for (x,w) in nodes)
return h*S
def __mul__(self, other):
if isinstance(other, matrix):
# dot multiplication TODO: use Strassen's method?
if self.__cols != other.__rows:
raise ValueError('dimensions not compatible for multiplication')
new = matrix(self.__rows, other.__cols)
for i in xrange(self.__rows):
for j in xrange(other.__cols):
new[i, j] = fdot((self[i,k], other[k,j])
for k in xrange(other.__rows))
return new
else:
# try scalar multiplication
new = matrix(self.__rows, self.__cols)
for i in xrange(self.__rows):
for j in xrange(self.__cols):
new[i, j] = other * self[i, j]
return new
def __mul__(self, other):
if isinstance(other, matrix):
# dot multiplication TODO: use Strassen's method?
if self.__cols != other.__rows:
raise ValueError('dimensions not compatible for multiplication')
new = matrix(self.__rows, other.__cols)
for i in xrange(self.__rows):
for j in xrange(other.__cols):
new[i, j] = fdot((self[i,k], other[k,j])
for k in xrange(other.__rows))
return new
else:
# try scalar multiplication
new = matrix(self.__rows, self.__cols)
for i in xrange(self.__rows):
for j in xrange(self.__cols):
new[i, j] = other * self[i, j]
return new
