def norm(x, p=2):
r"""
Gives the entrywise `p`-norm of an iterable *x*, i.e. the vector norm
`\left(\sum_k |x_k|^p\right)^{1/p}`, for any given `1 \le p \le \infty`.
Special cases:
If *x* is not iterable, this just returns ``absmax(x)``.
``p=1`` gives the sum of absolute values.
``p=2`` is the standard Euclidean vector norm.
``p=inf`` gives the magnitude of the largest element.
For *x* a matrix, ``p=2`` is the Frobenius norm.
For operator matrix norms, use :func:`mnorm` instead.
You can use the string 'inf' as well as float('inf') or mpf('inf')
to specify the infinity norm.
**Examples**
>>> from mpmath import *
>>> mp.dps = 15
>>> x = matrix([-10, 2, 100])
>>> norm(x, 1)
mpf('112.0')
>>> norm(x, 2)
mpf('100.5186549850325')
>>> norm(x, inf)
mpf('100.0')
"""
try:
iter(x)
except TypeError:
return absmax(x)
if type(p) is not int:
p = mpmathify(p)
if p == inf:
return max(absmax(i) for i in x)
elif p == 1:
return fsum(x, absolute=1)
elif p == 2:
return sqrt(fsum(x, absolute=1, squared=1))
elif p > 1:
return nthroot(fsum(abs(i)**p for i in x), p)
else:
raise ValueError('p has to be >= 1')
def norm(x, p=2):
r"""
Gives the entrywise `p`-norm of an iterable *x*, i.e. the vector norm
`\left(\sum_k |x_k|^p\right)^{1/p}`, for any given `1 \le p \le \infty`.
Special cases:
If *x* is not iterable, this just returns ``absmax(x)``.
``p=1`` gives the sum of absolute values.
``p=2`` is the standard Euclidean vector norm.
``p=inf`` gives the magnitude of the largest element.
For *x* a matrix, ``p=2`` is the Frobenius norm.
For operator matrix norms, use :func:`mnorm` instead.
You can use the string 'inf' as well as float('inf') or mpf('inf')
to specify the infinity norm.
**Examples**
>>> from mpmath import *
>>> mp.dps = 15
>>> x = matrix([-10, 2, 100])
>>> norm(x, 1)
mpf('112.0')
>>> norm(x, 2)
mpf('100.5186549850325')
>>> norm(x, inf)
mpf('100.0')
"""
try:
iter(x)
except TypeError:
return absmax(x)
if type(p) is not int:
p = mpmathify(p)
if p == inf:
return max(absmax(i) for i in x)
elif p == 1:
return fsum(x, absolute=1)
elif p == 2:
return sqrt(fsum(x, absolute=1, squared=1))
elif p > 1:
return nthroot(fsum(abs(i)**p for i in x), p)
else:
raise ValueError('p has to be >= 1')
def mnorm_oo(A):
"""
Calculate the oo-norm (maximal row sum) of a matrix.
"""
assert isinstance(A, matrix)
m, n = A.rows, A.cols
return max((sum((absmax(A[i, j]) for j in xrange(n))) for i in xrange(m)))
def mnorm_oo(A):
"""
Calculate the oo-norm (maximal row sum) of a matrix.
"""
assert isinstance(A, matrix)
m, n = A.rows, A.cols
return max((sum((absmax(A[i,j]) for j in xrange(n))) for i in xrange(m)))
def mnorm_1(A):
"""
Calculate the 1-norm (maximal column sum) of a matrix.
"""
assert isinstance(A, matrix)
m, n = A.rows, A.cols
return max(fsum(absmax(A[i,j]) for i in xrange(m)) for j in xrange(n))
def norm_p(x, p=2):
"""
Calculate the p-norm of a vector.
0 < p <= oo
Note: you may want to use float('inf') or mpmath's equivalent to specify oo.
"""
if p == inf:
return max((absmax(i) for i in x))
elif p > 1:
return nthroot(sum((abs(i)**p for i in x)), p)
elif p == 1:
return sum((abs(i) for i in x))
else:
raise ValueError('p has to be an integer greater than 0')
def norm_p(x, p=2):
"""
Calculate the p-norm of a vector.
0 < p <= oo
Note: you may want to use float('inf') or mpmath's equivalent to specify oo.
"""
if p == inf:
return max((absmax(i) for i in x))
elif p > 1:
return nthroot(sum((abs(i)**p for i in x)), p)
elif p == 1:
return sum((abs(i) for i in x))
else:
raise ValueError('p has to be an integer greater than 0')
