def triu(m, k=0):
"""
Upper triangle of an array.
Return a copy of a matrix with the elements below the `k`-th diagonal
zeroed.
Please refer to the documentation for `tril` for further details.
See Also
--------
tril : lower triangle of an array
Examples
--------
>>> np.triu([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 1, 2, 3],
[ 4, 5, 6],
[ 0, 8, 9],
[ 0, 0, 12]])
"""
m = asanyarray(m)
out = multiply((1 - tri(m.shape[0], m.shape[1], k - 1, dtype=m.dtype)), m)
return out
def kron(a,b):
"""kronecker product of a and b
Kronecker product of two arrays is block array
[[ a[ 0 ,0]*b, a[ 0 ,1]*b, ... , a[ 0 ,n-1]*b ],
[ ... ... ],
[ a[m-1,0]*b, a[m-1,1]*b, ... , a[m-1,n-1]*b ]]
"""
wrapper = get_array_wrap(a, b)
b = asanyarray(b)
a = array(a,copy=False,subok=True,ndmin=b.ndim)
ndb, nda = b.ndim, a.ndim
if (nda == 0 or ndb == 0):
return _nx.multiply(a,b)
as_ = a.shape
bs = b.shape
if not a.flags.contiguous:
a = reshape(a, as_)
if not b.flags.contiguous:
b = reshape(b, bs)
nd = ndb
if (ndb != nda):
if (ndb > nda):
as_ = (1,)*(ndb-nda) + as_
else:
bs = (1,)*(nda-ndb) + bs
nd = nda
result = outer(a,b).reshape(as_+bs)
axis = nd-1
for _ in xrange(nd):
result = concatenate(result, axis=axis)
if wrapper is not None:
result = wrapper(result)
return result
def tril(m, k=0):
"""
Lower triangle of an array.
Return a copy of an array with elements above the `k`-th diagonal zeroed.
Parameters
----------
m : array_like, shape (M, N)
Input array.
k : int, optional
Diagonal above which to zero elements. `k = 0` (the default) is the
main diagonal, `k < 0` is below it and `k > 0` is above.
Returns
-------
tril : ndarray, shape (M, N)
Lower triangle of `m`, of same shape and data-type as `m`.
See Also
--------
triu : same thing, only for the upper triangle
Examples
--------
>>> np.tril([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 0, 0, 0],
[ 4, 0, 0],
[ 7, 8, 0],
[10, 11, 12]])
"""
m = asanyarray(m)
out = multiply(tri(m.shape[0], m.shape[1], k=k, dtype=m.dtype),m)
return out
def triu(m, k=0):
""" returns the elements on and above the k-th diagonal of m. k=0 is the
main diagonal, k > 0 is above and k < 0 is below the main diagonal.
"""
m = asanyarray(m)
out = multiply((1-tri(m.shape[0], m.shape[1], k-1, int)),m)
return out
def tril(m, k=0):
""" returns the elements on and below the k-th diagonal of m. k=0 is the
main diagonal, k > 0 is above and k < 0 is below the main diagonal.
"""
m = asanyarray(m)
out = multiply(tri(m.shape[0], m.shape[1], k=k, dtype=int),m)
return out
def triu(m, k=0):
"""
Upper triangle of an array.
Construct a copy of a matrix with elements below the k-th diagonal zeroed.
Please refer to the documentation for `tril`.
See Also
--------
tril
Examples
--------
>>> np.triu([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 1, 2, 3],
[ 4, 5, 6],
[ 0, 8, 9],
[ 0, 0, 12]])
"""
m = asanyarray(m)
out = multiply((1-tri(m.shape[0], m.shape[1], k-1, int)),m)
return out
def kron(a,b):
"""
Kronecker product of two arrays.
Computes the Kronecker product, a composite array made of blocks of the
second array scaled by the first.
Parameters
----------
a, b : array_like
Returns
-------
out : ndarray
See Also
--------
outer : The outer product
Notes
-----
The function assumes that the number of dimenensions of `a` and `b`
are the same, if necessary prepending the smallest with ones.
If `a.shape = (r0,r1,..,rN)` and `b.shape = (s0,s1,...,sN)`,
the Kronecker product has shape `(r0*s0, r1*s1, ..., rN*SN)`.
The elements are products of elements from `a` and `b`, organized
explicitly by::
kron(a,b)[k0,k1,...,kN] = a[i0,i1,...,iN] * b[j0,j1,...,jN]
where::
kt = it * st + jt, t = 0,...,N
In the common 2-D case (N=1), the block structure can be visualized::
[[ a[0,0]*b, a[0,1]*b, ... , a[0,-1]*b ],
[ ... ... ],
[ a[-1,0]*b, a[-1,1]*b, ... , a[-1,-1]*b ]]
Examples
--------
>>> np.kron([1,10,100], [5,6,7])
array([ 5, 6, 7, 50, 60, 70, 500, 600, 700])
>>> np.kron([5,6,7], [1,10,100])
array([ 5, 50, 500, 6, 60, 600, 7, 70, 700])
>>> np.kron(np.eye(2), np.ones((2,2)))
array([[ 1., 1., 0., 0.],
[ 1., 1., 0., 0.],
[ 0., 0., 1., 1.],
[ 0., 0., 1., 1.]])
>>> a = np.arange(100).reshape((2,5,2,5))
>>> b = np.arange(24).reshape((2,3,4))
>>> c = np.kron(a,b)
>>> c.shape
(2, 10, 6, 20)
>>> I = (1,3,0,2)
>>> J = (0,2,1)
>>> J1 = (0,) + J # extend to ndim=4
>>> S1 = (1,) + b.shape
>>> K = tuple(np.array(I) * np.array(S1) + np.array(J1))
>>> c[K] == a[I]*b[J]
True
"""
b = asanyarray(b)
a = array(a,copy=False,subok=True,ndmin=b.ndim)
ndb, nda = b.ndim, a.ndim
if (nda == 0 or ndb == 0):
return _nx.multiply(a,b)
as_ = a.shape
bs = b.shape
if not a.flags.contiguous:
a = reshape(a, as_)
if not b.flags.contiguous:
b = reshape(b, bs)
nd = ndb
if (ndb != nda):
if (ndb > nda):
as_ = (1,)*(ndb-nda) + as_
else:
bs = (1,)*(nda-ndb) + bs
nd = nda
result = outer(a,b).reshape(as_+bs)
axis = nd-1
for _ in range(nd):
result = concatenate(result, axis=axis)
wrapper = get_array_prepare(a, b)
if wrapper is not None:
result = wrapper(result)
wrapper = get_array_wrap(a, b)
if wrapper is not None:
result = wrapper(result)
return result
def kron(a, b):
"""
Kronecker product of two arrays.
Computes the Kronecker product, a composite array made of blocks of the
second array scaled by the first.
Parameters
----------
a, b : array_like
Returns
-------
out : ndarray
See Also
--------
outer : The outer product
Notes
-----
The function assumes that the number of dimensions of `a` and `b`
are the same, if necessary prepending the smallest with ones.
If ``a.shape = (r0,r1,..,rN)`` and ``b.shape = (s0,s1,...,sN)``,
the Kronecker product has shape ``(r0*s0, r1*s1, ..., rN*SN)``.
The elements are products of elements from `a` and `b`, organized
explicitly by::
kron(a,b)[k0,k1,...,kN] = a[i0,i1,...,iN] * b[j0,j1,...,jN]
where::
kt = it * st + jt, t = 0,...,N
In the common 2-D case (N=1), the block structure can be visualized::
[[ a[0,0]*b, a[0,1]*b, ... , a[0,-1]*b ],
[ ... ... ],
[ a[-1,0]*b, a[-1,1]*b, ... , a[-1,-1]*b ]]
Examples
--------
>>> np.kron([1,10,100], [5,6,7])
array([ 5, 6, 7, ..., 500, 600, 700])
>>> np.kron([5,6,7], [1,10,100])
array([ 5, 50, 500, ..., 7, 70, 700])
>>> np.kron(np.eye(2), np.ones((2,2)))
array([[1., 1., 0., 0.],
[1., 1., 0., 0.],
[0., 0., 1., 1.],
[0., 0., 1., 1.]])
>>> a = np.arange(100).reshape((2,5,2,5))
>>> b = np.arange(24).reshape((2,3,4))
>>> c = np.kron(a,b)
>>> c.shape
(2, 10, 6, 20)
>>> I = (1,3,0,2)
>>> J = (0,2,1)
>>> J1 = (0,) + J # extend to ndim=4
>>> S1 = (1,) + b.shape
>>> K = tuple(np.array(I) * np.array(S1) + np.array(J1))
>>> c[K] == a[I]*b[J]
True
"""
b = asanyarray(b)
a = array(a, copy=False, subok=True, ndmin=b.ndim)
ndb, nda = b.ndim, a.ndim
if (nda == 0 or ndb == 0):
return _nx.multiply(a, b)
as_ = a.shape
bs = b.shape
if not a.flags.contiguous:
a = reshape(a, as_)
if not b.flags.contiguous:
b = reshape(b, bs)
nd = ndb
if (ndb != nda):
if (ndb > nda):
as_ = (1, ) * (ndb - nda) + as_
else:
bs = (1, ) * (nda - ndb) + bs
nd = nda
result = outer(a, b).reshape(as_ + bs)
axis = nd - 1
for _ in range(nd):
result = concatenate(result, axis=axis)
wrapper = get_array_prepare(a, b)
if wrapper is not None:
result = wrapper(result)
wrapper = get_array_wrap(a, b)
if wrapper is not None:
result = wrapper(result)
return result
def kron(a, b):
"""
Kronecker product of two arrays.
Computes the Kronecker product, a composite array made of blocks of the
second array scaled by the first.
Parameters
----------
a, b : array_like
Returns
-------
out : ndarray
See Also
--------
outer : The outer product
Notes
-----
The function assumes that the number of dimensions of `a` and `b`
are the same, if necessary prepending the smallest with ones.
If ``a.shape = (r0,r1,..,rN)`` and ``b.shape = (s0,s1,...,sN)``,
the Kronecker product has shape ``(r0*s0, r1*s1, ..., rN*SN)``.
The elements are products of elements from `a` and `b`, organized
explicitly by::
kron(a,b)[k0,k1,...,kN] = a[i0,i1,...,iN] * b[j0,j1,...,jN]
where::
kt = it * st + jt, t = 0,...,N
In the common 2-D case (N=1), the block structure can be visualized::
[[ a[0,0]*b, a[0,1]*b, ... , a[0,-1]*b ],
[ ... ... ],
[ a[-1,0]*b, a[-1,1]*b, ... , a[-1,-1]*b ]]
Examples
--------
>>> np.kron([1,10,100], [5,6,7])
array([ 5, 6, 7, ..., 500, 600, 700])
>>> np.kron([5,6,7], [1,10,100])
array([ 5, 50, 500, ..., 7, 70, 700])
>>> np.kron(np.eye(2), np.ones((2,2)))
array([[1., 1., 0., 0.],
[1., 1., 0., 0.],
[0., 0., 1., 1.],
[0., 0., 1., 1.]])
>>> a = np.arange(100).reshape((2,5,2,5))
>>> b = np.arange(24).reshape((2,3,4))
>>> c = np.kron(a,b)
>>> c.shape
(2, 10, 6, 20)
>>> I = (1,3,0,2)
>>> J = (0,2,1)
>>> J1 = (0,) + J # extend to ndim=4
>>> S1 = (1,) + b.shape
>>> K = tuple(np.array(I) * np.array(S1) + np.array(J1))
>>> c[K] == a[I]*b[J]
True
"""
# Working:
# 1. Equalise the shapes by prepending smaller array with 1s
# 2. Expand shapes of both the arrays by adding new axes at
# odd positions for 1st array and even positions for 2nd
# 3. Compute the product of the modified array
# 4. The inner most array elements now contain the rows of
# the Kronecker product
# 5. Reshape the result to kron's shape, which is same as
# product of shapes of the two arrays.
b = asanyarray(b)
a = array(a, copy=False, subok=True, ndmin=b.ndim)
is_any_mat = isinstance(a, matrix) or isinstance(b, matrix)
ndb, nda = b.ndim, a.ndim
nd = max(ndb, nda)
if (nda == 0 or ndb == 0):
return _nx.multiply(a, b)
as_ = a.shape
bs = b.shape
if not a.flags.contiguous:
a = reshape(a, as_)
if not b.flags.contiguous:
b = reshape(b, bs)
# Equalise the shapes by prepending smaller one with 1s
as_ = (1, ) * max(0, ndb - nda) + as_
bs = (1, ) * max(0, nda - ndb) + bs
# Insert empty dimensions
a_arr = expand_dims(a, axis=tuple(range(ndb - nda)))
b_arr = expand_dims(b, axis=tuple(range(nda - ndb)))
# Compute the product
a_arr = expand_dims(a_arr, axis=tuple(range(1, nd * 2, 2)))
b_arr = expand_dims(b_arr, axis=tuple(range(0, nd * 2, 2)))
# In case of `mat`, convert result to `array`
result = _nx.multiply(a_arr, b_arr, subok=(not is_any_mat))
# Reshape back
result = result.reshape(_nx.multiply(as_, bs))
return result if not is_any_mat else matrix(result, copy=False)