Python numpy.linalg 模块，eigvalsh() 实例源码
我们从Python开源项目中，提取了以下50个代码示例，用于说明如何使用numpy.linalg.eigvalsh()。
def test_UPLO(self):
Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
tgt = np.array([-1, 1], dtype=np.double)
rtol = get_rtol(np.double)
# Check default is 'L'
w = np.linalg.eigvalsh(Klo)
assert_allclose(w, tgt, rtol=rtol)
# Check 'L'
w = np.linalg.eigvalsh(Klo, UPLO='L')
assert_allclose(w, tgt, rtol=rtol)
# Check 'l'
w = np.linalg.eigvalsh(Klo, UPLO='l')
assert_allclose(w, tgt, rtol=rtol)
# Check 'U'
w = np.linalg.eigvalsh(Kup, UPLO='U')
assert_allclose(w, tgt, rtol=rtol)
# Check 'u'
w = np.linalg.eigvalsh(Kup, UPLO='u')
assert_allclose(w, tgt, rtol=rtol)
def test_UPLO(self):
Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
tgt = np.array([-1, 1], dtype=np.double)
rtol = get_rtol(np.double)
# Check default is 'L'
w = np.linalg.eigvalsh(Klo)
assert_allclose(w, tgt, rtol=rtol)
# Check 'L'
w = np.linalg.eigvalsh(Klo, UPLO='L')
assert_allclose(w, tgt, rtol=rtol)
# Check 'l'
w = np.linalg.eigvalsh(Klo, UPLO='l')
assert_allclose(w, tgt, rtol=rtol)
# Check 'U'
w = np.linalg.eigvalsh(Kup, UPLO='U')
assert_allclose(w, tgt, rtol=rtol)
# Check 'u'
w = np.linalg.eigvalsh(Kup, UPLO='u')
assert_allclose(w, tgt, rtol=rtol)
def test_UPLO(self):
Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
tgt = np.array([-1, 1], dtype=np.double)
rtol = get_rtol(np.double)
# Check default is 'L'
w = np.linalg.eigvalsh(Klo)
assert_allclose(w, tgt, rtol=rtol)
# Check 'L'
w = np.linalg.eigvalsh(Klo, UPLO='L')
assert_allclose(w, tgt, rtol=rtol)
# Check 'l'
w = np.linalg.eigvalsh(Klo, UPLO='l')
assert_allclose(w, tgt, rtol=rtol)
# Check 'U'
w = np.linalg.eigvalsh(Kup, UPLO='U')
assert_allclose(w, tgt, rtol=rtol)
# Check 'u'
w = np.linalg.eigvalsh(Kup, UPLO='u')
assert_allclose(w, tgt, rtol=rtol)
def test_UPLO(self):
Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
tgt = np.array([-1, 1], dtype=np.double)
rtol = get_rtol(np.double)
# Check default is 'L'
w = np.linalg.eigvalsh(Klo)
assert_allclose(w, tgt, rtol=rtol)
# Check 'L'
w = np.linalg.eigvalsh(Klo, UPLO='L')
assert_allclose(w, tgt, rtol=rtol)
# Check 'l'
w = np.linalg.eigvalsh(Klo, UPLO='l')
assert_allclose(w, tgt, rtol=rtol)
# Check 'U'
w = np.linalg.eigvalsh(Kup, UPLO='U')
assert_allclose(w, tgt, rtol=rtol)
# Check 'u'
w = np.linalg.eigvalsh(Kup, UPLO='u')
assert_allclose(w, tgt, rtol=rtol)
def test_UPLO(self):
Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
tgt = np.array([-1, 1], dtype=np.double)
rtol = get_rtol(np.double)
# Check default is 'L'
w = np.linalg.eigvalsh(Klo)
assert_allclose(w, tgt, rtol=rtol)
# Check 'L'
w = np.linalg.eigvalsh(Klo, UPLO='L')
assert_allclose(w, tgt, rtol=rtol)
# Check 'l'
w = np.linalg.eigvalsh(Klo, UPLO='l')
assert_allclose(w, tgt, rtol=rtol)
# Check 'U'
w = np.linalg.eigvalsh(Kup, UPLO='U')
assert_allclose(w, tgt, rtol=rtol)
# Check 'u'
w = np.linalg.eigvalsh(Kup, UPLO='u')
assert_allclose(w, tgt, rtol=rtol)
def test_UPLO(self):
Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
tgt = np.array([-1, 1], dtype=np.double)
rtol = get_rtol(np.double)
# Check default is 'L'
w = np.linalg.eigvalsh(Klo)
assert_allclose(w, tgt, rtol=rtol)
# Check 'L'
w = np.linalg.eigvalsh(Klo, UPLO='L')
assert_allclose(w, tgt, rtol=rtol)
# Check 'l'
w = np.linalg.eigvalsh(Klo, UPLO='l')
assert_allclose(w, tgt, rtol=rtol)
# Check 'U'
w = np.linalg.eigvalsh(Kup, UPLO='U')
assert_allclose(w, tgt, rtol=rtol)
# Check 'u'
w = np.linalg.eigvalsh(Kup, UPLO='u')
assert_allclose(w, tgt, rtol=rtol)
def test_UPLO(self):
Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
tgt = np.array([-1, 1], dtype=np.double)
rtol = get_rtol(np.double)
# Check default is 'L'
w = np.linalg.eigvalsh(Klo)
assert_allclose(w, tgt, rtol=rtol)
# Check 'L'
w = np.linalg.eigvalsh(Klo, UPLO='L')
assert_allclose(w, tgt, rtol=rtol)
# Check 'l'
w = np.linalg.eigvalsh(Klo, UPLO='l')
assert_allclose(w, tgt, rtol=rtol)
# Check 'U'
w = np.linalg.eigvalsh(Kup, UPLO='U')
assert_allclose(w, tgt, rtol=rtol)
# Check 'u'
w = np.linalg.eigvalsh(Kup, UPLO='u')
assert_allclose(w, tgt, rtol=rtol)
def do(self, a, b):
# note that eigenvalue arrays returned by eig must be sorted since
# their order isn't guaranteed.
ev = linalg.eigvalsh(a, 'L')
evalues, evectors = linalg.eig(a)
evalues.sort(axis=-1)
assert_allclose(ev, evalues, rtol=get_rtol(ev.dtype))
ev2 = linalg.eigvalsh(a, 'U')
assert_allclose(ev2, evalues, rtol=get_rtol(ev.dtype))
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
w = np.linalg.eigvalsh(x)
assert_equal(w.dtype, get_real_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def test_invalid(self):
x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
assert_raises(ValueError, np.linalg.eigvalsh, x, UPLO="lrong")
assert_raises(ValueError, np.linalg.eigvalsh, x, "lower")
assert_raises(ValueError, np.linalg.eigvalsh, x, "upper")
def do(self, a, b):
# note that eigenvalue arrays returned by eig must be sorted since
# their order isn't guaranteed.
ev = linalg.eigvalsh(a, 'L')
evalues, evectors = linalg.eig(a)
evalues.sort(axis=-1)
assert_allclose(ev, evalues, rtol=get_rtol(ev.dtype))
ev2 = linalg.eigvalsh(a, 'U')
assert_allclose(ev2, evalues, rtol=get_rtol(ev.dtype))
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
w = np.linalg.eigvalsh(x)
assert_equal(w.dtype, get_real_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def test_invalid(self):
x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
assert_raises(ValueError, np.linalg.eigvalsh, x, UPLO="lrong")
assert_raises(ValueError, np.linalg.eigvalsh, x, "lower")
assert_raises(ValueError, np.linalg.eigvalsh, x, "upper")
def do(self, a, b):
# note that eigenvalue arrays returned by eig must be sorted since
# their order isn't guaranteed.
ev = linalg.eigvalsh(a, 'L')
evalues, evectors = linalg.eig(a)
evalues.sort(axis=-1)
assert_allclose(ev, evalues, rtol=get_rtol(ev.dtype))
ev2 = linalg.eigvalsh(a, 'U')
assert_allclose(ev2, evalues, rtol=get_rtol(ev.dtype))
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
w = np.linalg.eigvalsh(x)
assert_equal(w.dtype, get_real_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def test_invalid(self):
x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
assert_raises(ValueError, np.linalg.eigvalsh, x, UPLO="lrong")
assert_raises(ValueError, np.linalg.eigvalsh, x, "lower")
assert_raises(ValueError, np.linalg.eigvalsh, x, "upper")
def do(self, a, b):
# note that eigenvalue arrays returned by eig must be sorted since
# their order isn't guaranteed.
ev = linalg.eigvalsh(a, 'L')
evalues, evectors = linalg.eig(a)
evalues.sort(axis=-1)
assert_allclose(ev, evalues, rtol=get_rtol(ev.dtype))
ev2 = linalg.eigvalsh(a, 'U')
assert_allclose(ev2, evalues, rtol=get_rtol(ev.dtype))
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
w = np.linalg.eigvalsh(x)
assert_equal(w.dtype, get_real_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def test_invalid(self):
x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
assert_raises(ValueError, np.linalg.eigvalsh, x, UPLO="lrong")
assert_raises(ValueError, np.linalg.eigvalsh, x, "lower")
assert_raises(ValueError, np.linalg.eigvalsh, x, "upper")
def do(self, a, b):
# note that eigenvalue arrays returned by eig must be sorted since
# their order isn't guaranteed.
ev = linalg.eigvalsh(a, 'L')
evalues, evectors = linalg.eig(a)
evalues.sort(axis=-1)
assert_allclose(ev, evalues, rtol=get_rtol(ev.dtype))
ev2 = linalg.eigvalsh(a, 'U')
assert_allclose(ev2, evalues, rtol=get_rtol(ev.dtype))
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
w = np.linalg.eigvalsh(x)
assert_equal(w.dtype, get_real_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def test_invalid(self):
x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
assert_raises(ValueError, np.linalg.eigvalsh, x, UPLO="lrong")
assert_raises(ValueError, np.linalg.eigvalsh, x, "lower")
assert_raises(ValueError, np.linalg.eigvalsh, x, "upper")
def do(self, a, b):
# note that eigenvalue arrays returned by eig must be sorted since
# their order isn't guaranteed.
ev = linalg.eigvalsh(a, 'L')
evalues, evectors = linalg.eig(a)
evalues.sort(axis=-1)
assert_allclose(ev, evalues, rtol=get_rtol(ev.dtype))
ev2 = linalg.eigvalsh(a, 'U')
assert_allclose(ev2, evalues, rtol=get_rtol(ev.dtype))
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
w = np.linalg.eigvalsh(x)
assert_equal(w.dtype, get_real_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def test_invalid(self):
x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
assert_raises(ValueError, np.linalg.eigvalsh, x, UPLO="lrong")
assert_raises(ValueError, np.linalg.eigvalsh, x, "lower")
assert_raises(ValueError, np.linalg.eigvalsh, x, "upper")
def genInvCov(size, low = 0 , upper = 0.6, portion = 0.05):
S = np.zeros((size,size))
G = GenRndGnm(PUNGraph, size, int((size*(size-1))*portion))
for EI in G.Edges():
value = (np.random.randint(2) - 0.5)*2*(low + (upper - low)*np.random.rand(1)[0])
# print value
S[EI.GetSrcNId(), EI.GetDstNId()] = value
S = S + S.T
vals = alg.eigvalsh(S)
S = S + (0.1 - vals[0])*np.identity(size)
return np.matrix(S)
def genInvCov(size, low = 0 , upper = 0.6, portion = 0.05):
S = np.zeros((size,size))
# low = abs(low)
# upper = abs(upper)
G = GenRndGnm(PUNGraph, size, int((size*(size-1))*portion))
for EI in G.Edges():
value = (np.random.randint(2) - 0.5)*2*(low + (upper - low)*np.random.rand(1)[0])
# print value
S[EI.GetSrcNId(), EI.GetDstNId()] = value
S = S + S.T
vals = alg.eigvalsh(S)
S = S + (0.1 - vals[0])*np.identity(size)
return np.matrix(S)
def genInvCov(size, low = 0 , upper = 0.6, portion = 0.05):
S = np.zeros((size,size))
G = GenRndGnm(PUNGraph, size, int((size*(size-1))*portion))
for EI in G.Edges():
value = (np.random.randint(2) - 0.5)*2*(low + (upper - low)*np.random.rand(1)[0])
S[EI.GetSrcNId(), EI.GetDstNId()] = value
S = S + S.T
vals = alg.eigvalsh(S)
S = S + (0.1 - vals[0])*np.identity(size)
return np.matrix(S)
def genInvCov(size, low = 0 , upper = 0.6, portion = 0.05):
S = np.zeros((size,size))
G = GenRndGnm(PUNGraph, size, int((size*(size-1))*portion))
for EI in G.Edges():
value = (np.random.randint(2) - 0.5)*2*(low + (upper - low)*np.random.rand(1)[0])
S[EI.GetSrcNId(), EI.GetDstNId()] = value
S = S + S.T
vals = alg.eigvalsh(S)
S = S + (0.1 - vals[0])*np.identity(size)
return np.matrix(S)
def genInvCov(size, low = 0 , upper = 0.6, portion = 0.05):
S = np.zeros((size,size))
G = GenRndGnm(PUNGraph, size, int((size*(size-1))*portion))
for EI in G.Edges():
value = (np.random.randint(2) - 0.5)*2*(low + (upper - low)*np.random.rand(1)[0])
S[EI.GetSrcNId(), EI.GetDstNId()] = value
S = S + S.T
vals = alg.eigvalsh(S)
S = S + (0.1 - vals[0])*np.identity(size)
return np.matrix(S)
def _estimate_kappa(self):
y, x, z = self._wy, self._wx, self._wz
is_exog = self._regressor_is_exog
e = c_[y, x[:, ~is_exog]]
x1 = x[:, is_exog]
if x1.shape[1] == 0:
# No exogenous regressors
return 1
ez = e - z @ (pinv(z) @ e)
ex1 = e - x1 @ (pinv(x1) @ e)
vpmzv_sqinv = inv_sqrth(ez.T @ ez)
q = vpmzv_sqinv @ (ex1.T @ ex1) @ vpmzv_sqinv
return min(eigvalsh(q))
def do(self, a, b):
# note that eigenvalue arrays returned by eig must be sorted since
# their order isn't guaranteed.
ev = linalg.eigvalsh(a, 'L')
evalues, evectors = linalg.eig(a)
evalues.sort(axis=-1)
assert_allclose(ev, evalues, rtol=get_rtol(ev.dtype))
ev2 = linalg.eigvalsh(a, 'U')
assert_allclose(ev2, evalues, rtol=get_rtol(ev.dtype))
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
w = np.linalg.eigvalsh(x)
assert_equal(w.dtype, get_real_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def test_invalid(self):
x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
assert_raises(ValueError, np.linalg.eigvalsh, x, UPLO="lrong")
assert_raises(ValueError, np.linalg.eigvalsh, x, "lower")
assert_raises(ValueError, np.linalg.eigvalsh, x, "upper")
def legcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Legendre basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
def hermcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Hermite basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-.5*c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
def legcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Legendre basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
def hermcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Hermite basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-.5*c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
def legcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Legendre basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
def hermcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Hermite basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-.5*c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
def legcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Legendre basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
def hermcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Hermite basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-.5*c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
def legcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Legendre basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
def hermcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Hermite basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-.5*c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
def legcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Legendre basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
def hermcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Hermite basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded::1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-.5*c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
def genMulCov(size, numberOfCov, low, upper, mode, portion = 0.05):
S_set = []
Cov_set = []
minEVal_set = []
m = size/3
mm = m/2
# print m, mm
S_init = np.zeros((size,size))
for k in range(numberOfCov):
S = np.zeros((size,size))
if k == 0:
S = genInvCov(size, low, upper, portion)
if mode == 5:
ind_zero = np.where(spy.sparse.rand(m, size-m, 0.5).todense() == 0)
value = np.multiply((np.random.randint(2, size = (m, size -m)) - 0.5)*2,(low + (upper - low)*np.random.rand(m,size -m)))
value[ind_zero] = 0
hub = value
S[:m, m:size] = hub
S[m:size, :m] = hub.T
minEVal_set.append(alg.eigvalsh(S)[0])
S_init = S
elif mode == 3: #'laplacian'
ind1 = range(m)
ind2 = np.random.permutation(m)
S = np.copy(S_init)
S[ind1, :] = S[ind2, :]
S[:, ind1] = S[:, ind2]
elif mode == 5: #'perturbation'
S = np.copy(S_init)
ind_zero = np.where(spy.sparse.rand(mm, size-mm, 0.5).todense() == 0)
pert = np.multiply((np.random.randint(2, size = (mm, size -mm)) - 0.5)*2,(low + (upper - low)*np.random.rand(mm,size -mm)))
pert[ind_zero] = 0
S[:mm, mm:size] = pert
S[mm:size, :mm] = pert.T
minEVal_set.append(alg.eigvalsh(S)[0])
else:
# print 'Activate normal mode'
S = genInvCov(size, low, upper, portion)
S_set.append(S)
for k in range(numberOfCov):
if mode == 5:
S_set[k] = S_set[k] + (0.1 - min(minEVal_set))*np.identity(size)
Cov_set.append(alg.inv(S_set[k]))
return S_set, Cov_set
def genMulCov(size, numberOfCov, low, upper, mode, portion = 0.05):
S_set = []
Cov_set = []
minEVal_set = []
# low = abs(low)
# upper = abs(upper)
m = size/3
mm = m/2
# print m, mm
S_init = np.zeros((size,size))
for k in range(numberOfCov):
S = np.zeros((size,size))
if k == 0:
S = genInvCov(size, low, upper, portion)
if mode == 5:
ind_zero = np.where(spy.sparse.rand(m, size-m, 0.5).todense() == 0)
value = np.multiply((np.random.randint(2, size = (m, size -m)) - 0.5)*2,(low + (upper - low)*np.random.rand(m,size -m)))
value[ind_zero] = 0
hub = value
S[:m, m:size] = hub
S[m:size, :m] = hub.T
minEVal_set.append(alg.eigvalsh(S)[0])
S_init = S
elif mode == 3: #'laplacian'
ind1 = range(m)
ind2 = np.random.permutation(m)
S = np.copy(S_init)
S[ind1, :] = S[ind2, :]
S[:, ind1] = S[:, ind2]
elif mode == 5: #'perturbation'
S = np.copy(S_init)
ind_zero = np.where(spy.sparse.rand(mm, size-mm, 0.5).todense() == 0)
pert = np.multiply((np.random.randint(2, size = (mm, size -mm)) - 0.5)*2,(low + (upper - low)*np.random.rand(mm,size -mm)))
pert[ind_zero] = 0
S[:mm, mm:size] = pert
S[mm:size, :mm] = pert.T
minEVal_set.append(alg.eigvalsh(S)[0])
else:
# print 'Activate normal mode'
S = genInvCov(size, low, upper, portion)
S_set.append(S)
for k in range(numberOfCov):
if mode == 5:
S_set[k] = S_set[k] + (0.1 - min(minEVal_set))*np.identity(size)
Cov_set.append(alg.inv(S_set[k]))
return S_set, Cov_set
def genMulCov(size, numberOfCov, low, upper, mode, portion = 0.05):
S_set = []
Cov_set = []
minEVal_set = []
m = size/3
mm = m/2
S_init = np.zeros((size,size))
for k in range(numberOfCov):
S = np.zeros((size,size))
if k == 0:
S = genInvCov(size, low, upper, portion)
if mode == 5:
ind_zero = np.where(spy.sparse.rand(m, size-m, 0.5).todense() == 0)
value = np.multiply((np.random.randint(2, size = (m, size -m)) - 0.5)*2,(low + (upper - low)*np.random.rand(m,size -m)))
value[ind_zero] = 0
hub = value
S[:m, m:size] = hub
S[m:size, :m] = hub.T
minEVal_set.append(alg.eigvalsh(S)[0])
S_init = S
elif mode == 3: #'laplacian'
ind1 = range(m)
ind2 = np.random.permutation(m)
S = np.copy(S_init)
S[ind1, :] = S[ind2, :]
S[:, ind1] = S[:, ind2]
elif mode == 5: #'perturbation'
S = np.copy(S_init)
ind_zero = np.where(spy.sparse.rand(mm, size-mm, 0.5).todense() == 0)
pert = np.multiply((np.random.randint(2, size = (mm, size -mm)) - 0.5)*2,(low + (upper - low)*np.random.rand(mm,size -mm)))
pert[ind_zero] = 0
S[:mm, mm:size] = pert
S[mm:size, :mm] = pert.T
minEVal_set.append(alg.eigvalsh(S)[0])
else:
# print 'Activate normal mode'
S = genInvCov(size, low, upper, portion)
S_set.append(S)
for k in range(numberOfCov):
if mode == 5:
S_set[k] = S_set[k] + (0.1 - min(minEVal_set))*np.identity(size)
Cov_set.append(alg.inv(S_set[k]))
return S_set, Cov_set
def genMulCov(size, numberOfCov, low, upper, mode, portion = 0.05):
S_set = []
Cov_set = []
minEVal_set = []
m = size/3
mm = m/2
S_init = np.zeros((size,size))
for k in range(numberOfCov):
S = np.zeros((size,size))
if k == 0:
S = genInvCov(size, low, upper, portion)
if mode == 5:
ind_zero = np.where(spy.sparse.rand(m, size-m, 0.5).todense() == 0)
value = np.multiply((np.random.randint(2, size = (m, size -m)) - 0.5)*2,(low + (upper - low)*np.random.rand(m,size -m)))
value[ind_zero] = 0
hub = value
S[:m, m:size] = hub
S[m:size, :m] = hub.T
minEVal_set.append(alg.eigvalsh(S)[0])
S_init = S
elif mode == 3: #'laplacian'
ind1 = range(m)
ind2 = np.random.permutation(m)
S = np.copy(S_init)
S[ind1, :] = S[ind2, :]
S[:, ind1] = S[:, ind2]
elif mode == 5: #'perturbation'
S = np.copy(S_init)
ind_zero = np.where(spy.sparse.rand(mm, size-mm, 0.5).todense() == 0)
pert = np.multiply((np.random.randint(2, size = (mm, size -mm)) - 0.5)*2,(low + (upper - low)*np.random.rand(mm,size -mm)))
pert[ind_zero] = 0
S[:mm, mm:size] = pert
S[mm:size, :mm] = pert.T
minEVal_set.append(alg.eigvalsh(S)[0])
else:
# print 'Activate normal mode'
S = genInvCov(size, low, upper, portion)
S_set.append(S)
for k in range(numberOfCov):
if mode == 5:
S_set[k] = S_set[k] + (0.1 - min(minEVal_set))*np.identity(size)
Cov_set.append(alg.inv(S_set[k]))
return S_set, Cov_set