def test_mb03rd_default(self):
# regression: mb03rd was failing with no third arg (X) supplied
A = np.array([[6, -1, -7, -2, 2], [-3, 4, 2, -7, 6],
[-6, -9, -3, -1, 10], [-2, -4, 1, 5, 7],
[-7, -5, -6, 6, 7]])
Aschur, Tschur = schur(A)
X = Tschur.copy()
Ar, Xr, blsize, W = mb03rd(Aschur.shape[0],
Aschur,
X,
'U',
'N',
pmax=1.0,
tol=0.0)
Ar2, Xr2, blsize2, W2 = mb03rd(Aschur.shape[0], Aschur)
assert_allclose(Ar, Ar2)
assert_allclose(Xr, Tschur.dot(Xr2))
def test_mb03rd(self):
""" Test for Schur form reduction.
RvP, 31 Jul 2019"""
test1_A = np.array([[1., -1., 1., 2., 3., 1., 2., 3.],
[1., 1., 3., 4., 2., 3., 4., 2.],
[0., 0., 1., -1., 1., 5., 4., 1.],
[0., 0., 0., 1., -1., 3., 1., 2.],
[0., 0., 0., 1., 1., 2., 3., -1.],
[0., 0., 0., 0., 0., 1., 5., 1.],
[0., 0., 0., 0., 0., 0., 0.99999999, -0.99999999],
[0., 0., 0., 0., 0., 0., 0.99999999, 0.99999999]])
test1_n = test1_A.shape[0]
test1_Ar = np.array([
[
1.0000, -1.0000, -1.2247, -0.7071, -3.4186, 1.4577, 0.0000,
0.0000
],
[1.0000, 1.0000, 0.0000, 1.4142, -5.1390, 3.1637, 0.0000, 0.0000],
[0.0000, 0.0000, 1.0000, -1.7321, -0.0016, 2.0701, 0.0000, 0.0000],
[0.0000, 0.0000, 0.5774, 1.0000, 0.7516, 1.1379, 0.0000, 0.0000],
[0.0000, 0.0000, 0.0000, 0.0000, 1.0000, -5.8606, 0.0000, 0.0000],
[0.0000, 0.0000, 0.0000, 0.0000, 0.1706, 1.0000, 0.0000, 0.0000],
[0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 1.0000, -0.8850],
[0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 1.0000],
])
test1_Xr = np.array(
[[1.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.9045, 0.1957],
[0.0000, 1.0000, 0.0000, 0.0000, 0.0000, 0.0000, -0.3015, 0.9755],
[
0.0000, 0.0000, 0.8165, 0.0000, -0.5768, -0.0156, -0.3015,
0.0148
],
[
0.0000, 0.0000, -0.4082, 0.7071, -0.5768, -0.0156, 0.0000,
-0.0534
],
[
0.0000, 0.0000, -0.4082, -0.7071, -0.5768, -0.0156, 0.0000,
0.0801
],
[0.0000, 0.0000, 0.0000, 0.0000, -0.0276, 0.9805, 0.0000, 0.0267],
[0.0000, 0.0000, 0.0000, 0.0000, 0.0332, -0.0066, 0.0000, 0.0000],
[0.0000, 0.0000, 0.0000, 0.0000, 0.0011, 0.1948, 0.0000, 0.0000]])
test1_W = np.array([
1 + 1j, 1 - 1j, 1 + 1j, 1 - 1j, 0.99999 + 0.99999j,
0.99999 - 0.99999j, 1., 1.
])
test1_pmax = 1e3
test1_tol = 0.01
# create schur form with scipy
A, X = schur(test1_A)
Ah, Xh = np.copy(A), np.copy(X)
# on this basis, get the transform
Ar, Xr, blsize, W = mb03rd(test1_n, A, X, 'U', 'S', test1_pmax,
test1_tol)
# ensure X and A are unchanged
assert_allclose(A, Ah)
assert_allclose(X, Xh)
# compare to test case results
assert_allclose(Ar, test1_Ar, atol=0.0001)
assert_allclose(Xr, test1_Xr, atol=0.0001)
assert_allclose(W, test1_W, atol=0.0001)
# Test that the non sorting options do not throw errors and that Xr is
# returned as None for jobx='N'
for sort in ['N', 'C', 'B']:
Ar, Xr, blsize, W = mb03rd(test1_n, A, X, 'N', sort, test1_pmax,
test1_tol)
assert Xr is None
def _bdschur_condmax_search(aschur, tschur, condmax):
"""Block-diagonal Schur decomposition search up to condmax
Iterates mb03rd with different pmax values until:
- result is non-defective;
- or condition number of similarity transform is unchanging despite large pmax;
- or condition number of similarity transform is close to condmax.
Parameters
----------
aschur: (N, N) real ndarray
Real Schur-form matrix
tschur: (N, N) real ndarray
Orthogonal transformation giving aschur from some initial matrix a
condmax: float
Maximum condition number of final transformation. Must be >= 1.
Returns
-------
amodal: (N, N) real ndarray
block diagonal Schur form
tmodal: (N, N) real ndarray
similarity transformation give amodal from aschur
blksizes: (M,) int ndarray
Array of Schur block sizes
eigvals: (N,) real or complex ndarray
Eigenvalues of amodal (and a, etc.)
Notes
-----
Outputs as for slycot.mb03rd
aschur, tschur are as returned by scipy.linalg.schur.
"""
try:
from slycot import mb03rd
except ImportError:
raise ControlSlycot("can't find slycot module 'mb03rd'")
# see notes on RuntimeError below
pmaxlower = None
# get lower bound; try condmax ** 0.5 first
pmaxlower = condmax**0.5
amodal, tmodal, blksizes, eigvals = mb03rd(aschur.shape[0],
aschur,
tschur,
pmax=pmaxlower)
if np.linalg.cond(tmodal) <= condmax:
reslower = amodal, tmodal, blksizes, eigvals
else:
pmaxlower = 1.0
amodal, tmodal, blksizes, eigvals = mb03rd(aschur.shape[0],
aschur,
tschur,
pmax=pmaxlower)
cond = np.linalg.cond(tmodal)
if cond > condmax:
msg = 'minimum cond={} > condmax={}; try increasing condmax'.format(
cond, condmax)
raise RuntimeError(msg)
pmax = pmaxlower
# phase 1: search for upper bound on pmax
for i in range(50):
amodal, tmodal, blksizes, eigvals = mb03rd(aschur.shape[0],
aschur,
tschur,
pmax=pmax)
cond = np.linalg.cond(tmodal)
if cond < condmax:
pmaxlower = pmax
reslower = amodal, tmodal, blksizes, eigvals
else:
# upper bound found; go to phase 2
pmaxupper = pmax
break
if _bdschur_defective(blksizes, eigvals):
pmax *= 2
else:
return amodal, tmodal, blksizes, eigvals
else:
# no upper bound found; return current result
return reslower
# phase 2: bisection search
for i in range(50):
pmax = (pmaxlower * pmaxupper)**0.5
amodal, tmodal, blksizes, eigvals = mb03rd(aschur.shape[0],
aschur,
tschur,
pmax=pmax)
cond = np.linalg.cond(tmodal)
if cond < condmax:
if not _bdschur_defective(blksizes, eigvals):
return amodal, tmodal, blksizes, eigvals
pmaxlower = pmax
reslower = amodal, tmodal, blksizes, eigvals
else:
pmaxupper = pmax
if pmaxupper / pmaxlower < _PMAX_SEARCH_TOL:
# hit search limit
return reslower
else:
raise ValueError(
'bisection failed to converge; pmaxlower={}, pmaxupper={}'.format(
pmaxlower, pmaxupper))