def svd(tensor,
row,
new,
col,
nmax=None,
tol=None,
return_truncation_err=False,
**karg):
'''
Perform the svd.
Parameters
----------
tensor : DTensor/STensor
The tensor to be svded.
row,col : list of Label or int
The labels or axes to be merged as the row/column during the svd.
new : Label
The label for the singular values.
nmax,tol,return_truncation_err :
Please refer to HamiltonianPy.Misc.Linalg.truncated_svd for details.
Returns
-------
U,S,V : DTensor/STensor
The result tensor.
err : np.float64, optional
The truncation error.
'''
assert len(row) + len(col) == tensor.ndim
row = [r if isinstance(r, Label) else tensor.label(r) for r in row]
col = [c if isinstance(c, Label) else tensor.label(c) for c in col]
if isinstance(tensor, STensor):
row_label, row_record = Label.union(row,
'__TENSOR_SVD_ROW__',
+1,
mode=2)
col_label, col_record = Label.union(col,
'__TENSOR_SVD_COL__',
-1,
mode=2)
m = tensor.merge((row, row_label, row_record),
(col, col_label, col_record)).data
us, ss, vs, qns = [], [], [], []
for (rowqn, colqn), block in m.iteritems():
assert rowqn == colqn
u, s, v = sl.svd(block, full_matrices=False,
lapack_driver='gesvd')[0:3]
us.append(u)
ss.append(s)
vs.append(v)
qns.append(rowqn)
temp = np.sort(np.concatenate([-s for s in ss]))
nmax = len(temp) if nmax is None else min(nmax, len(temp))
tol = temp[nmax - 1] if tol is None else min(-tol, temp[nmax - 1])
Us, Ss, Vs, contents = {}, [], {}, ([], [])
for u, s, v, qn in zip(us, ss, vs, qns):
cut = np.searchsorted(-s, tol, side='right')
if cut > 0:
Us[(qn, qn)] = u[:, 0:cut]
Ss.append(s[0:cut])
Vs[(qn, qn)] = v[0:cut, :]
contents[0].append(qn)
contents[1].append(cut)
new = new.replace(qns=QuantumNumbers('U',
contents,
protocol=QuantumNumbers.COUNTS),
flow=None)
U = STensor(Us, labels=[row_label, new.replace(flow=-1)]).split(
(row_label, row, row_record))
S = DTensor(np.concatenate(Ss), labels=[new])
V = STensor(Vs, labels=[new.replace(flow=+1), col_label]).split(
(col_label, col, col_record))
if return_truncation_err: err = (temp[nmax:]**2).sum()
elif tensor.qnon:
row_label, row_permutation = Label.union(row,
'__TENSOR_SVD_ROW__',
+1,
mode=1)
col_label, col_permutation = Label.union(col,
'__TENSOR_SVD_COL__',
-1,
mode=1)
m = tensor.merge((row, row_label, row_permutation),
(col, col_label, col_permutation)).data
row_od, col_od = row_label.qns.to_ordereddict(
), col_label.qns.to_ordereddict()
us, ss, vs, qns = [], [], [], []
for qn in it.ifilter(row_od.has_key, col_od):
u, s, v = sl.svd(m[row_od[qn], col_od[qn]],
full_matrices=False,
lapack_driver='gesvd')[0:3]
us.append(u)
ss.append(s)
vs.append(v)
qns.append(qn)
temp = np.sort(np.concatenate([-s for s in ss]))
nmax = len(temp) if nmax is None else min(nmax, len(temp))
tol = temp[nmax - 1] if tol is None else min(-tol, temp[nmax - 1])
Us, Ss, Vs, contents = [], [], [], ([], [])
for u, s, v, qn in zip(us, ss, vs, qns):
cut = np.searchsorted(-s, tol, side='right')
if cut > 0:
Us.append(u[:, 0:cut])
Ss.append(s[0:cut])
Vs.append(v[0:cut, :])
contents[0].append(qn)
contents[1].append(cut)
S = np.concatenate(Ss)
new = new.replace(qns=QuantumNumbers('U',
contents,
protocol=QuantumNumbers.COUNTS),
flow=None)
od = new.qns.to_ordereddict()
U = np.zeros((row_label.dim, new.dim), dtype=tensor.dtype)
V = np.zeros((new.dim, col_label.dim), dtype=tensor.dtype)
for u, v, qn in zip(Us, Vs, od):
U[row_od[qn], od[qn]] = u
V[od[qn], col_od[qn]] = v
U = DTensor(U, labels=[row_label, new.replace(flow=-1)]).split(
(row_label, row, np.argsort(row_permutation)))
S = DTensor(S, labels=[new])
V = DTensor(V, labels=[new.replace(flow=+1), col_label]).split(
(col_label, col, np.argsort(col_permutation)))
if return_truncation_err: err = (temp[nmax:]**2).sum()
else:
row_label = Label('__TENSOR_SVD_ROW__',
qns=np.product([label.dim for label in row]))
col_label = Label('__TENSOR_SVD_COL__',
qns=np.product([label.dim for label in col]))
m = tensor.merge((row, row_label), (col, col_label)).data
temp = hm.truncated_svd(m,
full_matrices=False,
nmax=nmax,
tol=tol,
return_truncation_err=return_truncation_err,
**karg)
u, s, v = temp[0], temp[1], temp[2]
new = new.replace(qns=len(s), flow=None)
U = DTensor(u, labels=[row_label, new.replace(flow=0)]).split(
(row_label, row))
S = DTensor(s, labels=[new])
V = DTensor(v, labels=[new.replace(flow=0), col_label]).split(
(col_label, col))
if return_truncation_err: err = temp[3]
return (U, S, V, err) if return_truncation_err else (U, S, V)
def partitioned_svd(tensor,
L,
new,
R,
mode='D',
nmax=None,
tol=None,
return_truncation_err=False):
'''
Partition a 1d-tensor according to L and R and then perform the Schmitt decomposition.
Parameters
----------
tensor : DTensor
The tensor to be partitioned_svded.
L,R : Label
The left/right part of the partition.
new : Label
The label for the singular values.
mode : 'D'/'S', optional
'D' for dense and 'S' for sparse.
nmax,tol,return_truncation_err :
Please refer to HamiltonianPy.Misc.Linalg.truncated_svd for details.
Returns
-------
U,S,V : DTensor/STensor
The Schmitt decomposition of the 1d tensor.
err : np.float64, optional
The truncation error.
'''
assert tensor.ndim == 1 and mode in 'DS'
if tensor.qnon:
data, qns = tensor.data, tensor.labels[0].qns
assert qns.num == 1 and sl.norm(qns.contents) < 10**-6
lod, rod = L.qns.to_ordereddict(), R.qns.to_ordereddict()
us, ss, vs, qns, count = [], [], [], [], 0
for qn in it.ifilter(lod.has_key, rod):
s1, s2 = lod[qn], rod[qn]
n1, n2 = s1.stop - s1.start, s2.stop - s2.start
u, s, v = sl.svd(data[count:count + n1 * n2].reshape((n1, n2)),
full_matrices=False,
lapack_driver='gesvd')[0:3]
us.append(u)
ss.append(s)
vs.append(v)
qns.append(qn)
count += n1 * n2
temp = np.sort(np.concatenate([-s for s in ss]))
nmax = len(temp) if nmax is None else min(nmax, len(temp))
tol = temp[nmax - 1] if tol is None else min(-tol, temp[nmax - 1])
if mode == 'D':
Us, Ss, Vs, contents = [], [], [], ([], [])
for u, s, v, qn in zip(us, ss, vs, qns):
cut = np.searchsorted(-s, tol, side='right')
if cut > 0:
Us.append(u[:, 0:cut])
Ss.append(s[0:cut])
Vs.append(v[0:cut, :])
contents[0].append(qn)
contents[1].append(cut)
new = new.replace(qns=QuantumNumbers('U', contents,
QuantumNumbers.COUNTS),
flow=None)
nod = new.qns.to_ordereddict()
U = np.zeros((L.dim, new.dim), dtype=tensor.dtype)
S = np.concatenate(Ss)
V = np.zeros((new.dim, R.dim), dtype=tensor.dtype)
for u, v, qn in zip(Us, Vs, nod):
U[lod[qn], nod[qn]] = u
V[nod[qn], rod[qn]] = v
U = DTensor(U, labels=[L, new.replace(flow=-1)])
S = DTensor(S, labels=[new])
V = DTensor(V, labels=[new.replace(flow=+1), R])
else:
Us, Ss, Vs, contents = {}, [], {}, ([], [])
for u, s, v, qn in zip(us, ss, vs, qns):
cut = np.searchsorted(-s, tol, side='right')
if cut > 0:
Us[(qn, qn)] = u[:, 0:cut]
Ss.append(s[0:cut])
Vs[(qn, qn)] = v[0:cut, :]
contents[0].append(qn)
contents[1].append(cut)
new = new.replace(qns=QuantumNumbers('U', contents,
QuantumNumbers.COUNTS),
flow=None)
U = STensor(Us, labels=[L, new.replace(flow=-1)])
S = DTensor(np.concatenate(Ss), labels=[new])
V = STensor(Vs, labels=[new.replace(flow=+1), R])
if return_truncation_err: err = (temp[nmax:]**2).sum()
else:
m = tensor.data.reshape((L.dim, R.dim))
data = hm.truncated_svd(m,
full_matrices=False,
nmax=nmax,
tol=tol,
return_truncation_err=return_truncation_err)
new = new.replace(qns=len(data[1]), flow=None)
U = DTensor(data[0], labels=[L, new.replace(flow=0)])
S = DTensor(data[1], labels=[new])
V = DTensor(data[2], labels=[new.replace(flow=0), R])
if return_truncation_err: err = data[3]
return (U, S, V, err) if return_truncation_err else (U, S, V)
