def LAPACK_solve_ls_with_QR(A, b):
# Ref: https://stackoverflow.com/questions/21970510/solving-a-linear-system-with-lapacks-dgeqrf
# The corresponding procedure in LAPACK is https://www.netlib.org/lapack/lug/node40.html
qr, tau, work, info = dgeqrf(A)
cq, work, info = dormqr('L', 'T', qr, tau, b, qr.shape[0])
x_qr, info = dtrtrs(qr, cq)
return x_qr[0:A.shape[1]]
def residual_variable_projection(matrix: np.ndarray, data: np.ndarray) \
-> typing.Tuple[typing.List[str], np.ndarray]:
"""Calculates the conditionaly linear parameters and residual with the variable projection
method.
Parameters
----------
matrix :
The model matrix.
data : np.ndarray
The data to analyze.
"""
# TODO: Reference Kaufman paper
# Kaufman Q2 step 3
qr, tau, _, _ = lapack.dgeqrf(matrix)
# Kaufman Q2 step 4
temp, _, _ = lapack.dormqr("L", "T", qr, tau, data, max(1, matrix.shape[1]),
overwrite_c=0)
clp, _ = lapack.dtrtrs(qr, temp)
for i in range(matrix.shape[1]):
temp[i] = 0
# Kaufman Q2 step 5
residual, _, _ = lapack.dormqr("L", "N", qr, tau, temp, max(1, matrix.shape[1]),
overwrite_c=0)
return clp[:matrix.shape[1]], residual
def residual_variable_projection(
matrix: np.ndarray,
data: np.ndarray) -> typing.Tuple[typing.List[str], np.ndarray]:
"""Calculates the conditionally linear parameters and residual with the variable projection
method.
Parameters
----------
matrix :
The model matrix.
data : np.ndarray
The data to analyze.
"""
# TODO: Reference Kaufman paper
# Kaufman Q2 step 3
qr, tau, _, _ = lapack.dgeqrf(matrix)
# Kaufman Q2 step 4
temp, _, _ = lapack.dormqr("L",
"T",
qr,
tau,
data,
max(1, matrix.shape[1]),
overwrite_c=0)
clp, _ = lapack.dtrtrs(qr, temp)
for i in range(matrix.shape[1]):
temp[i] = 0
# Kaufman Q2 step 5
residual, _, _ = lapack.dormqr("L",
"N",
qr,
tau,
temp,
max(1, matrix.shape[1]),
overwrite_c=0)
return clp[:matrix.shape[1]], residual
def calc_eq_g(l, B):
L = B.shape[0]
order = (np.arange(L) + l) % L
Q, jpvt, tau, work, info = dgeqp3(B[order[0]] if L % 2 ==
1 else np.dot(B[order[1]], B[order[0]]))
d = Q.diagonal().copy()
d[d == 0.0] = 1.0
T = ((np.triu(Q).T / d).T)[:, jpvt.argsort()]
for m in range((1 if L % 2 == 1 else 2), L, 2):
W, work, info = dormqr("R", "N", Q, tau,
np.dot(B[order[m + 1]], B[order[m]]),
work.shape[0])
W *= d
jpvt = (W * W).sum(0).argsort()[::-1]
Q, tau, work, info = dgeqrf(W[:, jpvt])
d[...] = Q.diagonal()
d[d == 0.0] = 1.0
T = np.dot((np.triu(Q).T / d).T, T[jpvt, :])
N = B.shape[1]
invDb = np.zeros((N, N))
for i in range(N):
invDb[i, i] = 1.0 / d[i] if np.abs(d[i]) > 1.0 else 1.0
invDbQT, work, info = dormqr("R", "T", Q, tau, invDb, work.shape[0])
for i in range(N):
if np.abs(d[i]) <= 1.0:
T[i, :] *= d[i]
T += invDbQT
T_LU, piv, info = dgetrf(T)
sign = 1
for i in range(N):
if (T_LU[i, i] < 0) ^ (piv[i] != i) ^ (invDb[i, i] < 0) ^ (tau[i] > 0):
sign *= -1
G, info = dgetrs(T_LU, piv, invDbQT)
return G, sign