def lstsq(a, b, rcond=-1):
"""
Return the least-squares solution to an equation.
Solves the equation `a x = b` by computing a vector `x` that minimizes
the norm `|| b - a x ||`.
Parameters
----------
a : array_like, shape (M, N)
Input equation coefficients.
b : array_like, shape (M,) or (M, K)
Equation target values. If `b` is two-dimensional, the least
squares solution is calculated for each of the `K` target sets.
rcond : float, optional
Cutoff for ``small`` singular values of `a`.
Singular values smaller than `rcond` times the largest singular
value are considered zero.
Returns
-------
x : ndarray, shape(N,) or (N, K)
Least squares solution. The shape of `x` depends on the shape of
`b`.
residues : ndarray, shape(), (1,), or (K,)
Sums of residues; squared Euclidian norm for each column in
`b - a x`.
If the rank of `a` is < N or > M, this is an empty array.
If `b` is 1-dimensional, this is a (1,) shape array.
Otherwise the shape is (K,).
rank : integer
Rank of matrix `a`.
s : ndarray, shape(min(M,N),)
Singular values of `a`.
Raises
------
LinAlgError
If computation does not converge.
Notes
-----
If `b` is a matrix, then all array results returned as
matrices.
Examples
--------
Fit a line, ``y = mx + c``, through some noisy data-points:
>>> x = np.array([0, 1, 2, 3])
>>> y = np.array([-1, 0.2, 0.9, 2.1])
By examining the coefficients, we see that the line should have a
gradient of roughly 1 and cuts the y-axis at more-or-less -1.
We can rewrite the line equation as ``y = Ap``, where ``A = [[x 1]]``
and ``p = [[m], [c]]``. Now use `lstsq` to solve for `p`:
>>> A = np.vstack([x, np.ones(len(x))]).T
>>> A
array([[ 0., 1.],
[ 1., 1.],
[ 2., 1.],
[ 3., 1.]])
>>> m, c = np.linalg.lstsq(A, y)[0]
>>> print m, c
1.0 -0.95
Plot the data along with the fitted line:
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'o', label='Original data', markersize=10)
>>> plt.plot(x, m*x + c, 'r', label='Fitted line')
>>> plt.legend()
>>> plt.show()
"""
import math
a, _ = _makearray(a)
b, wrap = _makearray(b)
is_1d = len(b.shape) == 1
if is_1d:
b = b[:, newaxis]
_assertRank2(a, b)
m = a.shape[0]
n = a.shape[1]
n_rhs = b.shape[1]
ldb = max(n, m)
if m != b.shape[0]:
raise LinAlgError, 'Incompatible dimensions'
t, result_t = _commonType(a, b)
real_t = _linalgRealType(t)
bstar = zeros((ldb, n_rhs), t)
bstar[:b.shape[0],:n_rhs] = b.copy()
a, bstar = _fastCopyAndTranspose(t, a, bstar)
s = zeros((min(m, n),), real_t)
nlvl = max( 0, int( math.log( float(min(m, n))/2. ) ) + 1 )
iwork = zeros((3*min(m, n)*nlvl+11*min(m, n),), fortran_int)
if isComplexType(t):
lapack_routine = lapack_lite.zgelsd
lwork = 1
rwork = zeros((lwork,), real_t)
work = zeros((lwork,), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
0, work, -1, rwork, iwork, 0)
lwork = int(abs(work[0]))
rwork = zeros((lwork,), real_t)
a_real = zeros((m, n), real_t)
bstar_real = zeros((ldb, n_rhs,), real_t)
results = lapack_lite.dgelsd(m, n, n_rhs, a_real, m,
bstar_real, ldb, s, rcond,
0, rwork, -1, iwork, 0)
lrwork = int(rwork[0])
work = zeros((lwork,), t)
rwork = zeros((lrwork,), real_t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
0, work, lwork, rwork, iwork, 0)
else:
lapack_routine = lapack_lite.dgelsd
lwork = 1
work = zeros((lwork,), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
0, work, -1, iwork, 0)
lwork = int(work[0])
work = zeros((lwork,), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
0, work, lwork, iwork, 0)
if results['info'] > 0:
raise LinAlgError, 'SVD did not converge in Linear Least Squares'
resids = array([], t)
if is_1d:
x = array(ravel(bstar)[:n], dtype=result_t, copy=True)
if results['rank'] == n and m > n:
resids = array([np.linalg.norm(ravel(bstar)[n:], 2)**2])
else:
x = array(transpose(bstar)[:n,:], dtype=result_t, copy=True)
if results['rank'] == n and m > n:
resids = array([np.linalg.norm(v[n:], 2)**2 for v in bstar])
st = s[:min(n, m)].copy().astype(_realType(result_t))
return wrap(x), wrap(resids), results['rank'], st
def svd_gesvd(a, full_matrices=True, compute_uv=True, check_finite=True):
"""svd with LAPACK's '#gesvd' (with # = d/z for float/complex).
Similar as :func:`numpy.linalg.svd`, but use LAPACK 'gesvd' driver.
Works only with 2D arrays.
Outer part is based on the code of `numpy.linalg.svd`.
Parameters
----------
a, full_matrices, compute_uv :
See :func:`numpy.linalg.svd` for details.
check_finite :
check whether input arrays contain 'NaN' or 'inf'.
Returns
-------
U, S, Vh : ndarray
See :func:`numpy.linalg.svd` for details.
"""
a, wrap = _makearray(a) # uses order='C'
if a.ndim != 2:
raise LinAlgError("array must be 2D!")
if a.size == 0 or np.product(a.shape) == 0:
raise LinAlgError("array cannot be empty")
if check_finite:
if not isfinite(a).all():
raise LinAlgError("Array must not contain infs or NaNs")
M, N = a.shape
# determine types
t, result_t = _commonType(a)
# t = type for calculation, (for my numpy version) actually always one of {double, cdouble}
# result_t = one of {single, double, csingle, cdouble}
is_complex = isComplexType(t)
real_t = _realType(t) # real version of t with same precision
# copy: the array is destroyed
a = _fastCopyAndTranspose(
t, a) # casts a to t, copy and transpose (=change to order='F')
lapack_routine = _get_gesvd(t)
# allocate output space & options
if compute_uv:
if full_matrices:
nu = M
lvt = N
option = b'A'
else:
nu = min(N, M)
lvt = min(N, M)
option = b'S'
u = np.zeros((M, nu), t, order='F')
vt = np.zeros((lvt, N), t, order='F')
else:
option = b'N'
nu = 1
u = np.empty((1, 1), t, order='F')
vt = np.empty((1, 1), t, order='F')
s = np.zeros((min(N, M), ), real_t, order='F')
INFO = c_int(0)
m = c_int(M)
n = c_int(N)
lu = c_int(u.shape[0])
lvt = c_int(vt.shape[0])
work = np.zeros((1, ), t)
lwork = c_int(-1) # first call with lwork=-1
args = [option, option, m, n, a, m, s, u, lu, vt, lvt, work, lwork, INFO]
if is_complex:
# differnt call signature: additional array 'rwork' of fixed size
rwork = np.zeros((5 * min(N, M), ), real_t)
args.insert(-1, rwork)
lapack_routine(
*args) # first call: just calculate the required `work` size
if INFO.value < 0:
raise Exception('%d-th argument had an illegal value' % INFO.value)
if is_complex:
lwork = int(work[0].real)
else:
lwork = int(work[0])
work = np.zeros((lwork, ), t, order='F')
args[11] = work
args[12] = c_int(lwork)
lapack_routine(*args) # second call: the actual calculation
if INFO.value < 0:
raise Exception('%d-th argument had an illegal value' % INFO.value)
if INFO.value > 0:
raise LinAlgError("SVD did not converge with 'gesvd'")
s = s.astype(_realType(result_t))
if compute_uv:
u = u.astype(result_t) # no repeated transpose: used fortran order
vt = vt.astype(result_t)
return wrap(u), s, wrap(vt)
else:
return s
def lstsq(a, b, rcond=-1):
"""
Return the least-squares solution to an equation.
Solves the equation `a x = b` by computing a vector `x` that minimizes
the norm `|| b - a x ||`.
Parameters
----------
a : array_like, shape (M, N)
Input equation coefficients.
b : array_like, shape (M,) or (M, K)
Equation target values. If `b` is two-dimensional, the least
squares solution is calculated for each of the `K` target sets.
rcond : float, optional
Cutoff for ``small`` singular values of `a`.
Singular values smaller than `rcond` times the largest singular
value are considered zero.
Returns
-------
x : ndarray, shape(N,) or (N, K)
Least squares solution. The shape of `x` depends on the shape of
`b`.
residues : ndarray, shape(), (1,), or (K,)
Sums of residues; squared Euclidian norm for each column in
`b - a x`.
If the rank of `a` is < N or > M, this is an empty array.
If `b` is 1-dimensional, this is a (1,) shape array.
Otherwise the shape is (K,).
rank : integer
Rank of matrix `a`.
s : ndarray, shape(min(M,N),)
Singular values of `a`.
Raises
------
LinAlgError
If computation does not converge.
Notes
-----
If `b` is a matrix, then all array results returned as
matrices.
Examples
--------
Fit a line, ``y = mx + c``, through some noisy data-points:
>>> x = np.array([0, 1, 2, 3])
>>> y = np.array([-1, 0.2, 0.9, 2.1])
By examining the coefficients, we see that the line should have a
gradient of roughly 1 and cuts the y-axis at more-or-less -1.
We can rewrite the line equation as ``y = Ap``, where ``A = [[x 1]]``
and ``p = [[m], [c]]``. Now use `lstsq` to solve for `p`:
>>> A = np.vstack([x, np.ones(len(x))]).T
>>> A
array([[ 0., 1.],
[ 1., 1.],
[ 2., 1.],
[ 3., 1.]])
>>> m, c = np.linalg.lstsq(A, y)[0]
>>> print m, c
1.0 -0.95
Plot the data along with the fitted line:
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'o', label='Original data', markersize=10)
>>> plt.plot(x, m*x + c, 'r', label='Fitted line')
>>> plt.legend()
>>> plt.show()
"""
import math
a, _ = _makearray(a)
b, wrap = _makearray(b)
is_1d = len(b.shape) == 1
if is_1d:
b = b[:, newaxis]
_assertRank2(a, b)
m = a.shape[0]
n = a.shape[1]
n_rhs = b.shape[1]
ldb = max(n, m)
if m != b.shape[0]:
raise LinAlgError, 'Incompatible dimensions'
t, result_t = _commonType(a, b)
real_t = _linalgRealType(t)
bstar = zeros((ldb, n_rhs), t)
bstar[:b.shape[0], :n_rhs] = b.copy()
a, bstar = _fastCopyAndTranspose(t, a, bstar)
s = zeros((min(m, n), ), real_t)
nlvl = max(0, int(math.log(float(min(m, n)) / 2.)) + 1)
iwork = zeros((3 * min(m, n) * nlvl + 11 * min(m, n), ), fortran_int)
if isComplexType(t):
lapack_routine = lapack_lite.zgelsd
lwork = 1
rwork = zeros((lwork, ), real_t)
work = zeros((lwork, ), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond, 0,
work, -1, rwork, iwork, 0)
lwork = int(abs(work[0]))
rwork = zeros((lwork, ), real_t)
a_real = zeros((m, n), real_t)
bstar_real = zeros((
ldb,
n_rhs,
), real_t)
results = lapack_lite.dgelsd(m, n, n_rhs, a_real, m, bstar_real, ldb,
s, rcond, 0, rwork, -1, iwork, 0)
lrwork = int(rwork[0])
work = zeros((lwork, ), t)
rwork = zeros((lrwork, ), real_t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond, 0,
work, lwork, rwork, iwork, 0)
else:
lapack_routine = lapack_lite.dgelsd
lwork = 1
work = zeros((lwork, ), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond, 0,
work, -1, iwork, 0)
lwork = int(work[0])
work = zeros((lwork, ), t)
results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond, 0,
work, lwork, iwork, 0)
if results['info'] > 0:
raise LinAlgError, 'SVD did not converge in Linear Least Squares'
resids = array([], t)
if is_1d:
x = array(ravel(bstar)[:n], dtype=result_t, copy=True)
if results['rank'] == n and m > n:
resids = array([np.linalg.norm(ravel(bstar)[n:], 2)**2])
else:
x = array(transpose(bstar)[:n, :], dtype=result_t, copy=True)
if results['rank'] == n and m > n:
resids = array([np.linalg.norm(v[n:], 2)**2 for v in bstar])
st = s[:min(n, m)].copy().astype(_realType(result_t))
return wrap(x), wrap(resids), results['rank'], st
