def generalized_inverse(a, rcond = 1.e-10):
u, s, vt = singular_value_decomposition(a, 0)
m = u.getshape()[0]
n = vt.getshape()[1]
cutoff = rcond*num.maximum.reduce(s)
for i in range(min(n,m)):
if s[i] > cutoff:
s[i] = 1./s[i]
else:
s[i] = 0.;
return num.dot(num.transpose(vt),
s[:, num.NewAxis]*num.transpose(u))
def cholesky_decomposition(a):
_assertRank2(a)
_assertSquareness(a)
t =_commonType(a)
a = _castCopyAndTranspose(t, a)
m = a.getshape()[0]
n = a.getshape()[1]
if _array_kind[t] == 1:
lapack_routine = lapack_lite2.zpotrf
else:
lapack_routine = lapack_lite2.dpotrf
results = lapack_routine('L', n, a, m, 0)
if results['info'] > 0:
raise LinAlgError, 'Matrix is not positive definite - Cholesky decomposition cannot be computed'
return copy.copy(num.transpose(mlab.triu(a,k=0)))
def linear_least_squares(a, b, rcond=1.e-10):
"""solveLinearLeastSquares(a,b) returns x,resids,rank,s
where x minimizes 2-norm(|b - Ax|)
resids is the sum square residuals
rank is the rank of A
s is an rank of the singual values of A in desending order
If b is a matrix then x is also a matrix with corresponding columns.
If the rank of A is less than the number of columns of A or greater than
the numer of rows, then residuals will be returned as an empty array
otherwise resids = sum((b-dot(A,x)**2).
Singular values less than s[0]*rcond are treated as zero.
"""
one_eq = len(b.getshape()) == 1
if one_eq:
b = b[:, num.NewAxis]
_assertRank2(a, b)
m = a.getshape()[0]
n = a.getshape()[1]
n_rhs = b.getshape()[1]
ldb = max(n,m)
if m != b.getshape()[0]:
raise LinAlgError, 'Incompatible dimensions'
t =_commonType(a, b)
real_t = _array_type[0][_array_precision[t]]
bstar = num.zeros((ldb,n_rhs),t)
bstar[:b.getshape()[0],:n_rhs] = copy.copy(b)
a,bstar = _castCopyAndTranspose(t, a, bstar)
s = num.zeros((min(m,n),),real_t)
nlvl = max( 0, int( math.log( float(min( m,n ))/2. ) ) + 1 )
iwork = num.zeros((3*min(m,n)*nlvl+11*min(m,n),), 'l')
if _array_kind[t] == 1: # Complex routines take different arguments
lapack_routine = lapack_lite2.zgelsd
lwork = 1
rwork = num.zeros((lwork,), real_t)
work = num.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 = num.zeros((lwork,),real_t)
a_real = num.zeros((m,n),real_t)
bstar_real = num.zeros((ldb,n_rhs,),real_t)
results = lapack_lite2.dgelsd( m, n, n_rhs, a_real, m, bstar_real,ldb , s, rcond,
0,rwork,-1,iwork,0 )
lrwork = int(rwork[0])
work = num.zeros((lwork,), t)
rwork = num.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_lite2.dgelsd
lwork = 1
work = num.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 = num.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 = num.array([],type=t)
if one_eq:
x = copy.copy(num.ravel(bstar)[:n])
if (results['rank']==n) and (m>n):
resids = num.array([num.sum((num.ravel(bstar)[n:])**2)])
else:
x = copy.copy(num.transpose(bstar)[:n,:])
if (results['rank']==n) and (m>n):
resids = copy.copy(num.sum((num.transpose(bstar)[n:,:])**2))
return x,resids,results['rank'],copy.copy(s[:min(n,m)])
def qr_decomposition(a, mode='full'):
""" calculates A=QR, Q orthonormal, R upper triangle matrix.
mode: 'full' ==> (Q,R) as return value
'r' ==> (None, R) as return value
'economic' ==> (None, A') where the diagonal + upper triangle
part of A' is R. This is faster if one only requires R. """
_assertRank2(a)
t=_commonType(a)
m = a.getshape()[0]
n = a.getshape()[1]
mn = min(m,n)
tau = num.zeros((mn,), t)
# a: convert num storing order to fortran storing order
a = _castCopyAndTranspose(t, a)
if _array_kind[t] == 1:
lapack_routine = lapack_lite2.zgeqrf
routine_name='ZGEQRF'
else:
lapack_routine = lapack_lite2.dgeqrf
routine_name='DGEQRF'
# calculate optimal size of work data 'work'
lwork = 1
work = num.zeros((lwork,), t)
results=lapack_routine(m, n, a, m, tau, work, -1, 0)
if results['info'] > 0:
raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
# do qr decomposition
lwork = int(abs(work[0]))
work = num.zeros((lwork,),t)
results=lapack_routine(m, n, a, m, tau, work, lwork, 0)
if results['info'] > 0:
raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
# atemp: convert fortrag storing order to num storing order
atemp = num.transpose(a)
# economic mode
if mode[0]=='e': return None, atemp
# generate r
r = num.zeros((mn,n), t)
for i in range(mn):
r[i, i:] = atemp[i, i:]
# 'r'-mode, that is, calculate only r
if mode[0]=='r': return None, r
# from here on: build orthonormal matrix q from a
if _array_kind[t] == 1:
lapack_routine = lapack_lite2.zungqr
routine_name = "ZUNGQR"
else:
lapack_routine = lapack_lite2.dorgqr
routine_name = "DORGQR"
# determine optimal lwork
lwork = 1
work=num.zeros((lwork,), t)
results=lapack_routine(m,mn,mn, a, m, tau, work, -1, 0)
if results['info'] > 0:
raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
# compute q
lwork = int(abs(work[0]))
work=num.zeros((lwork,), t)
results=lapack_routine(m,mn,mn, a, m, tau, work, lwork, 0)
if results['info'] > 0:
raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
q = num.transpose(a[:mn,:])
return q,r