def fit_polynomial_one_constraint(X, Y, d, x0, y0):
m = len(X)
assert (len(Y) == m)
A = matrix([[X**k] for k in xrange(d)])
G = matrix(0.0, (2, d))
G[0, range(d)] = matrix([[x0**k] for k in xrange(d)])
G[1, range(1, d)] = matrix([[k * x0**(k - 1)] for k in xrange(1, d)])
# LS fit
#
# minimize (1/2) * || A*x - Y ||_2^2
# subject to G*x = h
#
# Solve as a linear equation
#
# [ A'*A G' ] [ x ] [ A'*Y ]
# [ G 0 ] [ Y ] = [ 0 ].
K = matrix(0.0, (d + 2, d + 2))
K[:d, :d] = A.T * A
K[d:, :d] = G
xls = matrix(0.0, (d + 2, 1))
xls[:d] = A.T * Y
xls[d] = y0
lapack.sysv(K, xls)
xls = xls[:d]
return xls
def fit_polynomial_one_constraint(X, Y, d, x0, y0):
m = len(X)
assert(len(Y) == m)
A = matrix( [[X**k] for k in xrange(d)])
G = matrix(0.0, (2,d))
G[0, range(d)] = matrix([[x0**k] for k in xrange(d)])
G[1, range(1,d)] = matrix([[k*x0**(k-1)] for k in xrange(1,d)])
# LS fit
#
# minimize (1/2) * || A*x - Y ||_2^2
# subject to G*x = h
#
# Solve as a linear equation
#
# [ A'*A G' ] [ x ] [ A'*Y ]
# [ G 0 ] [ Y ] = [ 0 ].
K = matrix(0.0, (d+2,d+2))
K[:d,:d] = A.T * A
K[d:,:d] = G
xls = matrix(0.0, (d+2,1))
xls[:d] = A.T * Y
xls[d] = y0
lapack.sysv(K, xls)
xls = xls[:d]
return xls
def old_fit_polynomial_multiple_constraints(X, Y, d, x0, y0):
m = len(X)
assert (len(Y) == m)
A = matrix([[X**k] for k in xrange(d)])
ncst = len(x0)
assert (ncst == len(y0))
G = matrix(0.0, (2 * ncst, d))
for ii in xrange(ncst):
G[2 * ii + 0, range(d)] = matrix([[x0[ii]**k] for k in xrange(d)])
G[2 * ii + 1,
range(1, d)] = matrix([[k * x0[ii]**(k - 1)] for k in xrange(1, d)])
# LS fit
#
# minimize (1/2) * || A*x - Y ||_2^2
# subject to G*x = h
#
# Solve as a linear equation
#
# [ A'*A G' ] [ x ] [ A'*Y ]
# [ G 0 ] [ Y ] = [ 0 ].
K = matrix(0.0, (d + 2 * ncst, d + 2 * ncst))
K[:d, :d] = A.T * A
K[d:, :d] = G
xls = matrix(0.0, (d + 2 * ncst, 1))
xls[:d] = A.T * Y
for ii in xrange(ncst):
xls[d + ii * 2] = y0[ii]
lapack.sysv(K, xls)
xls = xls[:d]
return xls
def old_fit_polynomial_multiple_constraints(X, Y, d, x0, y0):
m = len(X)
assert(len(Y) == m)
A = matrix( [[X**k] for k in xrange(d)])
ncst = len(x0)
assert( ncst == len(y0))
G = matrix(0.0, (2*ncst,d))
for ii in xrange(ncst) :
G[2*ii+0, range(d)] = matrix([[x0[ii]**k] for k in xrange(d)])
G[2*ii+1, range(1,d)] = matrix([[k*x0[ii]**(k-1)] for k in xrange(1,d)])
# LS fit
#
# minimize (1/2) * || A*x - Y ||_2^2
# subject to G*x = h
#
# Solve as a linear equation
#
# [ A'*A G' ] [ x ] [ A'*Y ]
# [ G 0 ] [ Y ] = [ 0 ].
K = matrix(0.0, (d+2*ncst,d+2*ncst))
K[:d,:d] = A.T * A
K[d:,:d] = G
xls = matrix(0.0, (d+2*ncst,1))
xls[:d] = A.T * Y
for ii in xrange( ncst ) :
xls[d+ii*2] = y0[ii]
lapack.sysv(K, xls)
xls = xls[:d]
return xls
def lapack_sysv(a, b):
"""
Inverse using LaPaCK sysv algorithm.
:param a:
:param b:
:return:
"""
b_sysv = matrix(b)
lapack.sysv(matrix(a), b_sysv)
return b_sysv
def fit_polynomial_multiple_constraints(X, Y, d, x0, y0, x1=None, y1=None):
m = len(X)
assert (len(Y) == m)
A = matrix([[X**k] for k in xrange(d)])
ncst0 = len(x0)
assert (ncst0 == len(y0))
ncst1 = 0
if x1: ncst1 = len(x1)
G = matrix(0.0, (2 * ncst0 + ncst1, d))
for ii in xrange(ncst0):
G[2 * ii + 0, range(d)] = matrix([[x0[ii]**k] for k in xrange(d)])
G[2 * ii + 1,
range(1, d)] = matrix([[k * x0[ii]**(k - 1)] for k in xrange(1, d)])
if x1:
for ii in xrange(ncst1):
G[ii + 2 * ncst0,
range(d)] = matrix([[x1[ii]**k] for k in xrange(d)])
# LS fit
#
# minimize (1/2) * || A*x - Y ||_2^2
# subject to G*x = h
#
# Solve as a linear equation
#
# [ A'*A G' ] [ x ] [ A'*Y ]
# [ G 0 ] [ Y ] = [ 0 ].
K = matrix(0.0, (d + 2 * ncst0 + ncst1, d + 2 * ncst0 + ncst1))
K[:d, :d] = A.T * A
K[d:, :d] = G
xls = matrix(0.0, (d + 2 * ncst0 + ncst1, 1))
xls[:d] = A.T * Y
for ii in xrange(ncst0):
xls[d + ii * 2] = y0[ii]
if x1:
for ii in xrange(ncst1):
xls[d + 2 * ncst0 + ii] = y1[ii]
lapack.sysv(K, xls)
xls = xls[:d]
coefs = numpy.array(xls.T)[0]
coefs = coefs[::-1]
return coefs
def fit_polynomial_multiple_constraints(X, Y, d, x0, y0, x1 = None, y1 = None):
m = len(X)
assert(len(Y) == m)
A = matrix( [[X**k] for k in xrange(d)])
ncst0 = len(x0)
assert( ncst0 == len(y0))
ncst1 = 0
if x1 : ncst1 = len(x1)
G = matrix(0.0, (2*ncst0+ncst1,d))
for ii in xrange(ncst0) :
G[2*ii+0, range(d)] = matrix([[x0[ii]**k] for k in xrange(d)])
G[2*ii+1, range(1,d)] = matrix([[k*x0[ii]**(k-1)] for k in xrange(1,d)])
if x1 :
for ii in xrange(ncst1) :
G[ii+2*ncst0,range(d)] = matrix( [[x1[ii]**k] for k in xrange(d)] )
# LS fit
#
# minimize (1/2) * || A*x - Y ||_2^2
# subject to G*x = h
#
# Solve as a linear equation
#
# [ A'*A G' ] [ x ] [ A'*Y ]
# [ G 0 ] [ Y ] = [ 0 ].
K = matrix(0.0, (d+2*ncst0+ncst1,d+2*ncst0+ncst1))
K[:d,:d] = A.T * A
K[d:,:d] = G
xls = matrix(0.0, (d+2*ncst0+ncst1,1))
xls[:d] = A.T * Y
for ii in xrange( ncst0 ) :
xls[d+ii*2] = y0[ii]
if x1 :
for ii in xrange( ncst1 ) :
xls[d+2*ncst0+ii] = y1[ii]
lapack.sysv(K, xls)
xls = xls[:d]
coefs = numpy.array(xls.T)[0]
coefs = coefs[::-1]
return coefs
# LS fit
#
# minimize (1/2) * || A*x - y ||_2^2
# subject to G*x = h
#
# Solve as a linear equation
#
# [ A'*A G' ] [ x ] [ A'*y ]
# [ G 0 ] [ y ] = [ 0 ].
K = matrix(0.0, (n + 6, n + 6))
K[:n, :n] = A.T * A
K[n:, :n] = G
xls = matrix(0.0, (n + 6, 1))
xls[:n] = A.T * y
lapack.sysv(K, xls)
xls = xls[:n]
# Chebyshev fit
#
# minimize || A*x - y ||_inf
# subject to G*x = h
xcheb = variable(12)
op(max(abs(A * xcheb - y)), [G * xcheb == 0]).solve()
xcheb = xcheb.value
if pylab_installed:
pylab.figure(2, facecolor='w')
nopts = 100
ts = -1.0 + (1.0 - (-1.0)) / nopts * matrix(list(range(nopts)), tc='d')
# LS fit
#
# minimize (1/2) * || A*x - y ||_2^2
# subject to G*x = h
#
# Solve as a linear equation
#
# [ A'*A G' ] [ x ] [ A'*y ]
# [ G 0 ] [ y ] = [ 0 ].
K = matrix(0.0, (n+6,n+6))
K[:n,:n] = A.T * A
K[n:,:n] = G
xls = matrix(0.0, (n+6,1))
xls[:n] = A.T * y
lapack.sysv(K, xls)
xls = xls[:n]
# Chebyshev fit
#
# minimize || A*x - y ||_inf
# subject to G*x = h
xcheb = variable(12)
op( max(abs(A*xcheb - y)), [G*xcheb == 0]).solve()
xcheb = xcheb.value
if pylab_installed:
pylab.figure(2, facecolor='w')
nopts = 100
