def newton_solver(f,
x0,
lb=None,
ub=None,
infos=False,
verbose=False,
maxit=50,
tol=1e-8,
eps=1e-5,
numdiff=False):
'''Solves many independent systems f(x)=0 simultaneously using a simple gradient descent.
:param f: objective function to be solved with values p x N . The second output argument represents the derivative with
values in (p x p x N)
:param x0: initial value ( p x N )
:return: solution x such that f(x) = 0
'''
precision = x0.dtype # default tolerance should depend on precision
from dolo.numeric.serial_operations import serial_multiplication as stv, serial_solve
err = 1
it = 0
while err > tol and it <= maxit:
if not numdiff:
[res, dres] = f(x0)
else:
res = f(x0)
dres = numpy.zeros((res.shape[0], x0.shape[0], x0.shape[1]),
dtype=precision)
for i in range(x0.shape[0]):
xi = x0.copy()
xi[i, :] += eps
resi = f(xi)
dres[:, i, :] = (resi - res) / eps
try:
dx = -serial_solve(dres, res)
except:
dx = -serial_solve(dres, res, debug=True)
x = x0 + dx
err = abs(res).max()
x0 = x
it += 1
if not infos:
return x
else:
return [x, it]
def newton_solver(f, x0, lb=None, ub=None, infos=False, verbose=False, maxit=50, tol=1e-8, eps=1e-5, numdiff=False):
'''Solves many independent systems f(x)=0 simultaneously using a simple gradient descent.
:param f: objective function to be solved with values p x N . The second output argument represents the derivative with
values in (p x p x N)
:param x0: initial value ( p x N )
:return: solution x such that f(x) = 0
'''
precision = x0.dtype # default tolerance should depend on precision
from dolo.numeric.serial_operations import serial_multiplication as stv, serial_solve
err = 1
it = 0
while err > tol and it <= maxit:
if not numdiff:
[res,dres] = f(x0)
else:
res = f(x0)
dres = numpy.zeros( (res.shape[0], x0.shape[0], x0.shape[1]), dtype=precision )
for i in range(x0.shape[0]):
xi = x0.copy()
xi[i,:] += eps
resi = f(xi)
dres[:,i,:] = (resi - res)/eps
try:
dx = - serial_solve(dres,res)
except:
dx = - serial_solve(dres,res, debug=True)
x = x0 + dx
err = abs(res).max()
x0 = x
it += 1
if not infos:
return x
else:
return [x, it]
def newton_solver(f, x0, lb=None, ub=None, infos=False, backsteps=10, maxit=10):
'''Solves many independent systems f(x)=0 simultaneously using a simple gradient descent.
:param f: objective function to be solved with values p x N . The second output argument represents the derivative with
values in (p x p x N)
:param x0: initial value ( p x N )
:return: solution x such that f(x) = 0
'''
from dolo.numeric.serial_operations import serial_multiplication as stv, serial_solve
err = 1
tol = 1e-8
it = 0
while err > tol and it <= maxit:
[res,dres] = f(x0)
# res = f(x0)
# dres = df(x0)
fnorm = abs(res).max() # suboptimal
dx = - serial_solve(dres,res)
# x = x0 + dx
for i in range(backsteps):
xx = x0 + dx/(2**i)
if not ub==None:
xx = numpy.maximum(xx, lb)
xx = numpy.minimum(xx, ub)
new_res = f(xx)[0]
new_fnorm = abs(new_res).max()
if numpy.isfinite(new_fnorm) and new_fnorm < fnorm: # all right proceed to next iteration
x = xx
break
if i == backsteps -1:
if numpy.isfinite(new_fnorm):
x = xx
else:
raise Exception('Non finite value found')
err = abs(dx).max()
x0 = x
it += 1
# print (it <=maxit)
# print(err)
if not infos:
return x
else:
return [x, it]
def ncpsolve(f, a, b, x, tol=None, infos=False, verbose=False, serial=False):
'''
don't ask what ncpsolve can do for you...
:param f:
:param a:
:param b:
:param x:
:param tol:
:param serial:
:return:
'''
maxit = 100
if tol is None:
tol = sqrt(finfo(float64).eps)
maxsteps = 10
showiters = True
it = 0
if verbose:
headline = '|{0:^5} | {1:^12} | {2:^12} |'.format(
'k', ' backsteps', '||f(x)||')
stars = '-' * len(headline)
print(stars)
print(headline)
print(stars)
while it < maxit:
it += 1
[fval, fjac] = f(x)
[ftmp, fjac] = smooth(x, a, b, fval, fjac, serial=serial)
fnorm = norm(ftmp, ord=inf)
if fnorm < tol:
if verbose:
print(stars)
if infos:
return [x, it]
else:
return x
if serial:
from dolo.numeric.serial_operations import serial_solve
dx = -serial_solve(fjac, ftmp)
else:
dx = -solve(fjac, ftmp)
fnormold = inf
for backsteps in range(maxsteps):
xnew = x + dx
fnew = f(xnew)[0] # TODO: don't ask for derivatives
fnew = smooth(xnew, a, b, fnew, serial=serial)
fnormnew = norm(fnew, ord=inf)
if fnormnew < fnorm:
break
if fnormold < fnormnew:
dx = 2 * dx
break
fnormold = fnormnew
dx = dx / 2
x = x + dx
if verbose:
print('|{0:5} | {2:12.3e} | {2:12.3e} |'.format(
it, backsteps, fnormnew))
if verbose:
print(stars)
warnings.Warning('Failure to converge in ncpsolve')
fval = f(x)
return [x, fval]
def ncpsolve(f, a, b, x, tol=None, infos=False, verbose=False, serial=False):
'''
don't ask what ncpsolve can do for you...
:param f:
:param a:
:param b:
:param x:
:param tol:
:param serial:
:return:
'''
maxit = 100
if tol is None:
tol = sqrt( finfo( float64 ).eps )
maxsteps = 10
showiters = True
it = 0
if verbose:
headline = '|{0:^5} | {1:^12} | {2:^12} |'.format( 'k',' backsteps', '||f(x)||' )
stars = '-'*len(headline)
print(stars)
print(headline)
print(stars)
while it < maxit:
it += 1
[fval, fjac] = f(x)
[ftmp, fjac] = smooth(x, a, b, fval, fjac, serial=serial)
fnorm = norm( ftmp, ord=inf)
if fnorm < tol:
if verbose:
print(stars)
if infos:
return [x, it]
else:
return x
if serial:
from dolo.numeric.serial_operations import serial_solve
dx = - serial_solve( fjac, ftmp )
else:
dx = - solve( fjac, ftmp)
fnormold = inf
for backsteps in range(maxsteps):
xnew = x + dx
fnew = f(xnew)[0] # TODO: don't ask for derivatives
fnew = smooth( xnew, a, b, fnew, serial=serial)
fnormnew = norm(fnew, ord=inf)
if fnormnew < fnorm:
break
if fnormold < fnormnew:
dx = 2*dx
break
fnormold = fnormnew
dx = dx/2
x = x + dx
if verbose:
print('|{0:5} | {2:12.3e} | {2:12.3e} |'.format( it, backsteps, fnormnew) )
if verbose:
print(stars)
warnings.Warning('Failure to converge in ncpsolve')
fval = f(x)
return [x,fval]