def get_new_tvec(self, iev):
v1 = sp.ndarray(self.ndim, self.complex_precision) # v1 = M^{-1}*r
v2 = sp.ndarray(self.ndim, self.complex_precision) # v2 = M^{-1}*u
idx = 0
# First half of vector
for ii in range(self.nocc):
for jj in range(self.nvirt):
ediff = (self.evalai(ii, jj) - sp.real(self.teta[iev]))
if np.abs(ediff) > 1e-8:
dtmp = 1 / ediff
v1[idx] = self.r_vecs[idx, iev] * dtmp
v2[idx] = self.u_vecs[idx, iev] * dtmp
else:
print("Warning, (E_{a}-E_{i})<1e-8")
v1[idx] = self.complex_precision(
self.real_precision(0.0) + self.real_precision(0.0j))
v2[idx] = self.complex_precision(
self.real_precision(0.0) + self.real_precision(0.0j))
idx += 1
# Second half of vector
for ii in range(self.nocc):
for jj in range(self.nvirt):
ediff = -self.evalai(ii, jj) - sp.real(self.teta[iev])
if abs(ediff) > 1e-8:
dtmp = 1 / ediff
v1[idx] = self.r_vecs[idx, iev] * dtmp
v2[idx] = self.u_vecs[idx, iev] * dtmp
else:
print("Warning, (E_{a}-E_{i})<1e-8")
v1[idx] = self.complex_precision(
self.real_precision(0.0) + self.real_precision(0.0j))
v2[idx] = self.complex_precision(
self.real_precision(0.0) + self.real_precision(0.0j))
idx += 1
sp.savetxt("v1_c" + str(self.cycle) + "_i" + str(iev + 1) + ".txt", v1)
sp.savetxt("v2_c" + str(self.cycle) + "_i" + str(iev + 1) + ".txt", v2)
u_m1_u = sp.vdot(self.u_vecs[:, iev], v2)
print("u_m1_u = ", u_m1_u)
if abs(u_m1_u) > 1e-8:
u_m1_r = sp.vdot(self.u_vecs[:, iev], v1)
print("u_m1_r = ", u_m1_r)
factor = u_m1_r / sp.real(u_m1_u)
return factor * v2 - v1
else:
return -v1
def compute_loocv_gmm(variable,model,x,y,ids,K_u,alpha,beta,log_prop_u):
""" Function that computes the estimation of the loocv for the GMM model with variables ids + variable(i)
Inputs:
model : the GMM model
x,y : the training samples and the corresponding label
ids : the pool of selected variables
variable : the variable to be tested from the set of available variable
K_u : the initial prediction values computed with all the samples
alpha, beta and log_prop_u : constant that are computed outside of the loop to increased speed
Outputs:
loocv_temp : the loocv
Used in GMM.forward_selection()
"""
n = x.shape[0]
ids.append(variable) # Iteratively add one of the remaining variables
Kp = model.predict_gmm(x,ids=ids)[1]# Predict with all the samples with ids
loocv_temp=0.0; # Initialization of the temporary loocv
for j in range(n): # Predict the class with the model ids_t
Kloo = Kp[j,:] + K_u # Initialization of the decision rule for sample "j" #--- Change for only not C---#
c = int(y[j]-1) # Update of parameter of class c
m = (model.ni[c]*model.mean[c,ids] -x[j,ids])*alpha[c] # Update the mean value
xb = x[j,ids] - m # x centered
cov_u = (model.cov[c,ids,:][:,ids] - sp.outer(xb,xb)*alpha[c])*beta # Update the covariance matrix
logdet,rcond = safe_logdet(cov_u)
Kloo[c] = logdet - 2*log_prop_u[c] + sp.vdot(xb,mylstsq(cov_u,xb.T,rcond)) # Compute the new decision rule
del cov_u,xb,m,c
yloo = sp.argmin(Kloo)+1
loocv_temp += float(yloo==y[j]) # Check the correct/incorrect classification rule
ids.pop() # Remove the current variable
return loocv_temp/n # Compute loocv for variable
def FIRE(x0,
fprime,
fmax=0.005,
Nmin=5.,
finc=1.1,
fdec=0.5,
alphastart=0.1,
fa=0.99,
deltatmax=10.,
maxsteps=10**5):
Nmin, finc, fdec, alphastart, fa, deltatmax = (5., 1.1, 0.5, 0.1, 0.99,
10.)
alpha = alphastart
deltat = 0.1
pos = x0.copy()
v = sp.zeros_like(pos)
steps_since_negative = 0
def norm(vec):
return sp.sqrt(sp.sum(vec**2, 1))
def unitize(vec):
return ((vec.T) / norm(vec)).T
forces = fprime(pos)
step_num = 0
while max(norm(forces)) > fmax and step_num < maxsteps:
forces = fprime(pos)
power = sp.vdot(forces, v)
print "Step: {}, max_force: {}, power: {}".format(
step_num, max(norm(forces)), power)
v = (1.0 - alpha) * v + alpha * (norm(v) * unitize(forces).T).T
if power > 0.:
if steps_since_negative > Nmin:
deltat = min(deltat * finc, deltatmax)
alpha = alpha * fa
steps_since_negative += 1
else:
steps_since_negative = 0
deltat = deltat * fdec
v *= 0.
alpha = alphastart
v += forces * deltat
pos += v * deltat
step_num += 1
return pos
def GetProbabilities(n, Psi):
"""
Get probabilites by doing mod-squared of @Psi.
"""
density = sp.zeros(2**n)
for i in range(2**n):
density[i] = sp.vdot(Psi[i], Psi[i]).real
return density
def ConstructOverlapData(t, Psi, vec):
"""
Construct proper datapoint for overlap.
"""
datapoint = [t]
overlap = abs(sp.vdot(Psi, vec))**2
datapoint.append(overlap)
return datapoint
def ConstructOverlapData(t, Psi, vec):
"""
Construct proper datapoint for overlap.
"""
datapoint = [t]
overlap = abs(sp.vdot(Psi, vec)) ** 2
datapoint.append(overlap)
return datapoint
def GetProbabilities(n, Psi):
"""
Get probabilites by doing mod-squared of @Psi.
"""
density = sp.zeros(2**n)
for i in range(2**n):
density[i] = sp.vdot(Psi[i], Psi[i]).real
return density
def fire(self,fmax=0.1,
Nmin=5.,finc=1.1,fdec=0.5,alphastart=0.1,fa=0.99,deltatmax=10.,
maxsteps = 10**5):
""" Do a fire relaxation """
alpha = alphastart
deltat = 0.1
pos = self.get_pos_arr(force=True)
v = sp.zeros_like(pos)
self._update_vel(v)
v = self._get_vel_arr()
steps_since_negative = 0
def norm_arr(vec):
return sp.sqrt(sp.sum(vec**2,1))
def unitize_arr(vec):
return ((vec.T)/norm(vec)).T
forces = sp.nan_to_num(sp.array([ [sp.inf,sp.inf]]))
step_num = 0
print "Beginning FIRE Relaxation -- fmax={}".format(fmax)
while max(norm_arr(forces)) > fmax and step_num < maxsteps:
forces = self.forces
power = sp.vdot(forces,v)
print "Step: {}, max_force: {}, power: {}".format(step_num,max(norm_arr(forces)), power)
v = (1.0 - alpha)*v + alpha*(norm_arr(v)*unitize_arr(forces).T).T
if power>0.:
if steps_since_negative > Nmin:
deltat = min(deltat * finc, deltatmax)
alpha = alpha*fa
steps_since_negative += 1
else:
steps_since_negative = 0
deltat = deltat * fdec
v *= 0.
alpha = alphastart
v += forces*deltat
pos += v*deltat
self._update_pos(pos)
step_num += 1
self._update_pos(pos)
self._update_vel(v)
print "Relaxation finished..."
def fire(self,fmax=0.1,
Nmin=5.,finc=1.1,fdec=0.5,alphastart=0.1,fa=0.99,deltatmax=10.,
maxsteps = 10**5):
""" Do a fire relaxation """
alpha = alphastart
deltat = 0.1
pos = self.get_pos_arr(force=True)
v = sp.zeros_like(pos)
self._update_vel(v)
v = self._get_vel_arr()
steps_since_negative = 0
def norm_arr(vec):
return sp.sqrt(sp.sum(vec**2,1))
def unitize_arr(vec):
return ((vec.T)/norm(vec)).T
forces = sp.nan_to_num(sp.array([ [sp.inf,sp.inf]]))
step_num = 0
print "Beginning FIRE Relaxation -- fmax={}".format(fmax)
while max(norm_arr(forces)) > fmax and step_num < maxsteps:
forces = self.forces
power = sp.vdot(forces,v)
print "Step: {}, max_force: {}, power: {}".format(step_num,max(norm_arr(forces)), power)
v = (1.0 - alpha)*v + alpha*(norm_arr(v)*unitize_arr(forces).T).T
if power>0.:
if steps_since_negative > Nmin:
deltat = min(deltat * finc, deltatmax)
alpha = alpha*fa
steps_since_negative += 1
else:
steps_since_negative = 0
deltat = deltat * fdec
v *= 0.
alpha = alphastart
v += forces*deltat
pos += v*deltat
self._update_pos(pos)
step_num += 1
self._update_pos(pos)
self._update_vel(v)
print "Relaxation finished..."
def ConstructFidelityData(Psi, vecs, T, outdir):
"""
Output the fidelity for multi-T simulations by calculating \
the overlap between however many eigenstates we want.
"""
data = []
for i in range(0, len(vecs)):
overlap = abs(sp.vdot(Psi, vecs[i])) ** 2
data.append([T, overlap])
return data
def ConstructFidelityData(Psi, vecs, T, outdir):
"""
Output the fidelity for multi-T simulations by calculating \
the overlap between however many eigenstates we want.
"""
data = []
for i in range(0, len(vecs)):
overlap = abs(sp.vdot(Psi, vecs[i]))**2
data.append([T, overlap])
return data
def CheckNorm(t, nQubits, Psi, Hvecs, eps):
"""
Check for numerical error with normalization.
"""
norm = 0.0
for i in range(0, 2**nQubits):
norm += abs(sp.vdot(Psi, Hvecs[:,i]))**2
if (1.0 - norm) > eps:
print (str((1.0 - norm)) + " (norm error) > " + str(eps) +
" (eps) @ t = " + str(t))
def FIRE(x0,fprime,fmax=0.005,
Nmin=5.,finc=1.1,fdec=0.5,alphastart=0.1,fa=0.99,deltatmax=10.,
maxsteps = 10**5):
Nmin,finc,fdec,alphastart,fa,deltatmax=(5.,1.1,0.5,0.1,0.99,10.)
alpha = alphastart
deltat = 0.1
pos = x0.copy()
v = sp.zeros_like(pos)
steps_since_negative = 0
def norm(vec):
return sp.sqrt(sp.sum(vec**2,1))
def unitize(vec):
return ((vec.T)/norm(vec)).T
forces = fprime(pos)
step_num = 0
while max(norm(forces)) > fmax and step_num < maxsteps:
forces = fprime(pos)
power = sp.vdot(forces,v)
print "Step: {}, max_force: {}, power: {}".format(step_num,max(norm(forces)), power)
v = (1.0 - alpha)*v + alpha*(norm(v)*unitize(forces).T).T
if power>0.:
if steps_since_negative > Nmin:
deltat = min(deltat * finc, deltatmax)
alpha = alpha*fa
steps_since_negative += 1
else:
steps_since_negative = 0
deltat = deltat * fdec
v *= 0.
alpha = alphastart
v += forces*deltat
pos += v*deltat
step_num += 1
return pos
def get_reciprocal(positions, dihedral_atoms):
"""
In dihedral angle calculation, see if angle is the reciprocal or not.
"""
ii, jj, kk, ll = dihedral_atoms
# vector 0->1, 1->2, 2->3 and their normalized cross products:
a = positions[jj] - positions[ii]
b = positions[kk] - positions[jj]
c = positions[ll] - positions[kk]
bxa = sp.cross(b, a)
if sp.vdot(bxa, c) > 0:
return True
else:
return False
def compute_loocv_gmm(variable, model, x, y, ids, K_u, alpha, beta,
log_prop_u):
""" Function that computes the estimation of the loocv for the GMM model with variables ids + variable(i)
Inputs:
model : the GMM model
x,y : the training samples and the corresponding label
ids : the pool of selected variables
variable : the variable to be tested from the set of available variable
K_u : the initial prediction values computed with all the samples
alpha, beta and log_prop_u : constant that are computed outside of the loop to increased speed
Outputs:
loocv_temp : the loocv
Used in GMM.forward_selection()
"""
n = x.shape[0]
ids.append(variable) # Iteratively add one of the remaining variables
Kp = model.predict_gmm(x,
ids=ids)[1] # Predict with all the samples with ids
loocv_temp = 0.0
# Initialization of the temporary loocv
for j in range(n): # Predict the class with the model ids_t
Kloo = Kp[
j, :] + K_u # Initialization of the decision rule for sample "j" #--- Change for only not C---#
c = int(y[j] - 1) # Update of parameter of class c
m = (model.ni[c] * model.mean[c, ids] -
x[j, ids]) * alpha[c] # Update the mean value
xb = x[j, ids] - m # x centered
cov_u = (model.cov[c, ids, :][:, ids] - sp.outer(xb, xb) *
alpha[c]) * beta # Update the covariance matrix
logdet, rcond = safe_logdet(cov_u)
Kloo[c] = logdet - 2 * log_prop_u[c] + sp.vdot(
xb, mylstsq(cov_u, xb.T, rcond)) # Compute the new decision rule
del cov_u, xb, m, c
yloo = sp.argmin(Kloo) + 1
loocv_temp += float(
yloo == y[j]) # Check the correct/incorrect classification rule
ids.pop() # Remove the current variable
return loocv_temp / n # Compute loocv for variable
def build_subspace_matrix(self):
sb_dim = 0
for vs in [
self.vspace_r, self.vspace_l, self.vspace_rp, self.vspace_lp
]:
if vs is not None:
sb_dim = sb_dim + vs.shape[1]
submat = sp.zeros((sb_dim, sb_dim), self.complex_precision)
if self.vspace_r is not None:
print("self.vspace_r.shape = ", self.vspace_r.shape)
if self.vspace_l is not None:
print("self.vspace_l.shape = ", self.vspace_l.shape)
if self.vspace_rp is not None:
print("self.vspace_rp.shape = ", self.vspace_rp.shape)
if self.vspace_lp is not None:
print("self.vspace_lp.shape = ", self.vspace_lp.shape)
print("sb_dim = ", sb_dim)
vi = 0
for vs in [
self.vspace_r, self.vspace_l, self.vspace_rp, self.vspace_lp
]:
if vs is not None:
for ii in range(vs.shape[1]):
wj = 0
for ws in [
self.wspace_r, self.wspace_l, self.wspace_rp,
self.wspace_lp
]:
if ws is not None:
for jj in range(ws.shape[1]):
submat[vi + ii, wj + jj] = sp.vdot(
vs[:, ii], ws[:, jj])
wj = wj + ws.shape[1]
vi = vi + vs.shape[1]
return submat
def gexpmv(A, v, t, anorm, m=None, tol=0.0, w=None, verbose=False, mxstep=500, break_tol=None):
mxreject = 0 #matlab version has this set to 10
delta = 1.2
gamma = 0.9
if break_tol is None:
#break_tol = tol
break_tol = anorm*tol
n = A.shape[0]
if hasattr(A, "matvec"):
matvec = A.matvec
else:
matvec = lambda v: sp.dot(A,v)
if m is None:
m = min(20, n-1)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError("A is not a square matrix")
if m >= n or m <= 0:
raise ValueError("m is invalid")
k1 = 2
mh = m + 2
ibrkflag = 0
mbrkdwn = m
nmult = 0
nreject = 0
nexph = 0
nscale = 0
t_out = abs( t )
tbrkdwn = 0.0
step_min = t_out
step_max = 0.0
nstep = 0
s_error = 0.0
x_error = 0.0
t_now = 0.0
t_new = 0.0
eps = sp.finfo(A.dtype).eps
rndoff = eps*anorm
sgn = sp.sign(t)
if w is None: #allow supplying a starting vector
w = v.copy()
else:
w[:] = v
beta = la.norm(w)
vnorm = beta
hump = beta
#obtain the very first stepsize ...
SQR1 = sqrt( 0.1 )
xm = 1.0/m
p2 = tol*(((m+1)/sp.e)**(m+1)) * sqrt(2.0*sp.pi*(m+1))
t_new = (1.0/anorm) * (p2 / (4.0*beta*anorm))**xm
p1 = 10.0**(round( log10( t_new )-SQR1 )-1)
t_new = trunc( t_new/p1 + 0.55 ) * p1
#step-by-step integration ...
while t_now < t_out:
nstep = nstep + 1
t_step = min( t_out-t_now, t_new )
#initialize Krylov subspace
vs = sp.zeros((m+2,n), A.dtype)
vs[0,:] = w / beta
H = sp.zeros((mh,mh), A.dtype)
#Arnoldi loop ...
for j in range(1,m+1):
nmult = nmult + 1
vs[j,:] = matvec(vs[j-1,:])
for i in range(1,j+1):
#Compute overlaps of new vector Av with all other Kyrlov vectors
#(these are elements of an upper Hessenberg matrix)
hij = sp.vdot(vs[i-1,:], vs[j,:])
vs[j,:] -= hij * vs[i-1,:] #orthogonalize new vector. maybe switch to axpy
H[i-1,j-1] = hij #store matrix element
hj1j = la.norm( vs[j,:] )
#if the orthogonalized Krylov vector is zero, stop!
if hj1j <= break_tol:
if verbose:
print('breakdown: mbrkdwn =',j,' h =',hj1j)
k1 = 0
ibrkflag = 1
mbrkdwn = j
tbrkdwn = t_now
t_step = t_out-t_now
break
H[j,j-1] = hj1j
vs[j,:] *= 1.0/hj1j
if ibrkflag == 0: #if we didn't break down
nmult = nmult + 1
vs[m+1,:] = matvec(vs[m,:])
avnorm = la.norm( vs[m+1,:] )
#Orig: set 1 for the 2-corrected scheme
H[m+1, m] = 1.0
#loop while ireject<mxreject until the tolerance is reached
ireject = 0
#compute w = beta*V*exp(t_step*H)*e1 ...
#First compute expH for a good step size
while True:
nexph = nexph + 1
mx = mbrkdwn + k1 #max(mx) = m+2
#irreducible rational Pade approximation. scipy's implementation automatically chooses an order
expH = la.expm(sgn * t_step * H[:mx,:mx])
#nscale = nscale + ns #don't have this info
#local error estimation
if k1 == 0: #if breakdown has occured (the Krylov subspace is complete)
err_loc = tol #matlab uses break_tol
else:
p1 = abs( expH[m,0] ) * beta #wsp(iexph+m)
p2 = abs( expH[m+1,0] ) * beta * avnorm #avnorm is defined when k1 != 0
if p1 > 10.0*p2:
err_loc = p2
xm = 1.0/m
elif p1 > p2:
err_loc = (p1*p2)/(p1-p2)
xm = 1.0/m
else:
err_loc = p1
xm = 1.0/(m-1)
#reject the step-size if the error is not acceptable ...
if ( (k1 != 0) and (err_loc > delta*t_step*tol) and
(mxreject == 0 or ireject < mxreject) ):
t_old = t_step
t_step = gamma * t_step * (t_step*tol/err_loc)**xm
p1 = 10.0**(round( log10( t_step )-SQR1 )-1)
t_step = trunc( t_step/p1 + 0.55 ) * p1
if verbose:
print('t_step = ',t_old)
print('err_loc = ',err_loc)
print('err_required = ',delta*t_old*tol)
print('stepsize rejected, stepping down to: ',t_step)
ireject = ireject + 1
nreject = nreject + 1
if mxreject != 0 and ireject > mxreject:
print("Failure in gexpmv: ---")
print("The requested tolerance is too high.")
print("Rerun with a smaller value.")
iflag = 2
return
else:
break #step size OK (happens after a breakdown)
#now update w = beta*V*exp(t_step*H)*e1 and the hump ...
mx = mbrkdwn + max( 0, k1-1 ) #max(mx) = m+1
w = beta * vs[:mx,:].T.dot(expH[:mx,0])
beta = la.norm(w)
hump = max( hump, beta )
#suggested value for the next stepsize ...
t_new = gamma * t_step * (t_step*tol/err_loc)**xm
p1 = 10.0**(round( log10( t_new )-SQR1 )-1)
t_new = trunc( t_new/p1 + 0.55 ) * p1
err_loc = max( err_loc, rndoff )
#update the time covered ...
t_now = t_now + t_step
#display and keep some information ...
if verbose:
print('integration ', nstep, ' ---------------------------------')
#print('scale-square = ', nscale)
print('step_size = ', t_step)
print('err_loc = ',err_loc)
print('next_step = ',t_new)
step_min = min( step_min, t_step )
step_max = max( step_max, t_step )
s_error = s_error + err_loc
x_error = max( x_error, err_loc )
if mxstep == 0 or nstep < mxstep:
continue
iflag = 1
break
return w, nstep < mxstep, nstep, ibrkflag==1, mbrkdwn
def vlen(v):
return math.sqrt(scipy.vdot(v, v))
B = 1
r = 0.06
T = 1
K = 50
def f(x, strike):
return max(x - strike, 0)
m = scipy.array([[S * (1 - down), B * scipy.exp(r * T)],
[S * (1 + up), B * scipy.exp(r * T)]])
payoff = scipy.array([f(S * (1 - down), K), f(S * (1 + up), K)])
portfolio = scipy.linalg.solve(m, payoff)
value = scipy.vdot(portfolio, scipy.array([S, B]))
print('portfolio: phi=', portfolio[0], 'psi=', portfolio[1], '\n')
print('derivative value: ', value, '\n')
## NAME: singleperiod.py
## USAGE: From shell prompt: python3 singleperiod.py
## or within interactive python3 environment, import filename
## REQUIRED ARGUMENTS: none
## OPTIONS: none
## DESCRIPTION: Set up and solve for the replicating portfolio
## and the value of the corresponding derivative security in a
## single period binomial model.
## DIAGNOSTICS: none
## CONFIGURATION AND ENVIRONMENT:
## DEPENDENCIES: Scientific python
def vlen(v): return math.sqrt(scipy.vdot(v, v))
def gexpmv(A,
v,
t,
anorm,
m=None,
tol=0.0,
w=None,
verbose=False,
itrace=0,
mxstep=500,
break_tol=None):
mxreject = 0
delta = 1.2
gamma = 0.9
if break_tol is None:
#break_tol = tol
break_tol = anorm * tol
n = A.shape[0]
if hasattr(A, "matvec"):
matvec = A.matvec
else:
matvec = lambda v: sp.dot(A, v)
if m is None:
m = min(20, n - 1)
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError("A is not a square matrix")
if m >= n or m <= 0:
raise ValueError("m is invalid")
k1 = 2
mh = m + 2
ibrkflag = 0
mbrkdwn = m
nmult = 0
nreject = 0
nexph = 0
nscale = 0
t_out = abs(t)
tbrkdwn = 0.0
step_min = t_out
step_max = 0.0
nstep = 0
s_error = 0.0
x_error = 0.0
t_now = 0.0
t_new = 0.0
avnorm = 0.0 #I think the EXPOKIT source relied on this being initialized to zero by the compiler
#pretty sure this just computes machine epsilon
eps = 0.0
p1 = 4.0 / 3.0
while eps == 0.0:
p2 = p1 - 1.0
p3 = 3 * p2
eps = abs(p3 - 1.0)
if tol <= eps:
tol = sqrt(eps)
rndoff = eps * anorm
sgn = copysign(1.0, t)
if w is None: #allow supplying a starting vector
w = v.copy()
else:
w[:] = v
beta = la.norm(w)
vnorm = beta
hump = beta
#obtain the very first stepsize ...
SQR1 = sqrt(0.1)
xm = 1.0 / m
p2 = tol * (((m + 1) / sp.e)**(m + 1)) * sqrt(2.0 * sp.pi * (m + 1))
t_new = (1.0 / anorm) * (p2 / (4.0 * beta * anorm))**xm
p1 = 10.0**(round(log10(t_new) - SQR1) - 1)
t_new = trunc(t_new / p1 + 0.55) * p1
#step-by-step integration ...
while t_now < t_out:
nstep = nstep + 1
t_step = min(t_out - t_now, t_new)
#initialize Krylov subspace
vs = sp.zeros((m + 2, n), A.dtype)
vs[0, :] = w / beta
H = sp.zeros((mh, mh), A.dtype)
#Arnoldi loop ...
for j in range(1, m + 1):
nmult = nmult + 1
vs[j, :] = matvec(vs[j - 1, :])
for i in range(1, j + 1):
#Compute overlaps of new vector Av with all other Kyrlov vectors
#(these are elements of an upper Hessenberg matrix)
hij = sp.vdot(vs[i - 1, :], vs[j, :])
vs[j, :] -= hij * vs[
i - 1, :] #orthogonalize new vector. maybe switch to axpy
H[i - 1, j - 1] = hij #store matrix element
hj1j = la.norm(vs[j, :])
#if the orthogonalized Krylov vector is zero, stop!
if hj1j <= break_tol:
print('breakdown: mbrkdwn =', j, ' h =', hj1j)
k1 = 0
ibrkflag = 1
mbrkdwn = j
tbrkdwn = t_now
t_step = t_out - t_now
break
H[j, j - 1] = hj1j
vs[j, :] *= 1.0 / hj1j
if ibrkflag == 0: #if we didn't break down
nmult = nmult + 1
vs[m + 1, :] = matvec(vs[m, :])
avnorm = la.norm(vs[m + 1, :])
#Orig: set 1 for the 2-corrected scheme
H[m + 1, m] = 1.0
#loop while ireject<mxreject until the tolerance is reached
ireject = 0
#compute w = beta*V*exp(t_step*H)*e1 ...
#First compute expH for a good step size
while True:
nexph = nexph + 1
mx = mbrkdwn + k1 #max(mx) = m+2
#irreducible rational Pade approximation. scipy's implementation automatically chooses an order
expH = la.expm(sgn * t_step * H[:mx, :mx])
#nscale = nscale + ns #don't have this info
#local error estimation
if k1 == 0:
err_loc = tol
else:
p1 = abs(expH[m, 0]) * beta #wsp(iexph+m)
p2 = abs(
expH[m + 1, 0]
) * beta * avnorm #FIXME: avnorm is not always defined....
if p1 > 10.0 * p2:
err_loc = p2
xm = 1.0 / m
elif p1 > p2:
err_loc = (p1 * p2) / (p1 - p2)
xm = 1.0 / m
else:
err_loc = p1
xm = 1.0 / (m - 1)
#reject the step-size if the error is not acceptable ...
if ((k1 != 0) and (err_loc > delta * t_step * tol)
and (mxreject == 0 or ireject < mxreject)):
t_old = t_step
t_step = gamma * t_step * (t_step * tol / err_loc)**xm
p1 = 10.0**(round(log10(t_step) - SQR1) - 1)
t_step = trunc(t_step / p1 + 0.55) * p1
if verbose:
print('t_step = ', t_old)
print('err_loc = ', err_loc)
print('err_required = ', delta * t_old * tol)
print('stepsize rejected, stepping down to: ', t_step)
ireject = ireject + 1
nreject = nreject + 1
if mxreject != 0 and ireject > mxreject:
print("Failure in ZGEXPV: ---")
print("The requested tolerance is too high.")
print("Rerun with a smaller value.")
iflag = 2
return
else:
break #step size OK
#now update w = beta*V*exp(t_step*H)*e1 and the hump ...
mx = mbrkdwn + max(0, k1 - 1) #max(mx) = m+1
w = beta * vs[:mx, :].T.dot(expH[:mx, 0])
beta = la.norm(w)
hump = max(hump, beta)
#suggested value for the next stepsize ...
t_new = gamma * t_step * (t_step * tol / err_loc)**xm
p1 = 10.0**(round(log10(t_new) - SQR1) - 1)
t_new = trunc(t_new / p1 + 0.55) * p1
err_loc = max(err_loc, rndoff)
#update the time covered ...
t_now = t_now + t_step
#display and keep some information ...
if itrace != 0:
print('integration ', nstep, ' ---------------------------------')
#print('scale-square = ', nscale)
print('step_size = ', t_step)
print('err_loc = ', err_loc)
print('next_step = ', t_new)
step_min = min(step_min, t_step)
step_max = max(step_max, t_step)
s_error = s_error + err_loc
x_error = max(x_error, err_loc)
if mxstep == 0 or nstep < mxstep:
continue
iflag = 1
break
return w, nstep < mxstep, nstep, ibrkflag == 1, mbrkdwn