def predictor_func(ts, ys):
pdts = [f(ts) for f in p_dtn_funcs]
dynp1 = dy_funcs[0](ts, ys)
yn = y_funcs[1](ts, ys)
ynm1 = y_funcs[2](ts, ys)
ynm2 = y_funcs[3](ts, ys)
return ode.ibdf3_step(dynp1, pdts[0], yn, pdts[1], ynm1,
pdts[2], ynm2)
def predictor_func(ts, ys):
pdts = [f(ts) for f in p_dtn_funcs]
dynp1 = dy_funcs[0](ts, ys)
yn = y_funcs[1](ts, ys)
ynm1 = y_funcs[2](ts, ys)
ynm2 = y_funcs[3](ts, ys)
return ode.ibdf3_step(dynp1, pdts[0], yn, pdts[1], ynm1, pdts[2],
ynm2)
def generate_predictor_scheme(pt_infos, predictor_name, symbolic=None):
# Extract symbolic expressions from what we're given.
if symbolic is not None:
try:
symb_exact, symb_F = symbolic
except TypeError: # not iterable
symb_exact = symbolic
Sy = sympy.symbols('y', real=True)
symb_dy = sympy.diff(symb_exact, St, 1).subs(symb_exact, Sy)
symb_F = sympy.diff(symb_dy, Sy).subs(Sy, symb_exact)
dydt_func = sympy.lambdify(St, sympy.diff(symb_exact, St, 1))
f_y = sympy.lambdify(St, symb_exact)
else:
dydt_func = None
f_y = None
# To cancel errors we need the final step to be at t_np1
# assert pt_infos[0].time == 0
n_hist = len(pt_infos) + 5
# Create symbolic and python functions of (corrector) dts and ts
# respectively that give the appropriate dts for this predictor.
p_dtns = []
p_dtn_funcs = []
for pt1, pt2 in zip(pt_infos[:-1], pt_infos[1:]):
p_dtns.append(sum_dts(pt1.time, pt2.time))
p_dtn_funcs.append(generate_p_dt_func(p_dtns[-1]))
# For each time point the predictor uses construct symbolic error
# estimates for the requested estimate at that point and python
# functions to calculate the value of the estimate.
y_errors = map(ptinfo2yerr, pt_infos)
y_funcs = map(par(ptinfo2yfunc, y_of_t_func=f_y), pt_infos)
dy_errors = map(ptinfo2dyerr, pt_infos)
dy_funcs = map(par(ptinfo2dyfunc, dydt_func=dydt_func), pt_infos)
# Construct errors and function for the predictor
# ============================================================
if predictor_name == "ebdf2" or predictor_name == "wrong step ebdf2":
if predictor_name == "wrong step ebdf2":
temp_p_dtnm1 = p_dtns[1] + Sdts[0]
else:
temp_p_dtnm1 = p_dtns[1]
y_np1_p_expr = y_np1_exact - (
# Natural ebdf2 lte:
ebdf2_lte(p_dtns[0], temp_p_dtnm1, Sdddynph)
# error due to approximation to derivative at tn:
+ ode.ebdf2_step(p_dtns[0], 0, dy_errors[1], p_dtns[1], 0)
# error due to approximation to yn
+ ode.ebdf2_step(p_dtns[0], y_errors[1], 0, p_dtns[1], 0)
# error due to approximation to ynm1 (typically zero)
+ ode.ebdf2_step(p_dtns[0], 0, 0, p_dtns[1], y_errors[2]))
def predictor_func(ts, ys):
dtn = p_dtn_funcs[0](ts)
yn = y_funcs[1](ts, ys)
dyn = dy_funcs[1](ts, ys)
dtnm1 = p_dtn_funcs[1](ts)
ynm1 = y_funcs[2](ts, ys)
return ode.ebdf2_step(dtn, yn, dyn, dtnm1, ynm1)
elif predictor_name == "ab2":
y_np1_p_expr = y_np1_exact - (
# Natural ab2 lte:
ab2_lte(p_dtns[0], p_dtns[1], Sdddynph)
# error due to approximation to derivative at tn
+ ode.ab2_step(p_dtns[0], 0, dy_errors[1], p_dtns[1], 0)
# error due to approximation to derivative at tnm1
+ ode.ab2_step(p_dtns[0], 0, 0, p_dtns[1], dy_errors[2])
# error due to approximation to yn
+ ode.ab2_step(p_dtns[0], y_errors[1], 0, p_dtns[1], 0))
def predictor_func(ts, ys):
dtn = p_dtn_funcs[0](ts)
yn = y_funcs[1](ts, ys)
dyn = dy_funcs[1](ts, ys)
dtnm1 = p_dtn_funcs[1](ts)
dynm1 = dy_funcs[2](ts, ys)
return ode.ab2_step(dtn, yn, dyn, dtnm1, dynm1)
elif predictor_name == "ibdf2":
y_np1_p_expr = y_np1_exact - (
# Natural bdf2 lte:
bdf2_lte(p_dtns[0], p_dtns[1], Sdddynph)
# error due to approximation to derivative at tnp1
+ ode.ibdf2_step(p_dtns[0], 0, dy_errors[0], p_dtns[1], 0)
# errors due to approximations to y at tn and tnm1
+ ode.ibdf2_step(p_dtns[0], y_errors[1], 0, p_dtns[1], 0)
+ ode.ibdf2_step(p_dtns[0], 0, 0, p_dtns[1], y_errors[2]))
def predictor_func(ts, ys):
pdts = [f(ts) for f in p_dtn_funcs]
dynp1 = dy_funcs[0](ts, ys)
yn = y_funcs[1](ts, ys)
ynm1 = y_funcs[2](ts, ys)
tsin = [t_at_time_point(ts, pt.time)
for pt in reversed(pt_infos[1:])]
ysin = [ys[-(pt.time+1)] for pt in reversed(pt_infos[1:])]
dt = pdts[0]
return ode.ibdf2_step(pdts[0], yn, dynp1, pdts[1], ynm1)
elif predictor_name == "ibdf3":
y_np1_p_expr = y_np1_exact - (
# Natural bdf2 lte:
bdf3_lte(p_dtns[0], p_dtns[1], Sdddynph)
# error due to approximation to derivative at tnp1
+ ode.ibdf3_step(
dy_errors[0],
p_dtns[0],
0,
p_dtns[1],
0,
p_dtns[2],
0)
# errors due to approximations to y at tn, tnm1, tnm2
+ ode.ibdf3_step(
0,
p_dtns[0],
y_errors[1],
p_dtns[1],
0,
p_dtns[2],
0)
+ ode.ibdf3_step(
0,
p_dtns[0],
0,
p_dtns[1],
y_errors[2],
p_dtns[2],
0)
+ ode.ibdf3_step(0, p_dtns[0], 0, p_dtns[1], 0, p_dtns[2], y_errors[3]))
def predictor_func(ts, ys):
pdts = [f(ts) for f in p_dtn_funcs]
dynp1 = dy_funcs[0](ts, ys)
yn = y_funcs[1](ts, ys)
ynm1 = y_funcs[2](ts, ys)
ynm2 = y_funcs[3](ts, ys)
return ode.ibdf3_step(dynp1, pdts[0], yn, pdts[1], ynm1,
pdts[2], ynm2)
elif predictor_name == "ebdf3":
y_np1_p_expr = y_np1_exact - (
# Natural ebdf3 lte error:
ebdf3_lte(p_dtns[0], p_dtns[1], p_dtns[2], Sdddynph)
# error due to approximation to derivative at tn
+ ode.ebdf3_step(
p_dtns[0],
0,
dy_errors[1],
p_dtns[1],
0,
p_dtns[2],
0)
# errors due to approximation to y at tn, tnm1, tnm2
+ ode.ebdf3_step(p_dtns[0], y_errors[1], 0,
p_dtns[1], 0, p_dtns[2], 0)
+ ode.ebdf3_step(p_dtns[0], 0, 0,
p_dtns[1], y_errors[2], p_dtns[2], 0)
+ ode.ebdf3_step(p_dtns[0], 0, 0,
p_dtns[1], 0, p_dtns[2], y_errors[3]))
def predictor_func(ts, ys):
pdts = [f(ts) for f in p_dtn_funcs]
dyn = dy_funcs[1](ts, ys)
yn = y_funcs[1](ts, ys)
ynm1 = y_funcs[2](ts, ys)
ynm2 = y_funcs[3](ts, ys)
return ode.ebdf3_step(pdts[0], yn, dyn,
pdts[1], ynm1,
pdts[2], ynm2)
elif predictor_name == "use exact dddy":
symb_dddy = sympy.diff(symb_exact, St, 3)
f_dddy = sympy.lambdify(St, symb_dddy)
print symb_exact
print "dddy =", symb_dddy
# "Predictor" is just exactly dddy, error forumlated so that matrix
# turns out right. Calculate at one before last point (same as lte
# "location").
def predictor_func(ts, ys):
tn = t_at_time_point(ts, pt_infos[1].time)
return f_dddy(tn)
y_np1_p_expr = Sdddynph
elif predictor_name == "use exact Fddy":
symb_ddy = sympy.diff(symb_exact, St, 2)
f_Fddy = sympy.lambdify(St, symb_F * symb_ddy)
print "Fddy =", symb_F * symb_ddy
# "Predictor" is just exactly dddy, error forumlated so that matrix
# turns out right. Calculate at one before last point (same as lte
# "location").
def predictor_func(ts, ys):
tn = t_at_time_point(ts, pt_infos[1].time)
return f_Fddy(tn)
y_np1_p_expr = SFddynph
else:
raise ValueError("Unrecognised predictor name " + predictor_name)
return predictor_func, y_np1_p_expr
def generate_predictor_scheme(pt_infos, predictor_name, symbolic=None):
# Extract symbolic expressions from what we're given.
if symbolic is not None:
try:
symb_exact, symb_F = symbolic
except TypeError: # not iterable
symb_exact = symbolic
Sy = sympy.symbols('y', real=True)
symb_dy = sympy.diff(symb_exact, St, 1).subs(symb_exact, Sy)
symb_F = sympy.diff(symb_dy, Sy).subs(Sy, symb_exact)
dydt_func = sympy.lambdify(St, sympy.diff(symb_exact, St, 1))
f_y = sympy.lambdify(St, symb_exact)
else:
dydt_func = None
f_y = None
# To cancel errors we need the final step to be at t_np1
# assert pt_infos[0].time == 0
n_hist = len(pt_infos) + 5
# Create symbolic and python functions of (corrector) dts and ts
# respectively that give the appropriate dts for this predictor.
p_dtns = []
p_dtn_funcs = []
for pt1, pt2 in zip(pt_infos[:-1], pt_infos[1:]):
p_dtns.append(sum_dts(pt1.time, pt2.time))
p_dtn_funcs.append(generate_p_dt_func(p_dtns[-1]))
# For each time point the predictor uses construct symbolic error
# estimates for the requested estimate at that point and python
# functions to calculate the value of the estimate.
y_errors = map(ptinfo2yerr, pt_infos)
y_funcs = map(par(ptinfo2yfunc, y_of_t_func=f_y), pt_infos)
dy_errors = map(ptinfo2dyerr, pt_infos)
dy_funcs = map(par(ptinfo2dyfunc, dydt_func=dydt_func), pt_infos)
# Construct errors and function for the predictor
# ============================================================
if predictor_name == "ebdf2" or predictor_name == "wrong step ebdf2":
if predictor_name == "wrong step ebdf2":
temp_p_dtnm1 = p_dtns[1] + Sdts[0]
else:
temp_p_dtnm1 = p_dtns[1]
y_np1_p_expr = y_np1_exact - (
# Natural ebdf2 lte:
ebdf2_lte(p_dtns[0], temp_p_dtnm1, Sdddynph)
# error due to approximation to derivative at tn:
+ ode.ebdf2_step(p_dtns[0], 0, dy_errors[1], p_dtns[1], 0)
# error due to approximation to yn
+ ode.ebdf2_step(p_dtns[0], y_errors[1], 0, p_dtns[1], 0)
# error due to approximation to ynm1 (typically zero)
+ ode.ebdf2_step(p_dtns[0], 0, 0, p_dtns[1], y_errors[2]))
def predictor_func(ts, ys):
dtn = p_dtn_funcs[0](ts)
yn = y_funcs[1](ts, ys)
dyn = dy_funcs[1](ts, ys)
dtnm1 = p_dtn_funcs[1](ts)
ynm1 = y_funcs[2](ts, ys)
return ode.ebdf2_step(dtn, yn, dyn, dtnm1, ynm1)
elif predictor_name == "ab2":
y_np1_p_expr = y_np1_exact - (
# Natural ab2 lte:
ab2_lte(p_dtns[0], p_dtns[1], Sdddynph)
# error due to approximation to derivative at tn
+ ode.ab2_step(p_dtns[0], 0, dy_errors[1], p_dtns[1], 0)
# error due to approximation to derivative at tnm1
+ ode.ab2_step(p_dtns[0], 0, 0, p_dtns[1], dy_errors[2])
# error due to approximation to yn
+ ode.ab2_step(p_dtns[0], y_errors[1], 0, p_dtns[1], 0))
def predictor_func(ts, ys):
dtn = p_dtn_funcs[0](ts)
yn = y_funcs[1](ts, ys)
dyn = dy_funcs[1](ts, ys)
dtnm1 = p_dtn_funcs[1](ts)
dynm1 = dy_funcs[2](ts, ys)
return ode.ab2_step(dtn, yn, dyn, dtnm1, dynm1)
elif predictor_name == "ibdf2":
y_np1_p_expr = y_np1_exact - (
# Natural bdf2 lte:
bdf2_lte(p_dtns[0], p_dtns[1], Sdddynph)
# error due to approximation to derivative at tnp1
+ ode.ibdf2_step(p_dtns[0], 0, dy_errors[0], p_dtns[1], 0)
# errors due to approximations to y at tn and tnm1
+ ode.ibdf2_step(p_dtns[0], y_errors[1], 0, p_dtns[1], 0) +
ode.ibdf2_step(p_dtns[0], 0, 0, p_dtns[1], y_errors[2]))
def predictor_func(ts, ys):
pdts = [f(ts) for f in p_dtn_funcs]
dynp1 = dy_funcs[0](ts, ys)
yn = y_funcs[1](ts, ys)
ynm1 = y_funcs[2](ts, ys)
tsin = [
t_at_time_point(ts, pt.time) for pt in reversed(pt_infos[1:])
]
ysin = [ys[-(pt.time + 1)] for pt in reversed(pt_infos[1:])]
dt = pdts[0]
return ode.ibdf2_step(pdts[0], yn, dynp1, pdts[1], ynm1)
elif predictor_name == "ibdf3":
y_np1_p_expr = y_np1_exact - (
# Natural bdf2 lte:
bdf3_lte(p_dtns[0], p_dtns[1], Sdddynph)
# error due to approximation to derivative at tnp1
+ ode.ibdf3_step(dy_errors[0], p_dtns[0], 0, p_dtns[1], 0,
p_dtns[2], 0)
# errors due to approximations to y at tn, tnm1, tnm2
+ ode.ibdf3_step(0, p_dtns[0], y_errors[1], p_dtns[1], 0,
p_dtns[2], 0) +
ode.ibdf3_step(0, p_dtns[0], 0, p_dtns[1], y_errors[2], p_dtns[2],
0) + ode.ibdf3_step(0, p_dtns[0], 0, p_dtns[1], 0,
p_dtns[2], y_errors[3]))
def predictor_func(ts, ys):
pdts = [f(ts) for f in p_dtn_funcs]
dynp1 = dy_funcs[0](ts, ys)
yn = y_funcs[1](ts, ys)
ynm1 = y_funcs[2](ts, ys)
ynm2 = y_funcs[3](ts, ys)
return ode.ibdf3_step(dynp1, pdts[0], yn, pdts[1], ynm1, pdts[2],
ynm2)
elif predictor_name == "ebdf3":
y_np1_p_expr = y_np1_exact - (
# Natural ebdf3 lte error:
ebdf3_lte(p_dtns[0], p_dtns[1], p_dtns[2], Sdddynph)
# error due to approximation to derivative at tn
+ ode.ebdf3_step(p_dtns[0], 0, dy_errors[1], p_dtns[1], 0,
p_dtns[2], 0)
# errors due to approximation to y at tn, tnm1, tnm2
+ ode.ebdf3_step(p_dtns[0], y_errors[1], 0, p_dtns[1], 0,
p_dtns[2], 0) +
ode.ebdf3_step(p_dtns[0], 0, 0, p_dtns[1], y_errors[2], p_dtns[2],
0) + ode.ebdf3_step(p_dtns[0], 0, 0, p_dtns[1], 0,
p_dtns[2], y_errors[3]))
def predictor_func(ts, ys):
pdts = [f(ts) for f in p_dtn_funcs]
dyn = dy_funcs[1](ts, ys)
yn = y_funcs[1](ts, ys)
ynm1 = y_funcs[2](ts, ys)
ynm2 = y_funcs[3](ts, ys)
return ode.ebdf3_step(pdts[0], yn, dyn, pdts[1], ynm1, pdts[2],
ynm2)
elif predictor_name == "use exact dddy":
symb_dddy = sympy.diff(symb_exact, St, 3)
f_dddy = sympy.lambdify(St, symb_dddy)
print symb_exact
print "dddy =", symb_dddy
# "Predictor" is just exactly dddy, error forumlated so that matrix
# turns out right. Calculate at one before last point (same as lte
# "location").
def predictor_func(ts, ys):
tn = t_at_time_point(ts, pt_infos[1].time)
return f_dddy(tn)
y_np1_p_expr = Sdddynph
elif predictor_name == "use exact Fddy":
symb_ddy = sympy.diff(symb_exact, St, 2)
f_Fddy = sympy.lambdify(St, symb_F * symb_ddy)
print "Fddy =", symb_F * symb_ddy
# "Predictor" is just exactly dddy, error forumlated so that matrix
# turns out right. Calculate at one before last point (same as lte
# "location").
def predictor_func(ts, ys):
tn = t_at_time_point(ts, pt_infos[1].time)
return f_Fddy(tn)
y_np1_p_expr = SFddynph
else:
raise ValueError("Unrecognised predictor name " + predictor_name)
return predictor_func, y_np1_p_expr