def ptest(methods,ivps,tols=[1.e-1,1.e-2,1.e-4,1.e-6]):
"""
Runs a performance test, integrating a set of problems with a set
of methods using a sequence of error tolerances. Creates a plot
of the error achieved versus the amount of work done (number of
function evaluations) for each method.
**Input**:
* methods -- a list of ODEsolver instances
Note that all methods must have error estimators.
* ivps -- a list of IVP instances
* tols -- a specified list of error tolerances (optional)
**Example**::
import runge_kutta_method as rk
from ivp import *
bs5=rk.loadRKM('BS5')
myivp=load_ivp('nlsin')
work,err=ptest(bs5,myivp)
"""
pl.clf(); pl.draw(); pl.hold(True)
# In case just one method is passed in (and not as a list):
if not isinstance(methods,list): methods=[methods]
if not isinstance(ivps,list): ivps=[ivps]
err=np.ones([len(methods),len(tols)])
work=np.zeros([len(methods),len(tols)])
for ivp in ivps:
print "solving problem %s" % ivp
try:
exsol = ivp.exact(ivp.T)
except:
bs5=rk.loadRKM('BS5') #Use Bogacki-Shampine RK for fine solution
lowtol=min(tols)/100.
print 'solving for "exact" solution with tol= '+str(lowtol)
t,u=bs5(ivp,errtol=lowtol,dt=ivp.dt0)
print 'done'
exsol = u[-1].copy()
for imeth,method in enumerate(methods):
print 'Solving with method '+method.name
workperstep = len(method)-method.is_FSAL()
for jtol,tol in enumerate(tols):
t,u,rej,dt=method(ivp,errtol=tol,dt=ivp.dt0,diagnostics=True,controllertype='P')
print str(rej)+' rejected steps'
err[imeth,jtol]*= np.max(np.abs(u[-1]-exsol))
#FSAL methods save on accepted steps, but not on rejected:
work[imeth,jtol]+= len(t)*workperstep+rej*len(method)
for imeth,method in enumerate(methods):
for jtol,tol in enumerate(tols):
err[imeth,jtol]=err[imeth,jtol]**(1./len(ivps))
for imeth,method in enumerate(methods):
pl.semilogy(work[imeth,:],err[imeth,:],label=method.name,linewidth=3)
pl.xlabel('Function evaluations')
pl.ylabel('Error at $t_{final}$')
pl.legend(loc='best')
pl.hold(False)
pl.draw()
return work,err
def ctest(methods,ivp,grids=[20,40,80,160,320,640],verbosity=0,parallel=False):
"""
Runs a convergence test, integrating a single initial value problem
using a sequence of fixed step sizes and a set of methods.
Creates a plot of the resulting errors versus step size for each method.
**Inputs**:
- methods -- a list of ODEsolver instances
- ivp -- an IVP instance
- grids -- a list of grid sizes for integration.
optional; defaults to [20,40,80,160,320,640]
- parallel -- to exploit possible parallelization (optional)
**Example**::
>>> import nodepy.runge_kutta_method as rk
>>> from nodepy.ivp import load_ivp
>>> rk44=rk.loadRKM('RK44')
>>> myivp=load_ivp('nlsin')
>>> work, err=ctest(rk44,myivp)
TODO:
- Option to plot versus f-evals or dt
"""
import matplotlib.pyplot as pl
pl.clf(); pl.hold(True)
# In case just one method is passed in (and not as a list):
if not isinstance(methods,list): methods=[methods]
err=[]
try:
exsol = ivp.exact(ivp.T)
except:
bs5=rk.loadRKM('BS5')
bigN=grids[-1]*4
if verbosity>0: print('solving on fine grid with {} points'.format(bigN))
t,u=bs5(ivp,N=bigN)
if verbosity>0: print('done')
exsol = u[-1] + 0.
for method in methods:
if verbosity>0: print("solving with {}".format(method.name))
err0=[]
for i,N in enumerate(grids):
t,u=method(ivp,N=N)
err0.append(np.linalg.norm(u[-1]-exsol))
err.append(err0)
work=np.array([grid*len(method) for grid in grids])
if parallel:
speedup = len(method)/float(method.num_seq_dep_stages())
work = work/speedup
pl.loglog(work,err0,label=method.name,linewidth=3)
pl.xlabel('Function evaluations')
pl.ylabel('Error at $t_{final}$')
pl.legend(loc='best')
pl.hold(False)
pl.draw()
return work, err
def ctest(methods, ivp, grids=[20, 40, 80, 160, 320, 640]):
"""
Runs a convergence test, integrating a single initial value problem
using a sequence of fixed step sizes and a set of methods.
Creates a plot of the resulting errors versus step size for each method.
**Inputs**:
- methods -- a list of ODEsolver instances
- ivp -- an IVP instance
- grids -- a list of grid sizes for integration.
optional; defaults to [20,40,80,160,320,640]
**Example**::
import runge_kutta_method as rk
from ivp import *
rk44=rk.loadRKM('RK44')
myivp=load_ivp('nlsin')
err=ctest(rk44,myivp)
TODO:
- Option to plot versus f-evals or dt
"""
pl.clf()
pl.hold(True)
# In case just one method is passed in (and not as a list):
if not isinstance(methods, list): methods = [methods]
err = []
try:
exsol = ivp.exact(ivp.T)
except:
bs5 = rk.loadRKM('BS5')
bigN = grids[-1] * 4
print 'solving on fine grid with ' + str(bigN) + ' points'
t, u = bs5(ivp, N=bigN)
print 'done'
exsol = u[-1].copy()
for method in methods:
print "solving with %s" % method.name
err0 = []
for i, N in enumerate(grids):
t, u = method(ivp, N=N)
err0.append(np.linalg.norm(u[-1] - exsol))
err.append(err0)
work = [grid * len(method) for grid in grids]
pl.loglog(work, err0, label=method.name, linewidth=3)
pl.xlabel('Function evaluations')
pl.ylabel('Error at $t_{final}$')
pl.legend(loc='best')
pl.hold(False)
pl.draw()
return err
def ctest(methods,ivp,grids=[20,40,80,160,320,640]):
"""
Runs a convergence test, integrating a single initial value problem
using a sequence of fixed step sizes and a set of methods.
Creates a plot of the resulting errors versus step size for each method.
**Inputs**:
- methods -- a list of ODEsolver instances
- ivp -- an IVP instance
- grids -- a list of grid sizes for integration.
optional; defaults to [20,40,80,160,320,640]
**Example**::
import runge_kutta_method as rk
from ivp import *
rk44=rk.loadRKM('RK44')
myivp=load_ivp('nlsin')
err=ctest(rk44,myivp)
TODO:
- Option to plot versus f-evals or dt
"""
pl.clf(); pl.hold(True)
# In case just one method is passed in (and not as a list):
if not isinstance(methods,list): methods=[methods]
err=[]
try:
exsol = ivp.exact(ivp.T)
except:
bs5=rk.loadRKM('BS5')
bigN=grids[-1]*4
print 'solving on fine grid with '+str(bigN)+' points'
t,u=bs5(ivp,N=bigN)
print 'done'
exsol = u[-1].copy()
for method in methods:
print "solving with %s" % method.name
err0=[]
for i,N in enumerate(grids):
t,u=method(ivp,N=N)
err0.append(np.linalg.norm(u[-1]-exsol))
err.append(err0)
work=[grid*len(method) for grid in grids]
pl.loglog(work,err0,label=method.name,linewidth=3)
pl.xlabel('Function evaluations')
pl.ylabel('Error at $t_{final}$')
pl.legend(loc='best')
pl.hold(False)
pl.draw()
return err
def ptest(methods,
ivps,
tols=[1.e-1, 1.e-2, 1.e-4, 1.e-6],
verbosity=0,
parallel=False):
"""
Runs a performance test, integrating a set of problems with a set
of methods using a sequence of error tolerances. Creates a plot
of the error achieved versus the amount of work done (number of
function evaluations) for each method.
**Input**:
* methods -- a list of ODEsolver instances
Note that all methods must have error estimators.
* ivps -- a list of IVP instances
* tols -- a specified list of error tolerances (optional)
* parallel -- to exploit possible parallelization (optional)
**Example**::
>>> import nodepy.runge_kutta_method as rk
>>> from nodepy.ivp import load_ivp
>>> bs5=rk.loadRKM('BS5')
>>> myivp=load_ivp('nlsin')
>>> work,err=ptest(bs5,myivp)
"""
import matplotlib.pyplot as plt
plt.clf()
plt.draw()
# In case just one method is passed in (and not as a list):
if not isinstance(methods, list): methods = [methods]
if not isinstance(ivps, list): ivps = [ivps]
err = np.ones([len(methods), len(tols)])
work = np.zeros([len(methods), len(tols)])
for ivp in ivps:
if verbosity > 0: print("solving problem {}".format(ivp))
try:
exsol = ivp.exact(ivp.T)
except:
bs5 = rk.loadRKM('BS5') #Use Bogacki-Shampine RK for fine solution
lowtol = min(tols) / 100.
if verbosity > 0:
print(
'solving for "exact" solution with tol= {}'.format(lowtol))
t, u = bs5(ivp, errtol=lowtol, dt=ivp.dt0)
if verbosity > 0: print('done')
exsol = u[-1] + 0.
for imeth, method in enumerate(methods):
if verbosity > 0:
print('Solving with method {}'.format(method.name))
if parallel:
speedup = len(method) / float(method.num_seq_dep_stages())
else:
speedup = 1.
workperstep = len(method) - method.is_FSAL()
for jtol, tol in enumerate(tols):
t, u, rej, dt, errhist = method(ivp,
errtol=tol,
dt=ivp.dt0,
diagnostics=True,
controllertype='P')
if verbosity > 1: print('{} rejected steps'.format(rej))
err[imeth, jtol] *= np.max(np.abs(u[-1] - exsol))
#FSAL methods save on accepted steps, but not on rejected:
work[imeth, jtol] += (len(t) * workperstep +
rej * len(method)) / speedup
for imeth, method in enumerate(methods):
for jtol, tol in enumerate(tols):
err[imeth, jtol] = err[imeth, jtol]**(1. / len(ivps))
for imeth, method in enumerate(methods):
plt.semilogy(work[imeth, :],
err[imeth, :],
label=method.name,
linewidth=3)
plt.xlabel('Function evaluations')
plt.ylabel('Error at $t_{final}$')
plt.legend(loc='best')
plt.draw()
return work, err
def ctest(methods,
ivp,
grids=[20, 40, 80, 160, 320, 640],
verbosity=0,
parallel=False):
"""
Runs a convergence test, integrating a single initial value problem
using a sequence of fixed step sizes and a set of methods.
Creates a plot of the resulting errors versus step size for each method.
**Inputs**:
- methods -- a list of ODEsolver instances
- ivp -- an IVP instance
- grids -- a list of grid sizes for integration.
optional; defaults to [20,40,80,160,320,640]
- parallel -- to exploit possible parallelization (optional)
**Example**::
>>> import nodepy.runge_kutta_method as rk
>>> from nodepy.ivp import load_ivp
>>> rk44=rk.loadRKM('RK44')
>>> myivp=load_ivp('nlsin')
>>> work, err=ctest(rk44,myivp)
TODO:
- Option to plot versus f-evals or dt
"""
import matplotlib.pyplot as plt
plt.clf()
# In case just one method is passed in (and not as a list):
if not isinstance(methods, list): methods = [methods]
err = []
try:
exsol = ivp.exact(ivp.T)
except:
bs5 = rk.loadRKM('BS5')
bigN = grids[-1] * 4
if verbosity > 0:
print('solving on fine grid with {} points'.format(bigN))
t, u = bs5(ivp, N=bigN)
if verbosity > 0: print('done')
exsol = u[-1] + 0.
for method in methods:
if verbosity > 0: print("solving with {}".format(method.name))
err0 = []
for i, N in enumerate(grids):
t, u = method(ivp, N=N)
err0.append(np.linalg.norm(u[-1] - exsol))
err.append(err0)
work = np.array([grid * len(method) for grid in grids])
if parallel:
speedup = len(method) / float(method.num_seq_dep_stages())
work = work / speedup
plt.loglog(work, err0, label=method.name, linewidth=3)
plt.xlabel('Function evaluations')
plt.ylabel('Error at $t_{final}$')
plt.legend(loc='best')
plt.draw()
return work, err
def setUp(self):
self.RKs=rk.loadRKM()
def ptest(methods,ivps,tols=[1.e-1,1.e-2,1.e-4,1.e-6],verbosity=0,parallel=False):
"""
Runs a performance test, integrating a set of problems with a set
of methods using a sequence of error tolerances. Creates a plot
of the error achieved versus the amount of work done (number of
function evaluations) for each method.
**Input**:
* methods -- a list of ODEsolver instances
Note that all methods must have error estimators.
* ivps -- a list of IVP instances
* tols -- a specified list of error tolerances (optional)
* parallel -- to exploit possible parallelization (optional)
**Example**::
>>> import nodepy.runge_kutta_method as rk
>>> from nodepy.ivp import load_ivp
>>> bs5=rk.loadRKM('BS5')
>>> myivp=load_ivp('nlsin')
>>> work,err=ptest(bs5,myivp)
"""
import matplotlib.pyplot as pl
pl.clf(); pl.draw(); pl.hold(True)
# In case just one method is passed in (and not as a list):
if not isinstance(methods,list): methods=[methods]
if not isinstance(ivps,list): ivps=[ivps]
err=np.ones([len(methods),len(tols)])
work=np.zeros([len(methods),len(tols)])
for ivp in ivps:
if verbosity>0: print("solving problem {}".format(ivp))
try:
exsol = ivp.exact(ivp.T)
except:
bs5=rk.loadRKM('BS5') #Use Bogacki-Shampine RK for fine solution
lowtol=min(tols)/100.
if verbosity>0: print('solving for "exact" solution with tol= {}'.format(lowtol))
t,u=bs5(ivp,errtol=lowtol,dt=ivp.dt0)
if verbosity>0: print('done')
exsol = u[-1] + 0.
for imeth,method in enumerate(methods):
if verbosity>0: print('Solving with method {}'.format(method.name))
if parallel:
speedup = len(method)/float(method.num_seq_dep_stages())
else:
speedup = 1.
workperstep = len(method)-method.is_FSAL()
for jtol,tol in enumerate(tols):
t,u,rej,dt,errhist=method(ivp,errtol=tol,dt=ivp.dt0,diagnostics=True,controllertype='P')
if verbosity>1: print('{} rejected steps'.format(rej))
err[imeth,jtol]*= np.max(np.abs(u[-1]-exsol))
#FSAL methods save on accepted steps, but not on rejected:
work[imeth,jtol]+= (len(t)*workperstep+rej*len(method))/speedup
for imeth,method in enumerate(methods):
for jtol,tol in enumerate(tols):
err[imeth,jtol]=err[imeth,jtol]**(1./len(ivps))
for imeth,method in enumerate(methods):
pl.semilogy(work[imeth,:],err[imeth,:],label=method.name,linewidth=3)
pl.xlabel('Function evaluations')
pl.ylabel('Error at $t_{final}$')
pl.legend(loc='best')
pl.hold(False)
pl.draw()
return work,err
def ptest(methods, ivps, tols=[1.e-1, 1.e-2, 1.e-4, 1.e-6]):
"""
Runs a performance test, integrating a set of problems with a set
of methods using a sequence of error tolerances. Creates a plot
of the error achieved versus the amount of work done (number of
function evaluations) for each method.
**Input**:
* methods -- a list of ODEsolver instances
Note that all methods must have error estimators.
* ivps -- a list of IVP instances
* tols -- a specified list of error tolerances (optional)
**Example**::
import runge_kutta_method as rk
from ivp import *
bs5=rk.loadRKM('BS5')
myivp=load_ivp('nlsin')
work,err=ptest(bs5,myivp)
"""
pl.clf()
pl.draw()
pl.hold(True)
# In case just one method is passed in (and not as a list):
if not isinstance(methods, list): methods = [methods]
if not isinstance(ivps, list): ivps = [ivps]
err = np.ones([len(methods), len(tols)])
work = np.zeros([len(methods), len(tols)])
for ivp in ivps:
print "solving problem %s" % ivp
try:
exsol = ivp.exact(ivp.T)
except:
bs5 = rk.loadRKM('BS5') #Use Bogacki-Shampine RK for fine solution
lowtol = min(tols) / 100.
print 'solving for "exact" solution with tol= ' + str(lowtol)
t, u = bs5(ivp, errtol=lowtol, dt=ivp.dt0)
print 'done'
exsol = u[-1].copy()
for imeth, method in enumerate(methods):
print 'Solving with method ' + method.name
workperstep = len(method) - method.is_FSAL()
for jtol, tol in enumerate(tols):
t, u, rej, dt = method(ivp,
errtol=tol,
dt=ivp.dt0,
diagnostics=True,
controllertype='P')
print str(rej) + ' rejected steps'
err[imeth, jtol] *= np.max(np.abs(u[-1] - exsol))
#FSAL methods save on accepted steps, but not on rejected:
work[imeth, jtol] += len(t) * workperstep + rej * len(method)
for imeth, method in enumerate(methods):
for jtol, tol in enumerate(tols):
err[imeth, jtol] = err[imeth, jtol]**(1. / len(ivps))
for imeth, method in enumerate(methods):
pl.semilogy(work[imeth, :],
err[imeth, :],
label=method.name,
linewidth=3)
pl.xlabel('Function evaluations')
pl.ylabel('Error at $t_{final}$')
pl.legend(loc='best')
pl.hold(False)
pl.draw()
return work, err
import nodepy.runge_kutta_method as rk
import nodepy.convergence as cv
from nodepy import ivp
import matplotlib.pyplot as pl
#Load some methods:
rk4 = rk.loadRKM('RK44')
SSP2 = rk.loadRKM('SSP22')
SSP104 = rk.loadRKM('SSP104')
#Define an IVP:
#myivp=ivp.exp_fun(1.)
myivp = ivp.load_ivp('test')
#Start and end time:
T = [0., 5.]
cv.ctest([rk4, SSP2, SSP104], myivp)
pl.show()
import nodepy.runge_kutta_method as rk
import nodepy.convergence as cv
from nodepy import ivp
# Load some methods:
rk4 = rk.loadRKM("RK44")
SSP2 = rk.loadRKM("SSP22")
SSP104 = rk.loadRKM("SSP104")
# Define an IVP:
# myivp=ivp.exp_fun(1.)
myivp = ivp.load_ivp("test")
# Start and end time:
T = [0.0, 5.0]
cv.ctest([rk4, SSP2, SSP104], myivp)
import nodepy.runge_kutta_method as rk
import nodepy.convergence as cv
from nodepy import ivp
import matplotlib.pyplot as pl
#Load some methods:
rk4=rk.loadRKM('RK44')
SSP2=rk.loadRKM('SSP22')
SSP104=rk.loadRKM('SSP104')
#Define an IVP:
#myivp=ivp.exp_fun(1.)
myivp=ivp.load_ivp('test')
#Start and end time:
T=[0.,5.]
cv.ctest([rk4,SSP2,SSP104],myivp)
pl.show()
def setUp(self):
self.RKs = rk.loadRKM()
