Science > Physics > Is there a way to evaluate the quality of the numerical ODE solution?
| Topic: |
Science > Physics |
| User: |
"Vista" |
| Date: |
26 May 2007 01:13:57 PM |
| Object: |
Is there a way to evaluate the quality of the numerical ODE solution? |
Hi all,
I have a complicated first order non-linear ODE, and there is no way I can
find its closed-form solution.
I am using Matlab's ODE functions.
Is there a way to evaluate the quality of the numerical solutions found by
Matlab ODE functions?
In order to select from many of Matlab ODE solvers and those third-party ODE
solvers from Internet, for example, in order to improve the speed I may want
to use a F90 ODE solver obtained from online, I need to figure out a
systematic method to evaluate the quality of the numerical solutions, and
compare a few ODE solvers.
Could anybody give some suggestions so that I can know at the least the
solution is correct, up to a small approximation error?
------------------
Moreover, is there a way to evalute the quality of numerical integration?
Thanks!
.
|
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| User: "Hans Mittelmann" |
|
| Title: Re: Is there a way to evaluate the quality of the numerical ODE solution? |
26 May 2007 04:20:38 PM |
|
|
Hi,
you do not say of which type your problem is. Is it, for exampe,
stiff? The most important thing to know is that adaptive solvers such
as ode45 or ode87 (Matlab central) for non-stiff problems control the
estimated error and their estimates are pretty reliable. They work
with a default tolerance of 1e-6 but with odeset (for ode45) you can
change that. Do not use any solver without such built-in error
estimation. Check out the help for stiff solvers if you suspect your
problem of being stiff.
On May 26, 11:13 am, "Vista" <a...@gmai.com> wrote:
Hi all,
I have a complicated first order non-linear ODE, and there is no way I can
find its closed-form solution.
I am using Matlab's ODE functions.
Is there a way to evaluate the quality of the numerical solutions found by
Matlab ODE functions?
In order to select from many of Matlab ODE solvers and those third-party ODE
solvers from Internet, for example, in order to improve the speed I may want
to use a F90 ODE solver obtained from online, I need to figure out a
systematic method to evaluate the quality of the numerical solutions, and
compare a few ODE solvers.
Could anybody give some suggestions so that I can know at the least the
solution is correct, up to a small approximation error?
------------------
Moreover, is there a way to evalute the quality of numerical integration?
Thanks!
.
|
|
|
| User: "Vista" |
|
| Title: Re: Is there a way to evaluate the quality of the numerical ODE solution? |
26 May 2007 04:59:25 PM |
|
|
"Hans Mittelmann" <mittelmann@asu.edu> wrote in message
news:1180214438.067704.252380@z28g2000prd.googlegroups.com...
Hi,
you do not say of which type your problem is. Is it, for exampe,
stiff? The most important thing to know is that adaptive solvers such
as ode45 or ode87 (Matlab central) for non-stiff problems control the
estimated error and their estimates are pretty reliable. They work
with a default tolerance of 1e-6 but with odeset (for ode45) you can
change that. Do not use any solver without such built-in error
estimation. Check out the help for stiff solvers if you suspect your
problem of being stiff.
Thanks Hans.
I am trying to design an "automatic" system with varying ODE systems. Let's
say my ODE is of the form:
y'=c1*y + c2*y^2+ c3 +c4*exp(c5*y+ i * 2*pi* c6)
where "i" is the unit of imaginary numbers.
I want to vary the coefficients randomly, and use ode45 to solve for
numerical solutions.
There is no way for me tell manually if the problem is stiff or not.
I have to let the program to detect that automatically.
That's why I need an indicator of qualities. You mention that ODE45 has an
internal error estimate, is there a way I can get it outputted so I will be
able to take a look? It will be very useful.
When the quality is low, I know maybe the random combination of coefficients
give a stiff problem, then I might need to try next ODE solvers on the
ranking list.
Let's say the ranking list is as follows:
First try:
ODE45
ODE87
ODE113
....
....
----------------------
Is there an error estimate and quality evaluator for numerical integration?
.
|
|
|
| User: "Hans Mittelmann" |
|
| Title: Re: Is there a way to evaluate the quality of the numerical ODE solution? |
26 May 2007 08:08:27 PM |
|
|
Hi,
I strongly suggest you use (help) odeget together with ode45 to see if
the accuracy you want could be obtained. If not and if a lot of steps
were taken, you may want to try a stiff solver, not ode113 but ode23s
or such.
On May 26, 2:59 pm, "Vista" <a...@gmai.com> wrote:
"Hans Mittelmann" <mittelm...@asu.edu> wrote in message
news:1180214438.067704.252380@z28g2000prd.googlegroups.com...
Hi,
you do not say of which type your problem is. Is it, for exampe,
stiff? The most important thing to know is that adaptive solvers such
as ode45 or ode87 (Matlab central) for non-stiff problems control the
estimated error and their estimates are pretty reliable. They work
with a default tolerance of 1e-6 but with odeset (for ode45) you can
change that. Do not use any solver without such built-in error
estimation. Check out the help for stiff solvers if you suspect your
problem of being stiff.
Thanks Hans.
I am trying to design an "automatic" system with varying ODE systems. Let's
say my ODE is of the form:
y'=c1*y + c2*y^2+ c3 +c4*exp(c5*y+ i * 2*pi* c6)
where "i" is the unit of imaginary numbers.
I want to vary the coefficients randomly, and use ode45 to solve for
numerical solutions.
There is no way for me tell manually if the problem is stiff or not.
I have to let the program to detect that automatically.
That's why I need an indicator of qualities. You mention that ODE45 has an
internal error estimate, is there a way I can get it outputted so I will be
able to take a look? It will be very useful.
When the quality is low, I know maybe the random combination of coefficients
give a stiff problem, then I might need to try next ODE solvers on the
ranking list.
Let's say the ranking list is as follows:
First try:
ODE45
ODE87
ODE113
...
...
----------------------
Is there an error estimate and quality evaluator for numerical integration?
.
|
|
|
|
| User: "Michael Press" |
|
| Title: Re: Is there a way to evaluate the quality of the numerical ODE solution? |
31 May 2007 11:59:55 PM |
|
|
In article <f3aajt$94a$1@news.Stanford.EDU>,
"Vista" <abc@gmai.com> wrote:
"Hans Mittelmann" <mittelmann@asu.edu> wrote in message
news:1180214438.067704.252380@z28g2000prd.googlegroups.com...
Hi,
you do not say of which type your problem is. Is it, for exampe,
stiff? The most important thing to know is that adaptive solvers such
as ode45 or ode87 (Matlab central) for non-stiff problems control the
estimated error and their estimates are pretty reliable. They work
with a default tolerance of 1e-6 but with odeset (for ode45) you can
change that. Do not use any solver without such built-in error
estimation. Check out the help for stiff solvers if you suspect your
problem of being stiff.
Thanks Hans.
I am trying to design an "automatic" system with varying ODE systems. Let's
say my ODE is of the form:
y'=c1*y + c2*y^2+ c3 +c4*exp(c5*y+ i * 2*pi* c6)
where "i" is the unit of imaginary numbers.
I want to vary the coefficients randomly, and use ode45 to solve for
numerical solutions.
There is no way for me tell manually if the problem is stiff or not.
I have to let the program to detect that automatically.
That's why I need an indicator of qualities. You mention that ODE45 has an
internal error estimate, is there a way I can get it outputted so I will be
able to take a look? It will be very useful.
When the quality is low, I know maybe the random combination of coefficients
give a stiff problem, then I might need to try next ODE solvers on the
ranking list.
Let's say the ranking list is as follows:
First try:
ODE45
ODE87
ODE113
...
...
----------------------
Is there an error estimate and quality evaluator for numerical integration?
If I were serious about solving ODE's I would get LSODA.
<http://gams.nist.gov/serve.cgi/Package/ODEPACK/>
Adaptive initial value problem solver, switches
degrees, and between non-stiff and stiff methods on the
fly. Copious feedback on the state of the integration.
Linda Petzold wrote it, but her name is not on this
page. It is truly well written, and I should know,
having converted it to C.
--
Michael Press
.
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