• Re: Bulirsch???Stoer vs Runge-Kutta

    From Marco Amarante@21:1/5 to All on Fri Aug 5 04:50:07 2022
    Em domingo, 6 de março de 2011 às 06:18:29 UTC-3, glen herrmannsfeldt escreveu:
    nm...@cam.ac.uk wrote:
    (snip on integration algorithms)
    Rather longer ago that that, I was working on multi-dimensional
    scaling, which has the property that it has a set of fairly
    well-defined local minima, but where discontinuities are dense
    (but in the same direction as the slope). Steepest descents
    worked; little else did.
    There is a fairly general rule that the less well-behaved the
    problem you have, the cruder a method you must use, because all
    fancy methods need to make more assumptions. Runge-Kutta is a
    bit odd, but also blows up quite spectacularly given a hostile
    enough function. For some, there is little option but simple
    step-by-step approximation.
    I always liked Gaussian quadrature, though its assumptions
    likely cause problems with some functions. With no prediction,
    it shouldn't get too confused. The old trick of running with
    different order and checking that the result isn't too different
    should still work.
    As a side-note, IBM SSP contained the only matrix inversion code
    that I have ever seen that succeeded in inverting a singular 3x3
    matrix - AND returning elements of the same order as the input!
    Impressive.
    The code is still around, if anyone wants to try it.
    Harwell and NAG didn't like the matrix at all :-)
    -- glen

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  • From Marco Amarante@21:1/5 to All on Fri Aug 5 04:51:01 2022
    Em domingo, 6 de março de 2011 às 06:18:29 UTC-3, glen herrmannsfeldt escreveu:
    nm...@cam.ac.uk wrote:
    (snip on integration algorithms)
    Rather longer ago that that, I was working on multi-dimensional
    scaling, which has the property that it has a set of fairly
    well-defined local minima, but where discontinuities are dense
    (but in the same direction as the slope). Steepest descents
    worked; little else did.
    There is a fairly general rule that the less well-behaved the
    problem you have, the cruder a method you must use, because all
    fancy methods need to make more assumptions. Runge-Kutta is a
    bit odd, but also blows up quite spectacularly given a hostile
    enough function. For some, there is little option but simple
    step-by-step approximation.
    I always liked Gaussian quadrature, though its assumptions
    likely cause problems with some functions. With no prediction,
    it shouldn't get too confused. The old trick of running with
    different order and checking that the result isn't too different
    should still work.
    As a side-note, IBM SSP contained the only matrix inversion code
    that I have ever seen that succeeded in inverting a singular 3x3
    matrix - AND returning elements of the same order as the input!
    Impressive.
    The code is still around, if anyone wants to try it.
    Harwell and NAG didn't like the matrix at all :-)
    -- glen

    I would like to present my article to the group:

    https://brazilianjournals.com/ojs/index.php/BRJD/article/view/46840

    --


    PRESERVE A NATUREZA, EVITE O DESPERDÍCIO DE PAPEL E PENSE ANTES DE
    IMPRIMIR.

    "Responsabilidade com o MEIO AMBIENTE"

    Esta mensagem e
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  • From Arjen Markus@21:1/5 to Marco Amarante on Fri Aug 5 07:41:54 2022
    On Friday, August 5, 2022 at 1:51:03 PM UTC+2, Marco Amarante wrote:
    ...
    I would like to present my article to the group:

    https://brazilianjournals.com/ojs/index.php/BRJD/article/view/46840
    --

    You do not happen to have an English version of it? My understanding of Portuguese is rather limited, even though I can of course recognise the equations ;).

    Regards,

    Arjen

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  • From Marco Amarante@21:1/5 to All on Fri Aug 19 09:23:52 2022
    Em sexta-feira, 5 de agosto de 2022 às 11:41:56 UTC-3, arjen.m...@gmail.com escreveu:
    On Friday, August 5, 2022 at 1:51:03 PM UTC+2, Marco Amarante wrote:
    ...
    I would like to present my article to the group:

    https://brazilianjournals.com/ojs/index.php/BRJD/article/view/46840
    --

    You do not happen to have an English version of it? My understanding of Portuguese is rather limited, even though I can of course recognise the equations ;).

    Regards,

    Arjen

    --


    PRESERVE A NATUREZA, EVITE O DESPERDÍCIO DE PAPEL E PENSE ANTES DE
    IMPRIMIR.

    "Responsabilidade com o MEIO AMBIENTE"

    Esta mensagem e
    quaisquer arquivos em anexo podem conter informações confidenciais e/ou privilegiadas. Se você não for o destinatário ou a pessoa autorizada a receber esta mensagem, por favor não leia, copie, repasse, imprima, guarde, nem tome qualquer ação baseada nestas informações. Por favor notifique o remetente imediatamente por e-mail e apague a mensagem permanentemente.
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  • From Marco Amarante@21:1/5 to All on Fri Aug 19 09:24:41 2022
    Em sexta-feira, 5 de agosto de 2022 às 11:41:56 UTC-3, arjen.m...@gmail.com escreveu:
    On Friday, August 5, 2022 at 1:51:03 PM UTC+2, Marco Amarante wrote:
    ...
    I would like to present my article to the group:

    https://brazilianjournals.com/ojs/index.php/BRJD/article/view/46840
    --

    You do not happen to have an English version of it? My understanding of Portuguese is rather limited, even though I can of course recognise the equations ;).

    Regards,

    Arjen

    Arjen

    I can write an English version of it if participants are interested

    Regards, ,

    Marco Aurelio Amarante Ribeiro

    --


    PRESERVE A NATUREZA, EVITE O DESPERDÍCIO DE PAPEL E PENSE ANTES DE
    IMPRIMIR.

    "Responsabilidade com o MEIO AMBIENTE"

    Esta mensagem e
    quaisquer arquivos em anexo podem conter informações confidenciais e/ou privilegiadas. Se você não for o destinatário ou a pessoa autorizada a receber esta mensagem, por favor não leia, copie, repasse, imprima, guarde, nem tome qualquer ação baseada nestas informações. Por favor notifique o remetente imediatamente por e-mail e apague a mensagem permanentemente.
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  • From Arjen Markus@21:1/5 to Marco Amarante on Mon Aug 22 07:43:02 2022
    On Friday, August 19, 2022 at 6:24:43 PM UTC+2, Marco Amarante wrote:
    Em sexta-feira, 5 de agosto de 2022 às 11:41:56 UTC-3, arjen escreveu:
    On Friday, August 5, 2022 at 1:51:03 PM UTC+2, Marco Amarante wrote:
    ...
    I would like to present my article to the group:

    https://brazilianjournals.com/ojs/index.php/BRJD/article/view/46840
    --

    You do not happen to have an English version of it? My understanding of Portuguese is rather limited, even though I can of course recognise the equations ;).

    Regards,

    Arjen
    Arjen

    I can write an English version of it if participants are interested

    Regards, ,

    That might be interesting indeed :).

    Regards,

    Arjen

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  • From Arjen Markus@21:1/5 to Arjen Markus on Mon Aug 22 11:35:15 2022
    On Monday, August 22, 2022 at 4:43:04 PM UTC+2, Arjen Markus wrote:


    I can write an English version of it if participants are interested

    Regards, ,

    That might be interesting indeed :).

    Regards,

    Arjen

    On Wikipedia I found a short description of the method with several caveats: according to some authors there is little to gain with the techniques that Bulirsch-Stoer depends on. Do you share that conclusion?

    Regards,

    Arjen

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  • From Phillip Helbig (undress to reply@21:1/5 to Arjen Markus on Thu Sep 1 08:28:51 2022
    In article <bdfb06b3-90b3-40a8-94d3-42a0b99672bcn@googlegroups.com>,
    Arjen Markus <arjen.markus895@gmail.com> writes:

    On Monday, August 22, 2022 at 4:43:04 PM UTC+2, Arjen Markus wrote:


    I can write an English version of it if participants are interested

    Regards, ,

    That might be interesting indeed :).

    Regards,

    Arjen

    On Wikipedia I found a short description of the method with several
    caveats: according to some authors there is little to gain with the techniques that Bulirsch-Stoer depends on. Do you share that conclusion?

    I once coded a real-world application where Bulirsch-Stoer worked better
    than Runge-Kutta (though using polynomial rather than rational-function extrapolation).

    I think that Bulirsch-Stoer is one of the great ideas of numerical
    analysis.

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  • From gah4@21:1/5 to All on Thu Sep 1 02:07:39 2022
    On Thursday, September 1, 2022 at 1:28:55 AM UTC-7, Phillip Helbig (undress to reply) wrote:

    (snip)

    I think that Bulirsch-Stoer is one of the great ideas of numerical
    analysis.

    I used to know some of this, but I forget now.

    The only one I remember is that higher than 4th order Runge-Kutta
    isn't better, in terms of step size and number of function evaluations.

    With small enough step size, it doesn't make so much difference.

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  • From Thomas Koenig@21:1/5 to gah4@u.washington.edu on Thu Sep 1 16:50:50 2022
    gah4 <gah4@u.washington.edu> schrieb:
    On Thursday, September 1, 2022 at 1:28:55 AM UTC-7, Phillip Helbig (undress to reply) wrote:

    (snip)

    I think that Bulirsch-Stoer is one of the great ideas of numerical
    analysis.

    I used to know some of this, but I forget now.

    The only one I remember is that higher than 4th order Runge-Kutta
    isn't better, in terms of step size and number of function evaluations.

    This really depends.

    If you are trying to integrate x'' + x = 0, as smooth as you can
    get, then higher order will help.

    If your function is less well behaved, then lower order will be
    better, and if your system of equations is stiff, you will have
    to use quite different solvers.

    With small enough step size, it doesn't make so much difference.

    Unless you make it too small and get cancellation in your
    difference quotients :-)

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  • From Marco Aurelio Amarante Ribeiro@21:1/5 to All on Fri Oct 28 07:55:28 2022
    Em quinta-feira, 1 de setembro de 2022 às 13:50:53 UTC-3, Thomas Koenig escreveu:
    gah4 <ga...@u.washington.edu> schrieb:
    On Thursday, September 1, 2022 at 1:28:55 AM UTC-7, Phillip Helbig (undress to reply) wrote:

    (snip)

    I think that Bulirsch-Stoer is one of the great ideas of numerical
    analysis.

    I used to know some of this, but I forget now.

    The only one I remember is that higher than 4th order Runge-Kutta
    isn't better, in terms of step size and number of function evaluations.
    This really depends.

    If you are trying to integrate x'' + x = 0, as smooth as you can
    get, then higher order will help.

    If your function is less well behaved, then lower order will be
    better, and if your system of equations is stiff, you will have
    to use quite different solvers.
    With small enough step size, it doesn't make so much difference.
    Unless you make it too small and get cancellation in your
    difference quotients :-)

    hello everyone
    I have one plus one article using the
    Bulirsch–Stoer: https://gjeta.com/content/pollutant-dispersion-modeling-lakes-interconnected-channels-solution-using-bulirsch-stoer

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  • From Thomas Koenig@21:1/5 to Marco Aurelio Amarante Ribeiro on Fri Oct 28 17:30:42 2022
    Marco Aurelio Amarante Ribeiro <marcoaurelioamaranteribeiro@gmail.com> schrieb:
    Em quinta-feira, 1 de setembro de 2022 às 13:50:53 UTC-3, Thomas Koenig escreveu:
    gah4 <ga...@u.washington.edu> schrieb:
    On Thursday, September 1, 2022 at 1:28:55 AM UTC-7, Phillip Helbig (undress to reply) wrote:

    (snip)

    I think that Bulirsch-Stoer is one of the great ideas of numerical
    analysis.

    I used to know some of this, but I forget now.

    The only one I remember is that higher than 4th order Runge-Kutta
    isn't better, in terms of step size and number of function evaluations.
    This really depends.

    If you are trying to integrate x'' + x = 0, as smooth as you can
    get, then higher order will help.

    If your function is less well behaved, then lower order will be
    better, and if your system of equations is stiff, you will have
    to use quite different solvers.
    With small enough step size, it doesn't make so much difference.
    Unless you make it too small and get cancellation in your
    difference quotients :-)

    hello everyone
    I have one plus one article using the
    Bulirsch–Stoer: https://gjeta.com/content/pollutant-dispersion-modeling-lakes-interconnected-channels-solution-using-bulirsch-stoer

    This particular problem (basically three stirred tanks with
    volume flow between them, assumed to be perfectly mixed, with
    flows between them) lends itself to an analytical solution.
    The eigenvalues of a 3*3 matrix are not that hard to determine,
    and an analytical solution could also give you a frequency
    response (for example).

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  • From Marco Aurelio Amarante Ribeiro@21:1/5 to All on Fri Nov 11 04:03:47 2022
    Em sexta-feira, 28 de outubro de 2022 às 14:30:46 UTC-3, Thomas Koenig escreveu:
    Marco Aurelio Amarante Ribeiro <marcoaurelioa...@gmail.com> schrieb:
    Em quinta-feira, 1 de setembro de 2022 às 13:50:53 UTC-3, Thomas Koenig escreveu:
    gah4 <ga...@u.washington.edu> schrieb:
    On Thursday, September 1, 2022 at 1:28:55 AM UTC-7, Phillip Helbig (undress to reply) wrote:

    (snip)

    I think that Bulirsch-Stoer is one of the great ideas of numerical
    analysis.

    I used to know some of this, but I forget now.

    The only one I remember is that higher than 4th order Runge-Kutta
    isn't better, in terms of step size and number of function evaluations. >> This really depends.

    If you are trying to integrate x'' + x = 0, as smooth as you can
    get, then higher order will help.

    If your function is less well behaved, then lower order will be
    better, and if your system of equations is stiff, you will have
    to use quite different solvers.
    With small enough step size, it doesn't make so much difference.
    Unless you make it too small and get cancellation in your
    difference quotients :-)

    hello everyone
    I have one plus one article using the
    Bulirsch–Stoer: https://gjeta.com/content/pollutant-dispersion-modeling-lakes-interconnected-channels-solution-using-bulirsch-stoer
    This particular problem (basically three stirred tanks with
    volume flow between them, assumed to be perfectly mixed, with
    flows between them) lends itself to an analytical solution.
    The eigenvalues of a 3*3 matrix are not that hard to determine,
    and an analytical solution could also give you a frequency
    response (for example).

    Daniel(dcar...@gmail.com),
    sorry for the delay in responding. In my master's thesis, I answer your question. See the link.

    https://sucupira.capes.gov.br/sucupira/public/consultas/coleta/trabalhoConclusao/viewTrabalhoConclusao.jsf?popup=true&id_trabalho=11494524

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  • From Marco Aurelio Amarante Ribeiro@21:1/5 to All on Fri Nov 11 04:07:17 2022
    Em sexta-feira, 28 de outubro de 2022 às 14:30:46 UTC-3, Thomas Koenig escreveu:
    Marco Aurelio Amarante Ribeiro <marcoaurelioa...@gmail.com> schrieb:
    Em quinta-feira, 1 de setembro de 2022 às 13:50:53 UTC-3, Thomas Koenig escreveu:
    gah4 <ga...@u.washington.edu> schrieb:
    On Thursday, September 1, 2022 at 1:28:55 AM UTC-7, Phillip Helbig (undress to reply) wrote:

    (snip)

    I think that Bulirsch-Stoer is one of the great ideas of numerical
    analysis.

    I used to know some of this, but I forget now.

    The only one I remember is that higher than 4th order Runge-Kutta
    isn't better, in terms of step size and number of function evaluations. >> This really depends.

    If you are trying to integrate x'' + x = 0, as smooth as you can
    get, then higher order will help.

    If your function is less well behaved, then lower order will be
    better, and if your system of equations is stiff, you will have
    to use quite different solvers.
    With small enough step size, it doesn't make so much difference.
    Unless you make it too small and get cancellation in your
    difference quotients :-)

    hello everyone
    I have one plus one article using the
    Bulirsch–Stoer: https://gjeta.com/content/pollutant-dispersion-modeling-lakes-interconnected-channels-solution-using-bulirsch-stoer
    This particular problem (basically three stirred tanks with
    volume flow between them, assumed to be perfectly mixed, with
    flows between them) lends itself to an analytical solution.
    The eigenvalues of a 3*3 matrix are not that hard to determine,
    and an analytical solution could also give you a frequency
    response (for example).



    Thomas Koenig

    Really this specific problem can be solved by an EDO's that have analytical solutions, as the article showed. Thank you for your participation.

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