10:19 a.m.
To the best of my knowledge, we do not have in gretl a direct
root-finder for a non-linear equation (apart from "polroots" for
polynomials) , say
f(x,a) - b = 0
with function f continuous / non-linear, a,b given and with task to find
x. Assume that there indeed exists a unique solution x*.
I thought I'd trick the system by specifying
h(x,a,b) = [f(x,a) - b]^2
and ask the BFGScmin function to minimize h(x,a,b) (with numerical
derivatives).
Mathematically this looks sound, but is there something that lurks in
the software/computational basement?
--
Alecos Papadopoulos PhD
Affiliate Researcher
Dpt of Economics, Athens University of Economics and Business
Foundation for Economic and Industrial Research (IOBE)
web: alecospapadopoulos.wordpress.com/
ORCID:0000-0003-2441-4550
10:32 a.m.
On 17.11.2024 11:19, Alecos Papadopoulos wrote:
To the best of my knowledge, we do not have in gretl a direct
root-finder for a non-linear equation (apart from "polroots" for
polynomials) , say
f(x,a) - b = 0
with function f continuous / non-linear, a,b given and with task to
find x. Assume that there indeed exists a unique solution x*.
I thought I'd trick the system by specifying
h(x,a,b) = [f(x,a) - b]^2
and ask the BFGScmin function to minimize h(x,a,b) (with numerical
derivatives).
Mathematically this looks sound, but is there something that lurks in
the software/computational basement?
Well, there is a fzero() function, thought it works with only one variable.
Marcin
11:37 a.m.
Hi Marcin, thanks. It is a good thing that gretl's functionality has so
much expanded that one cannot really remember (or encounter) the whole
of it.
I tried both approaches (for a single unknown). The non-linear function
had the unknown in several places at the same time, either
exponentiated, or as an exponent itself, or in logs, or in the denominator.
REPORT
1. Run both methods 10,000 times on the same data (two series ranging in
(0,1)).
2. When they both succeed for the same data (see next for when they
don't), they provide identical results (max abs difference 1.8256e-007).
3. fzero issues: with the fzero function (where i provided only a
single 0.01 initial value), the limitation I see is that we cannot
constraint the value of the argmin (and I need it to be in (0,1) for
example). But still, in 10,000 attempts it provided an inadmissible
solution only for a few dozen cases (these were the cases were the
BFGScmin approach gave a solution below the initial value provided for
the fzero setup). It also appears that this initial value for fzero
cannot be very small because I get "the convergence criterion is not
met" and the loop stops (e.g. this happened when I tried to set it equal
to 0.001 instead of 0.01).
4. With the BFGScmin (with the (0,1) constraint in place and
bfgs_richardson "on"), for some 1.5% of the cases I got an obvious
failure in that the solution was admissible, but the value of the
objective function was away from zero. In all these cases the solution
given was the maximum permissible value, 1. Note that these observations
for which BFGScmin failed, /are totally different /from those that fzero
provided an inadmissible solution.
So, in all 10,000 cases, one method or the other gave a valid result,
and in 98% both succeeded.
Alecos Papadopoulos PhD
Affiliate Researcher
Dpt of Economics, Athens University of Economics and Business
Foundation for Economic and Industrial Research (IOBE)
web: alecospapadopoulos.wordpress.com/
ORCID:0000-0003-2441-4550
On 17/11/2024 12:32, Marcin Błażejowski wrote:
On 17.11.2024 11:19, Alecos Papadopoulos wrote:
> To the best of my knowledge, we do not have in gretl a direct
> root-finder for a non-linear equation (apart from "polroots" for
> polynomials) , say
>
> f(x,a) - b = 0
>
> with function f continuous / non-linear, a,b given and with task to
> find x. Assume that there indeed exists a unique solution x*.
>
> I thought I'd trick the system by specifying
>
> h(x,a,b) = [f(x,a) - b]^2
>
> and ask the BFGScmin function to minimize h(x,a,b) (with numerical
> derivatives).
>
> Mathematically this looks sound, but is there something that lurks in
> the software/computational basement?
>
Well, there is a fzero() function, thought it works with only one
variable.
Marcin
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11:49 a.m.
On 17.11.2024 12:37, Alecos Papadopoulos wrote:
Hi Marcin, thanks. It is a good thing that gretl's functionality has
so much expanded that one cannot really remember (or encounter) the
whole of it.
I tried both approaches (for a single unknown). The non-linear
function had the unknown in several places at the same time, either
exponentiated, or as an exponent itself, or in logs, or in the
denominator.
REPORT
1. Run both methods 10,000 times on the same data (two series ranging
in (0,1)).
2. When they both succeed for the same data (see next for when they
don't), they provide identical results (max abs difference 1.8256e-007).
3. fzero issues: with the fzero function (where i provided only a
single 0.01 initial value), the limitation I see is that we cannot
constraint the value of the argmin (and I need it to be in (0,1) for
example).
Hmm, taken from help:
<hansl>
x = fzero(sin(x), {3.1, 3.2})
eval x
x = fzero(sin(x), {3.0, 3.1})
eval x
</hansl>
So the error generated for the second example confirms that constraints
on x works fine :-)
Marcin
11:49 a.m.
On 17/11/2024 12:37, Alecos Papadopoulos wrote:
3. fzero issues: with the fzero function (where i provided only a
single 0.01 initial value), the limitation I see is that we cannot
constraint the value of the argmin (and I need it to be in (0,1) for
example).
The best solution in many cases is to map the constrained variable to an
unconstrained one via an invertible transformation: in the example below
I use the logistic function to find a root within there (0,1) interval,
but of course you have lots of alternatives:
<hansl>
set verbose off
clear
function scalar f(scalar z)
x = logistic(z)
### function whose solutions are 1/3 and 2
return (x-1/3)*(x-2)
end function
z_init = {-100, 100}
sol = fzero(f(z), z_init)
x = logistic(sol)
print sol x
</hansl>
-------------------------------------------------------
Riccardo (Jack) Lucchetti
Dipartimento di Scienze Economiche e Sociali (DiSES)
Università Politecnica delle Marche
(formerly known as Università di Ancona)
r.lucchetti(a)univpm.it
http://www2.econ.univpm.it/servizi/hpp/lucchetti
-------------------------------------------------------
38
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38
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4 comments
3 participants
participants (3)
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Alecos Papadopoulos -
Marcin Błażejowski -
Riccardo (Jack) Lucchetti