[R] solving integral equations with undefined parameters using multiroot

Fri May 7 04:02:16 CEST 2021

lm (cl$y ~ cl$x)$coef
(Intercept)        cl$x
  0.1817509  -1.0000000


On Fri, May 7, 2021 at 1:56 PM Abbs Spurdle <spurdle.a using gmail.com> wrote:
>
> #using vF1 function
> #from my previous posts
> u <- seq (0, 0.25,, 200)
> cl <- contourLines (u, u, outer (u, u, vF1),, 0)[[1]]
> plot (cl$x, cl$y, type="l")
>
>
> On Thu, May 6, 2021 at 10:18 PM Ursula Trigos-Raczkowski
> <utrigos using umich.edu> wrote:
> >
> > Thanks for your reply. Unfortunately the code doesn't work even when I change the parameters to ensure I have "different" equations.
> > Using mathematica I do see that my two equations form planes, intersecting in a line of infinite solutions but it is not very accurate, I was hoping R would be more accurate and tell me what this line is, or at least a set of solutions.
> >
> > On Thu, May 6, 2021 at 5:28 AM Abbs Spurdle <spurdle.a using gmail.com> wrote:
> >>
> >> Just realized five minutes after posting that I misinterpreted your
> >> question, slightly.
> >> However, after comparing the solution sets for *both* equations, I
> >> can't see any obvious difference between the two.
> >> If there is any difference, presumably that difference is extremely small.
> >>
> >>
> >> On Thu, May 6, 2021 at 8:39 PM Abbs Spurdle <spurdle.a using gmail.com> wrote:
> >> >
> >> > Hi Ursula,
> >> >
> >> > If I'm not mistaken, there's an infinite number of solutions, which
> >> > form a straight (or near straight) line.
> >> > Refer to the following code, and attached plot.
> >> >
> >> > ----begin code---
> >> > library (barsurf)
> >> > vF1 <- function (u, v)
> >> > {   n <- length (u)
> >> >     k <- numeric (n)
> >> >     for (i in seq_len (n) )
> >> >         k [i] <- intfun1 (c (u [i], v [i]) )
> >> >     k
> >> > }
> >> > plotf_cfield (vF1, c (0, 0.2), fb = (-2:2) / 10,
> >> >     main="(integral_1 - 1)",
> >> >     xlab="S[1]", ylab="S[2]",
> >> >     n=40, raster=TRUE, theme="heat", contour.labels=TRUE)
> >> > ----end code----
> >> >
> >> > I'm not familiar with the RootSolve package.
> >> > Nor am I quite sure what you're trying to compute, given the apparent
> >> > infinite set of solutions.
> >> >
> >> > So, for now at least, I'll leave comments on the root finding to someone who is.
> >> >
> >> >
> >> > Abby
> >> >
> >> >
> >> > On Thu, May 6, 2021 at 8:46 AM Ursula Trigos-Raczkowski
> >> > <utrigos using umich.edu> wrote:
> >> > >
> >> > > Hello,
> >> > > I am trying to solve a system of integral equations using multiroot. I have
> >> > > tried asking on stack exchange and reddit without any luck.
> >> > > Multiroot uses the library(RootSolve).
> >> > >
> >> > > I have two integral equations involving constants S[1] and S[2] (which are
> >> > > free.) I would like to find what *positive* values of S[1] and S[2] make
> >> > > the resulting
> >> > > (Integrals-1) = 0.
> >> > > (I know that the way I have the parameters set up the equations are very
> >> > > similar but I am interested in changing the parameters once I have the code
> >> > > working.)
> >> > > My attempt at code:
> >> > >
> >> > > ```{r}
> >> > > a11 <- 1 #alpha_{11}
> >> > > a12 <- 1 #alpha_{12}
> >> > > a21 <- 1 #alpha_{21}
> >> > > a22 <- 1 #alpha_{22}
> >> > > b1 <- 2  #beta1
> >> > > b2 <- 2 #beta2
> >> > > d1 <- 1 #delta1
> >> > > d2 <- 1 #delta2
> >> > > g <- 0.5 #gamma
> >> > >
> >> > >
> >> > > integrand1 <- function(x,S) {b1*g/d1*exp(-g*x)*(1-exp(-d1*
> >> > > x))*exp(-a11*b1*S[1]/d1*(1-exp(-d1*x))-a12*b2*S[2]/d2*(1-exp(-d2*x)))}
> >> > > integrand2 <- function(x,S) {b2*g/d2*exp(-g*x)*(1-exp(-d2*
> >> > > x))*exp(-a22*b2*S[2]/d2*(1-exp(-d2*x))-a21*b1*S[1]/d1*(1-exp(-d1*x)))}
> >> > >
> >> > > #defining equation we would like to solve
> >> > > intfun1<- function(S) {integrate(function(x) integrand1(x,
> >> > > S),lower=0,upper=Inf)[[1]]-1}
> >> > > intfun2<- function(S) {integrate(function(x) integrand2(x,
> >> > > S),lower=0,upper=Inf)[[1]]-1}
> >> > >
> >> > > #putting both equations into one term
> >> > > model <- function(S) c(F1 = intfun1,F2 = intfun2)
> >> > >
> >> > > #Solving for roots
> >> > > (ss <-multiroot(f=model, start=c(0,0)))
> >> > > ```
> >> > >
> >> > > This gives me the error Error in stode(y, times, func, parms = parms, ...) :
> >> > >   REAL() can only be applied to a 'numeric', not a 'list'
> >> > >
> >> > > However this simpler example works fine:
> >> > >
> >> > > ```{r}
> >> > > #Defining the functions
> >> > > model <- function(x) c(F1 = x[1]+ 4*x[2] -8,F2 = x[1]-4*x[2])
> >> > >
> >> > > #Solving for the roots
> >> > > (ss <- multiroot(f = model, start = c(0,0)))
> >> > > ```
> >> > >
> >> > > Giving me the required x_1= 4 and x_2 =1.
> >> > >
> >> > > I was given some code to perform a least squares analysis on the same
> >> > > system but I neither understand the code, nor believe that it is doing what
> >> > > I am looking for as different initial values give wildly different S values.
> >> > >
> >> > > ```{r}
> >> > > a11 <- 1 #alpha_{11}
> >> > > a12 <- 1 #alpha_{12}
> >> > > a21 <- 1 #alpha_{21}
> >> > > a22 <- 1 #alpha_{22}
> >> > > b1 <- 2  #beta1
> >> > > b2 <- 2 #beta2
> >> > > d1 <- 1 #delta1
> >> > > d2 <- 1 #delta2
> >> > > g <- 0.5 #gamma
> >> > >
> >> > >
> >> > > integrand1 <- function(x,S) {b1*g/d1*exp(-g*x)*(1-exp(-d1*
> >> > > x))*exp(-a11*b1*S[1]/d1*(1-exp(-d1*x))-a12*b2*S[2]/d2*(1-exp(-d2*x)))}
> >> > > integrand2 <- function(x,S) {b2*g/d2*exp(-g*x)*(1-exp(-d2*
> >> > > x))*exp(-a22*b2*S[2]/d2*(1-exp(-d2*x))-a21*b1*S[1]/d1*(1-exp(-d1*x)))}
> >> > >
> >> > > #defining equation we would like to solve
> >> > > intfun1<- function(S) {integrate(function(x)integrand1(x,
> >> > > S),lower=0,upper=Inf)[[1]]-1}
> >> > > intfun2<- function(S) {integrate(function(x)integrand2(x,
> >> > > S),lower=0,upper=Inf)[[1]]-1}
> >> > >
> >> > > #putting both equations into one term
> >> > > model <- function(S) if(any(S<0))NA else intfun1(S)**2+ intfun2(S)**2
> >> > >
> >> > > #Solving for roots
> >> > > optim(c(0,0), model)
> >> > > ```
> >> > >
> >> > > I appreciate any tips/help as I have been struggling with this for some
> >> > > weeks now.
> >> > > thank you,
> >> > > --
> >> > > Ursula
> >> > > Ph.D. student, University of Michigan
> >> > > Applied and Interdisciplinary Mathematics
> >> > > utrigos using umich.edu
> >> > >
> >> > >         [[alternative HTML version deleted]]
> >> > >
> >> > > ______________________________________________
> >> > > R-help using r-project.org mailing list -- To UNSUBSCRIBE and more, see
> >> > > https://stat.ethz.ch/mailman/listinfo/r-help
> >> > > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> >> > > and provide commented, minimal, self-contained, reproducible code.
> >
> >
> >
> > --
> > Ursula Trigos-Raczkowski (she/her/hers)
> > Ph.D. student, University of Michigan
> > Applied and Interdisciplinary Mathematics
> > 5828 East Hall
> > 530 Church St.
> > Ann Arbor, MI 48109-1085
> > utrigos using umich.edu
> >