[R] solving integral equations with undefined parameters using multiroot

[Abbs Spurdle]

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]]
> >
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