在R中的线性和非线性方程的交点处求x值

Question

I have two functions: one for a line (y) and another for a curve (hnc). I would like to determine the one x-value at which the two functions intersect

sigma = 0.075
mu = 0 
r=0.226 
theta=0.908 
H=0.16 

hnc <- function(x) (1/(sigma*sqrt(2*pi)))*(exp(-(x^2)/(2*(sigma^2))))
y <- function(x) 2*pi*x+(pi*r^2/((360/theta)/H))

curve(hnc,0,r,n=100,col="blue")
plot(y,0,r,add=T,col="red")

I have tried using the nleqslv package, but this results in two separate x-values that do not agree (perhaps because I am using it incorrectly)

int <- function(x){
z <- numeric(2) 
z[1] <- (1/(sigma*sqrt(2*pi)))*(exp(-(x[1]^2)/(2*(sigma^2))))
z[2] <- 2*pi*x[2]+(pi*r^2/((360/theta)/H))
z}

nleqslv(c(0.14,0.14),int,method="Broyden")

Any help would be much appreciated!

Thanks, Eric

Answer 1

Using optimize here to find the minimum of a function if a single variable seems to work well

xx <- optimize(function(x) abs(hnc(x)-y(x)), c(.10,.20))$minimum
abline(v=xx, lty=2)

Answer 2

You are not using nleqslv in the correct way. It is meant for solving a system of non linear equations with as many variables as there are equations.

You have two functions and you want to determine the intersection which in your case consists of a single value for x .

You need to define a new function like this

g <- function(x) hnc(x) - y(x)

Then you can use uniroot to find a zero of g(x) like this:

uniroot(g,c(0,1))

The root found will be 0.1417802 which corresponds with the graph in the first answer.

Minimizing won't always work to find a point of intersection; if there is no point of intersection you will get misleading results.