# findipl: Finds the s-left and s-right for a given internal point x[j] In inflection: Finds the Inflection Point of a Curve

## Description

From the definitions (1.3), (2.2) of references ,  it is necessary to find s_l and s_r in order to estimate the Extremum Surface Estimator (ESE) of the inflection point.

## Usage

 1 findipl(x, y, j) 

## Arguments

 x The numeric vector of x-abscissas, must be of length at least 4. y The numeric vector of the noisy or not y-ordinates, must be of length at least 4. j The data index j such that x=x_j

## Value

A list is returned that contains

 j The data index j such that x=x_j x=x_j The corresponding x-abscissa sl The value of s-left sr The value of s-right

## Note

This small function is used when we are scanning for the position of inflection point in ESE method.

## Author(s)

Demetris T. Christopoulos

## References

Demetris T. Christopoulos (2014). Developing methods for identifying the inflection point of a convex/concave curve. arXiv:1206.5478v2 [math.NA]. https://arxiv.org/pdf/1206.5478v2.pdf

Demetris T. Christopoulos (2016). On the efficient identification of an inflection point.International Journal of Mathematics and Scientific Computing, (ISSN: 2231-5330), vol. 6(1). https://veltech.edu.in/wp-content/uploads/2016/04/Paper-04-2016.pdf

## See Also

See also ese.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 # #Lets create some data based on the Fisher-Pry model, without noise: x<-seq(0,10,by=0.1);y<-5*(1+tanh(x-5)); tese=ese(x,y,0);tese; # j1 j2 chi # ESE 39 63 5 N<-length(x);N #  101 #We know that total symmetry exists, so for the middle point it is better to compute |sl|=|sr| j=(N-1)/2+1;j #  51 #Define the left and right chord: fl<-function(t){y + (y[j] - y) * (t - x) / (x[j] - x)} fr<-function(t){y[j] + (y[N] - y[j]) * (t - x[j]) / (x[N] - x[j])} #Find the s-left and s-right: LR<-findipl(x,y,j);LR; #  51.000000 5.000000 -9.031459 9.031459 #Show all results in a plot: plot(x,y,type="l",col="red") lines(c(x,x[j]),c(y,y[j]),type="l",col="green") lines(c(x[N],x[j]),c(y[N],y[j]),type="l",col="blue") points(x[j],y[j], type = "p",pch = 19,col="black") text(2.5,1,round(LR,digits=2)) text(6.5,7.5,round(LR,digits=2)) #The two surfaces are indeed absolutely equal |sl|=|sr| # 

### Example output j1 j2 chi
ESE 39 63   5
 101
 51
 51.000000  5.000000 -9.031459  9.031459


inflection documentation built on June 28, 2019, 5:03 p.m.