findroot {RootsExtremaInflections}

Find the root for a planar curve byn using Integration Root Finding Estimator (IEFE) algorithm

Description

Given a noisy or not planar curve as a set of discrete \left\{(x_i,y_i),i=1,2,\ldots n\right\} points we use Integration Root Finding Estimator (IRFE) algorithm as it is described at [1] in ordfer to find the root of it.

Usage

findroot(x, y, parallel = FALSE, silent = TRUE, tryfast = FALSE)

Arguments

x

A numeric vector for the independent variable without missing values

y

A numeric vector for the dependent variable without missing values

parallel

Logical input, if TRUE then parallel processing will be used (default=FALSE)

silent

Logical input, if TRUE then no details will be printed out during code execution (default=TRUE)

tryfast

Logical input, if TRUE then instead 'BEDE' will be used from IEFE algorithm instead of BESE (default=FALSE)

Details

The parallel=TRUE otpion must be used if length(x)>20000. The tryfast=TRUE can be used for big data sets, but BEDE is not so accuracy as BESE, so use it with caution.

Value

A named vector with next components is returned:

1. x1 the left endpoint of the final interval of BESE or BEDE iterations

2. x2 the right endpoint of the final interval of BESE or BEDE iterations

3. chi the estimation of extreme as x-abscissa

4. chi the estimation of extreme as y-abscissa taken from the interpolation polynomial of 2nd degree for the data points (x1,y1), (x2,y2), (chi,ychi)

Note

The 'yvalue' at output vector is an interpolation approxiamtion for the y-value of unknown function at its extreme point 'chi' and does not mean that it will be certainly accurate. Thta is the truth if underlying function can be well approximated by low order polynomials.

Author(s)

Demetris T. Christopoulos

References

[1]Demetris T. Christopoulos (2019). New methods for computing extremes and roots of a planar curve: introducing Noisy Numerical Analysis (2019). ResearchGate. http://dx.doi.org/10.13140/RG.2.2.17158.32324

See Also

scan_curve, scan_noisy_curve

Examples

#
## Legendre polynomial 5th order
f=function(x){(63/8)*x^5-(35/4)*x^3+(15/8)*x} 
x=seq(0.2,0.8,0.001);y=f(x);ya=abs(y)
plot(x,y,pch=19,cex=0.5,ylim=c(min(y),max(ya)))
abline(h=0);
lines(x,ya,lwd=4,col='blue')
rt=findroot(x,y)
rt
##           x1            x2           chi        yvalue 
## 5.370000e-01  5.400000e-01  5.385000e-01 -7.442574e-05 
abline(v=rt['chi'])
abline(v=rt[1:2],lty=2);abline(h=rt['yvalue'],lty=2)
points(rt[3],rt[4],pch=17,col='blue',cex=2)
#
## Same curve but with noise from U(-0.5,0.5)
#
set.seed(2019-07-24);r=0.05;y=f(x)+runif(length(x),-r,r)
ya=abs(y)
plot(x,y,pch=19,cex=0.5,ylim=c(min(y),max(ya)))
abline(h=0)
points(x,ya,pch=19,cex=0.5,col='blue')
rt=findroot(x,y)
rt
##         x1          x2         chi      yvalue 
## 0.53400000  0.53700000  0.53550000 -0.01762159 
abline(v=rt['chi'])
abline(v=rt[1:2],lty=2);abline(h=rt['yvalue'],lty=2)
points(rt[3],rt[4],pch=17,col='blue',cex=2)
#