Rational Point: R program

2025-01-10T04:00:29Z

R program

← Older revision	Revision as of 04:00, 10 January 2025
Line 1:	Line 1:
		R is not a symbolic algrebra program, but it does have that capability of manipulating expressions and polynomials.

		Merely a graphical demonstration.

		<syntaxhighlight lang="R" line="1">
		#! /usr/bin/R -f
		# Point doubling illustrated

		E <- y^2 ~ x^3 - x + 1
		xrange <- seq(-1.5,1.5,length=4001)
		point1 <- 1.0

		stopifnot(E[[1]] == "~", E[[2]] == quote(y^2))

		f <- call("sqrt", E[[3]])
		g <- D(f,"x")
		slope <- eval(g,list(x=point1))
		intercept <- eval(f,list(x=point1)) - point1 * slope

		r <- function(x){((intercept + slope * x)^2
		- eval(E[[3]],list(x=x)))/(x-point1)^2 }

		point2 <- uniroot(r,lower=min(xrange),upper=max(xrange))$root

		yrange <- eval(f,list(x=xrange))

		xvals <- c(rev(xrange),xrange)
		yvals <- c(rev(yrange),-yrange)

		svg("point-doubling.R.svg")
		plot(xvals,yvals,xlab="x",ylab="y",
		type="l",asp=1,lwd=3,lty="22",col="red",
		main=paste("point-doubling on the elliptic curve ",
		as.character(as.expression(E[[2]])),
		" = ", as.character(as.expression(E[[3]]))))
		abline(0,0,col="blue")
		abline(intercept,slope)
		lines(c(point2,point2),
		c(-intercept-slopepoint2,intercept+slopepoint2),
		col=c("green","black"),lwd=2)
		points(c(point1,point2,point2),
		c(eval(f,list(x=point1)),
		-intercept-slope*point2,
		intercept+slope*point2))
		text(point1,eval(f,list(x=point1)),"P_1",pos=4)
		text(point2,-intercept-slopepoint2,"P_2 = 2P_1",pos=4)

		</syntaxhighlight>

Rational Point at 03:08, 10 January 2025

2025-01-10T03:08:25Z

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