/*
 The Great Computer Language Shootout
 http://shootout.alioth.debian.org/

 contributed by Java novice Jarkko Miettinen
 modified ~3 lines of the original C#-version
 by Isaac Gouy
 */

import java.text.DecimalFormat;
import java.text.NumberFormat;

public class spectralnorm
{

    private static final NumberFormat formatter = new DecimalFormat("#.000000000");

    public static void main(String[] args) {
        int n = 100;
        if (args.length > 0) n = Integer.parseInt(args[0]);

        System.out.println(formatter.format(new spectralnorm().Approximate(n)));
    }

    private final double Approximate(int n) {
        // create unit vector
        double[] u = new double[n];
        for (int i=0; i<n; i++) u[i] =  1;

        // 20 steps of the power method
        double[] v = new double[n];
        for (int i=0; i<n; i++) v[i] = 0;

        for (int i=0; i<10; i++) {
            MultiplyAtAv(n,u,v);
            MultiplyAtAv(n,v,u);
        }

        // B=AtA         A multiplied by A transposed
        // v.Bv /(v.v)   eigenvalue of v
        double vBv = 0, vv = 0;
        for (int i=0; i<n; i++) {
            vBv += u[i]*v[i];
            vv  += v[i]*v[i];
        }

        return Math.sqrt(vBv/vv);
    }


    /* return element i,j of infinite matrix A */
    private final double A(int i, int j){
        return 1.0/((i+j)*(i+j+1)/2 +i+1);
    }

    /* multiply vector v by matrix A */
    private final void MultiplyAv(int n, double[] v, double[] Av){
        for (int i=0; i<n; i++){
            Av[i] = 0;
            for (int j=0; j<n; j++) Av[i] += A(i,j)*v[j];
        }
    }

    /* multiply vector v by matrix A transposed */
    private final void MultiplyAtv(int n, double[] v, double[] Atv){
        for (int i=0;i<n;i++){
            Atv[i] = 0;
            for (int j=0; j<n; j++) Atv[i] += A(j,i)*v[j];
        }
    }

    /* multiply vector v by matrix A and then by matrix A transposed */
    private final void MultiplyAtAv(int n, double[] v, double[] AtAv){
        double[] u = new double[n];
        MultiplyAv(n,v,u);
        MultiplyAtv(n,u,AtAv);
    }
}