21. Combining the two claims above reveals that the Perron-Frobenius eigenvalue " r " is simple root of the characteristic polynomial. 22. Then the number of sign changes ( in either chain ) around decreases by one, since is a simple root of. 23. The graph is tangent to this axis at the multiple roots of " f " and not tangent at the simple roots . 24. Although a given root system has more than one possible set of simple roots , the Weyl group acts transitively on such choices. 25. If is a simple root of the polynomial, then Laguerre's method converges cubically whenever the initial guess is close enough to the root. 26. Using one additional Horner evaluation, the value of the second derivative is used to obtain methods of cubic convergence order for simple roots . 27. If ? is a set of simple roots for ?, then ? ( " is a set of simple roots for ? ( ". 28. Note that while the Weyl group is abstractly isomorphic to the Coxeter group, a specific isomorphism depends on an ordered choice of simple roots . 29. Even though he spent most of his life outside Cephalonia, he knew the local culture intimately and he never forgot his own simple roots . 30. If ? is a set of simple roots for ?, then ? ( " is a set of simple roots for ? ( ".
