tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 14

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 452 Chapter 6. Solving Scalar Nonlinear Equations the coefficients of p z are not given as the original data it may be better to avoid computing them. An important example of this is the determination of eigenvalues of matrices the eigenvalues are zeros of the characteristic equation p A det A Al 0. Here the original data are the elements of the matrix A. Numerical values of p A can in general be evaluated much more accurately directly from the matrix elements see Chapter 9. Another example is given below. Example . The largest positive root of the equation p x x 2 x2 - I 6 - 3 10 6 X11 0 is to be computed. Here p z is a polynomial of degree 13. If the coefficients are computed using decimal floating point arithmetic with seven digits then the coefficient of X11 which is 12 3 10-6 will be rounded to . Thus the machine will treat the equation x 2 x2 I 6 0 whose exact positive root is 1. This is a poor result. One can get the root a to full accuracy for example by writing the equation in the form A 1 6 z - and solving this by the iteration Xo 1 Xfe I j xk . Hence the relative error in the previous result is greater then 5 . Simultaneous Determination of Roots We now consider iterative methods that under appropriate separation assumptions allows for the simultaneous determination of all the roots of a polynomial equation. Suppose that the numbers are a set of n distinct approximations to the zeros ữị i 1 n of p z . In Weierstrass method one computes a new set of approximations using the iteration formula n ep 1 -eịfc i l n. j l Wi Íl. Xí. KI . AK J1 _ r __1 _ _ 1 _ _ 1 _ J J_ _ I I V 1 . . . V I t h A Ì I o WA m T A I c o K zv ĨĨTV1 TT zxw VV1U11 y Ỉ V Si S9 77 kiiv 1O1 IllUiH may also Uv WTlLLvIi fc-f-l _ - A . - A F 7 F 7 F 7 I T F- I. Si Si PxFti 7 y vSi h which shows that to first approximation the method is identical to Newton s method. This relation can be used to prove that the asymptotic order of convergence