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Burden - Numerical Analysis 5e (PWS, 1993) Episode 2 Part 7

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) episode 2 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 7.1 Norms of Vectors and Matrices 393 ị Theorem 73 Cauchy Buniakowsky Schwarz Inequality for Sums Ị For each X x15 x2. . xf 1 and y yb y2 . . y ĩ in RF fdL i1 2 ÍJL i1 2 . 7.1 Ệ k-y.-l l y J Proof If Jfe 0 or X 0 the result is immediate since both sides of the inequality are zero. . Suppose y 0 0 and X 0 0. For each Ấ E R . j 0 x - Ay g X Xị - ẤyJ2 - 2 xĩ 2Ấ 2 w T A2 2 y - ụ 1 1 1 i i and 2Ấ 2 y - 2 A2 Sy IlK A2Ị y g . . Z I i l Since x 2 0 and II yg 0 we can let À x 2 y 2 .to give 2 INkẦ y II Ip 1 IMỈ II IP I IP 1 2 L H x3i I UK I 2 llylla 2llxll2 y Iiyih i i Iiyiiẽ u so 2 Ê xiyi 2 x g 2 llylb. yi Thus 2 3 UK llylỈ2 2 1 Replacing Xị by xf whenever x y - 0 and calling the vector X gives J i n r 77 1 1 2 I n I - 2 Kil - UK llyllz IKỉlylÍ2 1 s 4 f 1 s y f 1 . L 1 J U 1 J E E E With this result we see that for each x y E RF llx ylll 2 y 2 2 2 s 2 yl p.q i i j i 1 1 Í 1 1 so x y 2 x g 2 x 2 y 2 y 2 1 2 IIK y 2. Since the norm of a vector gives a measure for the distance between an arbitrary vector and the zero vector the distance between two vectors can be defined as the norm 3 of the difference of the vectors. Definition 7.4 If X x1 x2 . xf r and y yb y2 . yn 1 are vectors in ST the l2 and z distances between X and y are defined by r n i 1 2 llx ylỈ2 1 s - y 2 f and x - y - max xf - y . 394 CHAPTER 7 Iterative Techniques in Matrix Algebra EXAMPLE 2 The linear system 3.3330 15920 2 - 10.333 3 - 15913 2.2220 ị 16.710x2 9.6120 3 28.544 1.5611 4- 5.1791 2 1.6852 3 8.4254 has solution 2 x3 f 1.0000 1.0000 1.0000 . If Gaussian elimination is performed in five-digit arithmetic using maximal column pivoting Algorithm 6.2 the solution obtained is . X 1 2 x3 r 1.2001 0.99991 0.92538 Measurements of X - X are given by x - x U max 11.0000 - 1.20011 11.0000 - 0.999911 11.0000 - 0.92538 max 0.2001 0.00009 0.07462 0.2001 and llx - xị 2 1.0000 - 1.2001 2 1.0000 - 0.99991 2 1.0000 - 0.92538 2 172 0.200Ĩ 2 0.00009 2 0.07462 2 1 2 - 0.21356. Although the components 2 and 3 are good .