1 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. CHAPTER 7 7.1 In a fashion similar to Example 7.1, n = 4, a0 = 20, a1 = 3, a2 = 14.5, a3 = 7.5, a4 = 1 and t = 2. These can be used to compute r = a4 = 1 a4 = 0 For i = 3, s = a3 = 7.5 a3 = r = 1 r = s + rt = 7.5 + 1(2) = 5.5 For i = 2, s = a2 = 14.5 a2 = r = 5.5 r = s + rt = 14.55.5(2) = 3.5 For i = 1, s = a1 = 3 a1 = r = 3.5 r = s + rt = 3+3.5(2) = 10 For i = 0, s = a0 = 20 a0 = r = 10 r = s + rt = 20+10(2) = 0 Therefore, the quotient is x3 – 5.5x2 + 3.5x +10 with a remainder of zero. Thus, 2 is a root. This result can be easily verified with MATLAB, >> a = [1 -7.5 14.5 3 -20]; >> b = [1 -2]; >> [d,e] = deconv(a,b) d = 1.0000 -5.5000 3.5000 10.0000 e = 0 0 0 0 0 7.2 In a fashion similar to Example 7.1, n = 5, a0 = 10, a1 = 7, a2 = 6, a3 = 1, a4 = 5, a5 = 1, and t = 2. These can be used to compute r = a4 = 1 a4 = 0 For i = 4, s = a4 = 5 a3 = r = 1 r = s + rt = 5 + 1(2) = 3 For i = 3, s = a3 = 1 a3 = r = 32 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. r = s + rt = 1 3(2) = 5 For i = 2, s = a2 = 6 a2 = r = 5 r = s + rt = 6 5(2) = 16 For i = 1, s = a1 = 7 a1 = r = 16 r = s + rt = 7 16(2) = 39 For i = 0, s = a0 = 10 a0 = r = 39 r = s + rt = 10 39(2) = 68 Therefore, the quotient is x4 – 3x3 – 5x2 – 16x – 39 with a remainder of –68. Thus, 2 is not a root. This result can be easily verified with MATLAB, >> a=[1 -5 1 -6 -7 10]; >> b = [1 -2]; >> [d,e] = deconv(a,b) d = 1 -3 -5 -16 -39 e = 0 0 0 0 0 -68 7.3 (a) A plot indicates a root at about x = 2. -40-200204060-4 -2 0 2 4 Try initial guesses of x0 = 1, x1 = 1.5, and x2 = 2.5. Using the same approach as in Example 7.2, First iteration: f(1) = –6 f(1.5) = –4.375 f(2.5) = 7.875 h0 = 0.5 h1 = 1 0 = 3.25 1 = 12.25 65.0125.325.12a 25.1825.12)1(6b c = 7.875 979384.1)875.7)(6(425.1825.18)875.7(25.223x %30.26%100979384.15.2979384.1a The iterations can be continued as tabulated below:3 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. i x3 a 0 1.979384 26.3019% 1 1.999579 1.0100% 2 2 0.0210% 3 2 0.0000% (b) A plot indicates a root at about x = 0.5. -100-50050100-4 -3 -2 -1 0 1 2 3 4 Try initial guesses of x0 = 0.5, x1 = 1, and x2 = 1.5. Using the same approach as in Example 7.2, First iteration: f(0.5) = 0 f(1) = 2.5 f(1.5) = 6.25 h0 = 0.5 h1 = 0.5 0 = 5 1 = 7.5 5.25.05.055.7a 75.85.7)5.0(5.2b c = 6.25 5.0)25.6)(5.2(475.875.8)25.6(25.123x %200%1005.05.15.0a The iterations can be continued as tabulated below: i x3 a 0 0.5 200% 1 0.5 0% 7.4 Here are MATLAB sessions to determine the roots: (a) >> a=[1 -1 2 -2]; >> roots(a) ans = 0.0000 + 1.4142i 0.0000 - 1.4142i 1.0000 (b) >> a=[2 0 6 0 8]; >> roots(a) ans = -0.5000 + 1.3229i -0.5000 - 1.3229i 0.5000 + 1.3229i4 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 0.5000 - 1.3229i (c) >> a=[1 -2 6 -2 5]; >> roots(a) ans = 1.0000 + 2.0000i 1.0000 - 2.0000i -0.0000 + 1.0000i -0.0000 - 1.0000i 7.5 (a) A plot suggests 3 real roots: 0.44, 2 and 3.3. -4-202401234 Try r = 1 and s = –1, and follow Example 7.3 1st iteration: r = 1.085 s = 0.887 r = 2.085 s = –0.1129 2nd iteration: r = 0.4019 s = –0.5565 r = 2.487 s = –0.6694 3rd iteration: r = –0.0605 s = –0.2064 r = 2.426 s = –0.8758 4th iteration: r = 0.00927 s = 0.00432 r = 2.436 s = –0.8714 22)8714.0(4436.2436.2root21 4357.02)8714.0(4436.2436.2root22 The remaining root3 = 3.279. (b) Plot suggests 3 real roots at approximately 0.9, 1.2 and 2.3.5 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. -2-101230.511.522.5 Try r = 2 and s = –0.5, and follow Example 7.3 1st iteration: r = 0.2302 s = –0.5379 r = 2.2302 s = –1.0379 2nd iteration: r = –0.1799 s = –0.0422 r = 2.0503 s = –1.0801 3rd iteration: r = 0.0532 s = –0.01641 r = 2.1035 s = –1.0966 4th iteration: r = 0.00253 s = –0.00234 r = 2.106 s = –1.099 1525.12)099.1(4106.2106.2root21 9535.02)099.1(4106.2106.2root22 The remaining …
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