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 18 18.1 (a) 11.0791812 0.90309(10) 0.90309 (10 8) 0.99113612 8f 1 0.991136100% 0.886%1t (b) 11.0413927 0.9542425(10) 0.9542425 (10 9) 0.99781811 9f 1 0.997818100% 0.218%1t 18.2 First, order the points x0 = 9 f(x0) = 0.9542425 x1 = 11 f(x1) = 1.0413927 x2 = 8 f(x2) = 0.9030900 Applying Eq. (18.4) b0 = 0.9542425 Equation (18.5) yields 11.0413927 0.95424250.043575111 9b Equation (18.6) gives 20.9030900 1.04139270.04357510.0461009 0.04357518110.002525889 89b Substituting these values into Eq. (18.3) yields the quadratic formula 2( ) 0.9542425 0.0435751( 9) 0.0025258( 9)( 11)fx x x x which can be evaluated at x = 10 for 2(10) 0.9542425 0.0435751(10 9) 0.0025258(10 9)(10 11) 1.0003434f 18.3 First, order the points x0 = 9 f(x0) = 0.9542425 x1 = 11 f(x1) = 1.0413927 x2 = 8 f(x2) = 0.9030900 x3 = 12 f(x3) = 1.0791812 The first divided differences can be computed as2 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. 101.0413927 0.9542425[ , ] 0.043575111 9fx x 210.9030900 1.0413927[ , ] 0.0461009811fx x 321.0791812 0.9030900[ , ] 0.044022812 8fx x The second divided differences are 2100.0461009 0.0435751[ , , ] 0.002525889fx xx 3210.0440228 0.0461009[ , , ] 0.002078112 11fx x x The third divided difference is 32100.0020781 ( 0.0025258)[ , , , ] 0.0001492412 9fx x xx Substituting the appropriate values into Eq. (18.7) gives 3( ) 0.9542425 0.0435751( 9) 0.0025258( 9)( 11) 0.00014924( 9)( 11)( 8)fx x x xxx x which can be evaluated at x = 10 for 3( ) 0.9542425 0.0435751(10 9) 0.0025258(10 9)(10 11) 0.00014924(10 9)(10 11)(10 8) 1.0000449fx 18.4 18.1 (a): x0 = 8 f(x0) = 0.9030900 x1 = 12 f(x1) = 1.0791812 110 12 10 8(10) 0.9030900 1.0791812 0.991136812 128f 18.1 (b): x0 = 9 f(x0) = 0.9542425 x1 = 11 f(x1) = 1.0413927 110 11 10 9(10) 0.9542425 1.0413927 0.997818911 119f 18.2: x0 = 8 f(x0) = 0.9030900 x1 = 9 f(x1) = 0.9542425 x2 = 11 f(x2) = 1.04139273 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(10 9)(10 11) (10 8)(10 11)(10) 0.9030900 0.9542425(8 9)(8 11) (9 8)(9 11)(10 8)(10 9) 1.0413927 1.0003434(11 8)(11 9)f 18.3: x0 = 8 f(x0) = 0.9030900 x1 = 9 f(x1) = 0.9542425 x2 = 11 f(x2) = 1.0413927 x3 = 12 f(x3) = 1.0791812 3(10 9)(10 11)(10 12) (10 8)(10 11)(10 12)(10) 0.9030900 0.9542425(8 9)(8 11)(8 12) (9 8)(9 11)(9 12)(10 8)(10 9)(10 12) (10 8)(10 9)(10 11) 1.0413927(11 8)(11 9)(11 12) (12 8)(12 9)(12 11)f 1.0791812 1.0000449 18.5 First, order the points so that they are as close to and as centered about the unknown as possible x0 = 2.5 f(x0) = 14 x1 = 3.2 f(x1) = 15 x2 = 2 f(x2) = 8 x3 = 4 f(x3) = 8 x4 = 1.6 f(x4) = 2 Next, the divided differences can be computed and displayed in the format of Fig. 18.5, i xi f(xi)f[xi+1,xi] f[xi+2,xi+1,xi]f[xi+3,xi+2,xi+1,xi]f[xi+4,xi+3,xi+2,xi+1,xi]0 2.5 14 1.428571 -8.809524 1.011905 1.847718 1 3.2 15 5.833333 -7.291667 -0.651042 2 2 8 0 -6.25 3 4 8 2.5 4 1.6 2 The first through third-order interpolations can then be implemented as 1(2.8) 14 1.428571(2.8 2.5) 14.428571f 2(2.8) 14 1.428571(2.8 2.5) 8.809524(2.8 2.5)(2.8 3.2) 15.485714f 3(2.8) 14 1.428571(2.8 2.5) 8.809524(2.8 2.5)(2.8 3.2) 1.011905(2.8 2.5)(2.8 3.2)(2.8 2.) 15.388571f The error estimates for the first and second-order predictions can be computed with Eq. 18.19 as 115.485714 14.428571 1.057143R 215.388571 15.485714 0.097143R The error for the third-order prediction can be computed with Eq. 18.18 as 31.847718(2.8 2.5)(2.8 3.2)(2.8 2)(2.8 4) 0.212857R 18.6 First, order the points so that they are as close to and as centered about the unknown as possible4 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. x0 = 3 f(x0) = 19 x1 = 5 f(x1) = 99 x2 = 2 f(x2) = 6 x3 = 7 f(x3) = 291 x4 = 1 f(x4) = 3 Next, the divided differences can be computed and displayed in the format of Fig. 18.5, i xi f(xi)f[xi+1,xi] f[xi+2,xi+1,xi]f[xi+3,xi+2,xi+1,xi]f[xi+4,xi+3,xi+2,xi+1,xi]0 3 19 40 9 1 0 1 5 99 31 13 1 2 2 6 57 9 3 7 291 48 4 1 3 The first through fourth-order interpolations can then be implemented as 1(4) 19 40(4 3) 59f 2(4) 59 9(4 3)(4 5) 50f 3(4) 50 1(43)(45)(42) 48f 4(4) 48 0(4
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