现代数值计算方法习题解答肖筱南

1、? 1 ? ? ? ? ? ? ? ? ? ?E? r E? ? ?E? r E? ? ?E? r E? ? ? 7 22 ? ? ? ? ? ?E? r E? 14. 3 E ? 14. 3 0013. 0 ? ? 101 ? ? ? 1 ? ?)( * xEr? )1( 10 12 1 = n ? =?n? ?)()( 1 )()( 1 )( * 1 1 * 1 1 * xxx n xEx n xE nn n = ? ?)( 11)()(1)( )( * * * * * 1 1 * * * * xE nx xx n x xxx n x xE xE r n n n n n r = = = ?

2、 ?=20? ?=)( * xE? * xx ?47 . 4 20 ? ?= * x? ?20 ? ? ? 1 ? ?)( * xEr? )1( 10 42 1 = n ? =?= * x? ?x? 2 xy =?x? * x? ? ? ? ? * y ?y? ? 2 ?)(20)(20)(2)( * xExxxxxyE= = ? ?)( * xE? ?)()( *1 xxnxxE nn ? ?nxnE x xx n x xE xE r n n n r 01. 0)( )( )( * = =? ? ? ? ? ? ? ? ? ?)()()( * tgtEttgtSSSE=? ? t tE gt

3、 ttgt S SS SEr )(2 2/ )( )( 2 * = =? ? ? ? ? ? ?2 0= x?41. 1 * 0= x? * 00 xx ? =?= 2 10 2 1 ? ? ? * 11 xx ?110110 * 00 +xx? * 00 xx ? =10? ? * 22 xx ?110110 * 11 +xx? * 11 xx ? = 2 10? ? * 1010 xx? 0? 22 23 = 2 0? 301 022 123 = 4 0? ? A? ? ?A = L LT? ? 3 11= l 3 32 21= l 3 6 22= l 3 3 31= l 3 6 32 =

4、l 2 33= l ? L ? 2 3 6 3 3 0 3 6 3 32 003 ? ? ? LY = b ? ? Y = ? 3 35 ? 3 6 ?2?T? ? ? LTX = Y . ? X =?T? ? ?2? ? 5 ? 11 a? 3 0? 22 23 = 2 0? 1203 022 323 = 6 0? ? A? ?A = L LT? ? 3 11= l 3 32 21= l 3 6 22= l 3 31= l 6 32 =l 3 33= l ? L ? 363 0 3 6 3 32 003 ? ? ? ? LY = b ? ? Y = ? 3 35 ? 6 6 ? 3 3 ?

5、T ? ? ? LTX = Y . ? X = ?1? 2 1 ? 3 1?T ? ? ?1=i ? 2 111 = ad ? ?2=i?1 21 =t? 2 1 21 =l? 2 5 2 =d ? ?3=i?1 31= t? 2 7 32= t? 2 1 31= l? 5 7 32 =l? 5 27 3= d ? ? A ? = 5 27 5 7 2 1 0 2 5 2 1 002 A ? LY = b? ? Y = ?4?7? 5 69 ?T ? ? DLTX = Y? ? X = ? 9 10? 9 7 ? 9 23?T ? ? 6 5? ?1? A = LU = 1 01 000 00

6、 0 0 0 0 0 0 10 01 001 5 4 3 2 l l l l 5 4 3 2 1 0 6 000 000 0 0 0 6 0 0 00 60 06 u u u u u ? 5 1= u ? 5 1 2= l ? 19 5 2= u ? 19 5 3= l ? 19 65 3= u ? 65 19 4= l ? 65 211 4= u ? 211 65 5= l ? 211 665 5= u ? ? LY = d? Y = ?1? 5 1 ?19 1 ? 65 1 ? 211 212 ?T ? ? UX = Y? ? X = ? 665 1509? 665 1145? 665 7

7、03? 665 395 ?665 212 ?T ? ?2? A = LU = 10 01 001 3 2 l l 3 2 1 00 10 01 u u u ?5 1= u ? 5 1 2= l? 5 24 2= u ? 24 5 3= l ? 24 115 3= u ? ? LY = d ? ? Y = ?17? 5 53? 24 115?T? ? UX = Y ? ? X = ?T ? ? ? ? ? ? ? 7 10 7 2 7 1 )( 3 )( 2 )1( 1 += +kkk xxx? 1 4 1 4 1 )( 3 )( 1 )1( 2 += +kkk xxx? 3 2 9 2 9 2

8、 )( 2 )( 1 )1( 3 += +kkk xxx? ? 7 10 7 2 7 1 )( 3 )( 2 )1( 1 += +kkk xxx? 1 4 1 4 1 )( 3 )1( 1 )1( 2 += +kkk xxx? 3 2 9 2 9 2 )1( 2 )1( 1 )1( 3 += +kkk xxx? ? ? 7 5 4 5 1 5 2 )( 3 )( 2 )1( 1 += +kkk xxx? 5 2 5 3 5 1 )( 3 )( 1 )1( 2 += +kkk xxx? 5 11 5 1 5 2 )( 2 )( 1 )1( 3 += +kkk xxx? ? 5 4 5 1 5 2

9、 )( 3 )( 2 )1( 1 += +kkk xxx? 5 2 5 3 5 1 )( 3 )1( 1 )1( 2 += +kkk xxx? 5 11 5 1 5 2 )1( 2 )1( 1 )1( 3 += +kkk xxx? ? ? ? ? 5 12 5 1 5 2 )( 3 )( 2 )1( 1 = +kkk xxx? 5 2 1 4 1 )( 3 )( 1 )1( 2 += +kkk xxx? 10 3 10 3 5 1 )( 2 )( 1 )1( 3 += +kkk xxx? ? )0( x = ?3?1?1?T? ? ? )18( x = ?3.999?2.999?1.999?T

10、 . ? ? 5 12 5 1 5 2 )( 3 )( 2 )1( 1 = +kkk xxx? 5 2 1 4 1 )( 3 )1( 1 )1( 2 += +kkk xxx? 10 3 10 3 5 1 )1( 2 )1( 1 )1( 3 += +kkk xxx? ? )0( x = ?T? ? ? )8( x ?T? ? ? ? ? 8 ? =+= 0 3 2 3 1 001 100 )( 1 ULDBJ? 3 2 3 1 01 10 = J BI? ?1)(= J B? ? ? =+= 100 100 100 )( 1U LDM? 100 10 10 = MI? ?0 21 =?1 3=

11、?1)(=M? ? ? =+= 0 2 1 2 1 101 2 1 2 1 0 )( 1 ULDBJ? 2 1 2 1 11 2 1 2 1 = J BI? ?0 1= ?i 2 5 2= ?i 2 5 3 =?1)(= J B? ? ? ? =+= 4 3 4 1 0 2 1 2 1 0 2 1 2 1 0 )( 1U LDM? 4 3 4 1 0 2 1 2 1 0 2 1 2 1 = MI? ? 2 1 21 =?0 3= ?1)(M? ? a 0 ? 1 1 1 aa aa aa = (1 ? a)2(1 ?2a) 0? A ? ? 9 ? BJ ? 0 0 0 aa aa aa ?

12、|BJI | = aa aa aa = )2()(23 2233 aaaa+=+ ? |2|)(aBJ=? ?-0.5 a 0.5 ? 12? A? ? 1 A? ? F A?09 . 0 01 . 0 25 . 0 36 . 0 +? ATA = 3 . 05 . 0 1 . 06 . 0 3 . 01 . 0 5 . 06 . 0 = 34 . 0 33 . 0 33 . 0 37 . 0 |AAI T | = 34 . 0 33 . 0 33 . 0 37 . 0 = 2 ? 0.71 ? 0.0169 ? 0 ? max (ATA) = 0.685? 2 A? 685 . 0 = 0.

13、83? 13?(1)? = = = xnxxxxxx n i i n i i n i ii ni 11 11 1 1 1 maxmax ? xnxx 1 (2)? )()()()( 21max 2 2 AAAAAAAAA T n TTT +=LL 2 11 2 1 2 1 2 2 1 2 1 F n j n i ij n i in n i i n i i Aaaaa=+= = LL 2 21max 2 2 1 )()()( 1 )( F T n TTT A n AAAAAA n AAA=+=LL ? FF AAA n 2 1 ? 10 ? ? ? ? ? ? ?13)( 4 +=xxxf?0.

14、3,0.4?034)( 3 f?0)4 . 0(f?0.3,0.4? ? 1 2 +k? =? ? ?35 . 0 0= x? ? k?0)( 1 xf?35. 0 ,325. 0? ? k?0)( 3 xf?35. 0 ,3375. 0? ? k?0)( 4 += xxf?)(xf? ? ?0)2(f?0) 1 (f? ? ? 1 2 12 + k =? ? ? ? ?4/? 4 sincos 1 kk k xx x + = + ? ?)4(log2 1kk xx= + ? ?5 . 1 0= x? ?x?+= 32 1)(xx?)( x? =?L ? ? 3 2 1 1 kk xx+= +

15、 ? ?x?+= 2 /11)(xx?)( x? =?L ? ?1 1 = +kk xx? ?x?=1)( 3 xx?1 3 1 = +kk xx? ? ?5 . 1 0= x? ? 1 x? 2 x? 3 x ? 4 x? 5 x? 6 x ? ? 56 xx ? 3 10 2 1 ? ?* x? 6 x? ?2 . 0 5 += xx?)(x? 5 1 2 . 0+= +kk xx? ? 5 xy =?2 . 0+= xy? ?x?+= 5 2 . 0)(xx?)( x? =?L ? ? 3 2 1 1 kk xx+= + ? ?5 . 0 0= x? 1 x? 2 x? 3 x? 4 x

16、? 5 x? 6 x? 7 x? 8 x ? 9 x ? ? 89 xx ? 3 10 2 1 ? ?* x? 9 x? ?x?=10/ )2()( x ex?)( x? =?L ? ?10/ )2( 1 kx k ex= + ? ?5 . 0 0= x? ? 12 1 x? 2 x? 3 x? 4 x? 5 x? 6 x? ? 56 xx ? 4 10 2 1 ? ?* x? 6 x? ?)( x?5 . 0 * =x? 5 . 0 |)( = = x x ex?q? ? += = + + 607. 1/6 . 0607. 1/ 11 1 kkk x k xxx ex k ? ?5 . 0

17、0= x? ? 1 x? 2 x? 3 x? ? 23 xx ? 4 10 2 1 ? ?* x? 3 x? ? ? + = += += + + + + + + kkk k kk k kk kk xxx xxx x xx xx 12 1 2 2 1 3 12 3 1 2 1 1 ? ?5 . 1 0= x? ? 1 x? 2 x? 3 x? ? 23 xx ? 4 10 2 1 ? ?* x? 3 x? ?axxf n =)(? 1 )( = n nxxf? ? += = + 11 1 ) 1( 1 n k k n k n k kk x a xn nnx ax xx?=k? ? n x a x

18、f=1)(? 1 )( + = n x na xf? += + a x xn n x xf xf xx n k k k k k kk ) 1( )( )( 1 ?=k? ? 13 ? n n a n a 1 )( + =? ?0)(=axxf n ? n k n k n k a n xa xa 2 1 )( lim 2 1 = + ? ?01)(= n x a xf? n k n k n k a n xa xa 2 1 )( lim 2 1 + = + ? ?13)( 3 =xxxf?0) 1 (f? 033)( 2 = xxf?06)( =xxf? ?0)2()2( ff?2 0= x? ?

19、2 0= x? ) 1(3 12 33 13 2 3 2 3 1 + = = + k k k kk kk x x x xx xx? ? 1 x? 2 x? 3 x? ? 23 xx ?x?0)( xf?0)( xf?0 xf?0)( C? 3 0 Cx ?0)()( 0 0 xfxf? k x? 3C ? ?), 0( 3 0 Cx ?0)2( 3 )( 3 2 0 3 2 0 2 0 3 3 2 0 2 03 1 + = + =xC x xC C x Cx Cx? ? 3 1 Cx ? 1 x? k x?3C ? ?0C?0 3 0 xfxf? k x? ? 14 3C ? ?)0 ,(3

20、0 Cx ?0)2( 3 )( 3 2 0 3 2 0 2 0 3 3 2 0 2 03 1 + = + =xC x xC C x Cx Cx? ? 3 1 Cx ? 1 x? k x?3C ? ?0=C? k k k kk x x x xx 3 2 3 2 3 1 = + ? ?Rx 0 ? ?1 0= x? 1 x? 2 x? 3 x? 4 x? 5 x? 6 x? 7 x? ? 67 xx ? 6 10 2 1 ? ?* x? 7 x? ?xxxfsin1)(=?0 0= x?1 1= x? ? ? ? * x? 4 x? ? 15 ? ? ? ? ? ?)()()()()()()( 2

21、211002 xlxfxlxfxlxfxL+=? ? 3 7 2 3 6 5 ) 12)(12( ) 1)(1( 4 )21)(11( )2)(1( ) 3(0 2 += + + + +=xx xxxx ? ?)()()()()()()()()( 332211003 xlxfxlxfxlxfxlxfxL+=? 123 )2)(1( 0 )2() 1( ) 3)(2( 3 )3()2() 1( )3)(2)(1( 2 + + = xxxxxxxxx ? 6 )2)(1( 2 ) 3)(2(3 3 ) 3)(2)(1( + = xxxxxxxxx ? ?)()()()()()()()()( 332

22、211003 xlxfxlxfxlxfxlxfxL+=? ?1214. 0=? ? ?0=k? ? ? k x?)( 0 xlx i n i k i = ? k x? i x ?n? ? ?ax =?bx =?)(xf? 0)()()( 1 + = ab ax bf ba bx afxL? ? )(max)(max 2 1 )()( ! 2 1 )()( 1 bxaxfbxaxfxLxf bxabxa = ? )(max)( 8 1 2 fab bxa ? ? 16 ?4 . 1 0= x?5 . 1 1= x?6 . 1 2= x? )54. 1 ()6 . 1 ()54. 1 ()5 .

23、1 ()54. 1 ()4 . 1 ()54. 1 ( 2102 lflflfL+=? ?+ + = )6 . 15 . 1)(4 . 15 . 1 ( )6 . 154. 1)(4 . 154. 1 ( 837. 1 )6 . 14 . 1)(5 . 14 . 1 ( )6 . 154. 1)(5 . 154. 1 ( 602. 1? ?94472. 1 )5 . 16 . 1)(4 . 16 . 1 ( )5 . 154. 1)(4 . 154. 1 ( 121. 2= +? ?)()()()()()()()()( 332211003 xlxfxlxfxlxfxlxfxL+=? ? 369

24、 )2)(3( 10 )3(36 )6)(3( 20 )9()6()3( )6)(3( + + + + = xxxxxxxxx ? ?)3242346323( 162 1 23 +=xxx? ? ? ?)()()(xgxfxF+=?)()()( 101nn xxxxxxx= + L? ? ? = + = + + = n j jn jj n j jn j n x xgxf x xF xxxF 0 1 0 1 10 )( )()( )( )( , L? ? = + = + += n j jn j n j jn j x xg x xf 0 1 0 1 )( )( )( )( ? ?, 1010nn

25、xxxgxxxfLL+=? ?c=, 0? ?1=? ? ? = + = + = n j njjjjjj j n j jn j n xxxxxxxx xf x xf xxxf 0 110 0 1 10 )()()( )( )( )( , LL L ? ? 17 ? i? i x ? i y ? i y? i y 2 ? i y 3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?0 0= x?1=h?5 . 0=x?5 . 0 0 = = h xx t? ?5 . 0=t? 875. 018 6 )5 . 1)(5 . 0(5 . 0 14 2 )5 . 0(5

26、. 0 5 . 01)5 . 0( 3 = + +=N? ?5 . 2=x?5 . 0 0 = = h xx t?5 . 0=t? ? 375.3518 6 )5 . 1)(5 . 0(5 . 0 32 2 )5 . 0(5 . 0 5 . 04764)5 . 2( 3 = +=N? ? i? i x ? i y ? i y? i y 2 ? i y 3 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?-1,2? ? 5 . 1=t?5 . 0=x? ? 18 ? ? ? 1i x? i x ? 1+i x? ?)()()( ! 3 1 )( 11 2+ = ii

27、i xxxxxxfxR? ?),( 11+ ii xx?hxx ii = 1 ?hxx ii += +1 ? ? 3 4 34 11 4 2 393 1 3 2 6 1 )()(max 6 1 )( 11 h e hexxxxxxexR iii xxx ii = + + ? ? 53 4 10 39 h e ? 0658. 0h. ? 19 ? ? ? ? = 24 21 53 42 A? = 14 6 3 11 b? = 493 330 AAT? = 29 73 bAT? ?bAAXA TT =? = 4456 . 0 4555 . 2 X? ?xaaxy 1011 )(+=? ?8=n?2

28、6.15 8 1 = =i ix ?1556.30 8 1 2 = =i ix ?227.145 8 1 = =i iy ?93628.286 8 1 = =i iiy x? ? 1556.3026.15 26.158 1 0 a a ? 93628.286 227.145 ? ?916. 3 0= a?464. 7 1= a?916. 3464. 7)( 11 +=xxy? ? ? 2 bxay+=? ?5=n?5327 5 1 2 = =i ix ?7277699 5 1 4 = =i ix ?4 .271 5 1 = =i iy ?5 .369321 5 1 2 = =i iiy x?

29、? 72776995327 53275 b a ? 5 .369321 4 .271 ? ?973. 0=a?050. 0=b? 2 050 . 0 973 . 0 xy+=? ?t?s? ? 20 ? 2 ctbtas+=? 6=n?7 .14 6 1 = =i it ?63.53 6 1 2 = =i it ?= = 6 1 3 i it ?= = 6 1 4 i it ? 280 6 1 = =i is ?1078 6 1 = =i iis t?= = 6 1 2 i iis t? ? 63.53 63.537 .14 63.537 .146 c b a ? 1078 280 ? ?=a

30、?=b?=c? 2 ctbtas+=? ? ? ? at eII = 0 ?eatIIlglglg 0 =? ?Iulg=? 0 lg IA =?eaBlg=?BtAu+=? ?7=n?5 . 3 7 1 = =i it ?03. 2 7 1 2 = =i it ?8638. 0 7 1 = =i i u?08067. 0 7 1 = =i iiu t? ? 03 . 2 5 . 3 5 . 37 B A ? 08067 . 0 8638 . 0 ? ?7509. 0=A?2546. 1=B?635 . 5 0= I?889. 2=a? ? t eI 889. 2 635. 5 =? ? bx

31、 aey =?ebxaylglglg+=? ?yulg=?aAlg=?ebBlg=?BtAu+=? ? 21 ?5=n?5 . 7 5 1 = =i ix ?875.11 5 1 2 = =i ix ?0848. 4 5 1 = =i i y?2645. 6 5 1 = =i iiy x? ? 875.115 . 7 5 . 75 B A ? 2645 . 6 0848 . 4 ? ?4874. 0=A?2197. 0=B?072. 3=a?5057. 0=b? ? x ey 5057. 0 072 . 3 =? ?Xx =ln?bXaxy+=)(? ?4=n?178. 3 4 1 = =i

32、i X?60914. 3 4 1 2 = =i i X?4 .14 4 1 = =i iy ?9605.12 4 1 = =i iiy X? ? 60914 . 3 178 . 3 178 . 3 4 b a ? 9605.12 4 .14 ? ?496. 2=a?402. 1=b?xxyln402. 149. 2)(+=? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 ? ? ? ?68394. 0 2 1 01 1 0 1 =+= eedxeI x ? ?63233. 04 6 1 0 2 1 1 1 0 2 =+= eeedxeI x ? ?: 08333. 0 12 1 )( 12 )( 0 3 1 = =ef ab R. 00035. 0 2 1 90 1 )( 2 )( 90 1 0 5 )4(5 2 = =ef a