1,813被浏览209,207分享邀请回答上，摇滚和流行被列为同一分支，也就是说很难区分这两者的具体区别。尤其是音乐形式的长期演化，流行和摇滚在表现方式上也进行了互相的借鉴，也就更难具体区别两者的不同。当然，有些流行和摇滚还是非常好区分的，KTV里的口水歌必定是流行音乐，在LIVE HOUSE里演出的金属朋克之类定当属于摇滚无疑。但又有多少人能说清BEYOND的音乐到底属于流行还是摇滚呢，在某些人眼里，他们是华语摇滚的经典代表，而对于另外一些人，他们只是纯粹的流行音乐而已=====================================================================下面贴一个之前写的一篇关于国外经典摇滚乐的总结，对于刚听摇滚的朋友也许有所借鉴=====================================================================西方摇滚杂志（主要是美国和英国）很喜欢隔三岔五地搞个总结，比如“有史以来最好的100首歌(The Best 100 Singles of All Time)”，“有史以来最伟大的100张专辑(The Best 100 Albums of All Time)”。这样做主要是为了标榜自己的权威性，从而吸引读者，反过来读者又很喜欢这种榜单，于是各种各样名目繁多的排行榜便出现了。有排行榜的那期往往会比一般的卖的好些，甚至是大卖，于是榜单就成为了杂志的强心剂，什么时候销量下降了来那么一下保准管用现在的摇滚青年很幸福，我刚开始听摇滚的时候基本上是没有榜单的，好的音乐只能靠掏，有时掏到一张尖货还被万恶的打口贩子把价钱炒的极高。后来出现了若干本摇滚杂志，那些杂志对于那当时的我来说就像大海上的灯塔。可万事并不全如意，有时人家介绍的音乐我不喜欢，好不容易碰到介绍的我又喜欢又往往买不到打口带或者价钱被炒得很高。好在那时不像现在这么 “挑食”，一张只要还可以的专辑就能听很久，就像刚尝到禁果的小伙子，给个姑娘就能陶醉半天在互联网的时代，有各种摇滚网站介绍音乐，有各种P2P软件提供下载，可信息泛滥的时代过多的选择结果往往是无从选择，这时各种音乐榜单又成为了新的灯塔。这个世界音乐越来越多元化，出张专辑也比以前容易很多，榜单的作用也就越发明显，所以我认为找喜欢的音乐除了朋友推荐以外，从各种榜单上找是我认为效率最高的不像国内的大部分排行榜（那些榜单主要是唱片公司操纵的，所以基本上是关系榜）国外杂志的榜单是比较独立的，拿我经常看的《滚石》《Q》《NME》来说，他们基本上代表了目前主流摇滚的走向，虽然由于杂志所属地原因《滚石》偏向于美国而Q,NME偏向于英国，但自从摇滚乐诞生那天起就是以英美为主，所以鲜有其他国家的音乐在摇滚乐的舞台上有多大作为。榜单中公认比较权威的是《滚石》杂志在2003年推出的“史上最佳500张（首）专辑（单曲）”不过由于此榜单是音乐人和杂志编辑评选的，音乐更多的偏向于美国六七十年代的摇滚乐，不可否认那个年代是摇滚乐的黄金年代，而且确实出现了一大批相当经典的歌曲，可90年甚至80年以后的音乐有些经典都被忽略了，也许下一次的TOP500 ALL TIME会有所改变吧去年年底新浪做了个主题“全球经典摇滚100首”把这些歌的MV做成了集锦，刚看时很激动，可后来发现并不是所有歌都是我认为经典的，而且其中还有后街男孩及迈克尔杰克逊等流行音乐。原来这个榜单是从《滚石》的“历史上最流行单曲100首”拷贝过来的，而新浪是很有中国特色的把东西拿过来然后自己改个名字而已。之后我又查看了各种榜单，没有一个是完全合我意的，于是便开始酝酿这个摇滚经典100的榜单之所以称为经典，就是因为经过时间的沉淀，所以专辑基本上不考虑2000年以后的。而且由于工业，金属，硬核等音乐形式在摇滚乐里也不算主流，所以只收录了Metallica和Manson的总共两首歌，其他很多我喜欢的乐队的音乐只能割爱。所以这张精选集不能代表我所有的音乐喜好，它更像是一张摇滚乐的入门精选。每个乐队只选一两首音乐，Beatles除外，因为太经典了从 98年开始听摇滚到现在已经10多年了，买的打口磁带加唱片总共也快四五百张（盘）了，再加上后来上网下的MP3自认为对西方摇滚乐还是大概了解的。不过听音乐这东西总是很私人化的，每个人口味都不一样，就算是同一个人在不同时期听音乐的偏好也有不同，所以不要跟我说你觉得哪些歌应该入选或者哪个乐队该选哪首歌，一句话，这是我个人的榜单从开始做这个精选集到完成用了将近4个月的时间，其间听了我认为的经典乐队/歌曲累计几千首，参考了《滚石》杂志的“史上最佳单曲500首”《Q》《NME》的几个史上最佳单曲100首榜单，反复修改过多遍才完成。在编精选集的过程中，我又重温了很多以前喜欢乐队的音乐，还有很多以前“只闻其名，未闻其声”的乐队和以前听过但并不喜欢的音乐。这真是一个令人惊喜的过程，不但找回了初听摇滚乐时的那种激动也兴奋，也发现了原来Pink Floyd原来真是很经典，Dire Straits并不像我以前觉得的那么难听，甚至是相当好听的有些很不错的音乐我并没有选进去，比如“Hotel California” 和“ Yellow”，这两首歌都是好歌，但由于被播放次数过多导致我只能被动地就受它，也就是说如果我是在街边路过某个音像店或者服装店时听到他们我会觉得那是经典，但在家里让我连续听超过两遍以上我就会起鸡皮疙瘩，这就好比一般人在第N次听My Heart Will Go On时的感觉一样最后，向这张精选集中出现的所有乐队及音乐人致敬，正是他们的音乐使我们的世界更加精彩！The Beatles· Nirvana ·U2·Guns N'Roses ·R.E.M. · Santana ·The Police ·Eric Clapton·The Clash·Red Hot Chili Peppers·Van Halen·Bob Marley·Bob Dylan·Bon Jovi·The Velvet Underground·The Cure·The Smashing Pumpkins·John Lennon·Gorillaz·Oasis·Marilyn Manson·The Verve·Jamiroquai·Elvis Presley·The Offspring·The Mamas,The Papas·Run D.C.C.·James Brown·B.B.King·Blur·The Sex Pistols·Queen·Gary Moore·Mr.Big·Suede·Green Day·Radiohead·Tom Waits·Johnny Cash·Ben E.King·Pink Floyd·Chuck Berry·David Bowie·Scorpions·Black Sabbath·No Vacancy·Concrete Blonde·AC/DC·Jimi Hendrix·Cream·The Ramones·Joy Division·Metallica·Kid Rock·The Rolling Stones·Massive Attack·Dire Straits·4 Non Blondes·Aerosmith·Prince·Deep Purple·Bruce Springsteen·Beck·The Stooges·Jackson Browne·Fleetwood Mac·The Who·The Kinks·The Byrds·Jeff Buckley·The Doors·Iggy Pop·Buddy Holly·The Band·Led Zeppelin·Little Richard·Simon & Garfunkel·Lou Reed·Eagles·Mott The Hoople·The Strokes·Supergrass·Spencer Davis Group·Pulp·Grateful Dead前50首试听：后50首试听：01 Yesterday -- The Beatles02 Smells Like Teen Spi... -- Nirvana03 With or Without You -- U204 Sweet Child O' Mine -- Guns N' Roses05 Losing My Religion -- R.E.M.06 Smooth -- Santana07 Every Breath You Tak... -- The Police08 Tears in Heaven -- Eric Clapton09 London Calling -- The Clash10 Under the Bridge -- Red Hot Chili Peppers11 Jump -- Van Halen12 No Woman, No Cry -- Bob Marley13 Blowin' in the Wind -- Bob Dylan14 You Give Love a Bad ... -- Bon Jovi15 Hey Jude -- The Beatles16 Pale Blue Eyes -- The Velvet Underground17 Boys Don't Cry -- The Cure18 1979 -- The Smashing Pumpkins19 Imagine -- John Lennon20 Clint Eastwood -- Gorillaz21 Let It Be -- The Beatles22 Three Little Birds -- Bob Marley23 Wonderwall -- Oasis24 Rock Is Dead -- Marilyn Manson25 Bitter Sweet Symphon... -- The Verve26 Black Capricorn Day -- Jamiroquai27 Hound Dog -- Elvis Presley28 Why Don't You Get a ... -- The Offspring29 California Dreaming ... -- The MThe Papas30 Walk This Way -- Run-D.M.C.31 Livin'on a Prayer -- Bon Jovi32 Get Up (I Feel Like ... -- James Brown33 The Thrill Is Gone -- B.B. King34 Parklife -- Blur35 Anarchy in the U.K. -- The Sex Pistols36 We Will Rock You - ... -- Various Artists37 Oh, Pretty Woman -- Gary Moore38 To Be With You -- Mr. Big39 Beautiful Ones -- Suede40 Wake Me Up When Sept... -- Green Day41 Creep -- Radiohead42 You Really Got Me -- Van Halen43 I Hope That I Don't ... -- Tom Waits44 I Walk the Line -- Various Artists45 Stand By Me -- Ben E. King46 Another Brick in the... -- Pink Floyd47 Johnny B. Goode -- Chuck Berry48 I Want To Hold Your ... -- The Beatles49 Jean Genie, The -- David Bowie50 Wind of Change -- Scorpions51 Paranoid -- Black Sabbath52 Knockin' on Heaven's... -- Guns N' Roses53 Fight -- No Vacancy54 The Kids Aren't Alri... -- The Offspring55 Joey -- Concrete Blonde56 Highway to Hell -- AC/DC57 Purple Haze [*] -- Jimi Hendrix58 Sunshine Of Your Lov... -- Cream59 Blitzkrieg Bop -- The Ramones60 Love Will Tear Us Ap... -- Joy Division61 The Unforgiven II -- Metallica62 Comfortably Numb -- Pink Floyd63 Iris -- RJohnny64 Bohemian Rhapsody -- Queen65 Bawitdaba -- Kid Rock66 Like a Rolling Stone -- Bob Dylan67 Satisfaction -- The Rolling Stones68 Teardrop -- Massive Attack69 Sultans of Swing -- Dire Straits70 What's Up? -- 4 Non Blondes71 Cryin' -- Aerosmith72 Purple Rain -- Prince73 Smoke on the Water -- Deep Purple74 Born to Run -- Bruce Springsteen75 You Never Can Tell -- Jack Rabbit Slim's Twist Contest*/Chuck ...76 Loser -- Beck77 I Wanna Be Your Dog -- The Stooges78 Take It Easy -- Jackson Browne79 Go Your Own Way -- Fleetwood Mac80 My Generation -- The Who81 Waterloo Sunset -- The Kinks82 Spanish Harlem Incid... -- The Byrds83 Hallelujah -- Jeff Buckley84 Light My Fire -- The Doors85 Lust for Life -- Iggy Pop86 That'll Be the Day -- Buddy Holly & the Crickets87 The Weight -- The Band88 Whole Lotta Love -- Led Zeppelin89 Good Golly Miss Moll... -- Little Richard90 The Sound of Silence -- Simon & Garfunkel91 Walk on the Wild Sid... -- Lou Reed92 Desperado -- Eagles93 All The Young Dudes -- Mott The Hoople94 Where Did You Sleep ... -- Nirvana95 Is This It -- The Strokes96 Alright -- Supergrass97 Gimme Some Lovin’ -- Spencer Davis Group98 Common People -- Pulp99 Bank Robber -- The Clash100 Truckin' -- Grateful Dead14716 条评论分享收藏感谢收起1.5K143 条评论分享收藏感谢收起SGS是什么意思呀？_百度知道
SGS是什么意思呀？
在线等，谢谢
提示该问答中所提及的号码未经验证，请注意甄别。
我有更好的答案
阻燃，防火等等等等SGS是公司名字，油品，杂货的各种测试都能做，如果还有什么不明白的可以发我邮件Hannah.,要认准邮箱有sgs后缀的才是真正SGS公司的人，电话+86 41，我就是宁波SGS工业部高分子材料实验室的人员，我们部门能做的比如紫外UV，氙灯，臭氧老化，高低温老化，塑料，硬度，撕裂强度，海绵，建筑材料，橡胶，木塑材料，泡沫，薄膜，混凝土的各种物理性能，如拉伸，冲击，是通标标准技术服务有限公司的简称，做各方面检测的公司，由于以前ROHS报告做的比较多，所以ROHS报告有很多人称它为SGS报告，但是现在SGS不只做ROHS，其他各方面如工业，农业，食品
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我们会通过消息、邮箱等方式尽快将举报结果通知您。excel中“=ROUND($C17/($C$9+$C$19)*$I$8,2)”是什么意思?_百度知道
excel中“=ROUND($C17/($C$9+$C$19)*$I$8,2)”是什么意思?
我有更好的答案
c17/(c9+c19)*i8的值取2位小数
采纳率：27%
C17除以C9和C19的和,再乘以I8。对这个积保留2位小数。
固定列C17除以固定单元格C9与C19的和, 再乘以固定单元格I8, 最后四舍五入至小数后两位.
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我们会通过消息、邮箱等方式尽快将举报结果通知您。{VERSION 5 0 "IBM INTEL NT" "5.0" }
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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 53 "Solutions to the SIAM 100-
Dollar, 100-Digit Challenge" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 207 "Team: Gaston Gonnet, Informatik, ETH, Zurich, \+
and Robert Israel, Mathematics, UBC.\nMost of these solutions are the \+
ones that actually produced the results we sent in, but I (R.I.) have \+
changed a few of them." }}{PARA 0 "" 0 "" {TEXT -1 32 "All these solut
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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 99 "Each problem had an answer that was a single real number,
of which 10 correct digits were required." }}{PARA 0 "" 0 "" {TEXT
-1 28 "The complete results are at " }{URLLINK 17 "lab.o
x.ac.uk/oucl/work/nick.trefethen/hundred.html" 4 "lab.ox
.ac.uk/oucl/work/nick.trefethen/hundred.html" "" }{TEXT -1 55 ".\nOur \+
team was one of 20 first prize winners.
See our " }{URLLINK 17 "prize
" 4 "http://www.math.ubc.ca/~israel/challenge/Prize.jpg" "" }{TEXT -1
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 10 "P
roblem 1:" }}{PARA 0 "" 0 "" {TEXT -1 8 "What is " }{XPPEDIT 18 0 "lim
it(int(x^(-1)*cos(x^(-1)*log(x)), x=epsilon..1), epsilon = 0)" "6#-%&l
imitG6$-%$intG6$*&)%\"xG,$\"\"\"!\"\"F--%$cosG6#*&)F+,$F-F.F--%$logG6#
F+F-F-/F+;%(epsilonGF-/F:\"\"!" }{TEXT -1 2 " ?" }}}{SECT 1 {PARA 5 "
" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Us
e a change of variables " }{XPPEDIT 18 0 "y = - x^(-1)*log(x)" "6#/%\"
yG,$*&)%\"xG,$\"\"\"!\"\"F*-%$logG6#F(F*F+" }{TEXT -1 60 " to make the
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" }}}{EXCHG {PARA 0 ">
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{MPLTEXT 1 0 25 "ti:= time(): Digits:= 20:" }}}{EXCHG {PARA 0 "> " 0 "
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&%\"xG!\"\"-%$cosG6#*&
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We break it up into 3 pieces.
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1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG!\"\"-%$cosG6#*&F'F(-%#lnG6#F'\"\"
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" 0 "" {MPLTEXT 1 0 72 "map(factor,asympt(cos(t)*W(t
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{XPPMATH 20 "6#,2*,-%$cosG6#%\"tG\"\"\"-%\"WG6#,$*&%\"nGF)%#PiGF)\"\"#
F)F0!\"\",&F*F)F)F)F2F/F2#F)F1*&#F)\"\"%F)*0F%F)F(F)F*F1,&F*F)F1F)F)F0
!\"#F3!\"$F/F:F)F2*2#F)\"#;F)F%F)F(F1F*\"\"$,(*$)F*F1F)F1*&\"\")F)F*F)
F)\"\"*F)F)F0F;F3!\"&F/F;F)*&#F)\"#'*F)*0F%F)F(F?F*F7,**$)F*F?F)\"\"'*
&\"#OF)FBF)F)*&\"#zF)F*F)F)\"#kF)F)F0!\"%F3!\"(F/FTF)F2*2#F)\"$o(F)F%F
)F(F7F*\"\"&,,*$)F*F7F)\"#C*&\"$#>F)FMF)F)*&\"$A'F)FBF)F)*&\"$u*F)F*F)
F)\"$D'F)F)F0FFF3!\"*F/FFF)*&#F)\"%!o(F)*0F*FNF(FYF%F),.*$)F*FYF)\"$?
\"*&\"%+7F)FfnF)F)*&\"%E^F)FMF)F)*&\"&e " 0 "" {MPLTEXT 1 0 50 "J2:= map(factor,map(int, eva
l(%,O=0), t=-Pi..Pi));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#J2G,(*,%#
PiG!\"#-%\"WG6#,$*&%\"nG\"\"\"F'F/\"\"#\"\"$,(*$)F)F0F/F0*&\"\")F/F)F/
F/\"\"*F/F/,&F)F/F/F/!\"&F.!\"$#!\"\"\"\"%*&#F/\"#'*F/*.F)\"\"&,,*$)F)
F=F/\"#C*&\"$#>F/)F)F1F/F/*&\"$A'F/F4F/F/*&\"$u*F/F)F/F/\"$D'F/F/,&*$)
F'F0F/F/\"\"'F<F/F'!\"%F8!\"*F.F9F/F<*&#F/\"%!o(F/*.F)\"\"(,0*$)F)FBF/
\"%S')*&\"$?(F/)F)FRF/F/*&\"&cd%F/FEF/F/*&\"'GeDF/F)F/F/\"'\\w6F/*&\"'
]#[#F/F4F/F/*&\"'7v8F/FIF/F/F/,(FP!#?\"$?\"F/*$)F'F=F/F/F/F'!\"'F8!#8F
.!\"(F/F " 0 "" {MPLTEXT 1 0 42 "L:= convert(%,
list): evalf(eval(L,n=100));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%$!5?
JC/l7-$G+%!#F$!5'4FCaWtr'Rw!#K$!5+UMh'\\\\`=Z\"!#O" }}}{EXCHG {PARA 0
"" 0 "" {TEXT -1 14 "Apparently by " }{XPPEDIT 18 0 "n=100" "6#/%\"nG
\"$+\"" }{TEXT -1 18 " we only need the " }{XPPEDIT 18 0 "n^(-3)" "6#)
%\"nG,$\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n^(-5)" "6#)%\"
nG,$\"\"&!\"\"" }{TEXT -1 48 " terms.
We'll use these for n from 100 \+
to 1000." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "J3:= op(1,J2)+o
p(2,J2);\nr2:= add(evalf(J3),n=100..1000);" }}{PARA 12 "" 1 ""
{XPPMATH 20 "6#>%#J3G,&*,%#PiG!\"#-%\"WG6#,$*&%\"nG\"\"\"F'F/\"\"#\"\"
$,(*$)F)F0F/F0*&\"\")F/F)F/F/\"\"*F/F/,&F)F/F/F/!\"&F.!\"$#!\"\"\"\"%*
&#F/\"#'*F/*.F)\"\"&,,*$)F)F=F/\"#C*&\"$#>F/)F)F1F/F/*&\"$A'F/F4F/F/*&
\"$u*F/F)F/F/\"$D'F/F/,&*$)F'F0F/F/\"\"'F<F/F'!\"%F8!\"*F.F9F/F%#r2G$!5$4GeT 1000, use the asymptotics of the \+
Euler-Maclaurin form for the summation of the " }{XPPEDIT 18 0 "n^(-3)
" "6#)%\"nG,$\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n^(-5)" "
6#)%\"nG,$\"\"&!\"\"" }{TEXT -1 7 " terms." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 43 "J4:= map(factor,asympt(eulermac(J3,n),n)); " }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#J4G,,*,-%\"WG6#,$*&%\"nG\"\"\"%#PiG
F-\"\"#F/,&F'F-F/F-F-F.!\"#,&F'F-F-F-!\"$F,F1#F-\"\"%*.#F-\"\")F-F.F1F
'\"\"$,(*$)F'F/F-F/*&F8F-F'F-F-\"\"*F-F-F2!\"&F,F3F-*0#F-\"#KF-F'F5,**
$)F'F9F-\"\"'*&\"#OF-FF-F'\"\"&,,*$)F'F5F-\"#C*&FTF-FEF-F-*&\"$
A'F-F " 0 "" {MPLTEXT 1 0 36 "r3:=
-evalf(eval(J4,\{O=0, n=1001\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#r3G$!5;c:SwX$*3r@!#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
r:= r1+r2+r3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"5[;!zxnJuOB$
!#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Actual answer, as given by
Trefethen to 40 digits:" }}{PARA 0 "" 0 "" {TEXT -1 61 "
..." }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 20 "(time()-ti)*" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#,$%(secondsG$\"&z*[!\"$" }}}}{EXCHG {PARA 4 "" 0 ""
{TEXT -1 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 34 "A photon movin
g at speed 1 in the " }{XPPEDIT 18 0 "x-y" "6#,&%\"xG\"\"\"%\"yG!\"\"
" }{TEXT -1 17 " plane starts at " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!
" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "(x,y) = (0.5,0.1)" "6#/6$%\"xG%\"
yG6$-%&FloatG6$\"\"&!\"\"-F)6$\"\"\"F," }{TEXT -1 55 " heading due eas
Around every integer lattice point " }{XPPEDIT 18 0 "``(i,j);" "6#
-%!G6$%\"iG%\"jG" }{TEXT -1 43 " in the plane, a circular mirror of ra
dius " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 61 " ha
s been erected.
How far from the origin is the photon at " }{XPPEDIT
18 0 "t=10" "6#/%\"tG\"#5" }{TEXT -1 1 "?" }}}{SECT 1 {PARA 5 "" 0 ""
{TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "You have
to be careful to use enough digits here, because there are 13 reflect
ions, and each reflection from a curved surface amplifies the error.
We needed 11 additional digits.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
27 "The state of the photon is " }{XPPEDIT 18 0 "[x,y,dx,dy,s]" "6#7'%
\"xG%\"yG%#dxG%#dyG%\"sG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "[x,y]
" "6#7$%\"xG%\"yG" }{TEXT -1 18 " is the position, " }{XPPEDIT 18 0 "[
dx,dy]" "6#7$%#dxG%#dyG" }{TEXT -1 15 " the velocity, " }{XPPEDIT 18
0 "s" "6#%\"sG" }{TEXT -1 41 " the distance travelled.
We always have
" }{XPPEDIT 18 0 "dx^2 + dy^2 = 1" "6#/,&*$%#dxG\"\"#\"\"\"*$%#dyGF'F
(F(" }{TEXT -1 3 ".
" }}{PARA 0 "" 0 "" {TEXT -1 53 "Starting
s state, you hit a mirror centred at " }{XPPEDIT 18 0 "[mx, my]" "6#7$
%#mxG%#myG" }{TEXT -1 4 " if " }{XPPEDIT 18 0 "abs((mx-x)*dy-(my-y)*dx
) < 1/3;" "6#2-%$absG6#,&*&,&%#mxG\"\"\"%\"xG!\"\"F+%#dyGF+F+*&,&%#myG
F+%\"yGF-F+%#dxGF+F-*&F+F+\"\"$F-" }}{PARA 0 "" 0 "" {TEXT -1 90 "(ass
uming that mirror is in the forward direction, and you don't hit anyth
ing else first)." }}{PARA 0 "" 0 "" {TEXT -1 11 "If so, let " }
{XPPEDIT 18 0 "p = [px, py]" "6#/%\"pG7$%#pxG%#pyG" }{TEXT -1 38 " be \+
the point where it hits.
This is " }{XPPEDIT 18 0 "[x + t*dx, y + t*d
y]" "6#7$,&%\"xG\"\"\"*&%\"tGF&%#dxGF&F&,&%\"yGF&*&F(F&%#dyGF&F&" }
{TEXT -1 7 " where " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "(x-mx)^2 + (y-my
)^2 + 2*(dx*(x-mx) + dy*(y-my))*t + t^2 = 1/9" "6#/,**$,&%\"xG\"\"\"%#
mxG!\"\"\"\"#F(*$,&%\"yGF(%#myGF*F+F(*(F+F(,&*&%#dxGF(,&F'F(F)F*F(F(*&
%#dyGF(,&F.F(F/F*F(F(F(%\"tGF(F(*$F8F+F(*&F(F(\"\"*F*" }{TEXT -1 26 ".
Then the next state is " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "[px, py, d
x - h*(mx - px), dy - h*(my-py), s + sqrt((px-x)^2+(py-y)^2)] " "6#7'%
#pxG%#pyG,&%#dxG\"\"\"*&%\"hGF(,&%#mxGF(F$!\"\"F(F-,&%#dyGF(*&F*F(,&%#
myGF(F%F-F(F-,&%\"sGF(-%%sqrtG6#,&*$,&F$F(%\"xGF-\"\"#F(*$,&F%F(%\"yGF
-F " 0 "" {MPLTEXT 1 0 230 "plots[display]([plottools[c
ircle]([0,0],1), plots[arrow]([2,-2],[-1,2],shape=arrow,colour=blue),
\nplots[arrow]([1,0],[1,2],shape=arrow,colour=blue),\nplots[arrow]([1,
0],[2,0],shape=arrow,colour=red)\n],axes=none,scaling=constrained);" }
}{PARA 13 "" 1 "" {GLPLOT2D 255 255 255 {PLOTDATA 2 "6(-%'CURVESG6#7U7
$$\"\"\"\"\"!$F*F*7$$\"0yWJ,Z6#**!#:$\"00VcLKLD\"F/7$$\"0J'G6;$eo*F/$
\"0b[;())*o[#F/7$$\"0_#))e[w(H*F/$\"0xYo_X7o$F/7$$\"0kQ/!o1j()F/$\"09<
5uOv\"[F/7$$\"0[\\P%*p,4)F/$\"0tCH__y(eF/7$$\"079UF'o*G(F/$\"0)oGf5ZXo
F/7$$\"0!p[(*)RUP'F/$\"0*yvFC80xF/7$$\"0+!z\\zEe`F/$\"08?]DzKW)F/7$$\"
0y]c\"HzdUF/$\"00zi!#;$\"0s#G%Gn-)**F/7$$!04JH&>0ziFao
Fbo7$$!0?de98Q(=F/F\\o7$$!0U\\P%*p,4$F/$\"0b^H;l0^*F/7$$!0m]c\"HzdUF/$
\"0BgY_q#[!*F/7$$!0(**y\\zEe`F/$\"0:?]DzKW)F/7$$!0\"o[(*)RUP'F/$\"0'zv
FC80xF/7$$!059UF'o*G(F/$\"0\"pGf5ZXoF/7$$!0^\\P%*p,4)F/$\"0pCH__y(eF/7
$$!0hQ/!o1j()F/$\"0@<5uOv\"[F/7$$!0_#))e[w(H*F/$\"0wYo_X7o$F/7$$!0H'G6
;$eo*F/$\"0k[;())*o[#F/7$$!0yWJ,Z6#**F/$\"01VcLKLD\"F/7$$!0+++++++\"!#
9$!0s;mNP:w'!#H7$Fhr$!0*HkNBL`7F/7$$!0I'G6;$eo*F/$!0e[;())*o[#F/7$$!0b
#))e[w(H*F/$!0qYo_X7o$F/7$$!0kQ/!o1j()F/$!0:0ziFao$!
0r#G%Gn-)**F/7$$\"0xIH&>0ziFao$!0s#G%Gn-)**F/7$$\"0Fde98Q(=F/Ffv7$$\"0
R\\P%*p,4$F/$!0c^H;l0^*F/7$$\"0s]c\"HzdUF/F\\v7$$\"0')*y\\zEe`F/$!0A?]
DzKW)F/7$$\"0'o[(*)RUP'F/Fbu7$$\"099UF'o*G(F/F]u7$$\"0V\\P%*p,4)F/$!0z
CH__y(eF/7$F=Fct7$$\"0Z#))e[w(H*F/$!0)o%o_X7o$F/7$$\"0I'G6;$eo*F/Fis7$
$\"0wWJ,Z6#**F/$!0>VcLKLD\"F/7$$\"0+++++++\"F_s$!0cmnGDpZ'Fbs-F$6&7$7$
$\"\"#F*$!\"#F*F'7%7$$\"0++++++5\"F_s$!/++++++XF_sF'7$$\"0++++++I\"F_s
$!/++++++NF_s-%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBGF+F+$\"*++++\"
!\")-F$6&7$F'7$FdzFdz7%7$$\"0++++++q\"F_s$\"0++++++l\"F_sFa\\l7$$\"0++
++++!>F_s$\"0++++++b\"F_sFc[lFg[l-F$6&7$F'7$$\"\"$F*F+7%7$$\"0++++++g#
F_s$F]zF/F`]l7$Fe]l$F^sF/Fc[l-Fh[l6&Fj[lF[\\lF+F+-%(SCALINGG6#%,CONSTR
AINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 1 1.000000
45.. 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Global variables: " }{MPLTEXT 0
21 15 "x, y, dx, dy, s" }{TEXT -1 40 " \nif going m
ainly in " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 12 " direction, " }
{MPLTEXT 0 21 6 "iy = 0" }{TEXT -1 5 " and " }{MPLTEXT 0 21 2 "ix" }
{TEXT -1 21 " is integer to go in " }{XPPEDIT 18 0 "x" "6#%\"xG" }
{TEXT -1 26 " dire" }}{PARA 0 "" 0 "" {TEXT -1
19 "if going mainly in " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 12 " d
irection, " }{MPLTEXT 0 21 6 "ix = 0" }{TEXT -1 5 " and " }{MPLTEXT 0
21 2 "iy" }{TEXT -1 21 " is integer to go in " }{XPPEDIT 18 0 "y" "6#%
\"yG" }{TEXT -1 26 " direction to next mirror." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1164 "move:= proc()\n# return true if you hit the \+
mirror or end of path\n# and update state in this case,\n# false if yo
u don't (leaving state unchanged)\n
local mx, my, t, px, py,h,\n
global x,y,dx,dy,s,ix,\n
if iy = 0 then # going in x direction\n
mx:= round(x+ix-.001);\n
my:= round(y+dy/dx*ix);\n
ds:= sqrt(1
+(dy/dx)^2)*abs(mx-x);\n
else # going in y direction\n
my:= round(
mx:= round(x+dx/dy*iy);\n
ds:= sqrt(1+(dx/dy)^2)*abs(my
if abs((mx-x)*dy - (my-y)*dx) = 10 then\n
x:= x + (10-s)*\n
y:= y + (10-s)*\n
h:= 18*((mx-px)*dx+(my-py)*dy);\n
dx:= dx - h*(mx-px);\n
dy:= dy - h*(my-py);\n#
dx^2 + dy^2 should stay constant, but make sure\n
h:= sqrt(dx^2+
dx:= dx/h; dy:= dy/h;\n
# no hit, but maybe we'd go too far\n
elif s + ds
>= 10 then\n
x:= x + (10-s)*\n
y:= y + (10-s)*\n
0;\\\nfalse
\" }}{PARA 12 "" 1 ""
{XPPMATH 20 "6#>%%moveGf*6\"6)%#mxG%#myG%\"tG%#pxG%#pyG%\"hG%#dsGF&F&C
%@%/%#iyG\"\"!C%>8$-%&roundG6#,(%\"xG\"\"\"%#ixGF<$F8%-F86#
,&%\"yGF<*(%#dyGF<%#dxGF@F=F<F8**&-%%sqrtG6#,&F<F<*&FH\"\"#FI!\"#F<F
<-%$absG6#,&F6F<F;F@FFB-F86#,&FFF<F2FF6-F86#,&F;F<*(FIF<FHF@F2F<
FFK*&-FN6#,&F<F<*&FIFRFHFSF<F<-FU6#,&FBF<FFF@F<@&2-FU6#,&*&FWF<FHF<F
<*&FdoF<FIF<F@#F8&-%$minG6#-%'fsolveG6$/,&*$),(F;F<*&F`pF<FIF
<F<F6F@FRF<F<*$),(FFF<*&F`pF<FHF<F<FBF@FRF<F<#F8',&F;F<F\\qF
8(,&FFF<F`qFFK-FN6#,&*$),&FdqF<F;F@FRF<F<*$),&FgqF<FFF@FRF<F<@%1\"
#5,&%\"sGF<FKFF;,&F;F<*&,&FerF<FgrF@F<FIF<FFF,&FFF<*&F\\sF<FHF<F
FgrFerO%%trueGC,>8),&*&,&F6F<FdqF@F<FIF<\"#=*(FisF<,&FBF<FgqF@F<FHF<
FFgrFfr>FI,&FIF<*&FesF<FhsFFH,&FHF<*&FesF<F[tFFes-FN6#,&*$)F
IFRF<F<*$)FHFRF<FFI*&FIFFH*&FHFF;Fdq>FFFgqOFbsFdrC&>F;
Fjr>FFF^s>FgrFerOFbs%&falseGF&6)F;FFFIFHFgrF=F2F&" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 19 "Initial conditions." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 92 "x,y,dx,dy,s:= 0.5, 0.1, 1, 0, 0; path:= [x,y]; ix:= 1
; iy:= 0;\n_Envsignum0:= 0; Digits:= 50;" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>6'%\"xG%\"yG%#dxG%#dyG%\"sG6'$\"\"&!\"\"$\"\"\"F-F/\"\"!F0" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pathG7$$\"\"&!\"\"$\"\"\"F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ixG\"\"\"" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%#iyG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,_Envs
ignum0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#]" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "while s = abs(dy) then\n
:= signum(dx); iy:= 0\n
ix:= 0; iy:= signum(dy);\n
path:= path,[x,y];\n
else # go to next mirror in same directi
ix:= ix+signum(ix);\n
iy:= iy+signum(iy);\n
fi\nod:" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The final state:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x,y,dx,dy,s;" }}{PARA
12 "" 1 "" {XPPMATH 20 "6'$!Swb+!e*=X?<Rm=v8djjpU'p'F%$\"SpX!)QP$=#R]oQ,)='\\2=P+x`rLd$*F%$\"S&QK'=(
=5i_Xj1rSOoi`9nvjsq_$F%\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Th
e answer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ans:=sqrt(x^2+
y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ansG$\"S*)*HG@8H5i@nU4=J!*
3;aLW>HE&**!#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "with(plo
ts): display(\{plot([path]),seq(seq(plottools[circle]([i,j],1/3),i=-1.
.1),j=-1..2)\},scaling=constrained,axes=frame);" }}{PARA 7 "" 1 ""
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
{PARA 13 "" 1 "" {GLPLOT2D 494 396 396 {PLOTDATA 2 "63-%'CURVESG6#7U7$
$!Snmmmmmmmmmmmmmmmmmmmmmmmm!#]$\"\"!F,7$$!SuV.Vac^frSr!>BS$)HsS=c*4&H
p'F*$\"S1n)*[F(4)zfp@F`iqd%[ZV@Xux
;Rq/X2!pF*$\"S2jCw%GmpF?V;\\!\\cQ')fAGU=3F7F*7$$!S&f]?>J+AZ7$f`OhRcP@r
=Lx(*yqF*$\"SO-f'>>%oT&
owKI(F*$\"SrZD\"3bK&*edja^'o&*QwV#3VF*7$$!SR/`W4e(G'pC[O*)*3^*eh>
EvX5qvF*$\"Sd,4T1lHBktteyFudCi*GkoB=G#F*7$$!SHR2!4O)=o7muF1HWFjnV]nLDv
yF*$\"S$)RSA*)QU#Gf?87K+,OuH>fZx$oDF*7$$!SMI\"=#3)yRn'y52TIidgWLS$o5R@
)F*$\"SZQru>s7NQ*)))zo_G=E]+M=kU9GF*7$$!SZAFlvm+K%\\_3=D$e/<\"4$y%pN2e
)F*$\"S:9qj/olkecxf@cX!f<t')[/yGBN6QFJzx(zaq!>^]wQ)=qJF*7$$!S5WFG\"z`N%G!4&=U;e[c%eZ!=cRv$
*F*$\"Sn6A'4\\H@Z*3b4e!)=Og&Hid$3HuKF*7$$!SVk$*QQeibH7y%esguz?iN#o#)p!
z*F*$\"S^+*R<I76&o\"yG-cW<t=dF9wbnK$F*7$$!Scj5;;uV/x=_TFRD?zPk<t,$4-\"
!#\\Fbo7$$!SfDQ/Yi5FgoF]o7$$!SVj'>=nG'oR41R6Vn89
\\;zkc+.6FgoFhn7$$!SvFZVK$*zc]Z\">[nT&H)3p@0VE>9\"Fgo$\"S<9qj/olkecxf@
cX!f<t')[@gK8#*G*epPUR$*3'y6Fgo$\"S\\Qru>s7NQ*)
))zo_G=E]+M=kU9GF*7$$!S2E*4R;\"=tQ`AP4dDnBj&\\KmuC@\"Fgo$\"S&)RSA*)QU#
Gf?87K+,OuH>fZx$oDF*7$$!Scpa0>Cr.`<N1,\"*[5%Q!QZU&*)HC\"Fgo$\"Sf,4T1lH
BktteyFudCi*GkoB=G#F*7$$!S4Ij[Q`HN1ws0y4M!3eJe9Ls'p7Fgo$\"StZD\"3bK&*e
dja^'o&*QwV#3VF*7$$!SS\\z!)o*zFvoSYjQgViyG\"oEA5#H\"Fgo$\"SRV-L2a=9.A,Q3'H&\\D*48Fgo$\"S4jCw%G
mpF?V;\\!\\cQ')fAGU=3F7F*7$$!S8DMT+7Ygy:#=zAn\\1Pa4P0hGK\"Fgo$\"SpZ\")
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7$F^_lFhfm7$F[_lF[gm7$Ff_lF`gm7$Fc^lFcgm7$F`^lFfgm7$F]^lFigm7$Fd`lF^hm
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$F[zFfgm7$FfyFcgm7$FayFa[n7$F\\yF[gm7$FgxFhfm-%(SCALINGG6#%,CONSTRAINE
DG-%+AXESLABELSG6$Q!6\"F\\io-%*AXESSTYLEG6#%&FRAMEG-%%VIEWG6$%(DEFAULT
GFeio" 1 2 0 1 10 0 2 9 1 3 1 1... 0 "Curve \+
1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve \+
8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 47 "Just for comparison, here it is with 40 d
igits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 405 "x,y,dx,dy,s:= 0.
5, 0.1, 1, 0, 0: path:= [x,y]: ix:= 1: iy:= 0:\n_Envsignum0:= 0: Digit
s := 40:\nwhile s = abs(dy) then\n
ix:= signum(dx); iy:= 0\n
ix:= 0; iy:= signum(dy);\\n
path:= path,[x,y];\n
e # go to next mirror in same direction\n
ix:= ix+signum(ix);\n
iy:= iy+signum(iy);\n
fi\nod:\nans40:= sqrt(x^2+y^2);" }}{PARA 11 ""
1 "" {XPPMATH 20 "6#>%&ans40G$\"IkL4A0B%4=J!*3;aLW>HE&**!#S" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ans40 -" }}{PARA 11 ""
1 "" {XPPMATH 20 "6#$!,Fp6%pO!#S" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
69 "And here's what we would have got with a starting point different \+
by " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F$!\"\"" }{TEXT -1 20 " in th
e y direction." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 403 "path0:= \+
path:\nx,y,dx,dy,s:= 0.5, 0.1+1e-10, 1, 0, 0: path:= [x,y]: ix:= 1: iy
:= 0:\n_Envsignum0:= 0: Digits := 40:\nwhile s = abs(dy) then\n
ignum(dx); iy:= 0\n
ix:= 0; iy:= signum(dy);\\n
path:= path,[x,y];\n
else # go to next mirror in same direction\n
ix:= ix+signum(ix);\n
iy:= iy+signum(iy);\n
fi\nod:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "display(\{plot([path],colou
r=blue),plot([path0],colour=red),seq(seq(plottools[circle]([i,j],1/3),
i=-1..1),j=-1..2)\},scaling=constrained,axes=frame);" }}{PARA 13 "" 1
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u[KR%4DCa*FeoF`q" 1 5 4 1 10 0 2 9 1 4 2 1...000000
0 0 "Curve 1" }}}}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Problem 3:" }}
{PARA 0 "" 0 "" {TEXT -1 20 "The infinite matrix " }{XPPEDIT 18 0 "A"
"6#%\"AG" }{TEXT -1 14 " with entries " }{XPPEDIT 18 0 "a[1,1] = 1" "6
#/&%\"aG6$\"\"\"F'F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[1,2] = 1/2" "
6#/&%\"aG6$\"\"\"\"\"#*&F'F'F(!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "
a[2,1]=1/3" "6#/&%\"aG6$\"\"#\"\"\"*&F(F(\"\"$!\"\"" }{TEXT -1 2 ", "
}{XPPEDIT 18 0 "a[1,3] = 1/4" "6#/&%\"aG6$\"\"\"\"\"$*&F'F'\"\"%!\"\"
" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[2,2]=1/5" "6#/&%\"aG6$\"\"#F'*&\"
\"\"F)\"\"&!\"\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "a[3,1]=1/6" "6#/&%\"
aG6$\"\"$\"\"\"*&F(F(\"\"'!\"\"" }{TEXT -1 33 ", etc., is a bounded op
erator on " }{XPPEDIT 18 0 "l^2" "6#*$%\"lG\"\"#" }{TEXT -1 13 ".
t is ||" }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 4 "||?." }}}{SECT 1
{PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 57 "The norm is the square root of the largest eigenvalue of " }
{XPPEDIT 18 0 "A^T*A;" "6#*&)%\"AG%\"TG\"\"\"F%F'" }{TEXT -1 46 ", whi
ch can be computed by a simple iteration:" }}{PARA 0 "" 0 "" {TEXT -1
30 "Start with a \"typical\" vector " }{XPPEDIT 18 0 "x(0)" "6#-%\"xG6
#\"\"!" }{TEXT -1 17 ", normalized so
" }{XPPEDIT 18 0 "x(0)[1] = 1"
"6#/&-%\"xG6#\"\"!6#\"\"\"F*" }{TEXT -1 14 ", and repeat \n" }
{XPPEDIT 18 0 "y := A^T*A*x(n)" "6#>%\"yG*()%\"AG%\"TG\"\"\"F'F)-%\"xG
6#%\"nGF)" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "x(n+1) := y/y[1];" "6#>-%
\"xG6#,&%\"nG\"\"\"F)F)*&%\"yGF)&F+6#F)!\"\"" }}{PARA 0 "" 0 "" {TEXT
-1 5 "Then " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 585 " \+
converges to the largest eigenvalue (the second eigenvalue is much sma
ller, so it converges quite fast).
But we can only work with a finite
components, so we truncate to an m by m matrix.
I did it \+
in Maple for up to a 2500 by 2500 matrix (obtaining 1.7112
9126 which happens to have 10 correct digits, but I wasn't confident o
f the 10th), and Gaston used a C program to get the result for a 32768
by 32768 matrix (of course we don't actually construct the matrix, bu
t work directly with vectors).
His result was 1.109, whic
h has 13 correct digits." }}{PARA 0 "" 0 "" {TEXT -1 22 "Here's the Ma
ple code." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "mulAA:= proc(
V::Array,W::Array,n)\n# multiply the n-component vector V by A^T*A\n# \+
W is another vector of the same size used for\n# intermediate calculat
local i,j;\n
for i from 1 to n do\n
W[i]:= add(V[j]/((i+j-1)*(i+j-2)/2+i),j=1..n)\\n
for j from 1 to n do\n
V[j]:= add(W[i]/((i+j-1)*(i+j-
2)/2+i),i=1..n)\\n end:" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 532 "F:= proc(n)\n# get the approximate eigenvalue for an
n x n # truncation\n
local V,W,VP,i,\n
if not type(n,intege
r) then return 'procname(n)'\n
eps:= 1.0*10^(-Digits+4);\n
:= Array(1..n,datatype=float[8]);\n
W:= Array(1..n,datatype=float[8
V[1]:= 1.0;\n
for i from 2 to n do V[i]:= 0\n
VP:= copy(V);\n
mulAA(V,W,n);\n
if max(seq(abs(V[i]-V[1
]*VP[i]),i=1..n))
" 0 "" {MPLTEXT 1 0 95 "Digits:= 15: ti
:= time(): F(20); (time()-ti)*\nti:= time():\nF(200);\n(time()
-ti)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0(4aS)RQF\"!#9" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"%%y\"!\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"0s_8gBUF\"!#9" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#,$%(secondsG$\"'!Gu\"!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
62 "Some groups used various extrapolation techniques.
We didn't." }}
}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Problem 4:" }}{PARA 0 "" 0 ""
{TEXT -1 42 "What is the global minimum of the function" }}{PARA 0 ""
0 "" {XPPEDIT 18 0 "exp(sin(50*x))+sin(60*exp(y))+sin(70*sin(x))+sin(s
in(80*y))-sin(10*(x+y))+(x^2+y^2)/4;" "6#,.-%$expG6#-%$sinG6#*&\"#]\"
\"\"%\"xGF,F,-F(6#*&\"#gF,-F%6#%\"yGF,F,-F(6#*&\"#qF,-F(6#F-F,F,-F(6#-
F(6#*&\"#!)F,F4F,F,-F(6#*&\"#5F,,&F-F,F4F,F,!\"\"*&,&*$F-\"\"#F,*$F4FJ
F,F,\"\"%FFF," }{TEXT -1 1 "?" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "
Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Look at the function \+
for reasonably small values of x and y." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 122 "\nF := (x,y) -> exp(sin(50*x)) + sin(60*exp(
y)) + sin(70*sin(x)) +\n\011sin(sin(80*y)) - sin(10*(x+y)) + 1/4*(x^2+
y^2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6$%\"xG%\"yG6\"6$%)op
eratorG%&arrowGF),0-%$expG6#-%$sinG6#,$9$\"#]\"\"\"-F26#,$-F/6#9%\"#gF
7-F26#,$-F26#F5\"#qF7-F26#-F26#,$F=\"#!)F7-F26#,&F5\"#5*&FNF7F=F7F7!\"
\"*&#F7\"\"%F7)F5\"\"#F7F7*&FRF7)F=FUF7F7F)F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 87 "plot3d(F(x,y),x=-2..2,y=-2..2,axes=box,grid=[1
00,100],style=patchcontour,shading=ZHUE);" }}{PARA 13 "" 1 ""
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