This is a multi-part message in MIME format.
--------------315F529948125B85ECE1D484
Content-Type: text/plain; charset=us-ascii
Content-Transfer-Encoding: 7bit
Hi, everyone,
I need to find an algorithm to solve a least square fitting problem,
which is described below. If you have any advice on this problem, can
you help me? like, what articles or books to read, or whom to consult?
*Problem*
Given two sets of 3-element vectors:
a1, a2, a3 ... an
b1, b2, b3 ... bn
and a set of 3x3 matries:
F1,F2,F3,...Fn
I need a least-square fitting algorithm that can find a linear
transformation
T(T, p) that can minimize the following criterion function (T is also a
3 x 3
matrix, and p is a 3-element vector):
sum^{n}_{i=1}[a_{i}-t_{i}]^{t}F_{i}[a_{i}-t_{i}]
t1,t2,t3 ... tn are the transformed vectors of b1,b2,b3 ... bn, with
ti=bixT+p
I know there exists LSF algorithms that minimize criterion:
\sum^{n}_{i=1}w_{i}[a_{i}-t_{i}]^{2}
(w1,w2,w3 ... wn are a set of scalars.)
So now can we find such algorithms that can extend these scalars to
matrices?
Thanks in advance.
Sincerely,
Wei Ge
PS: if you find the above plain text hard to understand, I attached a ps
file which describes this problem, and the formulas are easier to read.
PS2: If you can give me any advice, please email me at
[EMAIL PROTECTED] (I am a CS & biology grad student.)
--------------315F529948125B85ECE1D484
Content-Type: application/postscript;
name="lsf_problem.ps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline;
filename="lsf_problem.ps"
%!PS-Adobe-2.0
%%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software
%%Title: lsf_problem.dvi
%%Pages: 1
%%PageOrder: Ascend
%%BoundingBox: 0 0 612 792
%%EndComments
%DVIPSCommandLine: dvips lsf_problem.dvi -o lsf_problem.ps
%DVIPSParameters: dpi=300, compressed, comments removed
%DVIPSSource: TeX output 2000.01.27:0255
%%BeginProcSet: texc.pro
/TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
/X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
forall round exch round exch]setmatrix}N /@landscape{/isls true N}B
/@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B
/FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{
/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N
string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N
end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{
/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]
N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup
length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{
128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub
get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data
dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N
/rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup
/base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx
0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff
setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff
.1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N
/cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id
gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp
add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add
/gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{
dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1
adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2
idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string
putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval
adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg}
{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{
adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2
chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{]
}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup
length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{
cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin
0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul
add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict
/eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook
known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X
/IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for
65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0
0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V
{}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}
ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1
false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform
round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg
rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail
{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}
B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{
4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{
p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p
a}B /bos{/SS save N}B /eos{SS restore}B end
%%EndProcSet
TeXDict begin 40258431 52099146 1000 300 300 (lsf_problem.dvi)
@start /Fa 1 89 df<B912C018E06CC81201EE001F6C6CED03F06C6C1500000F17386D
16186C6C160C6C6C1604A26C6C16026C6C1600137FA26D7E6D7E130F806D7E6D7EA26D7E
6D7E147E147F6E7E6E7EA2140F6E5A14034AC9FC140E5C5C5CA25C495A495A49CAFC130E
A24916024916045B49160C00011718491638484816F848C9EA01F0000E160F48ED01FF00
3FB812E0A25AB912C0373A7E7F3C>88 D E /Fb 2 3 df<B61280A219027D8A20>0
D<0040132000C01360006013C03830018038180300EA0C066C5A6C5AEA01B0EA00E0A2EA
01B0EA0318EA060C487E487E38300180386000C04813600040132013147A9320>2
D E /Fc 3 117 df<1206120712061200A41238124CA2128C12981218A212301232A212
64A2123808147F930C>105 D<EA30F8EA590C124E129C12981218A2EA301813191331A2
EA6032131C100D7F8C15>110 D<1204120CA35AEAFF80EA1800A25AA45A1261A2126212
64123809127F910D>116 D E /Fd 4 62 df<120C121C12EC120CAFEAFFC00A137D9211>
49 D<121FEA60C01360EAF07013301260EA0070A2136013C012011380EA02005AEA0810
1210EA2020EA7FE012FF0C137E9211>I<EA0FC0EA3070EA7038A31200133013E0EA0FC0
EA007013381318131C126012F01318EAE038EA6070EA1FC00E137F9211>I<387FFFE0B5
12F0C8FCA6B512F06C13E0140A7E8B19>61 D E /Fe 9 120 df<126012F0A212600404
7C830C>58 D<126012F0A212701210A41220A212401280040C7C830C>I<48B512F03800
3C00013813301520A35BA214081500495AA21430EBFFF03801C020A448485A91C7FCA348
C8FCA45AEAFFF01C1C7E9B1B>70 D<001FB512F0391C0380703930070030002014201260
1240130E1280A2000014005BA45BA45BA45BA41201EA7FFF1C1C7F9B18>84
D<EA01E3EA0717EA0C0F1218EA380E12301270A2485AA4EB3880A3EA607838319900EA1E
0E11127E9116>97 D<123F1207A2120EA45AA4EA39E0EA3A30EA3C1812381270131CA3EA
E038A313301370136013C01261EA2300121E0E1D7E9C12>I<380787803809C8603808D0
3013E0EA11C014381201A238038070A31460380700E014C0EB0180EB8300EA0E86137890
C7FCA25AA4123CB4FC151A819115>112 D<13C01201A3EA0380A4EAFFF0EA0700A3120E
A45AA4EA3820A21340A2EA1880EA0F000C1A80990F>116 D<001CEBC080392701C1C012
4714C03987038040A2120EA2391C070080A3EC0100EA1806A2381C0E02EB0F04380E1308
3803E1F01A127E911E>119 D E /Ff 48 123 df<137E3801C180EA0301380703C0120E
EB018090C7FCA5B512C0EA0E01B0387F87F8151D809C17>12 D<1380EA0100120212065A
A25AA25AA35AA412E0AC1260A47EA37EA27EA27E12027EEA0080092A7C9E10>40
D<7E12407E12307EA27EA27EA37EA41380AC1300A41206A35AA25AA25A12205A5A092A7E
9E10>I<1306ADB612E0A2D80006C7FCAD1B1C7E9720>43 D<126012F0A212701210A412
20A212401280040C7C830C>I<EAFFE0A20B0280890E>I<126012F0A2126004047C830C>
I<5A1207123F12C71207B3A5EAFFF80D1C7C9B15>49 D<EA07C0EA1830EA201CEA400C13
0EEAF00F12F81307A21270EA000F130EA2131CA213381370136013C0EA0180EA0300EA06
01120C1218EA1002EA3FFE127F12FF101C7E9B15>I<EA07E0EA1830EA201CA2EA781E13
0E131E1238EA001CA2131813301360EA07C0EA0030131CA2130E130FA2127012F8A3EAF0
0EEA401C1220EA1830EA07C0101D7E9B15>I<130CA2131C133CA2135C13DC139CEA011C
120312021204120C1208121012301220124012C0B512C038001C00A73801FFC0121C7F9B
15>I<EA300CEA3FF813F013C0EA2000A6EA23E0EA2C30EA3018131CEA200E1200130FA3
126012F0A3EA800EEA401E131CEA2038EA1870EA07C0101D7E9B15>I<13F0EA030CEA04
04EA0C0EEA181E1230130CEA7000A21260EAE3E0EAE430EAE818EAF00C130EEAE0061307
A51260A2EA7006EA300E130CEA1818EA0C30EA03E0101D7E9B15>I<126012F0A2126012
00AA126012F0A2126004127C910C>58 D<007FB512C0B612E0C9FCA8B612E06C14C01B0C
7E8F20>61 D<EA0FE0EA3038EA401CEAE00E12F0A3EA000C131C1330136013C013801201
1300A6C7FCA51203EA0780A2EA03000F1D7E9C14>63 D<B512F8380F007814181408A214
0C1404A213011400A35B13FF13037FA490C7FCA8EAFFF8161C7E9B1B>70
D<90381F8080EBE0613801801938070007000E13035A14015A00781300A2127000F01400
A6ECFFF0EC0F80007013071278A212387EA27E6C130B380180113800E06090381F80001C
1E7E9C21>I<EAFFF0EA0F00B3A8EAFFF00C1C7F9B0F>73 D<EAFFF8EA0F8090C7FCB014
08A31418A2141014301470EB01F0B5FC151C7E9B1A>76 D<3807E080EA1C19EA30051303
EA600112E01300A36C13007E127CEA7FC0EA3FF8EA1FFEEA07FFC61380130FEB07C01303
13011280A300C01380A238E00300EAD002EACC0CEA83F8121E7E9C17>83
D<007FB512C038700F010060130000401440A200C014201280A300001400B1497E3803FF
FC1B1C7F9B1E>I<12FEA212C0B3B312FEA207297C9E0C>91 D<12FEA21206B3B312FEA2
0729809E0C>93 D<EA1FC0EA3070EA78387F12301200A2EA01FCEA0F1C12381270126000
E01340A3EA603C38304E80381F870012127E9115>97 D<12FC121CAA137CEA1D87381E01
80381C00C014E014601470A6146014E014C0381E018038190700EA10FC141D7F9C17>I<
EA03F8EA0C0CEA181E1230EA700CEA600012E0A61260EA70021230EA1804EA0C18EA03E0
0F127F9112>I<EB1F801303AAEA03F3EA0E0BEA1807EA30031270126012E0A612601270
1230EA1807EA0E1B3803E3F0141D7F9C17>I<EA07E0EA0C30EA1818EA300CEA700EEA60
0612E0EAFFFEEAE000A41260EA70021230EA1804EA0C18EA03E00F127F9112>I<13F8EA
018CEA071E1206EA0E0C1300A6EAFFE0EA0E00B0EA7FE00F1D809C0D>I<EB03803807C4
C0EA1C783838388038301800EA701CA4EA3018EA38386C5AEA27C00020C7FCA21230EA3F
F86CB4FC1480EA2003386001C0EAC000A33860018038300300EA1C0EEA07F8121C7F9215
>I<12FC121CAA137C1387EA1D03001E1380121CAD38FF9FF0141D7F9C17>I<1218123CA2
1218C7FCA712FC121CB0EAFF80091D7F9C0C>I<12FC121CAAEB0FE0EB0780EB06005B13
105B5B13E0121DEA1E70EA1C781338133C131C7F130F148038FF9FE0131D7F9C16>107
D<12FC121CB3A9EAFF80091D7F9C0C>I<39FC7E07E0391C838838391D019018001EEBE0
1C001C13C0AD3AFF8FF8FF8021127F9124>I<EAFC7CEA1C87EA1D03001E1380121CAD38
FF9FF014127F9117>I<EA03F0EA0E1CEA1806487E00701380EA600100E013C0A6006013
80EA700300301300EA1806EA0E1CEA03F012127F9115>I<EAFC7CEA1D87381E0180001C
13C0EB00E0A21470A614E0A2EB01C0001E1380381D0700EA1CFC90C7FCA7B47E141A7F91
17>I<3803E080EA0E19EA1805EA3807EA7003A212E0A61270A2EA38071218EA0E1BEA03
E3EA0003A7EB1FF0141A7F9116>I<EAFCE0EA1D38EA1E78A2EA1C301300ACEAFFC00D12
7F9110>I<EA1F90EA2070EA4030EAC010A212E0EAF800EA7F80EA3FE0EA0FF0EA00F8EA
8038131812C0A2EAE010EAD060EA8FC00D127F9110>I<1204A4120CA2121C123CEAFFE0
EA1C00A91310A5120CEA0E20EA03C00C1A7F9910>I<38FC1F80EA1C03AD1307120CEA0E
1B3803E3F014127F9117>I<38FF07E0383C0380381C0100A2EA0E02A2EA0F06EA0704A2
EA0388A213C8EA01D0A2EA00E0A3134013127F9116>I<39FF3FC7E0393C0703C0001CEB
01801500130B000E1382A21311000713C4A213203803A0E8A2EBC06800011370A2EB8030
000013201B127F911E>I<38FF0FE0381E0700EA1C06EA0E046C5AEA039013B0EA01E012
007F12011338EA021C1204EA0C0E487E003C138038FE1FF014127F9116>I<EA7FFCEA70
381260EA407013F013E0EA41C012031380EA0700EA0F04120E121CEA3C0CEA3808127013
38EAFFF80E127F9112>122 D E /Fg 15 122 df<127012F812FCA212741204A41208A2
1210A212201240060F7C840E>44 D<EA01F0EA071CEA0C06487E00381380A2387001C0A4
00F013E0AE007013C0A3EA780300381380A2381C0700EA0C06EA071CEA01F013227EA018
>48 D<EA03F0EA0C1CEA100700201380384003C0A2008013E012F0EAF801A3EA20031200
14C0A2EB07801400130E5B13185B5B5B485A90C7FC000213205A5A00181360481340383F
FFC05AB5FC13217EA018>50 D<903807F00890383C0C18EBE0023901C001B839038000F8
48C71278481438121E15185AA2007C14081278A200F81400A7EC1FFF0078EB00F8157812
7C123CA27EA27E7E6C6C13B86C7E3900E0031890383C0C08903807F00020247DA226>71
D<3803FFE038001F007FB3A6127012F8A2130EEAF01EEA401C6C5AEA1870EA07C013237E
A119>74 D<3BFFF03FFC03FE3B1F8007E000F86C486C48137017206E7ED807801540A24A
7E2603C0021480A39039E004780100011600A2EC083CD800F01402A2EC101E01785CA2EC
200F013C5CA20260138890391E400790A216D090391F8003F0010F5CA2EC00016D5CA201
06130001025C2F237FA132>87 D<EA1FE0EA3038EA780C130EEA30071200A313FFEA07C7
EA1E07123C1278127000F01308A3130FEA7817383C2390380FC1E015157E9418>97
D<EA01FCEA0707380C0380381C01C01238007813E0EA700012F0B5FC00F0C7FCA5127000
78132012386C13406C138038070300EA00FC13157F9416>101 D<121C123EA3121CC7FC
A8120E127E121E120EB1EAFFC00A227FA10E>105 D<120E12FE121E120EB3ADEAFFE00B
237FA20E>108 D<380E1F8038FE60C0381E80E0380F0070A2120EAF38FFE7FF18157F94
1B>110 D<EA0E3CEAFE46EA1E8FEA0F0F13061300120EAD120FEAFFF010157F9413>114
D<EA0F88EA3078EA601812C01308A212E0EAF000127FEA3FE0EA0FF0EA01F8EA003CEA80
1C130CA212C01308EAE018EAD030EA8FC00E157E9413>I<000E137038FE07F0EA1E0000
0E1370AD14F0A238060170380382783800FC7F18157F941B>117
D<38FF80FE381E00781430000E1320A26C1340A2EB80C000031380A23801C100A2EA00E2
A31374A21338A31310A25BA35B12F05B12F10043C7FC123C171F7F941A>121
D E /Fh 18 118 df<B712E0A23907E0000F00031401ED00F016701630A21610A41608A3
15401600A415C0A2140390B5FCA2EBE0031400A21540A592C7FCAD487EB512C0A225317D
B02B>70 D<B512C0A2D807F0C8FC6C5AB3AA1620A41640A416C0A21501A21503ED078015
1F0007147FB7FCA223317DB029>76 D<B612E015FC3907E0003F0003EC0F80ED03C0ED01
E016F0ED00F8A21678167CA6167816F816F0150116E0ED03C0ED0F80ED3E0090B512F801
E0C8FCB3A3487EB57EA226317DB02D>80 D<90387F80203801FFE03907C07860380F001C
001EEB06E048130300381301007813001270156012F0A21520A37E1500127C127E7E13C0
EA1FF86CB47E6C13F06C13FCC613FF010F1380010013C0EC1FE01407EC03F01401140015
F8A26C1478A57E15706C14F015E07E6CEB01C000ECEB038000C7EB070038C1F01E38807F
FCEB0FF01D337CB125>83 D<13FE380303C0380C00E00010137080003C133C003E131C14
1EA21208C7FCA3EB0FFEEBFC1EEA03E0EA0F80EA1F00123E123C127C481404A3143EA212
78007C135E6CEB8F08390F0307F03903FC03E01E1F7D9E21>97 D<EA078012FFA2120F12
07AFEB81FC903886078090388801C090389000E001A0137001C07F49133C81A2151F81A2
1680A71600A25D151EA25D6D133801205B5D39061801C0D90607C7FC380401F821327EB1
25>I<EB3F80EBE0E0380380383807003C000E7F121E001C7F123C127C00781480140712
F8A2B6FC00F8C8FCA61278127C123C15807E000EEB0100000F13027E3801C00C3800F030
EB1FC0191F7E9E1D>101 D<15F090387F03083901C1C41C380380E8390700700848EB78
00001E7FA2003E133EA6001E133CA26C5B6C13706D5A3809C1C0D8087FC7FC0018C8FCA5
121C7E380FFFF86C13FF6C1480390E000FC00018EB01E048EB00F000701470481438A500
701470A26C14E06CEB01C00007EB07003801C01C38003FE01E2F7E9F21>103
D<120FEA1F80A4EA0F00C7FCABEA0780127FA2120F1207B3A6EA0FC0EAFFF8A20D307EAF
12>105 D<EA078012FFA2120F1207B3B3A7EA0FC0EAFFFCA20E327EB112>108
D<260780FEEB1FC03BFF83078060F0903A8C03C180783B0F9001E2003CD807A013E4DA00
F47F01C013F8A2495BB3A2486C486C133F3CFFFC1FFF83FFF0A2341F7E9E38>I<380780
FE39FF83078090388C03C0390F9001E0EA07A06E7E13C0A25BB3A2486C487E3AFFFC1FFF
80A2211F7E9E25>I<EB1FC0EBF0783801C01C38070007481480001EEB03C0001C130100
3C14E0A248EB00F0A300F814F8A8007814F0007C1301003C14E0A26CEB03C0A26CEB0780
3907800F003801C01C3800F078EB1FC01D1F7E9E21>I<90380FC01090387830303801E0
08380380043907000270481301001E14F0123E003C1300127CA2127812F8A71278127CA2
123C123E001E13017E6C1302380380043801C0083800F030EB1FC090C7FCAB4A7E91381F
FF80A2212D7E9E23>113 D<380783E038FF8418EB887CEA0F90EA07A01438EBC000A35B
B3487EEAFFFEA2161F7E9E19>I<3801FC10380E0330381800F048137048133012E01410
A37E6C1300127EEA3FF06CB4FC6C13C0000313E038003FF0EB01F813006C133CA2141C7E
A27E14186C1338143000CC136038C301C03880FE00161F7E9E1A>I<1340A513C0A31201
A212031207120F381FFFE0B5FC3803C000B01410A80001132013E000001340EB78C0EB1F
00142C7FAB19>I<D8078013F000FF131FA2000F130100071300B31401A300031302EBC0
040001497E3A007030FF80EB1FC0211F7E9E25>I E end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300dpi
TeXDict begin
%%EndSetup
%%Page: 1 1
1 0 bop 545 436 a Fh(Least)22 b(Square)f(Fitting)h(Problem)824
556 y Fg(W)l(ei)15 b(Julius)h(Ge)826 653 y(Jan)o(uary)l(,)g(2000)320
807 y Ff(I)d(need)i(to)f(\014nd)g(a)g(solution)f(to)g(the)i(follo)o
(wing)c(problem.)320 857 y(Giv)o(en)i(t)o(w)o(o)g(sets)i(of)f(3-elemen)
o(t)f(v)o(ectors:)813 998 y Fe(a)835 1004 y Fd(1)854
998 y Fe(;)7 b(a)895 1004 y Fd(2)913 998 y Fe(;)g(a)954
1004 y Fd(3)972 998 y Fe(:::a)1030 1004 y Fc(n)1636 998
y Ff(\(1\))830 1060 y Fe(b)848 1066 y Fd(1)866 1060 y
Fe(;)g(b)903 1066 y Fd(2)921 1060 y Fe(;)g(b)958 1066
y Fd(3)976 1060 y Fe(:::b)1030 1066 y Fc(n)1636 1060
y Ff(\(2\))320 1152 y(and)13 b(a)h(set)h(of)e(3)c Fb(\002)h
Ff(3)j(matries:)804 1293 y Fe(F)831 1299 y Fd(1)849 1293
y Fe(;)7 b(F)895 1299 y Fd(2)913 1293 y Fe(;)g(F)959
1299 y Fd(3)976 1293 y Fe(:::F)1039 1299 y Fc(n)1636
1293 y Ff(\(3\))320 1384 y(I)13 b(need)i(a)f(least-square)h(\014tting)e
(algorithm)e(that)j(can)g(\014nd)g(a)g(linear)f(transformation)257
1434 y Fe(T)6 b Ff(\()p Fe(T)s(;)h(p)p Ff(\))19 b(that)f(can)h(minim)o
(i)o(ze)e(the)h(follo)o(wing)e(criterion)j(function)f(\()p
Fe(T)24 b Ff(is)18 b(also)g(a)f(3)12 b Fb(\002)h Ff(3)257
1484 y(matrix,)f(and)i Fe(p)f Ff(is)h(a)g(3-elemen)o(t)f(v)o(ector\):)
756 1596 y Fc(n)736 1608 y Fa(X)739 1697 y Fc(i)p Fd(=1)796
1648 y Ff([)p Fe(a)830 1654 y Fc(i)852 1648 y Fb(\000)d
Fe(t)909 1654 y Fc(i)923 1648 y Ff(])935 1630 y Fc(t)949
1648 y Fe(F)976 1654 y Fc(i)989 1648 y Ff([)p Fe(a)1023
1654 y Fc(i)1046 1648 y Fb(\000)f Fe(t)1102 1654 y Fc(i)1116
1648 y Ff(])508 b(\(4\))320 1775 y Fe(t)335 1781 y Fd(1)353
1775 y Fe(;)7 b(t)387 1781 y Fd(2)405 1775 y Fe(;)g(t)439
1781 y Fd(3)457 1775 y Fe(:::t)508 1781 y Fc(n)543 1775
y Ff(are)15 b(the)f(transformed)f(v)o(ectors)i(of)f Fe(b)1123
1781 y Fd(1)1141 1775 y Fe(;)7 b(b)1178 1781 y Fd(2)1196
1775 y Fe(;)g(b)1233 1781 y Fd(3)1251 1775 y Fe(:::b)1305
1781 y Fc(n)1326 1775 y Ff(,)13 b(with)823 1916 y Fe(t)838
1922 y Fc(i)864 1916 y Ff(=)e Fe(b)925 1922 y Fc(i)939
1916 y Fe(T)k Ff(+)10 b Fe(p)595 b Ff(\(5\))320 2008
y(I)13 b(kno)o(w)h(there)h(exists)g(LSF)f(algorithms)d(that)j(minim)o
(ize)e(criterion:)818 2120 y Fc(n)798 2132 y Fa(X)801
2221 y Fc(i)p Fd(=1)865 2172 y Fe(w)895 2178 y Fc(i)909
2172 y Ff([)p Fe(a)943 2178 y Fc(i)965 2172 y Fb(\000)e
Fe(t)1022 2178 y Fc(i)1035 2172 y Ff(])1047 2155 y Fd(2)1636
2172 y Ff(\(6\))320 2302 y(\()p Fe(w)366 2308 y Fd(1)384
2302 y Fe(;)d(w)433 2308 y Fd(2)451 2302 y Fe(;)g(w)500
2308 y Fd(3)518 2302 y Fe(:::w)584 2308 y Fc(n)618 2302
y Ff(are)15 b(a)e(set)i(of)e(scalars.\))320 2351 y(So)c(no)o(w)g(can)g
(w)o(e)h(\014nd)g(suc)o(h)g(algorithms)d(that)i(can)h(extend)g(these)h
(scalars)f(to)f(matrices?)320 2401 y(Thanks.)963 2628
y(1)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
--------------315F529948125B85ECE1D484--
===========================================================================
This list is open to everyone. Occasionally, people lacking respect
for other members of the list send messages that are inappropriate
or unrelated to the list's discussion topics. Please just delete the
offensive email.
For information concerning the list, please see the following web page:
http://jse.stat.ncsu.edu/
===========================================================================
