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New Technique for Solving Sparse Equation Systems

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O((2

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δ(h) =

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ãç/H‘ó äç%76ä*í¢ä¡í™ã¡B®@/èç<BðH>9LLOèNí™LNDsä‘ýAõ?@?ä*í¢äã*7>ãêsBîë?äH>äí™èK9@?D?D?í¢D?Fîäí¢BðLK9LNBO76í¢D/ã*@?D?ñsD/HaóD‡÷e/H>9

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76ì¢FIH>9í¢äç?Bóí¢äçÊñMD/H‘óD 76ë?ë?9HI7>èç/Lã=äHðãH>ì¢5ILãêAãäLNBðãH6æãë%769ãL **HMH>ì™L‘76DÄLüz@%76äí™H>D/ãa÷

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0/ïèH>BðLöæ…9H>B äç/L

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÷ *ç/LãLä‰ó¡Hîë?9H>õ?ì™LNBðã769LëqH>ì¢êMD/H>Bîí–76ìZäí¢BðL®Lüs@?í¢5>76ì™LNDsä‘÷ ‡LNä

f

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f

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i

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1

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Z/¡íbæ176D/:ÊH>D?ì¢êéíbæí¢äí™ã7îãc76äí™ãSæ`êMí¢D?FO7>ããí¢F8D?BðLNDsäæXH>9äç/L­èH>D<_l@?D/èNäí¢5IL®D/H>9BO76ì]æXH>9B

F

f

= (x

a

1

11

. . .

x

a

1

k

k

)

. . .

(x

a

s

1

1

. . .

x

a

sk

k

),

óç/LN9L­ó¡L­:?LND/H>äL

x

a

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x,

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x,

íbæ

a

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x

a

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=

a

÷ í¢5ILND¨äç/LãêAãäLNB H6æKLüz@%76äí™H>D/ã

.

0/H>D/LéèH>D/ãä9@/èNäãå7

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F

óç?í™èçÓí™ã,7åèH>D<_l@?D/èNäí™H>D H6æ

F

f

i

÷

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.

0/íbæ•76D/: H>D?ì¢êÊíbæ1äç?í™ã=5ILèNäH>9í™ãK7éãV76äí™ãSæ…êsí¢D?F7>ããí¢F8D?BðLNDsä,æXH>9

F

÷ õz5Mí™H>@/ãì¢êIý]76Dzê

k

lJMù ë?9H>õ?ì™LNB BO7‘êïõL9LNë?9LãLNDsäL:ÊõzêÊ7åãêAãäLNB H6æ

(9)

k

lJMù

.

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k

3

/í™ãOH>D/L<H6æäç/LÊèNì–7>ããí™è‘76ì ‰èH>Bîë?ì™LNäLÄë?9H>õ?ì™LNBðãé76D/:Ñäç/LN9LÊí™ãO7ö587>ãä 9Lyæ…LN9LND/èLKì¢í™ãä¡H>Déäç?í™ã•ë?9H>õ?ì™LNBÊ÷A;,LèLNDsäì¢êðäç/LN9L,ç%7>ã•õLLNDï7®ì–769FIL=L]H>9ä£äH:?Lãí¢F8D:?LNäLN9Bîí¢D?í™ãäí™è

76D/:ð9V76D/:?H>Bîí-,‘L:éL?7>èNä•76ì¢FIH>9í¢äç?Bðã£æXH>9

k

lJMù ÷ *ç/LãLL]H>9äã19Lã@?ì¢äL:ðí¢Dé7Dz@?BõqLN9¡H6æ‡:?LLNëé76D/: ëqHaó•LN9Sæ…@?ì*äLèç?D?í™üs@/Lã®æXH>9ãVH>ì¢5sí¢D?F©JMù LOèNí™LNDzäì¢êI÷ =í™èLOL/76Bîë?ì™LãH6æ,ã@/èç¨äLèç?D?í™üs@/Lãð769Lðäç/L

75Mí™ã(@?äD%76B 76ì¢FIH>9í¢äç?BÊý1Jsç>D?í¢D?F ‡HMè‘76ì*JML‘769èVç 76D/:ò;=76D/:?H>B ò76ì¢ñMãa÷”JMù ôH>D

n

58769í–76õ?ì™Lã è‘76D(õLåä9í¢5sí–76ì¢ì¢êöãH>ì¢5IL:©í¢D©äí¢BðL

O(2

n

)

õsêÓä9êMí¢D?FÓ76ì¢ìëH8ããí¢õ?ì™Lð7>ããí¢F8D?BðLNDsäã‘ý]õ?@?ä®èH>D/ãä9@/èNäí¢D?F 76D

O(c

n

)

76ì¢FIH>9í¢äç?B æXH>9

c <

2

í™ã7ì™H>D?FðãäV76D/:Aí¢D?FðH>ëLNDë?9H>õ?ì™LNBÊ÷?3=Haó•LN5ILN9‘ýAæXH>9ãBO76ì¢ì5>76ì¢@/LãH6æ

k

äç/LN9Lî769LB®@/èVçöæX7>ãäLN976ì¢FIH>9í¢äç?BðãKæXH>9

k

lJMù óí¢äç

c

=

c

k

<

2

ýqãLLäç/Låã@?95ILNêÓ769äí™èNì™Lÿ

‰÷

Js@/èçÓõqH>@?D/:?ã,769L­ó•H>9ãä,è‘7>ãVLLãäí¢BO76äí™H>D/ã,äHåäç/L­ë?9H>õ?ì™LNBH6æãVH>ì¢5sí¢D?FOLüs@%76äí™H>D/ã

.

0/y÷

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76õqHa5IL­ó¡H>9ãä,è‘7>ãLõH>@?D/:?ã,76äì™L‘7>ãä*æXH>9,ãBO76ì¢ì

k

7>ãH>D/L®ãLLã*æ…9H>B äç/L­:/76äV7åäV76õ?@?ì–76äL:ÊõLNì™H‘ó

äç/Ló¡H>9ãä,è‘7>ãL Kì¢@?í¢D?F 8ý Kì¢@?í¢D?FAý

äç/L­7‘5ILN9V76FIL äç/L®75ILN9V76FIL

c

3

8÷(( 8÷

c

4

8÷) 8÷

( 8÷

(

c

5

8÷((" 8÷( 8÷

"

c

6

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. . .

. . .

. . .

. . .

*ç/L­èH>D/ãäV76Dsä

c

k

í™ãF8í¢5ILNDÄã@/èçÄäç%76ääç/L®76ì¢FIH>9í¢äç?B 9@?D/ã,í¢D<äí¢BðL

O(c

n

k

)

÷ EGD(äç/LOè‘7>ãVLOH6æ

n

**HsH>ì™L‘76D Lüz@%76äí™H>D/ãH6æ=76ì¢FILNõ?9V76í™èï:?LNF89LL

d

í¢D

n

58769í–76õ?ì™Lã:?L&/D?í¢D?Fö7ÄãH è‘76ì¢ì™L: ãLNBîíZ9LNF8@?ì–769ãêAãäLNBÊýäç/L<ëH>ë?@?ì–769 9>õ?D/LN9n*7>ãí™ãO76ì¢FIH>9í¢äç?B FILNäãðäç/LÊèH>Bîë?ì™LAí¢ä‰ê H6æ

O(

α

n

d

n

2

)

2

2

H

(

α

d)

n

õ?í¢äðH>ëLN9V76äí™H>D/ã‘ýóç/LN9L

H(α

d

)

:?LND/H>äLãðäç/L<õ?í¢D%769ê LNDsä9H>ëzê¥æ…@?D/èNäí™H>D‡÷

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α

d

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α

2

0.09

ý

α

3

0.15

ýŒ76D/:

α

4

0.2

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d

= 2

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O(1.7

n

)

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.

H>9 /76ë?ë?ì¢í™è‘76äí™H>D‡ý%ãLLOÿ%‰÷%*¡@?äæ…H>9

d

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äç/L5876ì¢@/L

2

2

H

(

α

d)

n

L?èLL:?ãäç/L èH8ãäåH6æ*äç/Léõ?9@?äLîæXH>9èL76ì¢FIH>9í¢äç?B!76D/:¨F8@/Lããí¢D?Fö76Dzê©Dz@?BõqLN9îH6æ58769í–76õ?ì™Lã:?HMLãD/H>ä®ç/LNì¢ë‡÷”E‰ä

í™ã,èNì™L‘769=äç%76ä‘ý+76ì¢äç/H>@?F8çÄäç/L76ì¢FILNõ?9V76í™è:?LNF89LLãKH6æ”9V76D/:?H>B2Lüs@%76äí™H>D/ãí¢D

.

0/,769LõqH>@?D/:?L:Äõzê

k

ý äç/L®ë?9H>õ%76õ?í¢ì¢í¢äGêÓäç%76ä7éèH>D/ãí™:?LN9V76õ?ì™Lë%769äKH6æ1äç/LNBþ769LH6æ*76ì¢FILNõ?9V76í™è:?LNF89LL*éH>9=ì¢í¢D/L‘769Kí™ã,5ILN9ê

ì™H‘ó­÷ ç/LND

k

4

ó•Lð769Låç/H‘ó¡LN5ILN976õ?ì™LåäH9LNë?ì–7>èLåäç/LîLüs@%76äí™H>D

f

i

(X

i

) = 0

õzê©76D(Lüs@%76äí™H>D

H6æ¡76ì¢FILNõ?9V76í™è:?LNF89LL

2

ý+äç/H>@?F8çÓóí¢äç ì™HsH8ãí¢D?FïãH>BðL®í¢DAæ…H>9BO76äí™H>D©H>D äç/L &/D%76ìãH>ì¢@?äí™H>D‡÷!*ç/LND ó•L76ë?ë?ì¢êîäç/L K9>õ?D/LN9*7>ãí™ã*76ì¢FIH>9í¢äç?BÊ÷Z*¡@?ä£äç/LK76õqHa5IL=Lãäí¢BO76äí™H>D/ã•í¢Bîë?ì¢êîäç%76ä¡H>@?9•ù=F89LLNí¢D?F

Kì¢@?í¢D?Fð76ì¢FIH>9í¢äç?B ãäí¢ì¢ì+F8í¢5ILã7®õqLNääLN9*õH>@?D/:Oí¢Dïäç?í™ã*è‘7>ãL8÷?JMH®í¢ä¡ì™HsH>ñAã•ë?ì–76@/ãí¢õ?ì™LKäç%76ä•äç/LKì–76ääLN9

ç%7>ãFILND/LN9V76ì¢ì¢êÊ7õLNääLN9õqLNç%7‘5sí™H>9,äç%76DÊäç/LæXH>9BðLN9‘÷

+j/Üsá()a%ÚjÛ

*ç/Lí™:?L‘7>ã:?LãèN9í¢õL:ç%75ILõLLND<äLãäL:<H>D<Lüs@%76äí™H>DÄãêAãäLNBðã*9LNë?9LãLNDzäí¢D?FO9L:A@/èL: Z9H>@?D/:Ê5ILN9

ãí™H>D/ã£H6æú•Jq÷ *ç/L=Lüz@%76äí™H>DéãêMãäLNBðãç%75IL,õqLLNDOèN9L‘76äL:O@/ãí¢D?F®äGó•H®ë%769V76BðLNäLN9ã

r

ý8äç/LDs@?B®õLN9 H6æ£9H>@?D/:?ã­76D/:

(10)

Lüz@%76äí™H>D/ãK76D/:<58769í–76õ?ì™Lãí¢DÄ7îãêAãäLNBÊý

m

76D/:

n

ýAí™ãäç/LNDÊF8í¢5ILNDÄõzê

n

= 56 + 32

·

ntxt

·

(r

2)

m

= 32

·

r

·

ntxt.

*ç/L­L/7>èNä,èH>D/ãä9@/èNäí™H>DÓH6æäç/LãêAãäLNBðãí™ãF8í¢5ILNDÄí¢DÊù,ë?ëLND/:AíÊù÷

JMH>BðL­èVç/H>í™èLã,9LNFz769:Aí¢D?Fïäç/Lí¢Bîë?ì™LNBðLNDsäV76äí™H>DöH6æäç/L­ãë?ì¢í¢ääí¢D?Fï76D/:ÊF8ì¢@?í¢D?FéBðLNäç/Hs:?ã,ç%7>:<äH

õqL­BO7>:?L8÷ *ç/Lãä9V76äLNF8í™Lãí¢Bîë?ì™LNBðLNDsäL:Äó¡LN9L7>ã*æXH>ì¢ì™H‘óãa÷

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ý/7®ì¢í¢ääì™Lç/LNì¢ëïæ…9H>B äç/Lãë?ì¢í¢ääí¢D?FðH>9*F8ì¢@?í¢D?FîBðLNäç/Hs:?ã,769LKD/LL:?L:]÷e(LKäLãäL: äç/LöBðLNäç/Hs:?ãïæXH>@?9äí¢BðLã<L‘7>èç‡ý*óí¢äç 7 :AíLN9LNDsä<ë?ì–76í¢DzäLAä6èNí¢ë?ç/LN9äLMäÓë%76í¢9<L‘7>èç äí¢BðL8÷ *ç/L

75ILN9V76FIL:?LNë?äçH6æ‡äç/L=ãVL‘769èçïä9LL,@/ãí¢D?Fäç/LKãë?ì¢í¢ääí¢D?FBðLNäç/HM:éó¡7>ã

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÷ *ç/L75ILN9V76FILãí-,‘LKH6æ‡äç/L ì–769FILãä,èH>D&/F8@?9V76äí™H>D ì¢í™ãä@/ãí¢D?Fðäç/LF8ì¢@?í¢D?FOBðLNäç/Hs:<ó*7>ã

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9L:A@/èLãäç/LèH>Bîë?ì™LAí¢äí™LãæXH>9äç/Lãë?ì¢í¢ääí¢D?Fé76D/:<F8ì¢@?í¢D?FOBðLNäç Hs:?ãa÷%3KLN9L®ó¡L76ì™ãH &/D/:Ääç/LãêAãäLNB9LyæXLN99L:ÄäHOí¢DÓäç/L®í¢Dzä9HM:A@/èNäí™H>D +æ…H>9

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äç%76äóç/LND

ntxt

= 8

ýAäç/L®ãí-,‘LH6æ”äç/L­ì–769FILãä,èH>D&/F8@?9V76äí™H>DÄì¢í™ãäõqLèH>BðLãì–769FILN9äç%76D

2

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÷ *ç/L<ãë?ì¢í¢ääí¢D?F©BðLNäç/HM:¥í™ãð76ì™ãHö5ILN9ê èH8ãäì¢ê¨äHö:?Höí¢DÑë?9V7>èNäí™èL8ýõ?@?äîç/LN9Ló¡LÊè‘76D Lãäí¢BO76äL

äç/L®:?LNë?äç H6æ”äç/LãL‘769èçÓä9LLóí¢äç/H>@?ä,ç%7‘5sí¢D?FéäHðó¡76í¢ä=@?Dzäí¢ìjäç/L®7>èNä@%76ìjãH>ì¢@?äí™H>DÓí™ã*æXH>@?D/:]÷ @?9

í¢Bîë?ì™LNBðLNDzäV76äí™H>DÄH6æ‡äç/LKãë?ì¢í¢ääí¢D?FåBðLNäç/HM:O9V76Dï@?Dzäí¢ì

1024

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ë?9H>F89V76Bä9V7‘5ILN9ãí¢D?FOäç/Lä9LL8ý%äç/L:AíLN9LNDsä=:?LNë?äç/ã=H6æì™L‘7‘5ILã=76ë?ëL‘769äHîõLLN5ILND?ì¢ê<:Aí™ãä9í¢õ?@?äL:]÷

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