ィ⟬≀㉁⛉Ꮫ䛾ᇶ┙䛸䛺䜛㉸኱つᶍ⣔䛾䛯䜑䛾㧗㏿ゎἲᙇ⤂Ⰻྡྂᒇ኱Ꮫ኱Ꮫ㝔ᕤᏛ◊✲⛉ィ⟬⌮ᕤᏛᑓᨷ䠎䠌䠍䠍ᖺ䠏᭶᭶䠑᪥1◊✲⤌⧊ᙇ⤂Ⰻ䠄ྡ኱䞉ᩍᤵ䠅௦⾲⪅䠈◊✲⥲ᣓ䛸ゎἲ㛤Ⓨ㜿㒊㑥⨾ 䠄ᒱ㜧⪷ᚨᏛᅬ኱䞉෸ᩍᤵ䠅ศᢸ⪅䠈᪂ᆺ๓ฎ⌮䛾◊✲䠄≉␗䛺⾜ิ䛻ᑐ䛧䛶ྍኚⓗ䛺๓ฎ⌮䜢ᥦ᱌䠅᭮ᡃ㒊▱ᗈ 䠄ឡ▱┴❧኱䞉෸ᩍᤵ䠅ศᢸ⪅䠈⥺ᙧ᪉⛬ᘧ䛻ᑐ䛩䜛㒊ศ✵㛫ἲ䛾㛤Ⓨ䠄ᙇ◊䛾௒಴ྩ䛸ᮭྩ䛸ඹྠ䛷㻸㻮㻙㻳㻹㻾㻱㻿㻔㼙㻕ἲ䛸㻵㻰㻾㻔㼟㻕ἲ䛾䝅䝣䝖∧䜢㛤Ⓨ䠅௒ᇼៅ἞ 䠄ྡ኱䞉ㅮᖌ䠅ศᢸ⪅䠈㞳ᩓ᭱㐺໬䛾䝯䝍ゎἲ䠄ྡ኱ᒣᮏ◊䛸ඹྠ䛷㻼㼃㻾⅔ᚰ䛻䛚䛡䜛⇞ᩱ㞟ྜయ⿦Ⲵ䝟䝍䞊䞁᭱㐺໬䠅ᐑ⏣⪃ྐ 䠄ྡ኱䞉ຓᩍ䠅ศᢸ⪅䠈ᅛ᭷್ၥ㢟䛻ᑐ䛩䜛㒊ศ✵㛫ἲ䛾㛤Ⓨ䠄ᮾ኱⸨ཎ◊䛸㫽ྲྀ኱ᫍ◊䛸ඹྠ䛷୍⯡໬ᅛ᭷್ၥ㢟䛾㻭㼞㼚㼛㼘㼐䠄㻹㻘㼃㻘㻳㻕ἲ䜢ᥦ᱌䠅ᒣᮏ᭷స 䠄⚄ᡞ኱䞉ᩍᤵ䠅ศᢸ⪅䠈ᅛ᭷್ၥ㢟䛻ᑐ䛩䜛┤᥋ἲ䛾㛤Ⓨ䠈㻴㻼㻯䠄ᙇ◊䛾῝㇂ྩ䛸ඹྠ䛷⮬ື䝏䝳䞊䝙䞁䜾ྍ⬟䛺䝤䝻䝑䜽㻽㻾ἲ䜢㛤ⓎÅືⓗィ⏬ἲ䜢฼⏝䠅ᮡཎṇ㢧䠄ᮾ኱䞉ᩍᤵ䠅༠ຊ⪅䠈㧗⢭ᗘᩘ್✚ศἲ䛾㛤Ⓨ䠄䝝䞊䝕䜱✵㛫䛻䛚䛡䜛᭱㐺㏆ఝබᘧ䛾㛤Ⓨ→ቃ⏺್ၥ㢟䚸✚ศ᪉⛬ᘧ䛾㧗㏿ゎἲ䠅2䝁䞁䝢䝳䞊䝔䜱䜽䝇䜈䛾ヨ䜏• 㒊ศ✵㛫ἲ䛾◊✲㛤Ⓨ㻳㻼㻮㼕㻙㻯㻳ἲ䠈㻮㼕㻙㻯㻾ἲ䠈㻸㼛㼛㼗㻙㻮㼍㼏㼗㻌㻳㻹㻾㻱㻿㻔㼙㻕ἲ䠈㻵㻰㻾㻔㼟㻕ἲ

Arnoldi(M,W,G)ἲ䠈୍⯡໬SSἲ䠈୪ิQRἲ㞳ᩓ᭱㐺໬䛾䝯䝍ᡓ␎䜰䝹䝂䝸䝈䝮䝝䞊䝕䜱✵㛫䛻䛚䛡䜛᭱㐺㏆ఝබᘧ䛾㛤Ⓨ• ᛂ⏝ศ㔝䛸䛾㐃ᦠ䛾䛯䜑䠖Ⓨ⾲ෆᐜィ⟬≀ᛶ⛉Ꮫ䠄䝅䝣䝖⥺ᙧ᪉⛬ᘧÅ୪ิ໬䛾䛯䜑䠈≉໬䛧䛯㻷㻿ἲ䞉㻵㻰㻾㻔㼟㻕ἲ䠅䠄୍⯡໬ᅛ᭷್ၥ㢟ÅArnoldi(M,W,G)ἲ䠈䝤䝻䝑䜽QRἲ䠅

3ᇶᮏၥ㢟䞊ゎἲ⌜䛾ㄢ㢟䞊≀⌮䠄≀㉁䝕䝄䜲䞁䠅䝁䞁䝢䝳䞊䝔䜱䜽䝇ǷȕȈ૾ᆉࡸ(A  DI) xD b, A  C NuN(A  DB) xDb, A, B  C‫׍‬ஊ͌բ᫆

O x,A  C NuN
Ax O㹀x, A, B  C NuN

AxNuNᮾ኱⸨ཎ◊䞉㫽ྲྀ኱ᫍ◊䛸䛾ඹྠ◊✲䜢⥅⥆୰2011/3/5

Df (x , D )0,f :C oCNᮾ኱Ώ㑓◊䛸䛾㐃ᦠ䜢ᶍ⣴୰N

2011/3/5

f (x, O )0,f : CN o CN㫽ྲྀ኱ᫍ◊䛸䛾ඹྠ◊✲䜢ィ⏬୰ᇶᮏၥ㢟䞊䛭䛾䠍䞊ᢸᙜ䠖᭮ᡃ㒊䞉ᙇǷȕȈ૾ᆉࡸ
D(A  DI) xb

ᶆ‽ၥ㢟b

୍⯡໬ၥ㢟

D(A  DB) x

D(㹀 A  DI) x
1

Df (x , D )㠀⥺ᙧၥ㢟0,1㹀 b ᶆ‽ၥ㢟䛻ኚ᥮ྍ⬟f :C oCNN⥺ᙧ໬䛜ᚲせ→㏆ఝ䛷ᶆ‽䞉୍⯡໬ၥ㢟䛻㏆ఝ→ၥ㢟䛻౫ᏑÅ㐃ᦠ䛾ᙉ໬ૠ͌̊• 㟁Ꮚᵓ㐀ィ⟬ᮾ኱⸨ཎ◊䞉㫽ྲྀ኱ᫍ◊䛸䛾ඹྠ◊✲Si 512 atoms, surface reconstruction simulation(A  DI) xD
D0D1b,0.400  0.001i
0.401  0.001i

(A  DI)  C2048u2048
1001 systems

D1000 1.400  0.001i
Cf. R. Takayama, et al., Phys. Rev. B 73 (2006)

5QNXGTDix
5JKHVGF%1%)
006

CPU time
Shifted COCG
Initial seedRe(α)

Time

Switch1.111

5.5 [s]00.9005.7 [s]10.7006.9 [s]2

COCG (1001 linear systems )150.4[s]䠄A + αB䠅䡔α =䠾䛻ᣑᙇ䛩䜛䛣䛸䛻䜒ᡂຌ䛧䛯
7䜽䝸䝻䝣㒊ศ✵㛫ἲ䛾฼Ⅼ( A  D1I) xD1

Dm( A  D m I) xb

KS methods
KS methodsb

KS methodsႺʩؕࡁƷನሰƴ࣏ᙲƳᘍЗȷșǯȈȫᆢƕᚘምƷȡǤȳᢿЎầσዴί(A  D1 I) xD1
Dm(A  D m I) xὸb┤஺ᇶᗏ䜢ᵓ⠏䛩䜛ؕࡁƷϐМဇb⾜ิ䞉䝧䜽䝖䝹✚䛾ィ⟬䛿୙せከ䛟䛾α䛻ᑐ䛧䛶୪ิฎ⌮䛿ᐜ᫆
8୍⯡໬ၥ㢟D(A  DB) xb
D(㹀 A  DI) x
1

1㹀 b

¾ᑐ⛠ᛶ➼䛾⾜ิᛶ㉁䜒ኚ໬¾䠧䠯ἲ䛿䛭䛾ኚ໬䛻䛹䛖ᑐᛂ¾䝷䞁䜽ⴠ䛱䛾ሙྜ䛻䛹䛖ᑐᛂÅせ◊✲¾䠞䛜ᑐ⛠䛷ṇ๎䛾䛸䛝䚸ᑐᛂྍ⬟
䠞䛾㏫⾜ิ䛾㒊ศ䛿ෆ㒊཯᚟䠄䠧䠯䠅䛷ฎ⌮ྍ⬟¾ᶆ‽ၥ㢟䛻ᑐ䛩䜛㛗䛔₞໬ᘧ䛾䠧䠯ἲ䛜ᚲせ䛭䛾㛤Ⓨ䛿ㄢ㢟䚹¾Look-Back GMRES(m)ἲ䠈IDR(s)ἲ䛾㛤Ⓨ䛿ᡂຌ䠄௒಴ྩ䠅䠄ᮭྩ䠅䝸䝇䝍䞊䝖௜䛝Krylov㒊ศ✵㛫ἲᢸᙜ䠖௒಴䞉᭮ᡃ㒊䞉ᙇἼἋἑὊἚỉ‫ݰ‬λỉᏑ୎ᵋᵋ ᵟᶐᶌᶍᶊᶂᶇҾྸỆؕỀẪᚐඥỊᵊᴾӒࣄ‫ׅ‬ૠ k Ệ‫ݣ‬Ẳ๫ም᣽ᾉ O(k2n), ᚡচܾ᣽ᾉ O(kn)Ể‫ٻف‬ẴỦẺỜᵊᴾӒࣄ‫ׅ‬ૠử‫ٶ‬ẪỂẨễẟᵌᵥᵫᵰᵣᵱᵆmᵇඥỉỴἽἆἼἌἲ ίlᾉ ἼἋἑὊἚ‫ׅ‬ૠὸ

Run m iterations
iterattions off GGMRESM
Input:, Output:
nput:

GMRES(m)ἲ䛻ᑐ䛩䜛ᨵⰋἲᵥᵫᵰᵣᵱᵆmᵇඥỆ‫ݣ‬ẴỦોᑣඥஜᄂᆮỂბႸᵥᵫᵰᵣᵱᵆmᵇඥỉỴἽἆἼἌἲ

Run m iiterations
terattions off GMRES
GM ESS
Input:
nput:, Output:

ࢼஹඥỉ
ბაໜ

ࢼஹỉોᑣඥᵋᵋ ἼἋἑὊἚԗ஖mửѣႎỆൿ‫ܭ‬ẴỦ૾ඥᵋᵋ ἙἧἾὊἉἹὅίᡈ˩‫׍‬ஊἫἁἚἽửМဇὸ

Look-Backᡓ␎ᵪᶍᶍᶉᵋᵠᵿᶁᶉ৆ဦỉბაໜὉཀྵẟസࠀ‫ٶ‬᪮ࡸỉϐನሰửܱྵẴỦẺỜỉỴἽἆἼἌἲ

ஜᄂᆮ
ỂბႸ

Run m iterations
iterattions off GGMRES
MR
Input:
nput:, Output:ᎋॾẴỔẨໜᵋᵋỉ‫ݰ‬λểസࠀ‫ٶ‬᪮ࡸỉ᧙ࣱ̞ᵋᵋỉൿ‫ܭ‬ࢼஹඥỉბაໜLook-Back GMRES(m)ἲ
ᵪᶍᶍᶉᵋᵠᵿᶁᶉ ᵥᵫᵰᵣᵱᵆmᵇඥཀྵẟᵋᵋ Ẑᢅӊẑỉസࠀ‫ٶ‬᪮ࡸửϐನሰẴỦỴἽἆἼἌἲᵹᵮᶐᶍᶎᶍᶑᶇᶒᶇᶍᶌᴾᵑᵌᵒᵻɦᚡỉ μ Ệấẟềҥᛦถ‫ࣱݲ‬ầ̬ᨦẰủỦRun m iiterations
terations ooff GGMRES
MR
Input:
nput:, Output:Numerical Results― GMRES(m)― Look-Back GMRES(m)The relative residual 2-norm history (m = 10)ADD20

DW2048

MEMPLUSRAEFSKY1

RDB2048L

WANG2

WATT__1

XENON1XENON214 / 23
2011/3/5

Restarted Shifted GMRESἲᵰᶃᶑᶒᵿᶐᶒᶃᶂᴾᵱᶆᶇᶄᶒᶃᶂᴾᵥᵫᵰᵣᵱඥᵹᵤᶐᶍᶋᶋᶃᶐ ᵿᶌᶂᴾᵥᶊᵿᶑᶑᶌᶃᶐᵊᴾᵏᵗᵗᵖᵻỴἽἆἼἌἲᵱᶃᶃᶂ ᶇᶌᶎᶓᶒσዴᵟᶂᶂ ᶇᶌᶎᶓᶒᵩᶐᶗᶊᶍᶔᢿЎᆰ᧓ỉನሰஇ‫ݱ‬സࠀவˑỆؕỀẨᡈ˩ᚐửဃ঺ᶍᶓᶒᶎᶓᶒσዴϐМဇஇ‫ݱ‬സࠀவˑỆؕỀẨσዴࣱவˑỆؕỀẨᡈ˩ᚐửဃ঺ᶍᶓᶒᶎᶓᶒ

ᵟᶂᶂ૾ᆉࡸỆσዴࣱவˑửᛢẴẮểỂἼἋἑὊἚửӧᏡỆ

Look-Backᡓ␎ᵰᶃᶑᶒᵿᶐᶒᶃᶂᴾᵱᶆᶇᶄᶒᶃᶂᴾᵥᵫᵰᵣᵱඥỆ‫ݣ‬ẴỦᵪᶍᶍᶉᵋᵠᵿᶁᶉ৆ဦỉᚨ‫ܭ‬ᵋᵋ ᵱᶃᶃᶂ૾ᆉࡸ ḵᴾᵪᶍᶍᶉᵋᵠᵿᶁᶉᴾᵥᵫᵰᵣᵱᵆmᵇඥᵋᵋ ᵟᶂᶂ૾ᆉࡸ

ḵᴾσዴࣱவˑửᛢẴᵹᵮᶐᶍᶎᶍᶑᶇᶒᶇᶍᶌᴾᵒᵌᵏᵻểầ࠯ᘍỂễẬủịᵊᴾσዴࣱவˑử฼ẺẴầ‫נ܍‬ẴỦᵌᵟᶂᶂ૾ᆉࡸỆ‫ݣ‬Ẳềờᵪᶍᶍᶉᵋᵠᵿᶁᶉ৆ဦửᢘဇӧᏡ

Look-Backᡓ␎੩కඥᾉ ᵱᶆᶇᶄᶒᶃᶂᴾᵪᶍᶍᶉᵋᵠᵿᶁᶉ ᵥᵫᵰᵣᵱᵆmᵇඥỴἽἆἼἌἲ
ᵱᶃᶃᶂ૾ᆉࡸ

ᵟᶂᶂ૾ᆉࡸ

ỊസࠀἠἽἲỉҥᛦถ‫ࣱݲ‬ử̬ᨦẴỦợạỆᚨ‫ܭ‬Ịσዴࣱவˑử฼ẺẴợạỆᚨ‫ܭ‬Run m iiterations
terations ooff GGMRES
MR
Input:
nput:, Output:

Run m iterations
terations ooff ccollinear-Krylov
ollinear
Input:
nput:, Output:᬴ܱኽௐᵥᵫᵰᵣᵱᵆᵑᵎᵇඥ ᵪᶍᶍᶉᵋᵠᵿᶁᶉᴾᵥᵫᵰᵣᵱᵆᵑᵎᵇඥ
ᵰᶃᶑᶒᵿᶐᶒᶃᶂᴾᵱᶆᶇᶄᶒᶃᶂᴾᵥᵫᵰᵣᵱඥ ੩కඥႻ‫ݣ‬സࠀἠἽἲᵡᵟᵴᵧᵲᵷᵎᵓᩘ್ᐇ㦂ኺᢅ଺᧓ ᵹᶑᶃᶁᵌᵻᇶᮏၥ㢟䞊䛭䛾䠎䞊ᢸᙜ䠖ᐑ⏣䞉᭮ᡃ㒊䞉ᙇ䠄཯᚟ἲ䠅ᐑ⏣䞉ᒣᮏ䞉ᙇ䠄┤᥋ἲ䠅• ᮏ◊✲䛷ᢅ䛖ၥ㢟– ୍⯡໬ᅛ᭷್ၥ㢟• ධຊ䠖 ⾜ิ• ฟຊ䠖 ᅛ᭷ᑐ ᅛ᭷್ᅛ᭷䝧䜽䝖䝹– 䛹䛾䜘䛖䛺ᅛ᭷್䛜ᚲせ䛛䠛• ᛂ⏝ศ㔝䛻ᛂ䛨䛶ከᵝ䛺䝙䞊䝈19ᅛ᭷್ィ⟬• ᅛ᭷್䛾䝙䞊䝈」⣲ᖹ㠃ୖ䛾ᅛ᭷್ศᕸ䠉 ᩘ䠈✀㢮 䠄ᛂ⏝ศ㔝䠅Im.

– 䛩䜉䛶䛾ᅛ᭷್䠄ex. 㟁Ẽᅇ㊰䠅Re.

– ᑡᩘ䠈ᣦᐃ䛥䜜䛯Ⅼ㏆ഐ
䠄ex. 㟁Ꮚᵓ㐀ィ⟬䠅– ᑡᩘ䠈ᣦᐃ䛥䜜䛯㡿ᇦෆ㒊
䠄ex. 㔞Ꮚ໬Ꮫィ⟬䠅20ග㞟✚ᅇ㊰䛻⌧䜜䜛ᅛ᭷್ၥ㢟• ᛂ⏝ศ㔝Im.

– ග㞟✚ᅇ㊰ [ Huang, 2008 ]䠄䝣䜷䝖䝙䜽䝇⤖ᬗ䛾ᛂ⏝䠅• ᅛ᭷್䛾䝙䞊䝈Re.

– ༢఩෇࿘ୖ䛾ᅛ᭷್– ᚲせ䛺ᅛ᭷್䛿ᑡᩘ༢఩෇ᚲせ䛺ᅛ᭷್୙せ䛺ᅛ᭷್– ከᩘ䛾୙せ䛺ᅛ᭷್䛜༢఩෇䛾ෆ㒊䞉እ㒊䛻Ꮡᅾ䛣䛾ၥ㢟䛻㐺䛧䛯㧗㏿ゎἲ䛿䠛21ᩘ್ゎἲ• 䛩䜉䛶䛾ᅛ᭷್ 䠉 ┤᥋ἲᢸᙜ䠖῝㇂䞉ᐑ⏣䞉ᒣᮏ䞉ᙇ䝤䝻䝑䜽QRศゎ– QZ ἲ [ Moler & Stewart, 1973 ]• ᑡᩘ䛾ᅛ᭷್ 䠉 ᑕᙳἲ– ᣦᐃ䛥䜜䛯Ⅼ㏆ഐᮾ኱⸨ཎ◊䞉㫽ྲྀ኱ᫍ◊䛸䛾ඹྠ◊✲• Krylov 㒊ศ✵㛫ἲ– Lanczos ἲ [ Lanczos, 1950 ]– Arnoldi ἲ [ Arnoldi, 1951 ]ᢸᙜ䠖ᒣୗ䞉ᐑ⏣䞉᭮ᡃ㒊䞉ᙇ• Jacobi-Davidson ἲ [ Sleijpen & Vorst, 1996 ] ᢸᙜ䠖ᐑ⏣䞉᭮ᡃ㒊䞉ᙇ– ᣦᐃ䛥䜜䛯㡿ᇦෆ㒊• Sakurai-Sugiura ἲ [ Sakurai & Sugiura, 2003 ]↓ ከ㔜㐃⤖㡿ᇦ䜈䛾ᣑᙇ– ᣦᐃ䛥䜜䛯⤯ᑐ್䜢᭷䛩䜛ᅛ᭷್ᢸᙜ䠖ᐑ⏣䞉᭮ᡃ㒊䞉ᙇ22཯᚟ᑕᙳἲึᮇ䝧䜽䝖䝹㒊ศ✵㛫䛾ᵓ⠏‫ ڼ‬䠄పḟඖ䠅㒊ศ✵㛫䛾ᇶᗏ䜢⏕ᡂ䛩䜛㏆ఝᅛ᭷ᑐ䛾ᵓᡂ‫ ڼ‬㒊ศ✵㛫䛾୰䛷ᅛ᭷ᑐ䜢㏆ఝ䛩䜛཰᮰ุᐃ‫ ڼ‬ṧᕪ䝜䝹䝮䛜༑ศᑠ䛥䛡䜜䜀ィ⟬䜢⤊஢‫ ڼ‬䛭䛖䛷䛺䛡䜜䜀䠈㒊ศ✵㛫䜢ᣑ኱䛧ྠᵝ䛾ィ⟬䜢཯᚟䛩䜛
23

㏆ఝᅛ᭷ᑐ䛾ᵓᡂ•ḟඖ㒊ศ✵㛫䛾ᇶᗏ䠖• ㏆ఝᅛ᭷ᑐ䠖z 㒊ศ✵㛫䛾୰䛷㏆ఝᅛ᭷䝧䜽䝖䝹z ṧᕪ䝧䜽䝖䝹䜢ồ䜑䜛

䛻ᑐ䛧䠈Petrov-Galerkin᮲௳䜢ㄢ䛩‫ ڼ‬ᑠつᶍ䛺ᅛ᭷್ၥ㢟䜢ゎ䛟䛣䛸䛻ᖐ╔24

Arnoldi ἲึᮇ䝧䜽䝖䝹㒊ศ✵㛫䛾ᵓ⠏‫ ڼ‬䠄పḟඖ䠅㒊ศ✵㛫䛾ᇶᗏ䜢⏕ᡂ䛩䜛㏆ఝᅛ᭷ᑐ䛾ᵓᡂ‫ ڼ‬ṧᕪ᮲௳䠖‫ ڼ‬ᑠつᶍ䛺ᅛ᭷್ၥ㢟䠖

཰᮰ุᐃ

25

㒊ศ✵㛫䛾ᵓ⠏z 㒊ศ✵㛫

‫ ڼ‬Krylov㒊ศ✵㛫䠖• ᶆ‽ᅛ᭷್ၥ㢟䛻ኚᙧ䛧䛶䜒ᅛ᭷ᑐ䛿୙ኚ

Advantagez 㒊ศ✵㛫䛾ᇶᗏ‫ ڼ‬ᇶᗏ䛾┤஺ᛶ䠖‫ ڼ‬ᇶᗏ䛾⏕ᡂἲ䠖 Arnoldiཎ⌮䠄䛾ୖHessenberg⾜ิ 䠅26

Arnoldi ἲ Disadvantageึᮇ䝧䜽䝖䝹䜢ಀᩘ䛸䛩䜛⥺ᙧ᪉⛬ᘧ䜢ゎ䛟䠄㻸㼁ศゎ䠅ィ⟬᫬㛫䛾ቑ኱㒊ศ✵㛫䛾ᵓ⠏‫ ڼ‬㒊ศ✵㛫䠖‫ ڼ‬ᇶᗏ䛾┤஺ᛶ䠖㏆ఝᅛ᭷ᑐ䛾ᵓᡂ‫ ڼ‬ṧᕪ᮲௳䠖‫ ڼ‬ᑠつᶍ䛺ᅛ᭷್ၥ㢟䠖཰᮰ุᐃ27

Arnoldi ἲ䛾ᣑᙇ
Arnoldi ἲ

Arnoldi(M,W,G) ἲ‫ ڼ‬㒊ศ✵㛫䠖‫ ڼ‬ᇶᗏ䛾┤஺ᛶ䠖‫ ڼ‬ṧᕪ᮲௳䠖‫ ڼ‬ᑠつᶍ䛺ᅛ᭷್ၥ㢟䠖z ⾜ิ䠄䛻௵ពᛶ䜢୚䛘䠈Arnoldiἲ䜢ᣑᙇ䠖ṇ๎䠈䠖ṇ್䜶䝹䝭䞊䝖 䠅28

⾜ิ M 䛻╔┠䛧䛯ゎἲ
ArnoldiἲArnoldi(M,W,G)ἲ

‫ ڼ‬㒊ศ✵㛫䛾ᵓ⠏‫ ڼ‬㒊ศ✵㛫䛾ᵓ⠏ᇶᗏ⏕ᡂẖ䛻䠈䜢ಀᩘ䛸䛩䜛⥺ᙧ᪉⛬ᘧ䜢ゎ䛟z ┤᥋ἲ‫ ڼ‬LUศゎ䠈Choleskyศゎz ᐃᖖ཯᚟ἲ 䠄㏆ఝⓗ䛻ゎ䛟䠅‫ ڼ‬Gauss-Seidelἲ䠈SORἲ䠈…
‫ ڼ‬೵Ṇ᮲௳䠖 ཯᚟ᅇᩘ• ྛᇶᗏ⏕ᡂ䛻䛚䛔䛶䛜ྠ䛨䛸䛿㝈䜙䛺䛔• ᐃᖖ཯᚟ἲ䛻ẚ䜉䛶ỗ⏝ᛶ䛿㧗䛔䛸ᮇᚅ䛷䛝䜛䠄ᅛᐃ䠅z Krylov㒊ศ✵㛫ἲ 䠄㏆ఝⓗ䛻ゎ䛟䠅‫ ڼ‬GMRESἲ䠈CGἲ䠈…
‫ ڼ‬೵Ṇ᮲௳䠖 䠄┦ᑐṧᕪ䠅29ᐇ㦂⤖ᯝ 1Arnoldiἲ౛㢟1 䠄㟁ẼᕤᏛ䠅 [Matrix Market]
⾜ิ A 䠄BFW782A䠅
• Size䠖 782• Nonzeros䠖 7,514• Type䠖 ᐇ㠀ᑐ⛠⾜ิ B 䠄BFW782B䠅

Its
39 ཯᚟
Time 0.76 ⛊ゎἲ1-1 䠄Gauss-Seidelἲ䠅

䠄

䛾ሙྜ䠅

Its
39 ཯᚟
Time 0.16 ⛊• Size䠖 782• Nonzeros䠖 5,982• Type䠖 ᐇᑐ⛠ゎἲ2-1 䠄ILU(0)௜䛝GMRES(10)ἲ䠅䠄

ItsTime䛾ሙྜ䠅39 ཯᚟0.21 ⛊300ᐇ㦂⤖ᯝ 2Lanczosἲ౛㢟2 䠄኱つᶍ㟁Ꮚᵓ㐀ィ⟬䠅[Iguchi et al., Phys. Rev. Lett. 99 (2007)]⾜ิ A• Size䠖 7,776• Nonzeros䠖 3,619,104• Type䠖 ᐇᑐ⛠⾜ิ B

Its38 ཯᚟Time 279 ⛊ゎἲ1-2 䠄Gauss-Seidelἲ䠅

䠄

䛾ሙྜ䠅

Its
38 ཯᚟
Time 11 ⛊• Size䠖 7,776• Nonzeros䠖 3,619,104• Type䠖 ᐇᑐ⛠ṇᐃ್ゎἲ2-2 䠄CGἲ䠅䠄

ItsTime

䛾ሙྜ䠅

38 ཯᚟
11 ⛊
311ᐇ㦂⤖ᯝ 3Arnoldiἲ౛㢟3 䠄஘ᩘ⾜ิ䠅⾜ิ A• Size䠖 2,000• Nonzeros䠖 400,000• Type䠖 」⣲㠀䜶䝹䝭䞊䝖⾜ิ BIts
Time15 ཯᚟20.9 ⛊ゎἲ1-1 䠄Gauss-Seidelἲ䠅• Size䠖 2,000• Nonzeros䠖 400,000• Type䠖」⣲㠀䜶䝹䝭䞊䝖ゎἲ2-1 䠄ILU(0)௜䛝GMRES(10)ἲ䠅䠄

Its-཯᚟Time -⛊ItsTime䠅

15 ཯᚟
16.8 ⛊322᭱㐺໬䛻䛴䛔䛶䛾ヰ㢟ᥦ౪

PWR⅔ᚰ䛻䛚䛡䜛⇞ᩱ㞟ྜయ⿦Ⲵ䝟䝍䞊䞁᭱㐺໬

PWR䠄ຍᅽỈᆺ㍍Ỉ⅔䠅⅔ᚰ䛸䛿ͤUnited States Nuclear Regulatory Commission䜘䜚ͤཎᏊຊ䞉䜶䝛䝹䜼䞊ᅗ㠃㞟2010ᖺ∧䜘䜚34

PWR㐠㌿䝃䜲䜽䝹
⿦Ⲵ吨吖呎
呉タィ

⣙୕䜹᭶⇞ᩱ叏⣙1|ᐃᮇ᳨ᰝ㛤ጞ⇞ᩱྲྀ叫ฟ厹呍⇞ᩱእほ᳨ᰝ

3可ྲྀ叫᥮厪⇞ᩱ⿦ⲴཎᏊ
⅔㐠㌿

ཎ
Ꮚ
⅔㉳
ືཎᏊ
⅔೵Ṇ⣙୍ᖺ㛫⿦Ⲵ䝟䝍䞊䞁タィ䛻䛛䛡䜙䜜䜛ᮇ㛫䛿䚸䠎᪥䡚䠍㐌㛫⛬ᗘ
35

PWR⅔ᚰ䛾⇞ᩱ⿦Ⲵ䝟䝍䞊䞁タィ• ⅔ᚰ
䞊 ┤ᚄ⣙3m䚸᭷ຠ㛗⣙3.5m• ⇞ᩱ㞟ྜయ
䞊 121䡚193య⛬ᗘ䞊 ᪂⇞ᩱ䚸୍ᅇ⇞↝䚸஧ᅇ⇞↝ etc.
• ⅔ᚰ⇞ᩱ䝃䜲䜽䝹䝁䝇䝖䞊 ᖺ㛫ᩘ༑൨෇

ͤཎᏊຊᏳ඲◊✲༠఍䜘䜚36

PWR⅔ᚰ䛾⇞ᩱ⿦Ⲵ䝟䝍䞊䞁タィ⤌ྜ䛫᭱㐺໬ၥ㢟278
䞊 ⇞ᩱ㞟ྜయ157→ 157!| 10㏻䜚
䞊 1/8ᑐ⛠ᛶ䜢⪃៖䛧䛶䜒⣙ 1019㏻䜚

⅔ᚰ䛾⤒῭ᛶ䞉Ᏻᐃᛶ䜢⪃៖⇞ᩱ㓄⨨䛾タィ䛻䜘䜚䚸㐠㌿୰䛾ฟຊ䜔⇞ᩱ䝁䝇䝖䛺䛹䛜኱ᖜ䛻ኚ໬ͤཎᏊຊᏳ඲◊✲༠఍䜘䜚37

PWR⅔ᚰ䛾⇞ᩱ⿦Ⲵ䝟䝍䞊䞁᭱㐺໬⤒῭ᛶ䛾㏣ồ䠄┠ⓗ䠅• 㞟ྜయྲྀ䜚ฟ䛧⇞↝ᗘ• 䝃䜲䜽䝹㛗
Ᏻ඲ᛶ䛾☜ಖ䠄ไ⣙᮲௳䠅• ฟຊ䝢䞊䜻䞁䜾ಀᩘ• ཯ᛂᗘಀᩘ
• ཯ᛂᗘ೵Ṇవ⿱• etc.

38

ICE-BURN䠄ྡ኱ᒣᮏ◊䠅
¾GUI䜢౑䛳䛯⅔ᚰタィ⏝䝋䝣䝖¾タィ䛧䛯⅔ᚰ䛷⇞↝ィ⟬䛜ྍ⬟39ၥ㢟䛾㞴䛧䛥• ไ⣙᮲௳䛜ከ䛔䞊 ฟຊ䝢䞊䜻䞁䜾ಀᩘ䚸㞟ྜయ⇞↝ᗘ䚸etc.
• 䠍䛴䛾ゎ䛾ホ౯䛻ከ䛟䛾᫬㛫䜢せ䛩䜛• ከ䛟䛾ᒁᡤ᭱㐺ゎ䜢䜒䛴• 㠀⥺ᙧᛶ䜢᭷䛩䜛z40ᢸᙜ䠖௒ᇼ䞉ᮧୖ䠄ྡ኱ᒣᮏ◊䛸䛾ඹྠ◊✲䠅䝯䝍ᡓ␎ศᯞ㝈ᐃἲ䠇⥺ᙧィ⏬ἲ㑇ఏⓗ䜰䝹䝂䝸䝈䝮ᒁᡤ᥈⣴ἲ䜰䝙䞊䝸䞁䜾ἲ┤᥋᥈⣴ἲx (1)x ( 2)

䝍䝤䞊᥈⣴䝝䜲䝤䝸䝑䝖䝯䝍ᡓ␎x ( 3)x ( 4)N ( x (1) )N ( x ( 2) )N ( x ( 3) )N ( x ( 4)41䛤㟼⫈䛹䛖䜒䛒䜚䛜䛸䛖䛤䛦䛔䜎䛧䛯

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