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鏈〉闈㈢郴缁熼槓杩扮嚎鎬ф柟绋嬬粍姹傝В鐨勬暟鍊兼柟娉曟灦鏋勶紝娑电洊鐩存帴娉曪紙楂樻柉娑堝厓銆丩U 鍒嗚В锛変笌杩唬娉曪紙闆呭彲姣旇凯浠ｏ級鐨勭悊璁哄熀纭€銆丮ATLAB 瀹炵幇涓庡伐绋嬪疄璺点€傞€氳繃瀵规瘮涓嶅悓绠楁硶鐨勯€傜敤鍦烘櫙銆佹敹鏁涚壒鎬т笌璁＄畻澶嶆潅搴︼紝涓哄ぇ瑙勬ā绋€鐤忕郴缁熺殑宸ョ▼姹傝В鎻愪緵鍐崇瓥渚濇嵁銆?

鏁板€兼柟娉曟灦鏋勬瑙? ​

绾挎€ф柟绋嬬粍姹傝В鏂规硶鎸夋槸鍚﹂€氳繃鏈夐檺姝ラ鐩存帴鑾峰緱瑙ｅ彲鍒嗕负**鐩存帴娉?涓?杩唬娉?*涓ゅぇ绫汇€傜洿鎺ユ硶閫氳繃鐭╅樀鍒嗚В绮剧‘姹傝В锛堝疄闄呭彈娴偣璇樊闄愬埗锛夛紝閫傚悎涓皬鍨嬬瀵嗙郴缁燂紱杩唬娉曢€氳繃閫愭閫艰繎鏀舵暃鍒拌В锛岄€傚悎澶у瀷绋€鐤忕郴缁熴€備互涓嬫灦鏋勫浘灞曠ず浜嗕笁绫绘牳蹇冩柟娉曠殑灞傛鍏崇郴涓庢暟鎹祦杞繃绋嬶細

mermaid
graph TB
    subgraph 杈撳叆灞?
        A[绯绘暟鐭╅樀 A<br/>n脳n]
        b[鍙崇椤?b<br/>n脳1]
    end
    
    subgraph 鐩存帴娉?
        direction LR
        G[楂樻柉娑堝厓娉?br/>Gaussian Elimination]
        LU[LU 鍒嗚В娉?br/>LU Decomposition]
        A1[A 鈫?涓夎鐭╅樀 U<br/>閫氳繃琛屽彉鎹
        A2[A = L 脳 U<br/>涓嬩笁瑙捗椾笂涓夎]
        S1[涓夎鍥炰唬<br/>Triangular Substitution]
        S2[鍓嶄唬+鍥炰唬<br/>Forward+Backward]
    end
    
    subgraph 杩唬娉?
        J[闆呭彲姣旇凯浠ｆ硶<br/>Jacobi Iteration]
        D[鎻愬彇瀵硅鐭╅樀 D]
        R[璁＄畻浣欑煩闃?R = A - D]
        Update[鏇存柊鍏紡:<br/>x^(k+1) = D鈦宦?b - Rx^(k))]
        Conv[鏀舵暃鍒ゆ柇:<br/>||b - Ax|| 鈮?tol]
    end
    
    subgraph MATLAB瀹炵幇
        Native[鍘熺敓绠楀瓙:<br/>x = A \ b]
        Custom[鑷畾涔夊嚱鏁?<br/>jacob.m]
    end
    
    subgraph 杈撳嚭灞?
        x[瑙ｅ悜閲?x<br/>n脳1]
        iter[杩唬娆℃暟]
        res[娈嬪樊鑼冩暟]
    end
    
    A --> G
    b --> G
    A --> LU
    b --> LU
    A --> J
    b --> J
    
    G --> A1 --> S1 --> x
    LU --> A2 --> S2 --> x
    J --> D --> R --> Update --> Conv
    Conv -->|鏈敹鏁泑 Update
    Conv -->|宸叉敹鏁泑 x
    Conv --> res
    Update --> iter
    
    G -.瀹炵幇鏂瑰紡.-> Native
    LU -.瀹炵幇鏂瑰紡.-> Native
    J -.瀹炵幇鏂瑰紡.-> Custom

璇ユ灦鏋勫浘鎻ず浜嗕笁涓叧閿璁″師鍒欙細棣栧厛锛岀洿鎺ユ硶閫氳繃鐭╅樀鍙樻崲灏嗗師绯荤粺杞寲涓虹瓑浠风殑涓夎绯荤粺锛岃€岃凯浠ｆ硶閫氳繃鍒嗛噺鏇存柊閫愭閫艰繎瑙ｏ紱鍏舵锛孧ATLAB 鍘熺敓绠楀瓙 \ 瀹為檯涓婇泦鎴愪簡澶氱鐩存帴娉曪紙鍖呮嫭 LU 鍒嗚В锛夛紝涓哄伐绋嬪簲鐢ㄦ彁渚涢粦鐩掑皝瑁咃紱鏈€鍚庯紝杩唬娉曢渶瑕佹槑纭殑鏀舵暃鍒ゅ畾鏈哄埗涓?*鍒濆€奸€夋嫨**绛栫暐锛岃繖鏄叾涓庣洿鎺ユ硶鐨勯噸瑕佸尯鍒€?

Sources: matlab/闆呭彲姣旇凯浠?jacob.m

闆呭彲姣旇凯浠ｆ硶锛氬師鐞嗕笌瀹炵幇 ​

闆呭彲姣旇凯浠ｆ硶鏄粡鍏哥殑鍥哄畾鐐硅凯浠?*鏂规硶锛屽叾鏍稿績鎬濇兂鏄皢绯绘暟鐭╅樀 A 鍒嗚В涓哄瑙掔煩闃?D 涓庝綑鐭╅樀 R锛屽嵆 A = D + R锛岀劧鍚庨€氳繃閫愬垎閲忔洿鏂版眰瑙ｃ€傝鏂规硶瑕佹眰瀵硅鍏冪礌闈為浂锛屼笖鐭╅樀鍏锋湁瀵硅鍗犱紭**鐗规€т互淇濊瘉鏀舵暃銆傝凯浠ｅ叕寮忎负锛?

x^(k+1)i = (b_i - 危 a_{ij} x^(k)j) / a

鍏朵腑 i = 1, 2, ..., n 琛ㄧず鍒嗛噺绱㈠紩锛宬 琛ㄧず杩唬娆℃暟銆傝鍏紡鐨勭墿鐞嗘剰涔夋槸锛氬湪璁＄畻绗?k+1 娆¤凯浠ｆ椂锛岀 i 涓垎閲忎粎渚濊禆浜庝笂涓€娆¤凯浠ｇ殑鍏朵粬鍒嗛噺鍊硷紝杩欐剰鍛崇潃鎵€鏈夊垎閲忓彲浠?*骞惰璁＄畻**锛岃繖鏄泤鍙瘮鏂规硶鍦ㄥ苟琛岃绠楃幆澧冧笅鐨勭嫭鐗逛紭鍔裤€?

MATLAB 瀹炵幇浣嶄簬 matlab/闆呭彲姣旇凯浠?jacob.m锛屽嚱鏁扮鍚嶄负锛?

[x, iter, res] = jacob(A, b, x0, tol, maxit)

杈撳叆鍙傛暟鍖呮嫭绯绘暟鐭╅樀 A銆佸彸绔」 b銆佸垵濮嬬寽娴?x0锛堥粯璁ら浂鍚戦噺锛夈€佸宸?tol锛堥粯璁?1e-6锛夊拰鏈€澶ц凯浠ｆ鏁?maxit锛堥粯璁?100锛夛紱杈撳嚭鍖呮嫭瑙ｅ悜閲?x銆佸疄闄呰凯浠ｆ鏁?iter 鍜屾渶缁堟畫宸寖鏁?res銆傚疄鐜拌繃绋嬩腑鍖呭惈澶氬眰杈撳叆楠岃瘉鏈哄埗锛氭鏌ョ煩闃垫槸鍚︿负鏂归樀銆佺淮搴︽槸鍚﹀尮閰嶃€佸瑙掔嚎鏄惁鍚浂鍏冪礌锛屽苟閫氳繃 nargin 鍑芥暟澶勭悊鍙€夊弬鏁扮殑榛樿鍊艰缃€?

Sources: matlab/闆呭彲姣旇凯浠?jacob.m

闆呭彲姣旇凯浠ｇ殑鏁版嵁娴佷笌鏀舵暃鏈哄埗 ​

浠ヤ笅娴佺▼鍥捐缁嗗睍绀轰簡闆呭彲姣旇凯浠ｇ殑瀹屾暣鎵ц杩囩▼锛屼粠杈撳叆楠岃瘉鍒版敹鏁涘垽瀹氱殑姣忎釜鐜妭锛?

mermaid
flowchart TD
    Start([鍑芥暟璋冪敤 jacob.m]) --> Check1{nargin妫€鏌
    Check1 -->|< 3 鎴?x0 涓虹┖| SetDefault1[x0 = zeros<br/>闆跺悜閲忓垵濮嬪€糫
    Check1 -->|鈮?3| Check2
    SetDefault1 --> Check2
    
    Check2{nargin妫€鏌 -->|< 4 鎴?tol 涓虹┖| SetDefault2[tol = 1e-6<br/>榛樿瀹瑰樊]
    Check2 -->|鈮?4| Check3
    SetDefault2 --> Check3
    
    Check3{nargin妫€鏌 -->|< 5 鎴?maxit 涓虹┖| SetDefault3[maxit = 100<br/>鏈€澶ц凯浠
    Check3 -->|鈮?5| DimCheck
    SetDefault3 --> DimCheck
    
    DimCheck[鑾峰彇鐭╅樀缁村害 n,m] --> IsSquare{n == m?}
    IsSquare -->|鍚 Error1[error: 鐭╅樀A蹇呴』鏄柟闃礭
    IsSquare -->|鏄瘄 DimMatch
    
    DimMatch{b缁村害 == n?} -->|鍚 Error2[error: 鍙崇椤筨蹇呴』涓嶢鍏煎]
    DimMatch -->|鏄瘄 ExtractD
    
    ExtractD[D = diagdiagA<br/>鎻愬彇瀵硅鐭╅樀] --> CheckZero{瀛樺湪闆跺瑙掑厓绱?}
    CheckZero -->|鏄瘄 Error3[error: 闆呭彲姣旀柟娉曟棤娉曠户缁璢
    CheckZero -->|鍚 ExtractR
    
    ExtractR[R = A - D<br/>浣欑煩闃礭 --> Init[x = x0<br/>杞崲涓哄垪鍚戦噺]
    Init --> CalcRes0[res = normb - A*x<br/>鍒濆娈嬪樊]
    
    CalcRes0 --> IterLoop{iter = 1:maxit}
    IterLoop --> Update[x_new = b - R*x ./ diagD<br/>鍒嗛噺骞惰鏇存柊]
    Update --> CalcRes[res = normb - A*x_new<br/>璁＄畻鏂版畫宸甝
    CalcRes --> Assign[x = x_new<br/>鏇存柊瑙ｅ悜閲廬
    Assign --> CheckConv{res 鈮?tol?}
    
    CheckConv -->|鏄瘄 Break[break<br/>閫€鍑哄惊鐜痌
    CheckConv -->|鍚 NextIter[iter = iter + 1]
    NextIter --> IterLoop
    
    Break --> FinalCheck{iter == maxit 涓?res > tol?}
    FinalCheck -->|鏄瘄 Warning[warning: 鏈敹鏁沒
    FinalCheck -->|鍚 Return
    
    Warning --> Return([杩斿洖 x, iter, res])
    Return --> End

璇ユ祦绋嬪浘鍑告樉浜嗛泤鍙瘮杩唬鐨?*瀹归敊璁捐**锛氫粠杈撳叆鍙傛暟鐨勬櫤鑳介粯璁ゅ€艰缃紝鍒扮煩闃电壒鎬х殑涓ユ牸妫€鏌ワ紙鏂归樀銆佺淮搴﹀尮閰嶃€侀潪闆跺瑙掞級锛屽啀鍒版敹鏁涘け璐ョ殑棰勮鏈哄埗锛屾瘡涓幆鑺傞兘浣撶幇浜嗘暟鍊艰绠楃殑宸ョ▼涓ヨ皑鎬с€傜壒鍒槸娈嬪樊鑼冩暟鐨勮绠楅噰鐢ㄦ鍑犻噷寰楄寖鏁?norm()锛屾彁渚涗簡瑙ｇ簿搴︾殑閲忓寲搴﹂噺銆?

鏀舵暃鎬т緷璧栦簬鐭╅樀鐨?璋卞崐寰? 蟻(D鈦宦筊) < 1锛屽嵆杩唬鐭╅樀鐨勬渶澶х壒寰佸€兼ā闀垮皬浜?1銆傚瑙掑崰浼樼煩闃碉紙姣忚瀵硅鍏冪礌缁濆鍊煎ぇ浜庤琛屽叾浠栧厓绱犵粷瀵瑰€间箣鍜岋級鏄敹鏁涚殑鍏呭垎鏉′欢銆傚疄闄呭簲鐢ㄤ腑锛屽綋绯绘暟鐭╅樀鍏锋湁瀵硅鍗犱紭鐗规€ф椂锛岄泤鍙瘮杩唬閫氬父鑳藉湪鍚堢悊杩唬娆℃暟鍐呮敹鏁涳紱鍙嶄箣锛屽浜庣梾鎬佺煩闃垫垨闆跺瑙掔煩闃碉紙濡傛祴璇曠敤渚?test_2锛夛紝鏂规硶灏嗙洿鎺ユ姤閿欑粓姝€?

Sources: matlab/闆呭彲姣旇凯浠?jacob.m, matlab/闆呭彲姣旇凯浠?data_create.m

楂樻柉娑堝厓娉曪細MATLAB 鍘熺敓瀹炵幇 ​

楂樻柉娑堝厓娉曟槸姹傝В绾挎€ф柟绋嬬粍鏈€缁忓吀鐨?鐩存帴娉?锛岄€氳繃鍒濈瓑琛屽彉鎹㈠皢澧炲箍鐭╅樀 [A|b] 杞崲涓轰笂涓夎褰㈠紡锛岀劧鍚庨€氳繃鍥炰唬姹傝В銆侻ATLAB 閫氳繃鍙嶆枩鏉犺繍绠楃 \ 鎻愪緵浜嗛粦鐩掑皝瑁咃紝璇ヨ繍绠楃浼氭牴鎹煩闃电壒鎬ф櫤鑳介€夋嫨绠楁硶锛氬浜庝竴鑸柟闃典娇鐢?LU 鍒嗚В锛屽浜庡绉版瀹氱煩闃典娇鐢?Cholesky 鍒嗚В锛屽浜庣煩褰㈢煩闃典娇鐢?QR 鍒嗚В銆?

鍦ㄩ」鐩疄鐜颁腑锛宍matlab/闆呭彲姣旇凯浠?matlab_native.m 鍜?matlab/绾挎€ф柟绋嬬粍绮剧‘瑙?matlab_native.m` 灞曠ず浜嗗師鐢熻В娉曠殑浣跨敤鏂瑰紡锛?

matlab
load('test.mat', '-mat')
res = test_1 \ b_1

杩欒浠ｇ爜瀹炵幇浜嗕笁闃剁嚎鎬ф柟绋嬬粍鐨勭簿纭眰瑙ｏ紝鍏朵腑 test_1 鏄瑙掑崰浼樼煩闃?[[2,1,1]; [2,4,1]; [2,1,2]]锛宍b_1鏄崟浣嶅彸绔」[1; 1; 1]`銆侻ATLAB 寮曟搸鍐呴儴鎵ц浠ヤ笅娴佺▼锛氶鍏堣繘琛?LU 鍒嗚В PA = LU锛圥 鏄疆鎹㈢煩闃碉紝澶勭悊闆朵富鍏冮棶棰橈級锛岀劧鍚庤В Ly = Pb锛堝墠浠ｏ級锛屾渶鍚庤В Ux = y锛堝洖浠ｏ級銆傜疆鎹㈢煩闃?P 鐨勫紩鍏ョ‘淇濅簡绠楁硶鐨?鏁板€肩ǔ瀹氭€?锛岄伩鍏嶅洜涓诲厓杩囧皬瀵艰嚧鐨勭簿搴︽崯澶便€?

Sources: matlab/闆呭彲姣旇凯浠?matlab_native.m, matlab/绾挎€ф柟绋嬬粍绮剧‘瑙?matlab_native.m, matlab/闆呭彲姣旇凯浠?data_create.m

LU 鍒嗚В娉曪細鐞嗚浼樺娍涓?MATLAB 闆嗘垚 ​

LU 鍒嗚В灏嗙郴鏁扮煩闃?A 鍒嗚В涓轰笅涓夎鐭╅樀 L 涓庝笂涓夎鐭╅樀 U 鐨勪箻绉紝鍗?A = LU銆傝繖绉嶅垎瑙ｇ殑鏍稿績浼樺娍鍦ㄤ簬锛氬綋闇€瑕佹眰瑙ｅ涓叿鏈夌浉鍚岀郴鏁扮煩闃典絾涓嶅悓鍙崇椤圭殑鏂圭▼缁勶紙濡?Ax = b鈧? Ax = b鈧? ..., Ax = b_k锛夋椂锛屽彧闇€鎵ц涓€娆?O(n鲁) 鐨勫垎瑙ｈ繃绋嬶紝鐒跺悗閽堝姣忎釜鍙崇椤规墽琛屼袱娆?O(n虏) 鐨勪笁瑙掔郴缁熸眰瑙ｏ紝澶у箙鎻愬崌璁＄畻鏁堢巼銆?

铏界劧椤圭洰涓殑 matlab/闆呭彲姣旇凯浠?lu.m 鏂囦欢褰撳墠涓虹┖锛屼絾 MATLAB 鍘熺敓 \ 绠楀瓙宸插畬鏁撮泦鎴愪簡 LU 鍒嗚В鍔熻兘銆傚浜庢祴璇曠煩闃?test_1锛屽垎瑙ｇ粨鏋滀负锛?

L = [[1, 0, 0]; [1, 1, 0]; [1, 0, 1]] U = [[2, 1, 1]; [0, 3, 0]; [0, 0, 1]]

楠岃瘉锛歀 脳 U = [[2,1,1]; [2,4,1]; [2,1,2]] = test_1銆傚垎瑙ｅ悗鐨勬眰瑙ｈ繃绋嬪垎涓轰袱姝ワ細

鍓嶄唬锛氭眰瑙?Ly = b y鈧?= b鈧?= 1 y鈧?= b鈧?- L鈧傗倎 y鈧?= 1 - 1脳1 = 0 y鈧?= b鈧?- L鈧冣倎 y鈧?= 1 - 1脳1 = 0

鍥炰唬锛氭眰瑙?Ux = y x鈧?= y鈧?/ U鈧冣們 = 0 / 1 = 0 x鈧?= (y鈧?- U鈧傗們 x鈧? / U鈧傗倐 = 0 / 3 = 0 x鈧?= (y鈧?- U鈧佲倐 x鈧?- U鈧佲們 x鈧? / U鈧佲倎 = 1 / 2 = 0.5

鍥犳绮剧‘瑙ｄ负 x = [0.5; 0; 0]锛岃繖鍙互閫氳繃 matlab/闆呭彲姣旇凯浠?test.m 涓皟鐢ㄨ嚜瀹氫箟闆呭彲姣斿嚱鏁拌繘琛岄獙璇佸姣斻€?

Sources: matlab/闆呭彲姣旇凯浠?data_create.m, matlab/闆呭彲姣旇凯浠?test.m

涓夌鏂规硶鐨勫姣斿垎鏋? ​

涓嬭〃浠庡涓淮搴︾郴缁熷姣斾簡楂樻柉娑堝厓娉曘€丩U 鍒嗚В娉曚笌闆呭彲姣旇凯浠ｆ硶鐨勭壒鎬э紝涓哄伐绋嬪簲鐢ㄤ腑鐨勭畻娉曢€夋嫨鎻愪緵鍐崇瓥渚濇嵁锛?

瀵规瘮缁村害楂樻柉娑堝厓娉?LU 鍒嗚В娉?闆呭彲姣旇凯浠ｆ硶
绠楁硶绫诲瀷鐩存帴娉?鐩存帴娉?杩唬娉?
*璁＄畻澶嶆潅搴?O(n鲁)锛堝崟娆★級O(n鲁) + k路O(n虏)锛坘 涓彸绔」锛?O(iter路n虏)锛坕ter 涓鸿凯浠ｆ鏁帮級
*鍐呭瓨闇€姹?O(n虏)锛堝彲鍘熷湴鎿嶄綔锛?O(n虏)锛堝瓨鍌?L 鍜?U锛?O(n虏)锛堜粎闇€鍘熺煩闃碉級
*鏁板€肩ǔ瀹氭€?涓瓑锛堥渶閫変富鍏冿級楂橈紙鍏ㄤ富鍏?LU 绋冲畾锛?渚濊禆鐭╅樀鏉′欢鏁?
*鏀舵暃鐗规€?鏈夐檺姝ョ簿纭В锛堟诞鐐硅宸級鏈夐檺姝ョ簿纭В锛堟诞鐐硅宸級鏃犻檺閫艰繎锛岄渶鏀舵暃鍒ゅ畾
骞惰娼滃姏浣庯紙涓茶鍥炰唬锛?浣庯紙涓茶鍥炰唬锛?楂橈紙鍒嗛噺鍙苟琛屾洿鏂帮級
閫傜敤鍦烘櫙涓皬鍨嬬瀵嗙郴缁?澶氬彸绔」绯荤粺銆侀噸澶嶆眰瑙?澶у瀷绋€鐤忕郴缁熴€佸瑙掑崰浼樼郴缁?
MATLAB 瀹炵幇A \ b锛堝唴閮ㄨ皟鐢?LU锛?A \ b 鎴?[L,U,P] = lu(A)鑷畾涔?jacob.m 鍑芥暟
澶辫触鏉′欢濂囧紓鐭╅樀銆佷弗閲嶇梾鎬佺煩闃?濂囧紓鐭╅樀銆佷弗閲嶇梾鎬佺煩闃?闆跺瑙掑厓绱犮€侀潪瀵硅鍗犲己銆佷笉鏀舵暃

浠庤〃涓彲瑙侊紝LU 鍒嗚В娉?*鍦ㄥ鍙崇椤瑰満鏅笅鍏锋湁鏄捐憲鐨勬椂闂村鏉傚害浼樺娍锛屽洜涓哄垎瑙ｆ楠ゅ彧闇€鎵ц涓€娆★紱闆呭彲姣旇凯浠ｆ硶**鍦ㄧ煩闃佃妯″ぇ涓旂█鐤忓害楂樻椂锛屽叾 O(iter路n虏) 鐨勫疄闄呭鏉傚害鍙兘杩滀綆浜庣洿鎺ユ硶鐨?O(n鲁)锛屼笖澶╃劧鏀寔骞惰璁＄畻锛?楂樻柉娑堝厓娉?浣滀负鏁欏鎰忎箟鏈€寮虹殑鐩存帴娉曪紝鍦ㄥ伐绋嬪疄璺典腑閫氬父琚洿绋冲畾鐨?LU 鍒嗚В鎵€鍙栦唬銆?

Sources: matlab/闆呭彲姣旇凯浠?jacob.m, matlab/闆呭彲姣旇凯浠?matlab_native.m

瀹炶返鎸囧崡锛氭祴璇曟暟鎹笌楠岃瘉 ​

椤圭洰鎻愪緵浜嗕袱濂楁祴璇曠煩闃电敤浜庨獙璇佺畻娉曠壒鎬э細

Test Case 1锛堝瑙掑崰寮虹煩闃碉級锛?

matlab
test_1 = [2, 1, 1; 2, 4, 1; 2, 1, 2];
b_1 = [1; 1; 1];

璇ョ煩闃垫瘡琛屽瑙掑厓绱犵粷瀵瑰€硷紙2, 4, 2锛夊潎澶т簬璇ヨ鍏朵粬鍏冪礌缁濆鍊间箣鍜岋紙2, 3, 3锛夛紝婊¤冻闆呭彲姣旇凯浠ｇ殑鏀舵暃鍏呭垎鏉′欢銆傞€氳繃 matlab/闆呭彲姣旇凯浠?test.m 鍙獙璇侀泤鍙瘮鏂规硶鐨勬敹鏁涙€т笌瑙ｇ殑姝ｇ‘鎬с€?

Test Case 2锛堥浂瀵硅鐭╅樀锛夛細

matlab
test_2 = [0, 11, 12; 13, 0, 15; 16, 17, 0];
b_2 = [1; 1; 1];

璇ョ煩闃靛瑙掔嚎鍏ㄤ负闆讹紝闆呭彲姣旀柟娉曠殑鐭╅樀鍒嗚В姝ラ D = diag(diag(A)) 灏嗕骇鐢熼浂瀵硅鐭╅樀锛屽鑷磋凯浠ｅ叕寮?x_new = (b - R*x) ./ diag(D) 鍑虹幇闄ら浂閿欒銆傝繖绉嶇煩闃甸渶瑕佷娇鐢ㄧ洿鎺ユ硶姹傝В锛屾垨鑰呴€氳繃**琛屽彉鎹?*灏嗛潪闆跺厓绱犵Щ鑷冲瑙掍綅缃€?

娴嬭瘯鏁版嵁閫氳繃 matlab/闆呭彲姣旇凯浠?data_create.m 鐢熸垚骞朵繚瀛樹负 test.mat 鏂囦欢锛屼娇鐢?load('test.mat', '-mat') 鍔犺浇銆侻ATLAB 鐨?.mat 鏂囦欢鏍煎紡鏀寔浜岃繘鍒跺瓨鍌紝鍔犺浇閫熷害蹇笖淇濈暀鏁版嵁绫诲瀷绮惧害锛岄€傜敤浜庡ぇ瑙勬ā鐭╅樀鏁版嵁鐨勬寔涔呭寲銆?

Sources: matlab/闆呭彲姣旇凯浠?data_create.m, matlab/闆呭彲姣旇凯浠?test.m

宸ョ▼搴旂敤寤鸿 ​

閫夋嫨绾挎€ф柟绋嬬粍姹傝В鏂规硶鏃堕渶缁煎悎鑰冭檻鐭╅樀瑙勬ā銆佺█鐤忔€с€佽绠楄祫婧愪笌绮惧害瑕佹眰锛?

  1. **涓皬鍨嬬瀵嗙郴缁燂紙n < 1000锛?*锛氫紭鍏堜娇鐢?MATLAB 鍘熺敓 A \ b 绠楀瓙锛屽唴閮ㄩ泦鎴愮殑 LU 鍒嗚В绠楁硶缁忚繃楂樺害浼樺寲锛屾暟鍊肩ǔ瀹氭€т笌璁＄畻鏁堢巼鍧囪揪鍒板伐涓氭爣鍑嗐€?

  2. 澶氬彸绔」绯荤粺锛氭樉寮忔墽琛?[L,U,P] = lu(A) 鍒嗚В锛岀劧鍚庨拡瀵规瘡涓彸绔」璋冪敤 x = U \ (L \ (P*b))锛岄伩鍏嶉噸澶嶅垎瑙ｇ殑鏃堕棿寮€閿€銆?

  3. **澶у瀷绋€鐤忕郴缁燂紙n 鈮?10,000锛?*锛氳嫢鐭╅樀鍏锋湁瀵硅鍗犲己鐗规€э紝浣跨敤闆呭彲姣旇凯浠ｆ硶锛涜嫢鏀舵暃鎬т笉瓒筹紝鑰冭檻鍏辫江姊害娉曪紙CG锛夋垨 GMRES 绛夋洿鍏堣繘鐨勮凯浠ｆ硶銆?

  4. 骞惰璁＄畻鐜锛氶泤鍙瘮杩唬鐨勫垎閲忔洿鏂板ぉ鐒舵敮鎸佸垎甯冨紡璁＄畻锛屽彲缁撳悎 MPI 鎴?GPU 鍔犻€熷疄鐜板ぇ瑙勬ā骞惰姹傝В銆?

  5. 鐥呮€佺郴缁?*锛氬浜庢潯浠舵暟鏋佸ぇ鐨勭煩闃碉紝鐩存帴娉曚笌杩唬娉曞潎鍙兘澶辨晥锛岄渶鑰冭檻浣跨敤棰勫鐞?鎶€鏈垨**姝ｅ垯鍖?鏂规硶鎻愬崌鏁板€肩ǔ瀹氭€с€?

Sources: matlab/闆呭彲姣旇凯浠?jacob.m, matlab/绾挎€ф柟绋嬬粍绮剧‘瑙?matlab_native.m

鎵╁睍闃呰涓庝笅涓€姝? ​

鎺屾彙绾挎€ф柟绋嬬粍姹傝В鏄暟鍊艰绠楃殑鍩虹鎶€鑳斤紝涓鸿繘涓€姝ユ彁鍗囧伐绋嬭兘鍔涳紝寤鸿鎸変互涓嬭矾寰勬繁鍏ュ涔狅細