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1 2009 4
2 LU QR Cholesky A: n n A : A = IEEE = : 1 / 36
3 A A κ(a) := A A 1. = κ(a) = Ax = b x := A 1 b Ay = b + b y := A 1 (b + b) x = y x x x κ(a) b b 2 / 36
4 IEEE 754 = 1 : u = κ(a) > u 1 1 = = = 3 / 36
5 Notation A = (a ij ), C = (c ij ) R n n A A A = ( a ij ) A C (i, j) a ij c ij notation 4 / 36
6 LU A R n n P A LU P R n n : L R n n : U R n n : Ax = b P Ax = P b LU x = b Ly = P b U x = y LU 5 / 36
7 P A LU LU P A LU P A LU A < ε = (Matlab demo.) P A LU O(u) L U O(u) A 6 / 36
8 LU P A = LÛ A Û κ(a) κ(û) (1) Û 1 X U κ(a) κ(ax U ) κ(ax U ) u 1 ÛX U I < 1 (1) = A Crout 7 / 36
9 A X U X L AX U I < 1 (2) AX U X L I < 1 (3) L X L = A X U 8 / 36
10 κ(û) u 1 X := Û 1 X 1 = fl( X) = X 1 = Û 1 + ij u Û 1 ij ÛX 1 I < 1 ÛX 1 I = Û(Û 1 + ) I = Û Û u κ(û) > 1 = Û 1 9 / 36
11 X 1 Û A Û = κ(ûx 1) = O(u) κ(û), u = = P AX U = LÛX 1X 2... X k L L P AX U L < ε tol 10 / 36
12 LU A R n n P A LU P R n n : L R n n : U R n n : ε tol < 1 L 1 P AU 1 I ε tol P AU 1 L ε tol (R A 1 RA I ε tol ) 11 / 36
13 = = = 12 / 36
14 x, y F n x T y = n i=1 x i y i 13 / 36
15 1965 Møller (BIT) 1970 Nickel (ZAMM) (Kahan-Babuška s algorithm) 1971 Malcolm (Comm. ACM) 1972 Pichat (Numer. Math.) 1973 Kie lbasziński (Math. Stos.) 1974 Neumaier (ZAMM) (improved Kahan-Babuška s algorithm) 1977 Bohlender (IEEE Trans. Comput.) 1982 Leuprecht, Oberaigner (Computing) 1986 Kulisch, Miranker (SIAM Review, originally 1970 s in Karlsruhe) 1991 Priest (Proc. IEEE Symposium) 1999 Anderson (SIAM J. Sci. Comput.) 2002 Li et al. (ACM Trans. Math. Softw., XBLAS) Demmel, Hida (SIAM J. Sci. Comput) 14 / 36
16 F: u: IEEE 754 u = A 1:p := p i=1 A i B 1:q := q i=1 B i A i, B i F n n A i u A i 1, B i u B i 1 15 / 36
17 C 1:L = [A 1:p B 1:q ] L K, C 1:L := L i=1 C i, C i F n n c 1, c 2 : O(1) C 1:l A 1:p B 1:q c 1 u L A 1:p B 1:q + c 2 u K A 1:p B 1:q. A 1:p B 1:q K L K L 16 / 36
18 Rump [unpublished, 1980 s], [JJIAM, to appear] function R = AccInv(A,m) % right inverse version n = dim(a); I = eye(n); R = A \ I; % pure fl-pt (Solve AR = I for R) for k=2:m C = AccMM(A,R,1); % accurate dot product T = C \ I; % pure fl-pt (Solve CT = I for T) R = AccMM(R,T,k+1); % accurate dot product end 17 / 36
19 LU [ ] 0: X 1:0 = I k = 1 1: B k [A X 1:k 1 ] 1 k [ / ] 2: B k LU P k B k L k U k 3: U 1 k T k 4: X 1:k [X 1:k 1 T k ] k k [ / ] 5: U k T k ε tol u 1 L := L k, X U := X 1:k P := P k 6: k k / 36
20 QR [ ] 0: X 1:0 = I k = 1 1: B k [A X 1:k 1 ] 1 k [ / ] 2: B k QR B k Q k R k 3: R 1 k T k 4: X 1:k [X 1:k 1 T k ] k k [ / ] 5: R k T k ε tol u 1 Q := Q k, X R := X 1:k 6: k k / 36
21 20 / 36
22 Proposition 1. A F n n X 1:k LU QR n n k 1 κ(ax 1:k ) = 1 + O(u k ) κ(a) (4) Remark 1. A = m i=1 A i, A i F n n 21 / 36
23 RX 1:k I = AX 1:k Q T < 1 k log[εtol κ(a) 1 ] k =. log u κ(a) k R k T k < ε tol u 1 (5) (5) = AX 1:k Q T < ε tol 22 / 36
24 BLAS LAPACK 4. 2 Level-3 BLAS 23 / 36
25 LU QR u = Rump INTLAB randmat(n,cnd) n = 100 cnd = A F κ(a) / 36
26 Table 1: Rump n = 100 and κ(a) LU k κ(u k ) κ(t k ) κ(ax 1:k ) u k κ(a) < < 1 25 / 36
27 Table 2: Rump n = 100 and κ(a) QR k κ(r k ) κ(t k ) κ(ax 1:k ) u k κ(a) < < 1 26 / 36
28 (1): 1. A T LU Crout A P A T X U L XU T AP L T 2. ỹ = [XU T b]1 m 3. L T z = ỹ z 4. x = P z 27 / 36
29 (1): ( )Hilbert H n H n n 21 n = 15 (κ(h 15 ) ) b = H 15 e F 15 e := (1,..., 1) T H 15 x = b H15 1 b = e = (1,..., 1)T (Matlab demo) 28 / 36
30 (2): Rump Rump randmat n = b = (1,..., 1) T ε tol = 10 6 GMP 29 / 36
31 function [p,rel_err] = test_gmp_lin(a,b,xt,tol) % xt: given exact solution of Ax = b % tol: tolerance for relative error d = 53; norm_xt = norm(double(xt)); while 1 xv = gmp_lin(a,b,d); % solve Ax=b using GMP % normwise relative error rel_err = norm(double(xt-xv))/norm_xt; if rel_err < tol, break, end d = 2*d; % d = 53, 106, 212,... end 30 / 36
32 Table 3: Rump n = 500 ε tol = 10 6 GMP-based GEPP κ(a) ε 1 t 1 k ε 2 t 2 d/ / 36
33 (2): Higham in ASNA LU U ii det(a) det(p T LU) = det(p ) det(u). 32 / 36
34 X L, X U : L, U B := X L P AX U det(b) = det(p ) det(a) det(x U ) det(a) = det(p ) det(b)/ det(x U ) (6) = B:, det(b) 1 = Gershgorin B = det(b) = (6) det(a) 33 / 36
35 A L U LU = B = X L P AX U = Gershgorin (Matlab demo) 34 / 36
36 Cholesky LDL T 35 / 36
37 36 / 36
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II 2016 7 21 computer-assisted proof 1 / 64 1. 2. 3. Siegfried M. Rump : [1] I,, 14:3 (2004), pp. 214 223. [2] II,, 14:4 (2004), pp. 346 359. 2 / 64 Risch 18 3 / 64 M n = 2 n 1 (n = 1, 2,... ) 2 2 1 1
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211 年ハイパフォーマンスコンピューティングと計算科学シンポジウム Computing Symposium 211 HPCS211 211/1/18 GPU 4 8 BLAS 4 8 BLAS Basic Linear Algebra Subprograms GPU Graphics Processing Unit 4 8 double 2 4 double-double DD 4 4 8 quad-double
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並列計算の数理とアルゴリズム サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/080711 このサンプルページの内容は, 初版 1 刷発行時のものです. Calcul scientifique parallèle by Frédéric Magoulès and François-Xavier
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