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- みりあ やたけ
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4 MPFR/GMP BNCpack (cf., Vol, 21, pp , 2011) Runge-Kutta (cf. arxiv preprint arxiv: , Vol.19, No.3, pp , 2009) Strassen (cf. JSIAM Letters, Vol.6, pp.81-84, 2014)
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6 + - ケチ表現 (1bit) 小数点 Precision _mpfr_prec 符号部 (1bit) 指数部仮数部 in bits + 8bits 1 23 bits ( + 1bit) _mpfr_sign Sign - 単精度 (single) 32 bits _mpfr_exp Exponent Practical multiple + Pointer to 11 bitsieee754(r) 1 standard 52 bits ( + 1bit) *_mpfr_d World record of π Mantissa - precision mantissa 倍精度 (double) 64 bits 1,030,700,000,000 in hex bits =4,122,800,000, bits ~65536 single double extented quadruple octuple double IEEE (CPU, GPU) Length of mantissa in bits (mpfr t ) IEEE754(r) standard 実用的な多倍長精度 ~65536 π の世界記録 ( 高橋, 2009) dec. digits = bits single double extented double quadruple octuple 仮数部の bit 数 1
7 IEEE754 ( ) vs. DD(4 ) QD(8 ) exflib MPFR/GMP, mpf(gmp), ARPREC(?) IEEE745 vs. DD QD, ARPREC (by Baily) exflib, MPFR/GMP, mpf(gmp) (GMP ) cf. MPACK (by ( )) = LAPACK / (DD QD + MPFR)
8 Year Category Integrated Computer Algebra Software Maple Mathematica Maxima Linear Computation LINPACK EISPACK LAPACK BLAS ScaLAPACK XBLAS ATLAS exflib CLN MPFR Over-quadruple Precision Floating-point Arithmetic Library GMP ARPREC Double-double QD/DD MP Floating-point Format and Arithmetic (Single and Double Precision) Binary Floating-point Arithmetic in Microsoft BASIC Hexadecimal Floating-point Arithmetic (IBM format) IEEE754 Binary Floating-point Arithmetic
9 MPFR/GMP GMP (GNU MP) (mpn) (mpz) (mpq) (mpf) Version 6.0.0a MPFR (GNU MPFR) GMP mpn IEEE754 Version mpz (integer) GNU MP(GMP) mpq (rational) mpf (real) GNU MPFR mpn(mp Natural number arithmetic) kernel generic (pure C codes) x86 x86_64 sparc arm Assembler codes
10 ( ) MPFR/GMP
11 MPFR/GMP GMP mpn basecase Karatsuba Toom-Cook FFT CPU SIMD milli-sec/mul 1.E+01 1.E #bits E-01 1.E-02 1.E-03 1.E-04 1.E-05 Mul of MPFR N^1.5 N^1.6 PentiumD Pentium4 Pentium4 Add milli-sec/div 1.E+02 1.E+01 1.E #bits E-01 N^1.4 1.E-02 N^1.5 PentiumD 1.E-03 Pentium4 1.E-04 Div of MPFR O(N 1.4 ) O(N 1.6 )
12 BNCpack BNCpack Basic Numerical Computation PACKage 2001 GMP mpf t MPFR/GMP mpfr t C (C++ ) MPI ( ) ( ( ) ) 3. ( ) 4. ( QR ) 5. (Newton Newton, Regula-Falsi ) 6. (DKA ) 7. ( Romberg ) 8. ( ) 9. ( )
13 A = [a ij ], B = [b ij ] C := AB a ij = 5 (i + j 1), b ij = 3 (n i + 1). l c ij := a ik b kj A = [A ik ], B = [B kj ] (1 i M, 1 k L, 1 j N) A, B C = [C ij ] L C ij := A ik B kj k=1 k=1
14 ( ): 128bits vs. Intel Math Kernel( ) H/W Intel Core i (3.6 GHz), 64 GB RAM S/W Scientific Linux 6.3 x86 64, Intel C Compiler Ver , BNCpack ver. 0.8, MPFR 3.1.2, GMP m n Simple Block(16) Block(32) Block(64) Double
15 : 1024bits m n Simple Block(16) Block(32) Block(64) bits alloc, free )
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17 CPU 12 mpfrbench:100 decimal digits 25 mpfrbench: decimal digits Speedup Ratio digits ION digits PentiumIII 100digits PentiumIV 100digits PentiumD 100digits Corei7 100digits Athlon64X2 100digits Core2Quad 100digits CeleronE digits PhenomII Speedup Ratio digits Corei digits ION digits PentiumIII 10000digits PentiumIV 10000digits PentiumD 10000digits Corei digits Athlon64X digits Core2Quad 10000digits CeleronE digits PhenomII 10000digits Corei CPU(1 core) MPI MPIBNCpack OpenMP( CPU) + BNCpack, CUDA(NVIDIA GPU) CUMP
18 MPIBNCpack MPFR/GMP or GMP (MPIGMP) _mpfr_size _mpfr_prec _mpfr_exp *_mpfr_d 符号部 指数部 仮数部 mpfr_t data type PE0 0 1 N (_mpfr_prec) パッキング処理 void * _mpfr_size _mpfr_prec _mpfr_exp 0 1 送受信 Tutorial
19 OpenMP( CPU) GPU CUMP ( GMP mpf t mpn generic C basecase ) (H/W) Intel Xeon E v2(2.10ghz) 2 = 12 cores, NVIDIA Tesla K20 (S/W) CentOS 6.5 x86 64, gcc-4.4.7, Intel C compiler , GMP 6.0.0a, MPFR 3.1.2, CUDA 6.5 CPU:12 threads, GPU: 8 8 = 64 threads
20 12 cores CPU vs. Tesla K20 GPU CPU / GPU CPU/GPU bits CPU GPU GPU
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22 [ ] Ax = b A R n n, b, x i nr n E(Ã), E( b) E( x) [ ] à x = b à = A + E(Ã), b = b + E( b), x = x + E( x) κ(a) = A A 1 ( ) E( x) κ(a) x 1 E(Ã) E(Ã) + E( b) A b A Strassen LU
23 1967 Moler n f(x) = 0, f : R n R n (1) Newton (1) f(x) = Ax b Jacobi A (L >> S) [L ] r k := b Ax k (2) [S ] Solve Az k = r k for z k (3) [L ] x k+1 := x k + z k (4) r k x k
24 (2) (4) A [S], b [L] S L A [L] := A, A [S] := A [L], b [L] := b, b [S] := b [L] A [S] := P [S] L [S] U [S] Solve (P [S] L [S] U [S] )x [S] 0 = b [S] for x [S] 0 x [L] 0 := x [S] 0 For k = 0, 1, 2,... r [L] k r [S] k := b [L] Ax [L] k := r [L] k Solve (P [S] L [S] U [S] )z [S] k z [L] k x [L] k+1 := z [S] k := x[l] k Exit if r [L] k + z[l] k = r [S] k for z [S] k 2 n ε R A F x [L] 2 + ε A k
25 ρ F (n)κ(a)ε S 1 ψ F (n)κ(a)ε S < 1 α F < 1 (5) lim x x k k β F 1 α F x (6) β F /(1 α F ) (cf. A.Buttari, J.Dogarra, Julie Langou, Julien Langou, P.Luszczek, and J.Karzak, Mixed precision iterative refinement techniques for the solution of dense linear system, The International Journal of High Performance Computing Applications, Vol. 21, No. 4, pp , 2007.) S-L κ(a)ε S << 1 (7) ε S, ε L S,L
26 Computational Time L digits Direct Method LU Decomposition Forward & Backward Substitions S digits Direct Method LU F&B Subst S-L Iterative Refinement LU Matrix-Vector Multiplication F&B Subst Matrix-Vector Multiplication F&B Subst Iteration 3 1. κ(a) < 10 7 = (S = 7)- (L = 15): SP-DP ( ) 2. κ(a) < = (S = 15)-4 or (L > 30): DP-MP 3. κ(a) > = 4 or - 8 or : MP-MP
27 DP-MP H/W AMD Athlon64X , 4GB S/W CentOS 5.2 x86 64, GCC 4.1.2, MPFR 2.3.2/GMP BNCpack 0.7b, LAPACK 3.2, ATLAS ^ Z DW DW W DW dd ^ Z / Z DW DW W DW dd dd dd dd dd ldde ldde lddd ldddd d d E >W< d>^ E >W< d>^ E >W< d>^ >ldd >lddd >lddd
28 (1/6): n n (ODE) { dy dt = f(t, y) Rn y(t 0 ) = y 0 :[t 0, α] (8) m IRK c 1,..., c m, a 11,..., a mm, b 1,..., b m m 2m Gauss c 1 a 11 a 1m... c m a m1 a mm b 1 b m = c A b T (9)
29 (2/6): m Runge-Kutta t 0, t 1 := t 0 + h 0,..., t k+1 := t k + h k... t k t k+1 y k+1 y(t k+1 ) 2 (A) Y = [Y 1... Y m ] T R mn Y 1 = y k + h m k j=1 a 1jf(t k + c j h k, Y j ). Y m = y k + h m k j=1 a mjf(t k + c j h k, Y j ) F(Y) = 0 (10) (B) Y y k+1 y k+1 := y k + h k m j=1 b j f(t k + c j h k, Y j )
30 (3/6):SPARK3 (1/2) W (Hairer & Wanner) (Gauss ) 1/2 ζ 1. X = W T ζ BAW = ζm 2 ζ m 2 0 ζ m 1 ζ m 1 0 W = [w ij ] = [ P j 1 (c i )] (i, j = 1, 2,..., m) ( P s (x) : s-th shifted Legendre polynomial) ( ζ i = 2 1 4i 2 1) (i = 1, 2,..., m 1) B = diag(b), I m = W T BW = diag(1 1 1)
31 (4/6):SPARK3 (2/4) Newton (W T B I n )(I m I n h k A J)(W I n ) E 1 F 1 G 1 E 2 F 2 = I m I n h k X J = G m 2 E m 1 F m 1 G m 1 E m E 1 = I n 1 2 h kj, E 2 = = E s = I n F i = h k ζ i J, G i = h k ζ i J (i = 1, 2,..., m 1)
32 (5/6):SPARK3 + Newton + 1. Y 1 := [y k y k... y k ] T R mn 2. For l = 0, 1, 2,... Newton Y l := [Y (l) 1 Y (l) 2... Y m (l) ] T Y (l) i = y 0 + h mj=1 k a ij f(t k + c i h k, Y (l 1) i ) C := I m I n h k X J, d := (W T B I n )( F(Y l )) (S) Solve Cx 0 = d for x 0 For ν = 0, 1, 2,... r ν := d Cx ν (S) r ν := r ν / r ν (S) Solve Cz = r ν for z x ν+1 := x ν + r ν z Check convergence x νstop Y l+1 := Y l + (W I n )x νstop Check convergence Y lstop 3. Y := Y lstop = [Y 1 Y 2... Y m ] T 4. y k+1 := y k + h k m j=1 b jf(t k + c j h k, Y j )
33 (6/6) Runge-Kutta (Gauss ) 128 ODE (10 50 ) Intel Core i7 920, 8GB RAM, CentOS 5.6 x86 64, gcc MPFR 3.1.1/GMP Comp.Time (s) m Relative Error 1.E+14 1.E+10 1.E+06 1.E+02 1.E-02 1.E-06 1.E-10 1.E-14 1.E-18 1.E-22 1.E-26 1.E-30 1.E-34 1.E-38 Iter.Ref-DM W-Trans. W-Iter.Ref-MM W-Iter.Ref-DM Max.Rel.Err SPARK3 + DP-MP
34 Strassen LU LU (A = LU) 1. A A 11 R K K A 12 R K (n K), A 21 R (n K) K, and A 22 R (n K) (n K) 2. A 11 L 11 U 11 (= A 11 ) LU, A 12 U 12 A 21 L A (1) 22 := A 22 L 21 U A := A (1) 22 n K 0
35 Winograd s variant (1/2) C := AB = [c ij ] C R m n, A = [a ij ] R m l B = [b ij ] R l n [ ] [ ] A11 A A = 12 B11 B, B = 12. (11) A 21 A 22 B 21 B 22 S 1 := A 21 + A 22, S 2 := S 1 A 11, S 3 := A 11 A 21, S 4 := A 12 S 2, S 5 := B 12 B 11, S 6 := B 22 S 5, S 7 := B 22 B 12, S 8 := S 6 B 21, (12) M 1 := S 2 S 6, M 2 := A 11 B 11, M 3 := A 12 B 21, M 4 := S 3 S 7, M 5 := S 1 S 5, M 6 := S 4 B 22, M 7 := A 22 S 8, (13)
36 Winograd s variant (2/2) T 1 := M 1 + M 2, T 2 := T 1 + M 4. (14) Through (12) (13) (14), we can obtain C as follows: [ ] M2 + M C := 3 T 1 + M 5 + M 6 T 2 M 7 T 2 + M 5 Winograd s variant involves the following arithmetical operations: Mul(m, l, n) = 7Mul(m/2, l/2, n/2), Addsub(m, l, n) = 4Addsub(m/2, l/2) + 4Addsub(l/2, n/2) + 7Addsub(m/2, n/2).
37 Table: Computation time: Strassen s and Winograd s algorithms (128 bits) m n min(simple, Block) Strassen Winograd
38 Table: Computation time: Strassen s and Winograd s algorithms (1024 bits) n n min(simple, Block) Strassen Winograd
39 Lotkin a ij = { 1 (i = 1) 1/(i + j 1) (i 2) A 1 A (n = 1024) x = [ n 1] T α K := 32α
40 Lotkin (n = 1024 ) 8650bits (458bits ) Comp.Time (s) Winograd 8192bits vs. 8650bits: Lotkin matrix, Normal LU: s min: s Normal LU: 3.3E log10(max.relative Error) α Winograd(8192 bits) Comp.Time Winograd(8650 bits) Comp.Time Winograd(8192 bits) Max.Rel.Err Winograd(8650 bits) Max.Rel.Err 32 %
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最新耐震構造解析 ( 第 3 版 ) 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/052093 このサンプルページの内容は, 第 3 版 1 刷発行時のものです. i 3 10 3 2000 2007 26 8 2 SI SI 20 1996 2000 SI 15 3 ii 1 56 6
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AVX を用いた倍々精度疎行列ベクトル積の高速化 菱沼利彰 1 藤井昭宏 1 田中輝雄 1 長谷川秀彦 2 1 工学院大学 2 筑波大学 1 目次 1. 研究背景 目的 2. 実装, 実験環境 3. 実験 - 倍々精度ベクトル演算 - 4. 実験 - 倍々精度疎行列ベクトル積 - 5. まとめ 多倍長精度計算フォーラム 2 目次 1. 研究背景 目的 2. 実装, 実験環境 3. 実験 - 倍々精度ベクトル演算
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6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
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PC 1 1 [1][2] [3][4] ( ) GPU(Graphics Processing Unit) GPU PC GPU PC ( 2 GPU ) GPU Harris Corner Detector[5] CPU ( ) ( ) CPU GPU 2 3 GPU 4 5 6 7 1 toyohiro@isc.kyutech.ac.jp 45 2 ( ) CPU ( ) ( ) () 2.1
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数論 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/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
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