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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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1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,
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2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)
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,, Department of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyo-ku, Tokyo 112 8551, JAPAN E-mail : [email protected] E-mail : [email protected] SATO KOGYO CO., LTD. 12-20, Nihonbashi-Honcho
2012年度HPCサマーセミナー_多田野.pptx
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1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac
1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
修士論文
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
微分積分 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. 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.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 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. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
106 4 4.1 1 25.1 25.4 20.4 17.9 21.2 23.1 26.2 1 24 12 14 18 36 42 24 10 5 15 120 30 15 20 10 25 35 20 18 30 12 4.1 7 min. z = 602.5x 1 + 305.0x 2 + 2
105 4 0 1? 1 LP 0 1 4.1 4.1.1 (intger programming problem) 1 0.5 x 1 = 447.7 448 / / 2 1.1.2 1. 2. 1000 3. 40 4. 20 106 4 4.1 1 25.1 25.4 20.4 17.9 21.2 23.1 26.2 1 24 12 14 18 36 42 24 10 5 15 120 30
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301 12 (2) : 1 (1) dx dt = f(x,t) ( (t 0,t 1,...,t N ) ) h k = t k+1 t k. h k k h. x(t k ) x k. : 2 (2) :1. step. 1 : explicit( ) : ξ k+1 = ξ k +h k Ψ(t k,ξ k,h k ) implicit( ) : ξ k+1 = ξ k +h k Ψ(t k,t
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ストリーミング SIMD 拡張命令2 (SSE2) を使用した SAXPY/DAXPY
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3 Signicant discrepancies (between the computed and the true result) are very rare, too rare to worry about all the time, yet not rare enough to ignore. by W.M.Kahan supercomputer 1 supercomputer supercomputer
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GPU クラスタによる格子 QCD 計算 広大理尾崎裕介 石川健一 1.1 Introduction Graphic Processing Units 1 チップに数百個の演算器 多数の演算器による並列計算 ~TFLOPS ( 単精度 ) CPU 数十 GFLOPS バンド幅 ~100GB/s コストパフォーマンス ~$400 GPU の開発環境 NVIDIA CUDA http://www.nvidia.co.jp/object/cuda_home_new_jp.html
HP High Performance Computing（HPC）
ACCELERATE HP High Performance Computing HPC HPC HPC HPC HPC 1000 HPHPC HPC HP HPC HPC HPC HP HPCHP HP HPC 1 HPC HP 2 HPC HPC HP ITIDC HP HPC 1HPC HPC No.1 HPC TOP500 2010 11 HP 159 32% HP HPCHP 2010 Q1-Q4
A Feasibility Study of Direct-Mapping-Type Parallel Processing Method to Solve Linear Equations in Load Flow Calculations Hiroaki Inayoshi, Non-member
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3 3 BASIC C++ 8 Tflop/s 8TB [] High precision symmetric eigenvalue computation through exact tridiagonalization by using rational number arithmetic Hikaru Samukawa Abstract: Since rational number arithmetic,
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II 6 [email protected] 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [
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On Applications of Rigorous Computing to Dynamical Systems (Zin ARAI) Department of Mathematics, Kyoto University email: [email protected] 1 [12, 13] Lorenz 2 Lorenz 3 4 2 Lorenz 2.1 Lorenz E. Lorenz
D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
![D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j](/thumbs/103/158001152.jpg "D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j")
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..
! 行行 CPUDSP PPESPECell/B.E. CPUGPU 行行 SIMD [SSE, AltiVec] 用 HPC CPUDSP PPESPE (Cell/B.E.) SPE CPUGPU GPU CPU DSP DSP PPE SPE SPE CPU DSP SPE 2
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! OpenCL [Open Computing Language] 言 [OpenCL C 言 ] CPU, GPU, Cell/B.E.,DSP 言 行行 [OpenCL Runtime] OpenCL C 言 API Khronos OpenCL Working Group AMD Broadcom Blizzard Apple ARM Codeplay Electronic Arts Freescale
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スパコンに通じる並列プログラミングの基礎
2016.06.06 2016.06.06 1 / 60 2016.06.06 2 / 60 Windows, Mac Unix 0444-J 2016.06.06 3 / 60 Part I Unix GUI CUI: Unix, Windows, Mac OS Part II 0444-J 2016.06.06 4 / 60 ( : ) 6 6 ( ) 6 10 6 16 SX-ACE 6 17
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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 [email protected] 45 2 ( ) CPU ( ) ( ) () 2.1
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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sdmp Maple - (Ver.2) ( ) September 27, 2011 1 (I) GotoBALS, http://www-is.amp.i.kyoto-u.ac.jp/ kkimur/charpoly.html 2 (II) Nehalem CPU GotoBLAS Intel CPU Nehalem CPU, GotoBLAS, Hyper-Thread technology