線形代数演算ライブラリBLASとLAPACKの 基礎と実践1
|
|
|
- みさえ たけすえ
- 9 years ago
- Views:
Transcription
1 .. BLAS LAPACK 1, 2013/05/23 CMSI A 1 / 43
2 BLAS LAPACK (I) BLAS, LAPACK BLAS : - LAPACK : 2 / 43
3 : 3 / 43
4 (wikipedia) V : f : V u, v u + v u + v V α K u V αu V V x, y f (x + y) = f (x) + f (y) V x K α f (αx) = α f (x) :BLAS, LAPACK 3 / 43
5 ( ) 1000 ( ; 1 2 ) :... 4 / 43
6 ( ) 1000 ( ; 1 2 ) :... 4 / 43
7 ( ) 1000 ( ; 1 2 ) :... 4 / 43
8 ( ) 1000 ( ; 1 2 ) :... 4 / 43
9 ( ) 1000 ( ; 1 2 ) :... 4 / 43
10 Google Page Rank Ax = λx 3D O AO ( ) ( ) U HU = diag(λ 1, λ 2, ) 5 / 43
11 Google Page Rank Ax = λx 3D O AO ( ) ( ) U HU = diag(λ 1, λ 2, ) 5 / 43
12 Google Page Rank Ax = λx 3D O AO ( ) ( ) U HU = diag(λ 1, λ 2, ) 5 / 43
13 Google Page Rank Ax = λx 3D O AO ( ) ( ) U HU = diag(λ 1, λ 2, ) 5 / 43
14 ( 1 2 ) : 6 / 43
15 ( 1 2 ) +Google trans. : : 9 1 4, 4 1 4, ( )... 7 / 43
16 ( 1 2 ) +Google trans. : 3x + 2y + z = 39 ( ) 2x + 3y + z = 34 ( ) x + 2y + 3z = 26 ( ) ( ) ( ) ( ) 3 ( ) 2 ( ) 3 ( ) ( ) 3(2x + 3y + z = 34) 2(3x + 2y + z = 39) 5y + z = 24 ( ) 3(x + 2y + 3z = 39) 3x + 2y + z = 39 4y + 8z = 39 ( ) ( ) 5 3x + 2y + z = 39 ( ) 5y + z = 24 ( ) 20y + 40z = 195 ( ) ( )-( )x4 36z = 99 8 / 43
17 ENIAC ENIAC( Electronic Numerical Integrator and Computer 1946 ) (Wikipedia ) 9 / 43
18 10 / 43
19 : = 1 a + (b + c) (a + b) + c 11 / 43
20 : = 1 a + (b + c) (a + b) + c 11 / 43
21 : = 1 a + (b + c) (a + b) + c 11 / 43
22 : = 1 a + (b + c) (a + b) + c 11 / 43
23 : = 1 a + (b + c) (a + b) + c 11 / 43
24 : IEEE Standard for Floating-Point Arithmetic binary64 ( ) :Core i7 920: 40GFLOPS; RADEON HD GFLOPS, 10PFLOPS) : 12 / 43
25 13 / 43
26 ... - :, / 43
27 ... - :, / 43
28 ... - :, / 43
29 ... - :, / 43
30 ? BLAS, LAPACK? 15 / 43
31 ? BLAS, LAPACK? 15 / 43
32 ? BLAS, LAPACK? 15 / 43
33 ? BLAS, LAPACK? 15 / 43
34 ? BLAS, LAPACK? 15 / 43
35 BLAS, LAPACK: BLAS+LAPACK ( OS ) ( ) BLAS, LAPACK! 16 / 43
36 BLAS, LAPACK: BLAS+LAPACK ( OS ) ( ) BLAS, LAPACK! 16 / 43
37 BLAS, LAPACK: BLAS+LAPACK ( OS ) ( ) BLAS, LAPACK! 16 / 43
38 BLAS, LAPACK: BLAS+LAPACK ( OS ) ( ) BLAS, LAPACK! 16 / 43
39 BLAS, LAPACK: BLAS+LAPACK ( OS ) ( ) BLAS, LAPACK! 16 / 43
40 BLAS, LAPACK: BLAS+LAPACK ( OS ) ( ) BLAS, LAPACK! 16 / 43
41 BLAS, LAPACK: BLAS+LAPACK ( OS ) ( ) BLAS, LAPACK! 16 / 43
42 BLAS? BLAS Basic Linear Algebra Subprograms FORTRAN77 (reference BLAS) BLAS! 17 / 43
43 BLAS? BLAS Basic Linear Algebra Subprograms FORTRAN77 (reference BLAS) BLAS! 17 / 43
44 BLAS? BLAS Basic Linear Algebra Subprograms FORTRAN77 (reference BLAS) BLAS! 17 / 43
45 BLAS? BLAS Basic Linear Algebra Subprograms FORTRAN77 (reference BLAS) BLAS! 17 / 43
46 BLAS? BLAS Basic Linear Algebra Subprograms FORTRAN77 (reference BLAS) BLAS! 17 / 43
47 Level 1 BLAS BLAS Level 1, 2, 3 Level 1: - (+ ) (DAXPY), (DDOT) y αx + y, (1) dot x T y, (2) 15,,,, / 43
48 Level 2 BLAS BLAS Level 1, 2, 3 Level 2: - - (DGEMV) y αax + βy, (3) (DTRSV) x A 1 b, (4) 25, 4 19 / 43
49 Level 3 BLAS BLAS Level 1, 2, 3 Level 3 BLAS - - (DGEMM), - DSYRK, C αab + βc (5) C αaa T + βc (6) DTRSM 9 B αa 1 B (7) 20 / 43
50 BLAS : s, d, c, z LEVEL1 BLAS zrotg zdcal drotg drot drotm zdrot zswap dswap zdscal dscal zcopy dcopy zaxpy daxpy ddot zdotc zdotu dznrm2 dnrm2 dasum izasum idamax dzabs1 LEVEL2 BLAS zgemv dgemv zgbmv dgbmv zhemv zhbmv zhpmv dsymv dsbmv ztrmv zgemv dgemv zgbmv dgemv zhemv zhbmv zhpmv dsymv dsbmv dspmv ztrmv dtrmv ztbmv ztpmv dtpmv ztrsv dtrsv ztbsv dtbsv ztpsv dger zgeru zgerc zher zhpr zher2 zhpr2 dsyr dspr dsyr2 dspr2 LEVEL3 BLAS zgemm dgemm zsymm dsymm zhemm zsyrk dsyrk zherk zsyr2k dsyr2k zher2k ztrmm dtrmm ztrsm dtrsm 21 / 43
51 LAPACK? LAPACK(Linear Algebra PACKage),. BLAS. : (LU,, QR,, Schur, Schur ),, CPU OS Fortran web ! 22 / 43
52 BLAS, LAPACK BLAS, LAPACK. Gaussian, Gamess, ADF, VASP CPLEX, NUOPT, GLPK.. Ruby, Python, Perl, Java, C, Mathematica, Maple, Matlab, R, octave, SciLab 23 / 43
53 Top 500 Top 500: Top 500, LINPACK., DGEMM -, 24 / 43
54 BLAS, LAPACK : BLAS, LAPACK Reference BLAS CPU GotoBLAS2: GotoBLAS2, BLAS, LAPACK. CPU, OS. ( ). BLAS, LAPACK SandyBridge OpenBLAS: Zhang Xianyi GotoBLAS2 SandyBridge ICT Loongson-3A, 3B 25 / 43
55 BLAS, LAPACK : BLAS, LAPACK ATLAS:R. Clint Whaley, BLAS 2001 BLAS, BLAS % 10% GPU BLAS, LAPACK: CPU,. CPU, 10,. nvidia GPU MAGMA BLAS, LAPACK: ScaLAPACK. 26 / 43
56 BLAS, LAPACK :Column major or Row major 2 1 ( ) A = column major, row major. 1, 4, 2, 5, 3, 6 column major FORTRAN Matlab, octave column major 27 / 43
57 BLAS, LAPACK :Column major or Row major A = ( 1 2 ) , 2, 3, 4, 5, 6 row major C, C++ row major C/C++ BLAS, LAPACK major!! 28 / 43
58 BLAS, LAPACK :leading dimension ( LU, Chokesky LAPACK ), leading dimension BLAS, LAPACK LDA, LDB. FORTRAN A(i + j m), A(i + j lda) M N A LDA N 29 / 43
59 BLAS, LAPACK : 0 1? FORTRAN 1, C, C++, 0. 1 N (FORTRAN), 0 n (C,C++). x i FORTRAN X(I), C x[i 1]. A i, j FORTRAN A(I, J), C column major A[i 1 + ( j 1) lda] 30 / 43
60 BLAS, LAPACK : OS BLAS BLAS (CBLAS? C FORTRAN ) Fortran integer 32bit 64bit? (64bit BLAS ) GPU Linux MacOSX Windows Ubuntu x86 31 / 43
61 BLAS LAPACK Ubuntu BLAS, LAPACK C / 43
62 BLAS LAPACK Ubuntu BLAS LAPACK $ sudo apt-get install gfortran g++ libblas-dev liblapack-dev $ sudo apt-get install gfortran g++ libblas-dev liblapack-dev... g++ gfortran libblas-dev liblapack-dev : 0 : 0 : 0 : / 43
63 - - DGEMM A = B = C = α = 3, β = 2, C αab + βc / 43
64 - :DGEMM CBLAS void F77_dgemm(const char *transa, const char *transb, int m, int n, int k, const double * alpha, const double *A, int lda, const double * B, int ldb, const double *beta, double *C, int ldc); transa, transb, transc A, B, C Row major M, N, K alpha, beta 35 / 43
65 - I #include <stdio.h> extern "C" { #define ADD_ #include <cblas_f77.h> } //Matlab/Octave format void printmat(int N, int M, double *A, int LDA) { double mtmp; printf("[ "); for (int i = 0; i < N; i++) { printf("[ "); for (int j = 0; j < M; j++) { mtmp = A[i + j * LDA]; printf("%5.2e", mtmp); if (j < M - 1) printf(", "); } if (i < N - 1) printf("]; "); else printf("] "); } printf("]"); } int main() { int n = 3; double alpha, beta; double *A = new double[n*n]; double *B = new double[n*n]; double *C = new double[n*n]; A[0+0*n]=1; A[0+1*n]= 8; A[0+2*n]= 3; A[1+0*n]=2; A[1+1*n]=10; A[1+2*n]= 8; A[2+0*n]=9; A[2+1*n]=-5; A[2+2*n]=-1; B[0+0*n]= 9; B[0+1*n]= 8; B[0+2*n]=3; B[1+0*n]= 3; B[1+1*n]=11; B[1+2*n]=2.3; B[2+0*n]=-8; B[2+1*n]= 6; B[2+2*n]=1; C[0+0*n]=3; C[0+1*n]=3; C[0+2*n]=1.2; C[1+0*n]=8; C[1+1*n]=4; C[1+2*n]=8; C[2+0*n]=6; C[2+1*n]=1; C[2+2*n]=-2; 36 / 43
66 - II printf("# dgemm demo...\n"); printf("a =");printmat(n,n,a,n);printf("\n"); printf("b =");printmat(n,n,b,n);printf("\n"); printf("c =");printmat(n,n,c,n);printf("\n"); alpha = 3.0; beta = -2.0; F77_dgemm("n", "n", &n, &n, &n, &alpha, A, &n, B, &n, &beta, C, &n); printf("alpha = %5.3e\n", alpha); printf("beta = %5.3e\n", beta); printf("ans="); printmat(n,n,c,n); printf("\n"); printf("#check by Matlab/Octave by:\n"); printf("alpha * A * B + beta * C =\n"); delete[]c; delete[]b; delete[]a; 37 / 43
67 - dgemm_demo.cpp $ g++ dgemm_demo.cpp -o dgemm_demo -lblas -lapack., Octave Matlab & $./dgemm_demo # dgemm demo... A =[ [ 1.00e+00, 8.00e+00, 3.00e+00]; [ 2.00e+00, 1.00e+01, 8.00e+00]; [ 9.00e+00, -5.00e+00, -1.00e+00] ] B =[ [ 9.00e+00, 8.00e+00, 3.00e+00]; [ 3.00e+00, 1.10e+01, 2.30e+00]; [ -8.00e+00, 6.00e+00, 1.00e+00] ] C =[ [ 3.00e+00, 3.00e+00, 1.20e+00]; [ 8.00e+00, 4.00e+00, 8.00e+00]; [ 6.00e+00, 1.00e+00, -2.00e+00] ] alpha = 3.000e+00 beta = e+00 ans=[ [ 2.10e+01, 3.36e+02, 7.08e+01]; [ -6.40e+01, 5.14e+02, 9.50e+01]; [ 2.10e+02, 3.10e+01, 4.75e+01] ] #check by Matlab/Octave by: alpha * A * B + beta * C 38 / 43
68 LAPACK : :DSYEV A = Av i = λ i v i (i = 1, 2, 3) λ 1, λ 2, λ 3 v 1, v 2, v , , v 1 = ( , , ) v 2 = ( , , ) v 3 = ( , , ) 39 / 43
69 DSYEV Fortran dsyev_f77(const char *jobz, const char *uplo, int *n, double *A, int *lda, double *w, double *work, int *lwork, int *info); jobz: uplo: A, lda: A leading dimension w: ( ) work, lwork: info: =0 <0: INFO=-i i >0: INFO=i 40 / 43
70 #include <iostream> #include <stdio.h> extern "C" int dsyev_(const char *jobz, const char *uplo, int *n, double *a, int *lda, double *w, double *work, int *lwork, int *info); //Matlab/Octave format void printmat(int N, int M, double *A, int LDA) { double mtmp; printf("[ "); for (int i = 0; i < N; i++) { printf("[ "); for (int j = 0; j < M; j++) { mtmp = A[i + j * LDA]; printf("%5.2e", mtmp); if (j < M - 1) printf(", "); } if (i < N - 1) printf("]; "); else printf("] "); } printf("]"); } int main() { int n = 3; int lwork, info; double *A = new double[n*n]; double *w = new double[n]; //setting A matrix A[0+0*n]=1;A[0+1*n]=2;A[0+2*n]=3; A[1+0*n]=2;A[1+1*n]=5;A[1+2*n]=4; A[2+0*n]=3;A[2+1*n]=4;A[2+2*n]=6; printf("a ="); printmat(n, n, A, n); printf("\n"); lwork = -1; double *work = new double[1]; dsyev_("v", "U", &n, A, &n, w, work, &lwork, &info); lwork = (int) work[0]; delete[]work; work = new double[std::max((int) 1, lwork)]; //get Eigenvalue dsyev_("v", "U", &n, A, &n, w, work, &lwork, &info); //print out some results. printf("#eigenvalues \n"); printf("w ="); printmat(n, 1, w, 1); printf("\n"); printf("#eigenvecs \n"); printf("u ="); printmat(n, n, A, n); printf("\n"); printf("#check Matlab/Octave by:\n"); printf("eig(a)\n"); printf("u *A*U\n"); delete[]work; delete[]w; delete[]a; 41 / 43
71 eigenvalue_demo.cpp $ g++ eigenvalue_demo.cpp -o eigenvalue_demo -lblas -lapack -lgfortran, Octave Matlab & A =[ [ 1.00e+00, 2.00e+00, 3.00e+00];\ [ 2.00e+00, 5.00e+00, 4.00e+00];\ [ 3.00e+00, 4.00e+00, 6.00e+00] ] #eigenvalues w =[ [ -4.10e-01]; [ 1.58e+00]; [ 1.08e+01] ] #eigenvecs U =[ [ -9.14e-01, 2.16e-01, 3.42e-01];\ [ 4.01e-02, -7.93e-01, 6.08e-01];\ [ 4.03e-01, 5.70e-01, 7.16e-01] ] #Check Matlab/Octave by: eig(a) U *A*U 42 / 43
72 BLAS, LAPACK BLAS, LAPACK BLAS, LAPACK 43 / 43
線形代数演算ライブラリBLASとLAPACKの 基礎と実践1
1 / 50 BLAS LAPACK 1, 2015/05/21 CMSI A 2 / 50 BLAS LAPACK (I) BLAS, LAPACK BLAS : - LAPACK : 3 / 50 ( ) 1000 ( ; 1 2 ) :... 3 / 50 ( ) 1000 ( ; 1 2 ) :... 3 / 50 ( ) 1000 ( ; 1 2 ) :... 3 / 50 ( ) 1000
線形代数演算ライブラリBLASとLAPACKの 基礎と実践1
1 / 56 BLAS LAPACK 1, 2017/05/25 CMSI A 2 / 56 BLAS LAPACK (I) BLAS, LAPACK BLAS : - LAPACK : 3 / 56 ( ) 1000 ( ; 1 2 ) :... 3 / 56 ( ) 1000 ( ; 1 2 ) :... 3 / 56 ( ) 1000 ( ; 1 2 ) :... 3 / 56 ( ) 1000
11020070-0_Vol16No2.indd
2552 チュートリアル BLAS, LAPACK 2 1 BLAS, LAPACKチュートリアル パート1 ( 簡 単 な 使 い 方 とプログラミング) 中 田 真 秀 1 読 者 の 想 定 BLAS [1], LAPACK [2] 2 線 形 代 数 の 重 要 性 について Google Page Rank 3D CPU 筆 者 紹 介 BLAS LAPACK http://accc.riken.jp/maho/
CMSI教育計算科学技術特論A_中田真秀
線形代数演算ライブラリBLAS とLAPACKの基礎と実践 (I) BLAS, LAPACK入門編 中田 真秀 理化学研究所 情報システム本部 2019/5/23 計算科学技術特論A BLAS, LAPACK入門編 講義目的 線形代数演算をコンピュータで行うには 必ずBLAS LAPACKのお世話になる 使うには(若干)知識がいる 実際にUbuntu Linuxで試せる形で提示し 使えるよう になる
倍々精度RgemmのnVidia C2050上への実装と応用
.. [email protected] http://accc.riken.jp/maho/,,, 2011/2/16 1 - : GPU : SDPA-DD 10 1 - Rgemm : 4 (32 ) nvidia C2050, GPU CPU 150, 24GFlops 25 20 GFLOPS 15 10 QuadAdd Cray, QuadMul Sloppy Kernel QuadAdd Cray,
untitled
A = QΛQ T A n n Λ Q A = XΛX 1 A n n Λ X GPGPU A 3 T Q T AQ = T (Q: ) T u i = λ i u i T {λ i } {u i } QR MR 3 v i = Q u i A {v i } A n = 9000 Quad Core Xeon 2 LAPACK (4/3) n 3 O(n 2 ) O(n 3 ) A {v i }
11050427-0_Vol16No3.indd
2599 チュートリアル BLAS, LAPACK 2 2 GPU BLAS, LAPACKチュートリアル パート2 (GPU 編 ) 中 田 真 秀 1 はじめに GPU Graphics Processing Unit BLAS, LAPACK GPU GPU NVIDIA AMD AMD RADEON HD NVIDIA NVIDIA GPU NVIDIA C2050 BLAS, LAPACK
4 倍精度基本線形代数ルーチン群 QPBLAS の紹介 [index] 1. Introduction 2. Double-double algorithm 3. QPBLAS 4. QPBLAS-GPU 5. Summary 佐々成正 1, 山田進 1, 町田昌彦 1, 今村俊幸 2, 奥田洋司
![4 倍精度基本線形代数ルーチン群 QPBLAS の紹介 [index] 1. Introduction 2. Double-double algorithm 3. QPBLAS 4. QPBLAS-GPU 5. Summary 佐々成正 1, 山田進 1, 町田昌彦 1, 今村俊幸 2, 奥田洋司](/thumbs/97/131936032.jpg "4 倍精度基本線形代数ルーチン群 QPBLAS の紹介 [index] 1. Introduction 2. Double-double algorithm 3. QPBLAS 4. QPBLAS-GPU 5. Summary 佐々成正 1, 山田進 1, 町田昌彦 1, 今村俊幸 2, 奥田洋司")
4 倍精度基本線形代数ルーチン群 QPBLAS の紹介 [index] 1. Introduction 2. Double-double algorithm 3. QPBLAS 4. QPBLAS-GPU 5. Summary 佐々成正 1, 山田進 1, 町田昌彦 1, 今村俊幸 2, 奥田洋司 3 1 1 日本原子力研究開発機構システム計算科学センター 2 理科学研究所計算科学研究機構 3 東京大学新領域創成科学研究科
14 2 Scilab Scilab GUI インタグラフ プリタ描画各種ライブラリ (LAPACK, ODEPACK, ) SciNOTES ハードウェア (CPU, GPU) 21 Scilab SciNotes 呼び出し 3 変数ブラウザ 1 ファイルブラウザ 2 コンソール 4 コマンド履歴
13 2 Scilab Scilab Scilab 21 Scilab Scilab[4] INRIA C/C++, Fortran LAPACK Matlab Scilab Web http://wwwscilaborg/ 2016 4 552 Windows 10/8x MacOS X, Linux Scilab Octave Matlab (Matlab Toolbox) (Mathematica
PC Windows 95, Windows 98, Windows NT, Windows 2000, MS-DOS, UNIX CPU
1. 1.1. 1.2. 1 PC Windows 95, Windows 98, Windows NT, Windows 2000, MS-DOS, UNIX CPU 2. 2.1. 2 1 2 C a b N: PC BC c 3C ac b 3 4 a F7 b Y c 6 5 a ctrl+f5) 4 2.2. main 2.3. main 2.4. 3 4 5 6 7 printf printf
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 HPCS /1/18 a a 1 a 2 a 3 a a GPU Graphics Processing Unit GPU CPU GPU GPGPU G
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
1 (bit ) ( ) PC WS CPU IEEE754 standard ( 24bit) ( 53bit)
GNU MP BNCpack [email protected] 2002 9 20 ( ) Linux Conference 2002 1 1 (bit ) ( ) PC WS CPU IEEE754 standard ( 24bit) ( 53bit) 10 2 2 3 4 5768:9:; = %? @BADCEGFH-I:JLKNMNOQP R )TSVU!" # %$ & " #
2008 ( 13 ) C LAPACK 2008 ( 13 )C LAPACK p. 1
2008 ( 13 ) C LAPACK LAPACK p. 1 Q & A Euler http://phase.hpcc.jp/phase/mppack/long.pdf KNOPPIX MT (Mersenne Twister) SFMT., ( ) ( ) ( ) ( ). LAPACK p. 2 C C, main Asir ( Asir ) ( ) (,,...), LAPACK p.
/* sansu1.c */ #include <stdio.h> main() { int a, b, c; /* a, b, c */ a = 200; b = 1300; /* a 200 */ /* b 200 */ c = a + b; /* a b c */ }
C 2: A Pedestrian Approach to the C Programming Language 2 2-1 2.1........................... 2-1 2.1.1.............................. 2-1 2.1.2......... 2-4 2.1.3..................................... 2-6
Untitled
VASP 2703 2006 3 VASP 100 PC 3,4 VASP VASP VASP FFT. (LAPACK,BLAS,FFT), CPU VASP. 1 C LAPACK,BLAS VASP VASP VASP VASP bench.hg VASP CPU CPU CPU northwood LAPACK lmkl lapack64, BLAS lmkl p4 LA- PACK liblapack,
理研スーパーコンピュータ・システム
線形代数演算ライブラリ BLAS と LAPACK の基礎と実践 2 理化学研究所情報基盤センター 2013/5/30 13:00- 大阪大学基礎工学部 中田真秀 この授業の目的 対象者 - 研究用プログラムを高速化したい人 - LAPACK についてよく知らない人 この講習会の目的 - コンピュータの簡単な仕組みについて - 今後 どうやってプログラムを高速化するか - BLAS, LAPACK
[1] #include<stdio.h> main() { printf("hello, world."); return 0; } (G1) int long int float ± ±
![[1] #include<stdio.h> main() { printf(hello, world.); return 0; } (G1) int long int float ± ±](/thumbs/99/143405735.jpg "[1] #include<stdio.h> main() { printf(hello, world.); return 0; } (G1) int long int float ± ±")
[1] #include printf("hello, world."); (G1) int -32768 32767 long int -2147483648 2147483647 float ±3.4 10 38 ±3.4 10 38 double ±1.7 10 308 ±1.7 10 308 char [2] #include int a, b, c, d,
A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18
2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1
(Basic Theory of Information Processing) 1
(Basic Theory of Information Processing) 1 10 (p.178) Java a[0] = 1; 1 a[4] = 7; i = 2; j = 8; a[i] = j; b[0][0] = 1; 2 b[2][3] = 10; b[i][j] = a[2] * 3; x = a[2]; a[2] = b[i][3] * x; 2 public class Array0
ストリーミング SIMD 拡張命令2 (SSE2) を使用した SAXPY/DAXPY
SIMD 2(SSE2) SAXPY/DAXPY 2.0 2000 7 : 248600J-001 01/12/06 1 305-8603 115 Fax: 0120-47-8832 * Copyright Intel Corporation 1999, 2000 01/12/06 2 1...5 2 SAXPY DAXPY...5 2.1 SAXPY DAXPY...6 2.1.1 SIMD C++...6
漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,
1 1.1 C 2 1 double a[ ][ ]; 1 3x x3 ( ) malloc() malloc 2 #include <stdio.h> #include
![1 1.1 C 2 1 double a[ ][ ]; 1 3x x3 ( ) malloc() malloc 2 #include <stdio.h> #include](/thumbs/98/138365120.jpg "1 1.1 C 2 1 double a[ ][ ]; 1 3x x3 ( ) malloc() malloc 2 #include <stdio.h> #include")
1 1.1 C 2 1 double a[ ][ ]; 1 3x3 0 1 3x3 ( ) 0.240 0.143 0.339 0.191 0.341 0.477 0.412 0.003 0.921 1.2 malloc() malloc 2 #include #include #include enum LENGTH = 10 ; int
Second-semi.PDF
PC 2000 2 18 2 HPC Agenda PC Linux OS UNIX OS Linux Linux OS HPC 1 1CPU CPU Beowulf PC (PC) PC CPU(Pentium ) Beowulf: NASA Tomas Sterling Donald Becker 2 (PC ) Beowulf PC!! Linux Cluster (1) Level 1:
bash on Ubuntu on Windows bash on Ubuntu on Windows bash on Ubuntu on Windows bash on Ubuntu on Windows bash on Ubuntu on Windows ˆ Windows10 64bit Wi
Windows bash on Ubuntu on Windows [Windows Creators Update(1703) ] TAKE 2017-10-06 bash on Ubuntu on Windows bash on Ubuntu on Windows bash on Ubuntu on Windows bash on Ubuntu on Windows bash on Ubuntu
C のコード例 (Z80 と同機能 ) int main(void) { int i,sum=0; for (i=1; i<=10; i++) sum=sum + i; printf ("sum=%d n",sum); 2
アセンブラ (Z80) の例 ORG 100H LD B,10 SUB A LOOP: ADD A,B DEC B JR NZ,LOOP LD (SUM),A HALT ORG 200H SUM: DEFS 1 END 1 C のコード例 (Z80 と同機能 ) int main(void) { int i,sum=0; for (i=1; i
. UNIX, Linux, KNOPPIX. C,.,., ( 1 ) p. 2
2009 ( 1 ) 2009 ( 1 ) p. 1 . UNIX, Linux, KNOPPIX. C,.,.,. 2009 ( 1 ) p. 2 , +, ( ), ( ), or PC orange2, knxm2008vm, iyokan-6 KNOPPIX/Math (DVD ) 2009 ( 1 ) p. 3 ,. Mathematica (20-30 /1 ), Maple (20 /1
インテル(R) Visual Fortran Composer XE
Visual Fortran Composer XE 1. 2. 3. 4. 5. Visual Studio 6. Visual Studio 7. 8. Compaq Visual Fortran 9. Visual Studio 10. 2 https://registrationcenter.intel.com/regcenter/ w_fcompxe_all_jp_2013_sp1.1.139.exe
1 4 2 EP) (EP) (EP)
2003 2004 2 27 1 1 4 2 EP) 5 3 6 3.1.............................. 6 3.2.............................. 6 3.3 (EP)............... 7 4 8 4.1 (EP).................... 8 4.1.1.................... 18 5 (EP)
tuat1.dvi
( 1 ) http://ist.ksc.kwansei.ac.jp/ tutimura/ 2012 6 23 ( 1 ) 1 / 58 C ( 1 ) 2 / 58 2008 9 2002 2005 T E X ptetex3, ptexlive pt E X UTF-8 xdvi-jp 3 ( 1 ) 3 / 58 ( 1 ) 4 / 58 C,... ( 1 ) 5 / 58 6/23( )
untitled
II yacc 005 : 1, 1 1 1 %{ int lineno=0; 3 int wordno=0; 4 int charno=0; 5 6 %} 7 8 %% 9 [ \t]+ { charno+=strlen(yytext); } 10 "\n" { lineno++; charno++; } 11 [^ \t\n]+ { wordno++; charno+=strlen(yytext);}
comment.dvi
( ) (sample1.c) (sample1.c) 2 2 Nearest Neighbor 1 (2D-class1.dat) 2 (2D-class2.dat) (2D-test.dat) 3 Nearest Neighbor Nearest Neighbor ( 1) 2 1: NN 1 (sample1.c) /* -----------------------------------------------------------------
ex01.dvi
,. 0. 0.0. C () /******************************* * $Id: ex_0_0.c,v.2 2006-04-0 3:37:00+09 naito Exp $ * * 0. 0.0 *******************************/ #include int main(int argc, char **argv) { double
1.ppt
/* * Program name: hello.c */ #include int main() { printf( hello, world\n ); return 0; /* * Program name: Hello.java */ import java.io.*; class Hello { public static void main(string[] arg)
I ASCII ( ) NUL 16 DLE SP P p 1 SOH 17 DC1! 1 A Q a q STX 2 18 DC2 " 2 B R b
I 4 003 4 30 1 ASCII ( ) 0 17 0 NUL 16 DLE SP 0 @ P 3 48 64 80 96 11 p 1 SOH 17 DC1! 1 A Q a 33 49 65 81 97 113 q STX 18 DC " B R b 34 50 66 8 98 114 r 3 ETX 19 DC3 # 3 C S c 35 51 67 83 99 115 s 4 EOT
2 2.1 Mac OS CPU Mac OS tar zxf zpares_0.9.6.tar.gz cd zpares_0.9.6 Mac Makefile Mekefile.inc cp Makefile.inc/make.inc.gfortran.seq.macosx make
Sakurai-Sugiura z-pares 26 9 5 1 1 2 2 2.1 Mac OS CPU......................................... 2 2.2 Linux MPI............................................ 2 3 3 4 6 4.1 MUMPS....................................
/* do-while */ #include <stdio.h> #include <math.h> int main(void) double val1, val2, arith_mean, geo_mean; printf( \n ); do printf( ); scanf( %lf, &v
1 http://www7.bpe.es.osaka-u.ac.jp/~kota/classes/jse.html [email protected] /* do-while */ #include #include int main(void) double val1, val2, arith_mean, geo_mean; printf( \n );
ex01.dvi
,. 0. 0.0. C () /******************************* * $Id: ex_0_0.c,v.2 2006-04-0 3:37:00+09 naito Exp $ * * 0. 0.0 *******************************/ #include int main(int argc, char **argv) double
r1.dvi
2006 1 2006.10.6 ( 2 ( ) 1 2 1.5 3 ( ) Ruby Java Java Java ( Web Web http://lecture.ecc.u-tokyo.ac.jp/~kuno/is06/ / ( / @@@ ( 3 ) @@@ : ( ) @@@ (Q&A) ( ) 1 http://www.sodan.ecc.u-tokyo.ac.jp/cgi-bin/qbbs/view.cgi
新・明解Ｃ言語で学ぶアルゴリズムとデータ構造
第 1 章 基本的 1 n 141 1-1 三値 最大値 algorithm List 1-1 a, b, c max /* */ #include int main(void) { int a, b, c; int max; /* */ List 1-1 printf("\n"); printf("a"); scanf("%d", &a); printf("b"); scanf("%d",
…J…−†[†E…n…‘†[…hfi¯„^‚ΛžfiüŒå
[email protected] II 2009 6 11 [A] D B A B A B A B DVD y = 2x + 5 x = 3 y = 11 x = 5 y = 15. Google Web (2 + 3) 5 25 2 3 5 25 Windows Media Player Media Player (typed lambda calculus) (computer
(2 Linux Mozilla [ ] [ ] [ ] [ ] URL 2 qkc, nkc ~/.cshrc (emacs 2 set path=($path /usr/meiji/pub/linux/bin tcsh b
![(2 Linux Mozilla [ ] [ ] [ ] [ ] URL 2 qkc, nkc ~/.cshrc (emacs 2 set path=($path /usr/meiji/pub/linux/bin tcsh b](/thumbs/92/108952818.jpg "(2 Linux Mozilla [ ] [ ] [ ] [ ] URL 2 qkc, nkc ~/.cshrc (emacs 2 set path=($path /usr/meiji/pub/linux/bin tcsh b")
II 5 (1 2005 5 26 http://www.math.meiji.ac.jp/~mk/syori2-2005/ UNIX (Linux Linux 1 : 2005 http://www.math.meiji.ac.jp/~mk/syori2-2005/jouhousyori2-2005-00/node2. html ( (Linux 1 2 ( ( http://www.meiji.ac.jp/mind/tool/internet-license/
KBLAS[7] *1., CUBLAS.,,, Byte/flop., [13] 1 2. (AT). GPU AT,, GPU SYMV., SYMV CUDABLAS., (double, float) (cu- FloatComplex, cudoublecomplex).,, DD(dou
![KBLAS[7] *1., CUBLAS.,,, Byte/flop., [13] 1 2. (AT). GPU AT,, GPU SYMV., SYMV CUDABLAS., (double, float) (cu- FloatComplex, cudoublecomplex).,, DD(dou](/thumbs/71/64446865.jpg "KBLAS[7] *1., CUBLAS.,,, Byte/flop., [13] 1 2. (AT). GPU AT,, GPU SYMV., SYMV CUDABLAS., (double, float) (cu- FloatComplex, cudoublecomplex).,, DD(dou")
Vol.214-HPC-146 No.14 214/1/3 CUDA-xSYMV 1,3,a) 1 2,3 2,3 (SYMV)., (GEMV) 2.,, mutex., CUBLAS., 1 2,. (AT). 2, SYMV GPU., SSYMV( SYMV), GeForce GTXTitan Black 211GFLOPS( 62.8%)., ( ) (, ) DD(double-double),
Ｃ言語によるアルゴリズムとデータ構造
Algorithms and Data Structures in C 4 algorithm List - /* */ #include List - int main(void) { int a, b, c; int max; /* */ Ÿ 3Ÿ 2Ÿ 3 printf(""); printf(""); printf(""); scanf("%d", &a); scanf("%d",
Microsoft Word - no14.docx
ex26.c #define MAX 20 int max(int n, int x[]); int num[max]; int i, x; printf(" "); scanf("%d", &x); if(x > MAX) printf("%d %d \n", MAX, MAX); x = MAX; for(i = 0; i < x; i++) printf("%3d : ", i + 1); scanf("%d",
Microsoft Word - C.....u.K...doc
C uwêííôöðöõ Ð C ÔÖÐÖÕ ÐÊÉÌÊ C ÔÖÐÖÕÊ C ÔÖÐÖÕÊ Ç Ê Æ ~ if eíè ~ for ÒÑÒ ÌÆÊÉÉÊ ~ switch ÉeÍÈ ~ while ÒÑÒ ÊÍÍÔÖÐÖÕÊ ~ 1 C ÔÖÐÖÕ ÐÊÉÌÊ uê~ ÏÒÏÑ Ð ÓÏÖ CUI Ô ÑÊ ÏÒÏÑ ÔÖÐÖÕÎ d ÈÍÉÇÊ ÆÒ Ö ÒÐÑÒ ÊÔÎÏÖÎ d ÉÇÍÊ
3.2 Linux root vi(vim) vi emacs emacs 4 Linux Kernel Linux Git 4.1 Git Git Linux Linux Linus Fedora root yum install global(debian Ubuntu apt-get inst
1 OS Linux OS OS Linux Kernel 900 1000 IPA( :http://www.ipa.go.jp/) 8 12 ( ) 16 ( ) 4 5 22 60 2 3 6 Linux Linux 2 LKML 3 3.1 Linux Fedora 13 Ubuntu Fedora CentOS 3.2 Linux root vi(vim) vi emacs emacs 4
アセンブラ入門（CASL II） 第３版
CASLDV i COMET II COMET II CASL II COMET II 1 1 44 (1969 ) COMETCASL 6 (1994 ) COMETCASL 13 (2001 ) COMETCASL COMET IICASL II COMET IICASL II CASL II 2001 1 3 3 L A TEX 2 CASL II COMET II 6 6 7 Windows(Windows
C¥×¥í¥°¥é¥ß¥ó¥° ÆþÌç
C (3) if else switch AND && OR (NOT)! 1 BMI BMI BMI = 10 4 [kg]) ( [cm]) 2 bmi1.c Input your height[cm]: 173.2 Enter Input your weight[kg]: 60.3 Enter Your BMI is 20.1. 10 4 = 10000.0 1 BMI BMI BMI = 10
01_OpenMP_osx.indd
OpenMP* / 1 1... 2 2... 3 3... 5 4... 7 5... 9 5.1... 9 5.2 OpenMP* API... 13 6... 17 7... 19 / 4 1 2 C/C++ OpenMP* 3 Fortran OpenMP* 4 PC 1 1 9.0 Linux* Windows* Xeon Itanium OS 1 2 2 WEB OS OS OS 1 OS
ARM gcc Kunihiko IMAI 2009 1 11 ARM gcc 1 2 2 2 3 3 4 3 4.1................................. 3 4.2............................................ 4 4.3........................................
2 1. Ubuntu 1.1 OS OS OS ( OS ) OS ( OS ) VMware Player VMware Player jp/download/player/ URL VMware Plaeyr VMware
1 2010 [email protected] http://www.jsk.t.u-tokyo.ac.jp/~k-okada/lecture/ 2010 4 5 Linux 1 Ubuntu Ubuntu Linux 1 Ubuntu Ubuntu 3 1. 1 Ubuntu 2. OS Ubuntu OS 3. OS Ubuntu https://wiki.ubuntulinux.jp/ubuntutips/install/installdualboot
1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th
1 n A a 11 a 1n A = a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = ( x ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 11 Th9-1 Ax = λx λe n A = λ a 11 a 12 a 1n a 21 λ a 22 a n1 a n2
Krylov (b) x k+1 := x k + α k p k (c) r k+1 := r k α k Ap k ( := b Ax k+1 ) (d) β k := r k r k 2 2 (e) : r k 2 / r 0 2 < ε R (f) p k+1 :=
127 10 Krylov Krylov (Conjugate-Gradient (CG ), Krylov ) MPIBNCpack 10.1 CG (Conjugate-Gradient CG ) A R n n a 11 a 12 a 1n a 21 a 22 a 2n A T = =... a n1 a n2 a nn n a 11 a 21 a n1 a 12 a 22 a n2 = A...
‚æ2›ñ C„¾„ê‡Ìš|
I 8 10 10 I ( 6 ) 10 10 1 / 23 1 C ( ) getchar(), gets(), scanf() ( ) putchar(), puts(), printf() 1 getchar(), putchar() 1 I ( 6 ) 10 10 2 / 23 1 (getchar 1 1) 1 #include 2 void main(void){ 3 int
II ( ) prog8-1.c s1542h017%./prog8-1 1 => 35 Hiroshi 2 => 23 Koji 3 => 67 Satoshi 4 => 87 Junko 5 => 64 Ichiro 6 => 89 Mari 7 => 73 D
II 8 2003 11 12 1 6 ( ) prog8-1.c s1542h017%./prog8-1 1 => 35 Hiroshi 2 => 23 Koji 3 => 67 Satoshi 4 => 87 Junko 5 => 64 Ichiro 6 => 89 Mari 7 => 73 Daisuke 8 =>. 73 Daisuke 35 Hiroshi 64 Ichiro 87 Junko
1 911 9001030 9:00 A B C D E F G H I J K L M 1A0900 1B0900 1C0900 1D0900 1E0900 1F0900 1G0900 1H0900 1I0900 1J0900 1K0900 1L0900 1M0900 9:15 1A0915 1B0915 1C0915 1D0915 1E0915 1F0915 1G0915 1H0915 1I0915
1 C STL(1) C C C libc C C C++ STL(Standard Template Library ) libc libc C++ C STL libc STL iostream Algorithm libc STL string vector l
C/C++ 2007 6 18 1 C STL(1) 2 1.1............................................... 2 1.2 stdio................................................ 3 1.3.......................................... 10 2 11 2.1 sizeof......................................
OpenMP (1) 1, 12 1 UNIX (FUJITSU GP7000F model 900), 13 1 (COMPAQ GS320) FUJITSU VPP5000/64 1 (a) (b) 1: ( 1(a))
OpenMP (1) 1, 12 1 UNIX (FUJITSU GP7000F model 900), 13 1 (COMPAQ GS320) FUJITSU VPP5000/64 1 (a) (b) 1: ( 1(a)) E-mail: {nanri,amano}@cc.kyushu-u.ac.jp 1 ( ) 1. VPP Fortran[6] HPF[3] VPP Fortran 2. MPI[5]
joho07-1.ppt
0xbffffc5c 0xbffffc60 xxxxxxxx xxxxxxxx 00001010 00000000 00000000 00000000 01100011 00000000 00000000 00000000 xxxxxxxx x y 2 func1 func2 double func1(double y) { y = y + 5.0; return y; } double func2(double*
I. Backus-Naur BNF S + S S * S S x S +, *, x BNF S (parse tree) : * x + x x S * S x + S S S x x (1) * x x * x (2) * + x x x (3) + x * x + x x (4) * *
2015 2015 07 30 10:30 12:00 I. I VI II. III. IV. a d V. VI. 80 100 60 1 I. Backus-Naur BNF S + S S * S S x S +, *, x BNF S (parse tree) : * x + x x S * S x + S S S x x (1) * x x * x (2) * + x x x (3) +
:30 12:00 I. I VI II. III. IV. a d V. VI
2018 2018 08 02 10:30 12:00 I. I VI II. III. IV. a d V. VI. 80 100 60 1 I. Backus-Naur BNF N N y N x N xy yx : yxxyxy N N x, y N (parse tree) (1) yxyyx (2) xyxyxy (3) yxxyxyy (4) yxxxyxxy N y N x N yx