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- たつや あみおか
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1 y () 5 C Fortran () Fortran 32bit 64bit bit bit bit bit byte 8bit 1byte byte Fortran A A 8byte double presicion y ( REAL*8) A 64bit 4byte REAL 32bit REAL A(1001,1001)! 4byte DOUBLE PRECISION B(1001,1001)! 8byte A byte 3:82MB 1 B A 7:65MB REAL A(1001,1001) REAL A(1000,1000) REAL A(1024,1024) 1 MB MegaByte 1MB byte 1
2 128 2n 4n (n ) (bank conict) [1] 2n +1 4n +1 2 (oating point number) M-1800/20 Fortran 24bit 56bit 3 M-1800/20 IBM 4 4 bit =10 x =10 x x IEEE bit 4 IBM International Business Machines 5 6 IEEE Institute of Electrical and Electronics Engineers CRAY 1 1: IBM IEEE (7:22 ) (6:92 ) (16:86 ) (15:65 ) ( ) 10 IEEE IBM IBM ( [2] [3]) [4] IEEE IBM 1 \" \ " 4 [5], [6] I (64bit) (32bit) 4 4 Fortran 2
3 4 7 FUJITSU M-1800/20 FU- JITSU VP2600/ VMGGM, DVMGGM VLAX, DVLAX 1 \D" SSL II, SSL II/VP 8 [8] n n = 16, 32, 64, 128, 256, 512, >< >: A = B =(b ij ) b ij =r 2 n +1 f =(f i ) f i = nx j=1 b ij sin ij n +1 (1) C = A 2 B Ax = f 7 JIS Fortran 90 ([7]) Fortran 4 8 x = (x i ) Fortran CLOCK ([9]) 9 CALL CLOCK(T1,2,2) CALL DVMGGM(A,MAXN,B,MAXN,C,MAXN,N,N,N,ICON) CALL CLOCK(T2,2,2) CPU=(T1-T2)*1.0D-3 CLOCK : M-1800/ (: ) n = :1 CLOCK 10 n = FORTRAN77 EX CLOCK 3
4 2: I M-1800/20 VP2600/10 OS OS IV/MSP V10L10 OS IV/MSP V10L10 FORTRAN77 EX V12L10 FORTRAN77 EX/VP V12L10 SSL II V12L10 SSL II/VP V12L10 VMGGM, DVMGGM ( ) VLAX, DVLAX ( 1 ) 4: VP2600/ (: ) MYLU 2 OPT(E) :2 1:9 3, 4 SSL II Fortran n 4000 A f (1) Ax = f x =(1; 1; 111; 1) 1 Ax = b Gauss MYLU MYLU MYLU x =(x i ) max jx i 0 1j 1in 5: Fortran M-1800/20 VP2600/ E E E E-8 (: ) 1: :2 189 M-1800/20 \1.02E+2" VP2600/
5 7 VP2600/ DO \ " I FORTRAN Fortran Formula Translation Fortran \FORTRAN" Fortran 90 F Fortran 90 \FORTRAN" \Fortran" \FORTRAN77 EX" \Fortran" FACOM M-1800/20 VP2600/10 \FUJITSU" M VP \FUJITSU" \FACOM" FACOM Factory Automation COMputer [10] FACOM Fuck on 11 FUJITSU M-1800/20 5
6 I DVLAX DVMGGM [11] SSL II Fortran II FLOPS FLOPS Floating Point Operations per Second 1 FLOPS MFLOPS, GFLOPS 12 \raison 13 () FLOPS do 10 i=1,n do 10 j=1,n do 10 k=1,n c(i,j)=c(i,j)+a(i,k)*b(k,j) 10 continue DO i, j, k 2 2 n 2 n 2 n = 2n 3 FLOPS FLOPS = 2n3 ( ) i, j, k c 6 DO IJK do 10 i=1,n do 10 j=1,n do 10 k=1,n c(i,j)=c(i,j)+a(i,k)*b(k,j) 10 continue do 10 j=1,n do 10 i=1,n do 10 k=1,n JIK c(i,j)=c(i,j)+a(i,k)*b(k,j) 10 continue 13 d'hatetre"( ) 6
7 do 10 k=1,n KIJ do 10 i=1,n do 10 j=1,n c(i,j)=c(i,j)+a(i,k)*b(k,j) 10 continue do 10 k=1,n do 10 j=1,n do 10 i=1,n KJI c(i,j)=c(i,j)+a(i,k)*b(k,j) 10 continue do 10 i=1,n do 10 k=1,n IKJ do 10 j=1,n c(i,j)=c(i,j)+a(i,k)*b(k,j) 10 continue do 10 j=1,n do 10 k=1,n do 10 i=1,n JKI c(i,j)=c(i,j)+a(i,k)*b(k,j) 10 continue Fortran [1] II I M-1800/20 VP2600/10 DEC AlphaStation Sun Microsystems SPARCstation (optimization) 14 : : : : : : : M-1800/20 VP2600/10 OPT(E) VP2600/ ( A = B =(b ij ) (2) b ij = n +10max(i; j) 7
8 1: (MFLOPS) [1] JIK IJK KJI KIJ IKJ Fortran JKI VP2600/10 VP-200 JKI IJK 1988 VP-200 VP2600/10 FOR- TRAN77 EX/VP DO VP2600/10 8
9 2: ( ) [11] II 18 black box SSL II/VP, NUMPAC/VP
10 (benchmark) Fortran C Dhrystone, Whetstone, Livermore Fortran Kernels, NAS Kernel, Linpack, SPEC, PERFECT Benchmark FUJITSU VP2600/10 10
11 1 1 A \ " 1 (linear equation) ( ) 1 Ax = b A 0 (sparse matrix) 0 (band matrix) A b x A Gauss Gauss (Gaussian elimination) 1 Gauss- Jordan Gauss-Seidel \Gauss" Gauss- Jordan Gauss-Jordan Gauss Gauss 1:5 [13] Gauss A L U LU (LU decomposition) L U x 22 A LU 0 [2] Gauss A 0 (dense matrix) n 1 Gauss 22 () 11
12 Gauss n 3n 3 =2 LU A b LU 1 FLOPS III Gauss 1 Gauss A (pivot) 0 23 (partial pivoting) 24 (complete pivoting) Gauss LU Gauss SUBROUTINE MYLU(A,B,N,NX,P,VW) DOUBLE PRECISION A(NX,1),B(1),VW(1),D INTEGER N,NX,I,J,K,L,P(1)! PARTIAL PIVOTING DO 10 K=1,N P(K)=K 10 CONTINUE DO 20 K=1,N D=0.0D0 DO 30 I=k,N IF(ABS(A(P(I),K)).GE.D) THEN D=ABS(A(P(I),K)) L=I END IF 30 CONTINUE IF(K.NE.L) THEN J=P(K) P(K)=P(L) P(L)=J END IF! LU DECOMPOSITION A(P(K),K)=1.0D0/A(P(K),K) DO 40 I=K+1,N A(P(I),K)=A(P(I),K)*A(P(K),K) DO 50 J=K+1,N A(P(I),J)=A(P(I),J)-A(P(I),K)*A(P(K),J) 50 CONTINUE 40 CONTINUE 20 CONTINUE! FORWARD ELIMINATION DO 60 I=1,N VW(I)=B(P(I)) DO 70 J=1,I-1 VW(I)=VW(I)-A(P(I),J)*VW(J) 70 CONTINUE 60 CONTINUE! BACKWARD SUBSTITUTION DO 80 I=N,1,-1 B(I)=VW(I) DO 90 J=I+1,N B(I)=B(I)-A(P(I),J)*B(J) 90 CONTINUE B(I)=B(I)*A(P(I),I) 80 CONTINUE RETURN END MYLU 12
13 Gauss 8: III DLAX DVLAX LEQLUW DGEFA & DGESL SSL II/VP SSL II/VP( ) NUMPAC/VP Linpack (64bit) Linpack LU DGEFA DGESL DGEFA DGESL DO BLAS CALL DO [14] DGEFA & DGESL Fortran exact [15] n I I ( ) 25 VP2600/10 I, II DLAX, DVLAX, LEQLUW DGEFA & DGESL MYLU OPT(E) Linpack Linpack Argonne Dongarra >< >: A =(a ij ) a ij = a ji = n +10 i (if i j) b =(b i ) X n0i+1 b i = k=1 n 0 k +1 n (3) Ax = b x =(1=n; 1=n; 111; 1=n) II II (1) 8>< A =(a ij ) a ij =r 2 ij sin (i; j =1; 2; 111;n) n +1 n +1 b =(f i ) >: b i = nx j=1 b ij (i =1; 2; 111;n) Ax = b x = (1; 1; 111; 1) CLOCK error ^x i 13
14 4: Gauss (MFLOPS) VP2600/10 4:9GFLOPS DVLAX \" DVLAX DLAX LEQLUW Fortran DLAX Fortran MYLU 4 9, 10 n SSL II/VP DLAXR II ( 14
15 9: Gauss ( I) n =1000 n = 7000 DLAX E E-07 DVLAX E E-07 LEQLUW E E-07 MYLU E E-07 DGEFA&DGESL E E : Gauss ( II) n =1000 n = 7000 DLAX E E-11 DVLAX E E-11 LEQLUW E E-11 MYLU E E-12 DGEFA&DGESL E E-11 ) III Carl Friedrich Gauss Gauss Gauss Gauss Gauss Gauss ( ) Gauss Gauss ( ) Gauss Gauss Carl Friedrich Gauss \Gauss" Gauss Carl Friedrich Gauss Braunschweig Gauss 22 Euclid 2 Gauss Braunschweig Gauss
16 Fortran C, C++ ( ) (JIS) sin, cos, log 7 15 black box FORTRAN CABS( ) ( ) 3 Fortran [9] 28 Tailor 4 sqrt 4.16E D-17 exp 4.56E D-16 log 4.68E D-16 sin 3.28E D-17 cos 4.18E D-17 tan 8.17E D-15 arctan 7.01E D-16 FORTRAN77 EX FORTRAN77 EX ATAN [16] arctan = 4 (arctan arctan 1 3 ) Euler = 4 (2 arctan arctan 1 7 ) Clausen = 4 (4 arctan arctan ) Machin = 4 (4 arctan arctan 3 79 ) Euler-Vega = 4 (arctan arctan arctan 1 8 )Dase = 4 (4 arctan arctan arctan 1 99 ) Rutherford M-1800/ M-1800/20 IBM 1 29 = 4 arctan(1) 16
17 5: A n order arctan arctan II nx r=1 arctan 1 r 2 + r +1 = arctan n n +2 ( 1 1 ) 4 n = 2 1 ; 2 2 ; 111; n n 1 n 17
18 8: D NUMPAC DCOTHP COT 2 NUMPAC 2 cos 18
19 C Fortran 9: E 9 \ " n =2 30 = M-1800/20 IBM IEEE IV David Chudnovsky 19
20 C Fortran Fortran, C, C++, Pascal C Fortran \ " 35 C Fortran [7] 30 Fortran Fortran Fortran M. Metcalf, J.Reid Fortran ISDN Pascal C Fortran Fortran Fortran 36 Fortran C C Fortran 90 Fortran 95 Fortran Fortran 90 C (cf.[7]) C Fortran Fortran C IV Fortran C VP2600/
21 10: C Fortran () VP2600/10 SPARCstation C 21
22 11: C Fortran ( ) SPARCstation AlphaStation Fortran VP2600/10 Fortran VP2600/10 M-1800/20 AlphaStation SPARCstation Fortran () Fortran 10 (2) 20 c 30 C = A 2 B JKI 40 C 0 A 2 B + A T 2 B T parameter(maxn=401,n=maxn-1) real*8 a(maxn,maxn),b(maxn,maxn), & c(maxn,maxn) integer i,j,k do 10 j=1,n do 10 i=1,n a(i,j)=dble(n+1-max(i,j)) b(i,j)=a(i,j) 10 continue do 20 j=1,n do 20 i=1,n c(i,j)=0.0d0 20 continue do 30 j=1,n do 30 k=1,n do 30 i=1,n c(i,j)=c(i,j)+a(i,k)*b(k,j) 30 continue do 40 j=1,n do 40 k=1,n do 40 i=1,n c(i,j)=c(i,j)-a(i,k)*b(k,j)+a(k,i)*b(j,k) 40 continue END Fortran 2 C #include <stdio.h> #define maxn 401 #define n maxn-1 main() { double a[maxn][maxn],b[maxn][maxn], c[maxn][maxn]; } int i,j,k,z; for(j=0; j<n; j++) { for(i=0; i<n; i++) { } z = (i>j)? (i+1):(j+1) ; a[i][j]=(double)(n+1-z); b[i][j]=a[i][j]; } for(j=0; j<n; j++) { for(i=0; i<n; i++) { c[i][j]=0.0; } } for(j=0; j<n; j++) { } for(k=0; k<n; k++) { for(i=0; i<n; i++) { c[i][j] += a[i][k]*b[k][j]; } } for(j=0; j<n; j++) { for(k=0; k<n; k++) { for(i=0; i<n; i++) { } } c[i][j] += -a[i][k]*b[k][j] +a[k][i]*b[j][k] ; } C Fortran 2 22
23 13: C Fortran ( ) 13 Fortran 1 2 C M-1800/20 SPARCstation10/ AlphaStation 2 VP2600/10 2 C
24 VP2600/10 Fortran VP2600/10 M-1800/20 AlphaStation SPARCstation SPARCstation Fortran C V Fortran Fortran C f2c C ( ) Fortran C () 3 Navier-Stokes illustration : Hidaki Naoko and SPARCstation 24
25 [1] :, 9, (1988). [2] :, 7, (1984). [18] II, (1957). [19] : e 2,, Vol.24, No.1, 34{42 (1993). [3] :, 12, (1993). [4] :,, Vol.27, No.2, 93{110 (1994). [5], Vol.31, Np.9 (1990). [6] :,,Vol.27, No.5, 556{580 (1994). [7] M.Metcalf,J.Reid(,,, ) : Fortran90, bit, (1993). [8] SSL II ( ), 99SP{4070-2, (1991) [9] OS IV/MSP FORTRAN77 EX V12, 79SP-5031 (1991). [10] :, (1994). [11] :, 9, (1989). [12] OS IV/MSP FORTRAN77 EX/VP V12, 79SP-5041 (1991). [13], :, (1985). [14] :,, Vol.31, No.3, pp.313{320 (1990). [15] Gregory, R. T. and Karney, D. L.: A collection of matrices for testing computational algorithms, John Wiley & Sons, New York (1969). [16] :, (1991). [17], :, (1983)
main.dvi
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