Linux Linux Linux

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2 Linux Linux Linux MPI Linux MPICH MPI NFS NIS rsh MPICH MPICH Cannon LU LU Cholesky Cholesky i

3 ii Cholesky MPI 65 MPI MPI MPI MPI MPI MPI (MPI IO) MPI MPI

4 iii atomicity MPI MPI MPI Poisson MPI MPI A.1 Linpack A A.1.2 HPL A.1.3 Linux BLAS A.1.4 HPL

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11 Linux MPI RedHat 9 Linux Linux 3.1 Linux ( ) TOP500 Supercomputer Sites Cluster@TOP500 : cluster howto Linux Linux : RedHat, Debian, SuSE, Mandrake,... ( ) : Myrinet, QsNet, Dolphin SCI, Infiniband,... : gcc/g++/g77, PGI, Intel,... MPI : MPICH, LAM-MPI,... : NFS, PVFS, Lustre... : PBS, Condor, LSF,... : BLAS MKL, ATLAS, Kazushige Goto s BLAS ( ) FFTW LAPACK ScaLAPACK : PETSc UG 37

12 38 Linux 3.3 Linux Linux, C Fortran, MPICH LAM-MPI : : TCP/IP 3.3.2, Linux,, MPICH LAM-MPI, (NFS), NIS LDAP Internet IP ( x), (NTP) NIS/LDAP (NFS) IP (IP Masquerading) (NAT, Network Address Translation) Internet (ipchains iptables) LVS (Linux Virtual Server) UPS : CRack, LSSC I, LSSC II 3.1: Linux

13 3.4 MPI MPI Linux Linux, RedHat-9 ( RedHat rpm Maximum RPM ) gcc GNU C, C gcc-g77 GNU Fortran 77, Fortran rsh rsh-server MPI rsh-server, root /etc/hosts.equiv, ( ) rsh : /sbin/chkconfig rsh on rshd ( root) : rsh /bin/hostname /var/log/messages shell (.cshrc,.profile,.bashrc ) stdout stderr MPI MPICH MPICH doc MPICH, mpich-1.x.x.tar.gz, 1.x.x MPICH ( 1.2.5) MPICH 1.x.x MPICH : tar xzpvf mpich-1.x.x.tar.gz MPICH: cd mpich-1.x.x./configure -prefix=/usr/local/mpi --disable-weak-symbols make MPICH ch_p4 --disable-weak-symbols MPICH bug, MPI_File_xxxx ( PMPI_File_xxx ) MPICH: make install root, MPICH /usr/local/mpi MPICH /usr/local/mpi/bin /usr/local/mpi/man PATH MANPATH /etc/profile.d mpich.sh mpich.csh, Bourne shell

14 40 Linux C shell, : /etc/profile.d/mpich.sh : #!/bin/bash export MANPATH=${MANPATH}:/usr/local/mpi/man export PATH=${PATH}:/usr/local/mpi/bin /etc/profile.d/mpich.csh : #!/bin/csh if ( $?MANPATH == 0 ) then setenv MANPATH :/usr/local/mpi/man else setenv MANPATH ${MANPATH}:/usr/local/mpi/man endif setenv PATH ${PATH}:/usr/local/mpi/bin MPICH: C Fortran cp /usr/local/mpi/examples/cpi.c. mpicc -o cpi cpi.c. mpirun -np 1 cpi mpirun -np 2 cpi cp /usr/local/mpi/examples/pi3.f. mpif77 -o pi3 pi3.f. mpirun -np 1 pi3 mpirun -np 2 pi3 ( ) 3.5 MPI Linux TCP/IP, /etc/hosts /etc/hosts : localhost.localdomain localhost node1.mydomain node node2.mydomain node n noden.mydomain noden ( IP ) NIS (Network Information Service, SUN Yellow Pages) NFS (Network File System) NIS NFS

15 3.5 MPI NFS : mkdir -p /home/local /usr/local /home/local: /bin/rm -rf /usr/local ln -s /home/local /usr/. ( : /usr/local, ) nfs-utils NFS : /sbin/chkconfig nfs on /sbin/chkconfig nfslock on /etc/init.d/nfslock restart /etc/init.d/nfs restart /etc/exports : /home *(rw,no_root_squash) /home ( :, no_root_squash root_squash man 5 exports ) (/home): exportfs -a () : mkdir -p /home /etc/fstab : < >:/home /home nfs defaults 0 0 ( < > IP ) : /sbin/chkconfig netfs on /home : mount /home () /usr/local /home/local: /bin/rm -rf /usr/local ln -s /home/local /usr/. ( : /usr/local, ) root /home /usr/local df NFS NFS

16 42 Linux NIS CLUSTER NIS : ypserv, ypbind, yp-tools /etc/sysconfig/network : NISDOMAIN=CLUSTER NIS : /sbin/chkconfig ypserv on /etc/init.d/ypserv start NIS : /usr/lib/yp/ypinit -m Ctrl-D, y NIS No rule to make target... NIS : /sbin/chkconfig ypbind on /etc/init.d/ypbind start NIS ypwhich ypcat passwd ( ) : ypbind, yp-tools /etc/sysconfig/network : NISDOMAIN=CLUSTER NIS : /sbin/chkconfig ypbind on /etc/init.d/ypbind start NIS ypwhich ypcat passwd NIS /etc/nsswitch.conf : passwd: files nis shadow: files nis group: files nis hosts: files nis dns NIS ( ypcat passwd ) /etc/passwd /etc/shadow NIS ( ypcat group ) /etc/group NIS ( /home

17 3.5 MPI 43 ), cd /var/yp; make NIS NIS passwd yppasswd root NIS NIS rsh /etc/hosts.equiv rsh : /sbin/chkconfig rsh on : root rsh /etc/hosts.equiv /root/.rhosts, /etc/securetty rsh root, () : NIS : rsh /bin/hostname /var/log/messages shell (.cshrc,.profile,.bashrc ) stdout stderr MPI MPICH MPICH /usr/local /etc/profile.d/mpich.sh /etc/profile.d/mpich.csh MPICH MPICH mpicc (C), mpif77 (Fortran), mpicc (C++) MPICH shell C/C++/Fortran MPICH MPICH driver ch_p4 driver, MPI, : mpirun -machinefile -np 4 MPI [MPI] mpirun ( 4) mpirun mpirun -help mpirun./mpi -p4pg [MPI] MPI

18 44 Linux ( Master/Slave ) : n 1 n 1, 1 MPI ( /home/zlb/test/cpi) node1 /home/zlb/test MPI cpi p4file : node1 0 /home/zlb/test/cpi node2 1 /home/zlb/test/cpi node1 1 /home/zlb/test/cpi node2 1 /home/zlb/test/cpi./cpi -p4pg p4file node1, node2, 0 2 node1, 1 3 node2 3.6, I/O : IP RedHat Linux,

19 /etc/sysconfig/network eth0 IP /etc/sysconfig/network-scripts/ifcfg-eth0 DHCP IP, : MAC /etc/hosts, MAC IP /etc/hosts : localhost.localdomain localhost node0 node0.cluster 00:90:27:57:42: node1 node1.cluster 00:90:27:57:45:F node2 node2.cluster 00:90:27:57:42:B node3 node3.cluster 00:90:27:57:45:E node4 node4.cluster 00:90:27:57:3F:7E ( MAC MAC /sbin/ifconfig eth0 ) /etc/sysconfig/network : NETWORKING=yes hwaddr= /sbin/ifconfig eth0 /bin/grep -i HWaddr awk {print $5} HOSTNAME= grep ^[0-9] /etc/hosts grep -w $hwaddr awk {print $2} unset hwaddr GATEWAY= NISDOMAIN=CLUSTER /etc/sysconfig/network-scripts/ifcfg-eth0 : DEVICE=eth0 BOOTPROTO=static IPADDR= grep ^[0-9] /etc/hosts grep -w "$HOSTNAME" awk {print $1} BROADCAST= NETMASK= NETWORK= ONBOOT=yes, (PXE boot) 3.7 : : CPU : ( ) MPI rsh

20 46 Linux

21 ,,. MMP,,., [2, 3]. [4],.,, p, P j j, P myid, send(x, j) recv(x, j) P myid x P j P j x, P myid., i mod p i p.,,,.,,.,,.,, S p p, : S p = /p,,,,. S p = p /,,,.,,., : T c = α + β N, α, β, N. 4.1,.,. C = A B (4.1) A B m k k n, C m n., m = m p, k = k p n = n p, A B A B : [ ] T [ ] A = A T 0 A T 1... A T p 1, B = B 0 B 1... B p 1 (4.2) 47

22 48 C = (C i,j ) = (A i B j ), C i,j m n. A i, B i C i,j, j = 0,..., p 1 P i,.,. p, C i,j, C p. C i,j, : for i = 0 to p 1 do l i+myid mod p C l = A B, mp1 myid+1 mod p, mm1 myid-1 mod p if i p 1, send(b, mm1), recv(b, mp1) end{f or}, C l = C myid,l, A = A myid, B, k n, p 1. DTA , CA , DTA 1 = 2 k (n n), CA 1 = m k n/p A B, A (4.2), B : [ ] T B = B0 T B1 T... Bp 1 T (4.3) C i A i C i, A i B, [ ] A i = A i0 A i1... A i,p 1 p 1 C i = A i B = A i,j B j A, B C 4.1.1, P myid for i = 0 to p 1 do l i+myid mod p C = C + A l B, mp1 myid+1 mod p, mm1 myid-1 mod p if i p 1, send(b, mm1), recv(b, mp1) end{f or} 4.1.1, C, A l = A myid,l j=0 A B, B (4.2), A : [ ] A = A 0 A 1... A p 1 (4.4)

23 C B,, B i A [ ] T, B i = Bi0 T Bi1 T... Bi,p 1 T, C. p 1 C j = A B j = A i B i,j A B, : for i = 0 to p 1 do l i+myid mod p C = C + A B l, mp1 myid+1 mod p, mm1 myid-1 mod p if i p 1, send(a, mm1), recv(a, mp1) end{f or} , DTA 3 = 2 m (k k). m n, DTA 1 DTA 3. m n, m n, DTA 3 DTA 1., A B, A (4.4), B (4.3). p 1 C = A B = A i B i C p C,,,,.,, C C,,, C C = A B for i = 1 to p 1 do l i+myid mod p k myid-i mod p send(c l, l), recv(t, k) C myid = C myid + T end{f or},, DTA 4 = 2 (m m) n. C, 4.1.4, DTA 4 = 2 m (n n), m n m n, C Cannon A, B C m m,, A = (A ij ) m m, B = (B ij ) m m, C = i=0 i=0

24 50 (C ij ) m m, A ij, B ij C ij n n, p = m m. Cannon, Q = (Q ij ) Q ij = { In, j i + 1 mod m 0 n, I n 0 n n. D (l) A A = m 1 l=0 D(l) A Ql. C = A B = m 1 l=0 m 1 D (l) A Ql B = l=0 D (l) A = diag(d(l) i ) = diag(a i,i+l mod m ), B(l) B (l) = Q l B = Q B (l 1).,, P myid = P myrow,mycol, A ij,b ij, C ij P ij, P myid C = 0 for i = 0 to m 1 do k myrow+i mod m; mp1 mycol+1 mod m; mm1 mycol-1 mod m; if mycol=k then else send(a, (myrow, mp1)); copy(a, tmpa); recv(tmpa, (myrow, mm1)); if mycol mp1, send(tmpa, (myrow, mp1)); end{if} C = C+tmpA B; mp1 myrow+1 mod m; mm1 myrow-1 mod m; if i m 1 then send(b, (mm1, mycol)); recv(b, (mp1, mycol)); end{if} end{f or}, A, C, B. DTA 5 = m 2 n 2 + (m 1) 2 n 2 = 2(2m 1)n 2 = 4m 2 n 2 / p 2m 2 n 2 /p.,.,, p 4, Cannon.,,. 4.2,.,,., Linpack.,

25 4.2 51,. Ax = b, A LU, L,U, Q, AQ = LU.,, Ly = b Ux = y LU n n A = (a ij ) LU., Gauss,, L. a k A k for j = 0 to n 2 do find l: a lj = max{ a ij, i = j,..., n 1} if l j, swap A j and A l if a jj = 0, A is singular and return a ij = a ij /a jj, i = j + 1,..., n 1 for k = j + 1 to n 1 do a ik = a ik a ij a jk, i = j + 1,..., n 1 end{f or} end{f or} 4.2.1, A, a ik a ij a jk, A. LU, (wrap) A, A i P i mod p. n = p m, A i A i p + myid, P myid icol= 0 for j = 0 to n 2 do if myid=j mod p then find l: a l,icol = max{ a i,icol, i = j,..., n 1} if l j, swap a j,icol and a l,icol if a j,icol = 0, A is singular and kill all processes a i,icol = a i,icol /a j,icol, i = j + 1,..., n 1 f i j 1 = a i,icol, i = j + 1,..., n 1 send(l, myid+1) and send(f, myid+1) icol+1 icol else recv(l, myid-1) and recv(f, myid+1) if myid+1 j mod p, send(l, myid+1) and send(f, myid+1)

26 52 end{if} if l j, swap A j and A l for k=icol to m 1 do a ik = a ik f i a jk, i = j + 1,..., n 1 end{f or} end{f or} LU,,.,,.,., if myid=j mod p if myid=j + 1 mod p,,,.,, LU,.,. LU,,, A.,, gfactor, gl, lfactor l, flag gfactor. : icol= 0 for j = 0 to n 2 do if myid=j mod p then else find l: a l,icol = max{ a i,icol, i = j,..., n 1} if l j, swap a j,icol and a l,icol if a j,icol = 0, A is singular and kill all processes a i,icol = a i,icol /a j,icol, i = j + 1,..., n 1 lfactor i j 1 = a i,icol, i = j + 1,..., n 1 if p 1 k=0 flag k p, wait copy(lfactor, gfactor), copy(l, gl), flag k = 0, k = 0,..., p 1 icol+1 icol if flag myid 0, wait copy(gfactor, lfactor), copy(gl, l), flag myid = 1 end{if} if l j, swap A j and A l for k =icol to m 1 do a ik = a ik lfactor i a jk, i = j + 1,..., n 1 end{f or} end{f or}

27 , copy send recv.,., , 0, : icol=0 if myid=0, icol=1 for j = 0 to n 2 do cid=j mod 2p, next=j + 1 mod p if flag cid 0, wait copy(gfactor, lfactor), copy(gl, l) if myid=next then if l j, swap a l,icol and a j,icol a i,icol = a i,icol lfactor i a j,icol, i = j + 1,..., n 1 find gl: a gl,icol = max{ a i,icol, i = j + 1,..., n 1} if gl j + 1, swap a j+1,icol and a gl,icol if a j+1,icol = 0, A is singular and kill all processes a i,icol = a i,icol /a j,icol, i = j + 2,..., n 1 gfactor i j 2 = a i,icol, i = j + 2,..., n 1 flag cid+1 mod 2p = 0, flag cid+2 mod 2p = 1 icol+1 icol end{if} if l j, swap A j and A l for k =icol to m 1 do a ik = a ik lfactor i a jk, i = j + 1,..., n 1 end{f or} end{f or},,,, LU, [5] , Lx = b.,, for i = 0 to n 1 do x i = b i /l ii for j = i + 1 to n 1 do b j = b j l ji x i

28 54 end{f or} end{f or} b L, b,., L, b,.,.,,,,.. LU, flag gb, : k = 0 for i = 0 to n 1 do if myid i mod p then gb i = b k /l ki, flag i = 1, k + 1 k end{if} if flag i 1, wait for j = k to m 1 do b j = b j l ji gb i end{f or} end{f or},, b,,.,.,,,.,.,,,,.,, : k = 0 if myid=0, then u i = b i, i = 0,..., n 1, v i = 0, i = 0,..., p 2 else u i = 0, i = 0,..., n 1 for i = 0 to n 1 do if i > 0, recv(v, i 1 mod p) x k = (u i + v 0 )/l ik v j = v j+1 + u i+1+j l i+1+j,k x k, j = 0,..., p 3

29 v p 2 = u i+p 1 l i+p 1,k x k send(v, i + 1 mod p) u j = u j l jk x k, j = i + p,..., n 1 k + 1 k end{f or} [6]. 4.3 A LU, Cholesky, A = R T R, R., Cholesky. Cholesky,,. Cholesky Cholesky Cholesky, Cholesky, [7]. Cholesky for j = 0 to n 1 do a jj = a jj j 1 a jk a jk, a jj = a jj k=0 for i = j + 1 to n 1 a ij = (a ij j 1 a jk a ik )/a jj end{f or} end{f or} k=0, R T A, j, R T,. j j, R., F, p 1, (p 1) n, A, P myid : 2 for i = 0 to m 1 do k = i p + myid, l = k p + 1 if k > 0 then, recieve G from P myid 1 for j = 0 to p 2 do a i,j+l = (a i,j+l j+l 1 a it g jt )/g j,j+l F j = G j+1 t=0

30 56 end{f or} a ik = a ik k 1 a it a it t=0 a ik = a ik, F p 2 = A i Send F to P myid+1 for e = i + 1 to m 1 do for j = 0 to p 2 do a e,j+l = (a e,j+l j+l 1 a et g jt )/g j,j+l a ek = (a ek k 1 end{f or} end{f or} t=0 t=0 a et a it )/a ik R T p,, R T p,,. [7] Cholesky Cholesky Cholesky : [ ] coshφ sinhφ H = sinhφ coshφ, : [ 1 ρ ] H = (1 ρ 2 ) 1 2 ρ 1 ρ = tanh( φ). A R T R,. A = D + U T + U, D,U, W = D 1/2 U, V = D 1/2 + W. A = V T V W T W : [ R T 0 ] [ I 0 0 I ] [ R 0 ] [ = V T W T ] [ I 0 0 I ] [ V W ] I n n, Cholesky, m 2m Θ : [ ] [ I 0 I 0 Θ T Θ = 0 I 0 I ] I m m, (pseudo-orthogonal).

31 4.3 57, Q [ ] [ ] V R Q = W 0 A = R T R, Q. [8] R S n n, R T R S T S, R, : s kk r 1 kk < 1, 1 k n R S n n, R T R S T S, ρ k s kk r 1 k n, ˆQ = Q (n) Q(n 1) Q (1), Q (k) :, [ R S q (k) ij = ] = ˆQ [ R S 1, i = j k i = j n + k (1 ρ 2 k ) 1 2, i = j = k i = j = n + k (1 ρ 2 k ) 1 2 ρ k, (i, j) = (k, n + k) (i, j) = (n + k, k) 0, ], ˆQ, R, S. Q (k), R S k, [ ] H k = (1 ρ 2 k) ρk ρ k 1 ] ] [ [ Rk H k Q R (k) k k + n. Q (k). S k S Q n n, p 1,n = 1 p i,i 1 = 1, 2 i n. 2, Cholesky : V 0 = V, W 0 = W, A = V T V W T W for i = 0 to n 1 do [ ] [ ] [ V i+1 I 0 V i = ˆQ (i) W i+1 0 Q end { for } W i ] kk, 1 ˆQ (i) ˆQ, ˆQ (i) W i 0. A β, i β A Cholesky,, [8]. H k,,.

32 Cholesky 3 ˆQ (i) Q (k), Q (k) V i W i k, 2 l, M 2 l, H. M = HM m 21 = 0. m 21 = 0, H, ρ = m 21 m l., M = KB, K = diag(k 1, K 2 ) 2 2, K 1 > 0 K 2 > 0. G = K 1 HK, K 2 2. B = GB, M = K B. K,. K,. L = K 2, L = K 2, L α = L2 L 1, β = b21 b 11. L = (1 αβ 2 ) 1 L, G = [ 1 αβ β 1 ] Proof. H ρ = m 21 m 1 11 = K2b21 K 1b 11. HK = (1 ρ 2 ) 1 2 [ K 1 K2 2 b21 K 1b 11 K2b21 b 11 K 2 ] = (1 αβ 2 ) 1 2 K [ 1 αβ β 1 ] R S n n, E F, R T ER S T F S, α k = F k E k, β k = s kk r kk, [ R S ] = ˆQ [ R ˆQ = Q (n) Q(n 1) Q (1), Q (k), E F : 1, i = j q (k) α k β k, i = k, j = n + k ij = β k, i = n + k, j = k 0, S ] E k Ẽ k = 1 α k βk 2, Fk = 1 α k βk 2, 1 k n F k R T Ẽ R S T F S = R T ER S T F S, R, S.,. A = V T EV W T F W, Cholesky : V 0 = V, W 0 = W, E 0 = E, F 0 = F for i = 0 to n 1 do

33 [ V i+1 W i+1 ] [ I 0 = 0 P ] ˆQ (i) [ V i W i ] E i+1 k = F1 i+1 = end { for } E i k 1 α i k βi k 2, F i+1 k = F i k 1 αk i βi 2, 1 k n k i+1 F n, F i+1 i+1 k+1 = F k, 1 k n 1 ˆQ (i) ˆQ ,, [9] [10].,,. Ax = d, a 0 b 0 c 1 a 1 b 1 A = T (c, a, b) = c n 2 a n 2 b n 2 c n 1 a n , A LU, A, LU : 1 e 1 1 A = e n 2 1 e n 1 1 f 0 b 0 f 1 b f n 2 b n 2 f n 1 : f 0 = a 0, f i 1 e i = c i, f i = a i e i b i 1, i = 1,..., n 1 : f i = a i c i b i 1 /f i 1, i = 1,..., n 1 f i = u i /u i 1, u 0 = f 0, u 1 = 1, : [ ] [ ] [ ] ui ai t i ui 1 =, t i = c i b i 1, i = 1,..., n u i 1, : x 0 = 0, x i = a i x i 1 + b i, 1 i n u i 2

34 60 x 1 = x 2 = x 3 = x 4 = x 5 = x 6 = x 7 = x 8 = Q(1, 1) Q(2, 1) Q(3, 1) Q(4, 1) Q(5, 1) Q(6, 1) Q(7, 1) Q(8, 1) Q(3, 1) Q(4, 1) Q(5, 2) Q(6, 3) Q(7, 4) Q(8, 7) a 3 a 2 a 1 a 4 a 3 a 2 a 1 a 5 a 4 a 3 a 2 a 6 a 5 a 4 a 3 a 7 a 6 a 5 a 4 a 8 a 7 a 6 a 5 Q(2, 1) Q(3, 2) Q(4, 3) Q(5, 4) Q(6, 5) Q(7, 6) Q(8, 7) a 2 a 1 a 3 a 2 a 4 a 3 a 5 a 4 a 6 a 5 a 7 a 6 a 8 a 7 Q(1, 1) Q(2, 2) Q(3, 3) Q(4, 4) Q(5, 5) Q(6, 6) Q(7, 7) Q(8, 8) a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 4.1: [11], Q(t, s) = t ( t j=s r=j+1 a r )b j t r=t+1 a r = 1, 1 s, t n, s t. Q(t, s) : (1) Q(i, i) = b i, 1 i n (2) l 0, s + l < t, : t Q(t, s) = ( a r )Q(s + l, s) + Q(t, s + l + 1) r=s+l+1 (3) : x i = Q(i, 1), 1 i n,. n = 2 m x n = Q(n, 1), x n = Q(n, 1) = ( 2 m r=2 m 1 +1 a r )Q(2 m 1, 1) + Q(2 m, 2 m 1 + 1),Q(2 m 1, 1) Q(2 m, 2 m 1 + 1),, 2 m r=2 m 1 +1 a r. x 1,..., x n logn,, O( logn), O(n), O(n/ logn). n,,. 4.1 n = 8 x 1,..., x 8, Q(i, i) = b i, i = 1,..., 8.,, LU.,,.

35 ,, [12, 10]. : A 0 B 0 B T 0 A 1 B 1 A = Bp 3 T A p 2 B p 2 B T p 2 A p 1 B i,a i. A i LDL T, D,L. A i = L i D i L T i, L = diag(l i), L 1 AL T = D 0 B0 B T 0 D 1 B B T p 3 D p 2 Bp 2 B T p 2 D p 1 B i = L 1 i B i L T i+1 = B il T i+1, 0. D i, B i 0, D i B i, D i B i B i T.,,., LDL T, R T R,, LDL T., : (1) L i, D i, A i = L i D i L T i ; (2), B i ; (3), ; (4).,.,,,.,.,. 4.5,,. 60, [13] x = Bx + c, ρ( B ) < 1,.,

36 62 [14] x = F x, F P -,. P - (P-contraction) [15] x = F x, [14]. A A, x x, A 0 A 0. F x i f i (x) f i (x 1,..., x n ), x (j), j = 0, 1,..., N F R n R n, F x (0) x (j) R n, j = 1, 2,..., J = {J j j = 1, 2,...} S = {(s (j) 1,..., s(j) n ) j = 1, 2,...} x (j) i = x (j 1) i, i / J j f i (x (s(j) 1 ) 1,..., x n (s(j) n ) ), i J j, J {1, 2,..., n}, S N n., i = 1,..., n, J S : (1) s (j) i j 1, j = 1, 2,...; (2) s (j) i j ; (3) i J j (j = 1, 2,...). [14], F x F y A x y, A ρ(a) < 1. x = F x , Ax = b., A, A = B C, : x = B 1 Cx B 1 b (4.5) B , ρ( B 1 C ) < 1,. A M, M H, [15]. [15] A, B C, A = B C, B 1 0 B 1 C 0, A = B C A A = B C, ρ(b 1 C) < 1 A 1 A 1 0.

37 B C n n, C B, ρ(c) ρ(b) A, a ij 0 (i j), a ii > 0, A M. [16] A = B C A A M, (4.5)., Jacobi Gauss-Seidel., A A = D L U, D,L U A,A = B C, B = D αl, C = (1 α)l + U, 0 α 1, (4.5).,,,.

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41 MPI MPI Message Passing Interface,., SPCs (Scalable Parallel Computers) NOWs (Networks of Workstations) COWs (Clusters of Workstations).,. MPI PVM,,,,,. 4, MPI, MPI,, MPI PVM. SPCs, SPMD MPI. MPI, MPI MPI_. C,,, MPI_,,, Fortran, MPI,., MPI (Process Group), MPI MPI_COMM_WORLD,,., MPI (communicator). MPI,, Fortran, MPI, MPI [1]. 5.1 MPI MPI, MPI_Init( ) MPI_Finalize( ). : MPI INIT C int MPI_Init( int *argc, char **argv ) Fortran MPI_INIT( IERROR ) INTEGER IERROR ANSI C main argc argv, MPI_Init. Fortran, MPI_INIT. MPI, MPI ( MPI_INITIALIZED), MPI,. MPI FINALIZE C int MPI_Finalize( void ) Fortran MPI_FINALIZE( IERROR ) INTEGER IERROR 67

42 68 MPI MPI., MPI, MPI_INIT., MPI_INIT, MPI_INITIALIZED, MPI_INIT. MPI INITIALIZED C int MPI_Initialized( int flag ) Fortran MPI_INITIALIZED( FLAG, IERROR ) LOGICAL FLAG INTEGER IERROR OUT FLAG, MPI_INIT, TRUE, FALSE. MPI MPI_ERROR_STRING,. MPI ERROR STRING C int MPI_Error_string( int errorcode, char *string, int *len ) Fortran MPI_ERROR_STRING( ERRORCODE, STRING, LEN, IERROR ) INTEGER ERRORCODE, LEN, IERROR CHARACTER*(*) STRING IN ERRORCODE, MPI. OUT STRING, ERRORCODE. OUT LEN, STRING., STRING MPI_MAX_ERROR_STRING. 5.2,,,. 5.3 MPI MPI COMM GROUP C int MPI_Comm_group( MPI_Comm comm, MPI_Group *group ) Fortran MPI_COMM_GROUP( COMM, GROUP, IERROR ) INTEGER COMM, GROUP, IERROR IN COMM,. OUT GROUP, COMM.

43 5.3 MPI 69 MPI_COMM_GROUP,. MPI GROUP FREE C int MPI_group_free( MPI_Group *group ) Fortran MPI_GROUP_FREE( GROUP, IERROR ) INTEGER GROUP, IERROR INOUT GROUP, MPI_GROUP_NULL. MPI_GROUP_FREE,. MPI GROUP SIZE C int MPI_Group_size( MPI_Group group, int *size ) Fortran MPI_GROUP_SIZE( GROUP, SIZE, IERROR ) INTEGER GROUP, SIZE, IERROR IN GROUP,. OUT SIZE,. MPI_GROUP_EMPTY, SIZE 0. MPI GROUP RANK C int MPI_Group_rank( MPI_Group group, int *rank ) Fortran MPI_GROUP_RANK( GROUP, RANK, IERROR ) INTEGER GROUP, RANK, IERROR IN GROUP,. OUT RANK,., RANK MPI_UNDEFINED. MPI GROUP TRANSLATE RANKS C int MPI_Group_translate_ranks( MPI_Group group1, int n, int *ranks1, MPI_Group group2, int *ranks2) Fortran MPI_GROUP_TRANSLATE_RANKS( GROUP1, N, RANKS1, GROUP2, RANKS2, IERROR ) INTEGER GROUP1, N, RANKS1(*), GROUP2, RANKS2(*), IERROR

44 70 MPI IN GROUP1, 1. IN N, RANKS1 RANKS2. IN RANKS1, 1. IN GROUP2, 2. OUT RANKS2, RANKS RANKS1, 2, RANKS2 RANKS1 MPI_UNDEFINED. MPI GROUP INCL C int MPI_Group_incl( MPI_Group group, int n, int *ranks, MPI_Group newgroup) Fortran MPI_GROUP_INCL( GROUP, N, RANKS, NEWGROUP, IERROR ) INTEGER GROUP, N, RANKS(*), NEWGROUP, IERROR IN GROUP,. IN N, RANKS. IN RANKS,. OUT NEWGROUP, RANKS. MPI GROUP EXCL C int MPI_Group_excl( MPI_Group group, int n, int *ranks, MPI_Group newgroup) Fortran MPI_GROUP_EXCL( GROUP, N, RANKS, NEWGROUP, IERROR ) INTEGER GROUP, N, RANKS(*), NEWGROUP, IERROR IN GROUP,. IN N, RANKS. IN RANKS,. OUT NEWGROUP, RANKS. MPI GROUP UNION C int MPI_Group_union( MPI_Group group1, MPI_Group group2, MPI_Group newgroup) Fortran MPI_GROUP_UNION( GROUP1, GROUP2, NEWGROUP, IERROR ) INTEGER GROUP1, GROUP2, NEWGROUP, IERROR

45 5.4 MPI 71 IN GROUP1, 1. IN GROUP2, 2. OUT NEWGROUP, 1 2. MPI GROUP INTERSECTION C int MPI_Group_intersection( MPI_Group group1, MPI_Group group2, MPI_Group newgroup) Fortran MPI_GROUP_INTERSECTION( GROUP1, GROUP2, NEWGROUP, IERROR ) INTEGER GROUP1, GROUP2, NEWGROUP, IERROR IN GROUP1, 1. IN GROUP2, 2. OUT NEWGROUP, 1 2. MPI GROUP DIFFERENCE C int MPI_Group_difference( MPI_Group group1, MPI_Group group2, MPI_Group newgroup) Fortran MPI_GROUP_DIFFERENCE( GROUP1, GROUP2, NEWGROUP, IERROR ) INTEGER GROUP1, GROUP2, NEWGROUP, IERROR IN GROUP1, 1. IN GROUP2, 2. OUT NEWGROUP, 1 2.,. 5.4 MPI. MPI COMM SIZE C int MPI_Comm_size( MPI_Comm comm, int *size ) Fortran MPI_COMM_SIZE( COMM, SIZE, IERROR ) INTEGER COMM, SIZE, IERROR IN COMM,. OUT SIZE,.

46 72 MPI MPI COMM RANK C int MPI_Comm_rank( MPI_Comm comm, int *rank ) Fortran MPI_COMM_RANK( COMM, RANK, IERROR ) INTEGER COMM, RANK, IERROR IN COMM,. OUT RANK,. MPI COMM DUP C int MPI_Comm_dup( MPI_Comm comm, MPI_Comm *newcomm ) Fortran MPI_COMM_DUP( COMM, NEWCOMM, IERROR ) INTEGER COMM, NEWCOMM, IERROR IN COMM,. OUT NEWCOMM, COMM. MPI COMM CREATE C int MPI_Comm_create( MPI_Comm comm, MPI_Group group, MPI_Comm *newcomm ) Fortran MPI_COMM_CREATE( COMM, GROUP, NEWCOMM, IERROR ) INTEGER COMM, GROUP, NEWCOMM, IERROR IN COMM,. IN GROUP, COMM. OUT NEWCOMM, GROUP. MPI COMM SPLIT C int MPI_Comm_split(MPI_Comm comm, int color, int key, MPI_Comm *newcomm) Fortran MPI_COMM_SPLIT( COMM, COLOR, KEY, NEWCOMM, IERROR ) INTEGER COMM, COLOR, KEY, NEWCOMM, IERROR IN COMM,. IN COLOR,. IN KEY,. OUT NEWCOMM,.

47 COMM, COLOR,, COLOR. MPI COMM FREE C int MPI_Comm_free( MPI_Comm *comm ) Fortran MPI_COMM_FREE( COMM, IERROR ) INTEGER COMM, IERROR INOUT COMM,. 5.5,,. MPI,. 5.6 MPI SEND C int MPI_Send( void* buf, int count, MPI_Datatype datatype, int dest, int tag, MPI_Comm comm ) Fortran MPI_SEND(BUF, COUNT, DATATYPE, DEST, TAG, COMM, IERROR ) <type> BUF(*) INTEGER COUNT, DATATYPE, DEST, TAG, COMM, IERROR IN BUF,. IN COUNT,. IN DATATYPE,. IN DEST,. IN TAG,. IN COMM,. COUNT BUF. MPI RECV C int MPI_Recv( void* buf, int count, MPI_Datatype datatype, int source, int tag, MPI_Comm comm, MPI_Status *status ) Fortran MPI_RECV(BUF, COUNT, DATATYPE, SOURCE, TAG, COMM, STATUS, IERROR ) <type> BUF(*) INTEGER COUNT, DATATYPE, SOURCE, TAG,COMM,STATUS(MPI_STATUS_SIZE), IERROR

48 74 MPI OUT BUF,. IN COUNT,. IN DATATYPE,. IN SOURCE,. IN TAG,. IN COMM,. OUT STATUS,., SOURCE TAG, MPI_ANY_SOURCE MPI_ANY_TAG,. STATUS C, Fortran, MPI_SOURCE, MPI_TAG MPI_ERROR. STATUS,, MPI_GET_COUNT. MPI Fortran C : Fortran MPI MPI datatype Fortran datatype MPI_INTEGER MPI_REAL MPI_DOUBLE_PRECISION MPI_COMPLEX MPI_LOGICAL MPI_CHARACTER MPI_BYTE MPI_PACKED INTEGER REAL DOUBLE PRECISION COMPLEX LOGICAL CHARACTER(1) C MPI MPI datatype Fortran datatype MPI_CHAR MPI_SHORT MPI_INT MPI_LONG MPI_UNSIGNED_CHAR MPI_UNSIGNED_SHORT MPI_UNSIGNED_INT MPI_UNSIGNED_LONG MPI_FLOAT MPI_DOUBLE MPI_LONG_DOUBLE MPI_BYTE signed char signed short int signed int signed long int unsigned char unsigned short int unsigned int unsigned long int float double long double MPI_PACKED MPI_BYTE MPI_PACKED Fortran C, MPI_BYTE, MPI_PACKED., MPI, MPI, MPI. MPI

49 MPI_INIT, MPI_COMM_RANK, MPI_COMM_SIZE, MPI MPI_FINALIZE, MPI_SEND MPI_RECV p, myid (myid= 0,..., p 1) m, m (myid + 1) mod p. program ring c c The header file mpif.h must be included when you use MPI fuctions. c include mpif.h integer myid, p, mycomm, ierr, m, status(mpi_status_size), & next, front, mod, n c c Create MPI parallel environment and get the necessary data. c call mpi_init( ierr ) call mpi_comm_dup( mpi_comm_world, mycomm, ierr) call mpi_comm_rank( mycomm, myid, ierr ) call mpi_comm_size( mycomm, p, ierr ) c c Beginning the main parallel work on each process. c m = myid front = mod(p+myid-1, p) next = mod(myid+1, p) c c Communication with each other. c if(myid.eq. 0) then call mpi_recv(n, 1, mpi_integer, front, 1, mycomm, status, ierr) call mpi_send(m, 1, mpi_integer, next, 1, mycomm, ierr) m = n else call mpi_send(m, 1, mpi_integer, next, 1, mycomm, ierr) call mpi_recv(m, 1, mpi_integer, front, 1, mycomm, status, ierr) endif c c Ending of parallel work. c print *, The value of m is, m, on Process, myid call mpi_comm_free(mycomm, ierr) c c Remove MPI parallel environment. c call mpi_finalize(ierr) end C MPI GET COUNT int MPI_Get_count( MPI_Status status, MPI_Datatype datatype, int *count) Fortran MPI_GET_COUNT( STATUS, DATATYPE, COUNT, IERROR ) INTEGER STATUS(MPI_STATUS_SIZE), DATATYPE, COUNT, IERROR

50 76 MPI IN STATUS,. IN DATATYPE,. OUT COUNT,.,,. C Fortran MPI SENDRECV int MPI_Sendrecv( void *sendbuf, int sendcount, MPI_Datatype sendtype, int dest, int sendtag, void *recvbuf, int recvcount, MPI_Datatype recvtype, int source, int recvtag, MPI_Comm comm, MPI_Status *status ) MPI_SENDRECV(SENDBUF, SENDCOUNT, SENDTYPE, DEST, SENDTAG, RECVBUF, RECVCOUNT, RECVTYPE, SOURCE, RECVTAG, COMM, STATUS, IERROR ) <type> SENDBUF(*), RECVBUF(*) INTEGER SENDCOUNT, SENDTYPE, DEST, SENDTAG, RECVCOUNT, RECVTYPE SOURCE, RECVTAG, COMM, STATUS(MPI_STATUS_SIZE), IERROR IN SENDBUF,. IN SENDCOUNT,. IN SENDTYPE,. IN DEST,. IN SENDTAG,. OUT RECVBUF,. IN RECVCOUNT,. IN RECVTYPE,. IN SOURCE,. IN RECVTAG,. IN COMM,. OUT STATUS,., : call mpi_sendrecv(m, 1, mpi_integer, next, 1, n, 1, mpi_integer, & front, 1, mycomm, status, ierr) m = n MPI_SENDRECV,,.

51 C Fortran MPI SENDRECV REPLACE int MPI_Sendrecv_replace(void *sendbuf, int count, MPI_Datatype datatype, int dest, int sendtag, int source, int recvtag, MPI_Comm comm, MPI_Status *status ) MPI_SENDRECV_REPLACE(BUF, COUNT, DATATYPE, DEST, SENDTAG, SOURCE, RECVTAG, COMM, STATUS, IERROR ) <type> BUF(*) INTEGER COUNT, DATATYPE, DEST, SENDTAG, SOURCE, RECVTAG, COMM, STATUS(MPI_STATUS_SIZE), IERROR OUT BUF,. IN COUNT,. IN DATATYPE,. IN DEST,. IN SENDTAG,. IN SOURCE,. IN RECVTAG,. IN COMM,. OUT STATUS,., BUF : & call mpi_sendrecv_replace(m, 1, mpi_integer, next, 1, front, 1, mycomm, status, ierr), MPI_SENDRECV_REPLACE., MPI MPI_PROC_NULL, , p 1 0 m, :... if(myid.eq. 0) front = MPI_PROC_NULL if(myid.eq. p-1) next = MPI_PROC_NULL call mpi_sendrecv_replace(m, 1, mpi_integer, next, 1, & front, 1, mycomm, status, ierr) MPI PROBE C int MPI_Probe(int source, int tag, MPI_Comm comm, MPI_Status *status ) Fortran MPI_PROBE( SOURCE, TAG, COMM, STATUS, IERROR ) INTEGER SOURCE, TAG, COMM, STATUS(MPI_STATUS_SIZE), IERROR IN SOURCE,. IN TAG,. IN COMM,. OUT STATUS,.

52 78 MPI MPI_IPROBE, MPI_PROBE. MPI IPROBE C int MPI_Iprobe(int source, int tag, MPI_Comm comm, int* flag, MPI_Status *status ) Fortran MPI_IPROBE( SOURCE, TAG, COMM, FLAG, STATUS, IERROR ) LOGICAL FLAG INTEGER SOURCE, TAG, COMM, STATUS(MPI_STATUS_SIZE), IERROR IN SOURCE,. IN TAG,. IN COMM,. OUT FLAG,, FLAG TRUE. OUT STATUS,. SOURCE, TAG COMM,. SOURCE TAG MPI_RECV, (wildcard) MPI_ANY_SOURCE MPI_ANY_TAG. 5.7,,.,,. MPI ISEND C int MPI_Isend( void* buf, int count, MPI_Datatype datatype, int dest, int tag, MPI_Comm comm, MPI_Request *request ) Fortran MPI_ISEND(BUF, COUNT, DATATYPE, DEST, TAG, COMM, REQUEST, IERROR ) <type> BUF(*) INTEGER COUNT, DATATYPE, DEST, TAG, COMM, REQUEST, IERROR IN BUF,. IN COUNT,. IN DATATYPE,. IN DEST,. IN TAG,. IN COMM,. OUT REQUEST,.

53 MPI IRECV C int MPI_Irecv( void* buf, int count, MPI_Datatype datatype, int source, int tag, MPI_Comm comm, MPI_Request *request ) Fortran MPI_IRECV(BUF, COUNT, DATATYPE, SOURCE, TAG, COMM, REQUEST, IERROR ) <type> BUF(*) INTEGER COUNT, DATATYPE, SOURCE, TAG, COMM, REQUEST, IERROR OUT BUF,. IN COUNT,. IN DATATYPE,. IN SOURCE,. IN TAG,. IN COMM,. OUT REQUEST,. MPI_ISEND MPI_IRECV,, : MPI WAIT C int MPI_Wait( MPI_Request *request, MPI_Status *status ) Fortran MPI_WAIT( REQUEST, STATUS, IERROR ) INTEGER REQUEST, STATUS(MPI_STATUS_SIZE), IERROR INOUT REQUEST,. OUT STATUS,. REQUEST, MPI_WAIT. MPI TEST C int MPI_Test( MPI_Request *request, int* flag, MPI_Status *status ) Fortran MPI_TEST( REQUEST, FLAG, STATUS, IERROR ) LOGICAL FLAG INTEGER REQUEST, STATUS(MPI_STATUS_SIZE), IERROR INOUT REQUEST,. OUT FLAG, REQUEST TRUE. OUT STATUS,. STATUS, STATUS MPI_TEST_CANCELLED.

54 80 MPI MPI REQUEST FREE C int MPI_Request_free( MPI_Request *request ) Fortran MPI_REQUEST_FREE( REQUEST, IERROR ) INTEGER REQUEST, IERROR INOUT REQUEST,, MPI_REQUEST_NULL. REQUEST,,. MPI WAITANY C int MPI_Waitany( int count, MPI_Request *array_of_requests, int *index, MPI_Status *status ) Fortran MPI_WAITANY( COUNT, ARRAY_OF_REQUESTS, INDEX, STATUS, IERROR ) INTEGER COUNT, ARRAY_OF_REQUESTS(*), INDEX, STATUS(MPI_STATUS_SIZE), IERROR IN COUNT,. INOUT ARRAY_OF_REQUESTS,. OUT INDEX,. OUT STATUS,.,,, MPI_WAITANY. MPI TESTANY C int MPI_Testany( int count, MPI_Request *array_of_requests, int *index, int *flag, MPI_Status *status ) Fortran MPI_TESTANY( COUNT, ARRAY_OF_REQUESTS, INDEX, FLAG, STATUS, IERROR ) LOGICAL FLAG INTEGER COUNT, ARRAY_OF_REQUESTS(*), INDEX, STATUS(MPI_STATUS_SIZE), IERROR IN COUNT,. INOUT ARRAY_OF_REQUESTS,. OUT INDEX,. OUT FLAG,, FLAG=TRUE. OUT STATUS,..

55 MPI WAITALL C int MPI_Waitall( int count, MPI_Request *array_of_requests, MPI_Status *array_of_statuses ) Fortran MPI_WAITALL( COUNT, ARRAY_OF_REQUESTS, ARRAY_OF_STATUSES, IERROR ) INTEGER COUNT, ARRAY_OF_REQUESTS(*), ARRAY_OF_STATUSES(MPI_STATUS_SIZE, *), IERROR IN COUNT,. INOUT ARRAY_OF_REQUESTS,. INOUT ARRAY_OF_STATUSES,.,. MPI TESTALL C int MPI_Testall( int count, MPI_Request *array_of_requests, int *flag, MPI_Status *array_of_statuses ) Fortran MPI_TESTALL( COUNT, ARRAY_OF_REQUESTS, FLAG, ARRAY_OF_STATUSES, IERROR ) LOGICAL FLAG INTEGER COUNT, ARRAY_OF_REQUESTS(*), ARRAY_OF_STATUSES(MPI_STATUS_SIZE, *), IERROR IN COUNT,. INOUT ARRAY_OF_REQUESTS,. OUT FLAG,, FLAG=FALSE. INOUT ARRAY_OF_STATUSES,.. MPI CANCEL C int MPI_Cancel( MPI_Request *request ) Fortran MPI_CANCEL( REQUEST, IERROR ) INTEGER REQUEST, IERROR INOUT REQUEST,., MPI_WAIT MPI_TEST,. REQUEST, :

56 82 MPI MPI TEST CANCELLED C int MPI_Test_cancelled( MPI_Status *status, int *flag ) Fortran MPI_TEST_CANCELLED( STATUS, FLAG, IERROR ) LOGICAL FLAG INTEGER REQUEST, IERROR IN STATUS,. OUT FLAG,, FLAG=TRUE. 5.8 MPI SEND INIT C int MPI_Send_init(void* buf, int count, MPI_Datatype datatype, int dest, int tag, MPI_Comm comm, MPI_Request *request ) Fortran MPI_SEND_INIT(BUF, COUNT, DATATYPE, DEST, TAG, COMM, REQUEST, IERROR ) <type> BUF(*) INTEGER COUNT, DATATYPE, DEST, TAG, COMM, REQUEST, IERROR IN BUF,. IN COUNT,. IN DATATYPE,. IN DEST,. IN TAG,. IN COMM,. OUT REQUEST,. MPI RECV INIT C int MPI_Recv_init(void* buf, int count, MPI_Datatype datatype,int source, int tag, MPI_Comm comm, MPI_Request *request ) Fortran MPI_RECV_INIT(BUF, COUNT, DATATYPE, SOURCE, TAG, COMM, REQUEST, IERROR ) <type> BUF(*) INTEGER COUNT, DATATYPE, SOURCE, TAG, COMM, REQUEST, IERROR OUT BUF,. IN COUNT,. IN DATATYPE,. IN SOURCE,. IN TAG,. IN COMM,. OUT REQUEST,.

57 MPI_SEND_INIT MPI_RECV_INIT,,,., MPI_START MPI_STARTALL. MPI START C int MPI_Start( MPI_Request *request ) Fortran MPI_START( REQUEST, IERROR ) INTEGER REQUEST, IERROR INOUT REQUEST,. MPI_START, MPI_SEND_INIT MPI_RECV_INIT. MPI STARTALL C int MPI_Startall( int count, MPI_Request *array_of_request ) Fortran MPI_STARTALL( COUNT, ARRAY_OF_REQUEST, IERROR ) INTEGER COUNT, ARRAY_OF_REQUEST, IERROR IN COUNT,. INOUT ARRAY_OF_REQUEST, , m,, : program mring include mpif.h c integer iter parameter(iter = 3) integer myid, p, mycomm, ierr, m, status(mpi_status_size, 2), & next, front, mod, n, i, requests(2) c c call mpi_init( ierr ) call mpi_comm_dup( mpi_comm_world, mycomm, ierr) call mpi_comm_rank( mycomm, myid, ierr ) call mpi_comm_size( mycomm, p, ierr ) print *, Process, myid, of, p, is running m = myid front = mod(p+myid-1, p) next = mod(myid+1, p) call mpi_send_init(m, 1, mpi_integer, next, 1, mycomm, & requests(1), ierr) call mpi_recv_init(n, 1, mpi_integer, front, 1, mycomm, & requests(2), ierr) do 10 i=1, iter

58 84 MPI call mpi_startall(2, requests, ierr) call mpi_waitall(2, requests, status, ierr) m = n 10 continue print *, The value of m is, m, on Process, myid c call mpi_comm_free(mycomm, ierr) call mpi_finalize(ierr) stop end 5.9 MPI, MPI, (buffered-mode) (synchronous-mode), MPI (ready-mode).,.,,. MPI_BSEND,,,,,. MPI_SSEND,,,. MPI_RSEND,,. MPI_IBSEND, MPI_BSEND. MPI_ISSEND, MPI_SSEND. MPI_IRSEND, MPI_RSEND. MPI_BSEND_INIT, MPI_SEND_INIT. MPI_SSEND_INIT, MPI_SEND_INIT. MPI_RSEND_INIT, MPI_SEND_INIT., : MPI BUFFER ATTACH C int MPI_Buffer_attach( void *buffer, int size ) Fortran MPI_BUFFER_ATTACH( BUFFER, SIZE, IERROR ) <TYPE> BUFFER(*) INTEGER SIZE, IERROR IN BUFFER,. IN SIZE,..

59 MPI BUFFER DETACH C int MPI_Buffer_detach( void *buffer, int *size ) Fortran MPI_BUFFER_DETACH( BUFFER, SIZE, IERROR ) <TYPE> BUFFER(*) INTEGER SIZE, IERROR OUT BUFFER,. OUT SIZE, MPI C,, MPI,., : ; (displacement)., (type map). : T ypemap = {(type 0, disp 0 ),..., (type n 1, disp n 1 )}, disp i type i, i = 0,..., n 1.. MPI TYPE CONTIGUOUS C int MPI_Type_contiguous( int count, MPI_Datatype oldtype, MPI_Datatype *newtype ) Fortran MPI_TYPE_CONTIGUOUS( COUNT, OLDTYPE, NEWTYPE, IERROR ) INTEGER COUNT, OLDTYPE, NEWTYPE, IERROR IN COUNT, OLDTYPE. IN OLDTYPE,. OUT NEWTYPE,., COUNT,. COUNT=5, OLDTYPE=INTEGER, NEWTYPE 5.

60 86 MPI MPI TYPE VECTOR C int MPI_Type_vector( int count, int blocklength, int stride, MPI_Datatype oldtype, MPI_Datatype *newtype ) Fortran MPI_TYPE_VECTOR( COUNT, BLOCKLENGTH, STRIDE, OLDTYPE, NEWTYPE, IERROR ) INTEGER COUNT, BLOCKLENGTH, STRIDE, OLDTYPE, NEWTYPE, IERROR IN COUNT, OLDTYPE. IN BLOCKLENGTH,. IN STRIDE, OLDTYPE. IN OLDTYPE,. OUT NEWTYPE,. COUNT=3, BLOCKLENGTH=2, STRIDE=3, OLDTYPE=INTEGER, NEWTYPE , 2, 2. MPI TYPE HVECTOR C int MPI_Type_hvector( int count, int blocklength, int stride, MPI_Datatype oldtype, MPI_Datatype *newtype ) Fortran MPI_TYPE_HVECTOR( COUNT, BLOCKLENGTH, STRIDE, OLDTYPE, NEWTYPE, IERROR ) INTEGER COUNT, BLOCKLENGTH, STRIDE, OLDTYPE, NEWTYPE, IERROR IN COUNT, OLDTYPE. IN BLOCKLENGTH,. IN STRIDE,. IN OLDTYPE,. OUT NEWTYPE,. MPI_TYPE_VECTOR,,. C Fortran MPI TYPE INDEXED int MPI_Type_indexed( int count, int *array_of_blocklengths, int *array_of_displacements, MPI_Datatype oldtype, MPI_Datatype *newtype ) MPI_TYPE_INDEXED( COUNT, ARRAY_OF_BLOCKLENGTHS, ARRAY_OF_DISPLACEMENTS, OLDTYPE, NEWTYPE, IERROR ) INTEGER COUNT, ARRAY_OF_BLOCKLENGTHS(*), ARRAY_OF_DISPLACEMENTS(*), OLDTYPE, NEWTYPE, IERROR

61 IN COUNT, OLDTYPE. IN ARRAY_OF_BLOCKLENGTHS,. IN ARRAY_OF_DISPLACEMENTS,. IN OLDTYPE,. OUT NEWTYPE,. ARRAY_OF_BLOCKLENGTHS ARRAY_OF_DISPLACEMENTS OLDTYPE, ARRAY_OF_DISPLACEMENTS. MPI_TYPE_VECTOR,. MPI_TYPE_VECTOR( COUNT, BLOCKLENGTH, STRIDE, OLDTYPE, NEWTYPE, IERROR ) MPI_TYPE_INDEXED( COUNT, LENGTHS, DISPLACEMENTS, OLDTYPE, NEWTYPE, IERROR ) LENGTHS(i)=BLOCKLENGTH, DISPLACEMENTS(i)=i*STRIDE, i=0,..., COUNT-1. MPI_TYPE_INDEXED OLDTYPE,. C Fortran MPI TYPE HINDEXED int MPI_Type_hindexed( int count, int *array_of_blocklengths, int *array_of_displacements, MPI_Datatype oldtype, MPI_Datatype *newtype ) MPI_TYPE_HINDEXED( COUNT, ARRAY_OF_BLOCKLENGTHS, ARRAY_OF_DISPLACEMENTS, OLDTYPE, NEWTYPE, IERROR ) INTEGER COUNT, ARRAY_OF_BLOCKLENGTHS(*), ARRAY_OF_DISPLACEMENTS(*), OLDTYPE, NEWTYPE, IERROR IN COUNT, OLDTYPE. IN ARRAY_OF_BLOCKLENGTHS,. IN ARRAY_OF_DISPLACEMENTS,. IN OLDTYPE,. OUT NEWTYPE,.,. C Fortran MPI TYPE STRUCT int MPI_Type_struct( int count, int *array_of_blocklengths, int *array_of_displacements, MPI_Datatype *array_of_types, MPI_Datatype *newtype) MPI_TYPE_STRUCT( COUNT, ARRAY_OF_BLOCKLENGTHS, ARRAY_OF_DISPLACEMENTS, ARRAY_OF_TYPES, NEWTYPE, IERROR ) INTEGER COUNT, ARRAY_OF_BLOCKLENGTHS(*), ARRAY_OF_DISPLACEMENTS(*), ARRAY_OF_TYPES(*), NEWTYPE, IERROR

62 88 MPI IN COUNT,. IN ARRAY_OF_BLOCKLENGTHS,. IN ARRAY_OF_DISPLACEMENTS,. IN ARRAY_OF_TYPES,. OUT NEWTYPE,. MPI, MPI_TYPE_STRUCT,.,,,,. MPI TYPE COMMIT C int MPI_Type_commit( MPI_Datatype *datatype ) Fortran MPI_TYPE_COMMIT( DATATYPE, IERROR ) INTEGER DATATYPE, IERROR INOUT DATATYPE,. MPI, MPI_TYPE_COMMIT. MPI TYPE FREE C int MPI_Type_free( MPI_Datatype *datatype ) Fortran MPI_TYPE_FREE( DATATYPE, IERROR ) INTEGER DATATYPE, IERROR INOUT DATATYPE,. MPI TYPE EXTENT C int MPI_Type_extent( MPI_Datatype datatype, MPI_Aint *extent ) Fortran MPI_TYPE_EXTENT( DATATYPE, EXTENT, IERROR ) INTEGER DATATYPE, EXTENT, IERROR IN DATATYPE,. OUT EXTENT,. MPI ADDRESS C int MPI_Address( void* location, MPI_Aint *address ) Fortran MPI_ADDRESS( LOCATION, ADDRESS, IERROR ) <type> LOCATION(*) INTEGER ADDRESS, IERROR

63 IN LOCATION,. OUT ADDRESS,. MPI_TYPE_EXTENT MPI_ADDRESS, C,, MPI. #include <stdio.h> #include "mpi.h" typedef struct { double value; char str; } data; void main(argc, argv) int argc; char **argv; { int p, myid, lens[3]={1, 1, 1}, i; MPI_Comm mycomm; data tst[3]; MPI_Datatype new, type[3] = {MPI_DOUBLE, MPI_CHAR, MPI_UB}; MPI_Aint disp[3], size; MPI_Status status; } MPI_Init( &argc, &argv ); MPI_Comm_dup( MPI_COMM_WORLD, &mycomm); MPI_Comm_rank( mycomm, &myid); MPI_Comm_size( mycomm, &p); if (myid == 0) for(i=0; i<3; i++) { tst[i].value = 1.0; tst[i].str= a ; } printf("\nthe Process %1d of %1d is running.\n", myid, p); MPI_Address(&tst[0], &disp[0]); MPI_Address(&tst[0].str, &disp[1]); for (i=2; i>=0; i--) disp[i] -= disp[0]; MPI_Type_struct(3, lens, disp, type, &new); MPI_Type_commit(&new); MPI_Type_extent(new, &size); if(myid == 0) printf("\nthe Extent of Struct Data Type is %d.\n", size); if(myid == 0) MPI_Ssend(tst, 3, new, 1, 1, mycomm); else if (myid == 1) MPI_Recv(tst, 3, new, 0, 1, mycomm, &status); MPI_Type_free(&new); if(myid == 1) printf("\nthe values are %f and %c\n", tst[1].value, tst[1].str); MPI_Comm_free(&mycomm); MPI_Finalize(); C, MPI,. MPI (pseudo datatype) MPI_UB, MPI MPI_LB. MPI +new+ MPI_TYPE_EXTENT 4 8, 4, 12, 16. MPI_TYPE_SIZE, 9.

64 90 MPI MPI TYPE SIZE C int MPI_Type_size( MPI_Datatype datatype, int *size ) Fortran MPI_TYPE_SIZE( DATATYPE, SIZE, IERROR ) INTEGER DATATYPE, SIZE, IERROR IN DATATYPE,. OUT SIZE,. MPI GET ELEMENTS C int MPI_Get_elements(MPI_Status *status,mpi_datatype datatype,int *count) Fortran MPI_GET_ELEMENTS( STATUS, DATATYPE, COUNT, IERROR ) INTEGER STATUS(MPI_STATUS_SIZE), DATATYPE, COUNT, IERROR IN STATUS,. IN DATATYPE,. OUT COUNT, MPI. MPI_GET_COUNT, MPI_GET_COUNT, MPI_GET_ELEMENTS MPI. MPI, C MPI_INT (Fortran MPI_INTEGER), MPI_GET_COUNT MPI_GET_ELEMENTS..... call mpi_type_contiguous(2, mpi_integer, type, ierr) call mpi_type_commit(type, ierr) if(myid.eq. 0.and. p.gt. 1) then call mpi_send(ia, 3, mpi_integer, 1, 1, mycomm, ierr) elseif(myid.eq. 1) then call mpi_recv(ia, 2, type, 0, 1, mycomm, status, ierr) call mpi_get_count(status, type, count1, ierr) call mpi_get_elements(status, type, count2, ierr) endif...,,, MPI_GET_COUNT., MPI,, MPI,, MPI MPI_BOTTOM, MPI, MPI PVM (Parallel Virtual Machine), MPI MPI..

65 5.12 MPI 91 C Fortran MPI PACK int MPI_Pack( void* inbuf, int incount, MPI_Datatype datatype, void* outbuf, int outsize, int *position, MPI_Comm comm ) MPI_PACK( INBUF, INCOUNT, DATATYPE, OUTBUF, OUTSIZE, POSITION, COMM, IERROR ) <type> INBUF(*), OUTBUF(*) INTEGER INCOUNT, DATATYPE, OUTSIZE, POSITION, COMM, IERROR IN INBUF,. IN INCOUNT,. IN DATATYPE,. OUT OUTBUF,. IN OUTSIZE,. INOUT POSITION,. IN COMM,. C Fortran MPI UNPACK int MPI_Unpack( void* inbuf, int insize, int *position, void* outbuf, int outcount, MPI_Datatype datatype, MPI_Comm comm ) MPI_UNPACK( INBUF, INSIZE, POSITION, OUTBUF, OUTCOUNT, DATATYPE, COMM, IERROR ) <type> INBUF(*), OUTBUF(*) INTEGER INSIZE, POSITION, OUTCOUNT, DATATYPE, COMM, IERROR IN INBUF,. IN INSIZE,. INOUT POSITION,. OUT OUTBUF,. IN OUTCOUNT,. IN DATATYPE,. IN COMM,., MPI_PACKED., MPI : MPI PACK SIZE C int MPI_Pack_size( int incount, MPI_Datatype datatype, MPI_Comm comm, int *size ) Fortran MPI_PACK_SIZE( INCOUNT, DATATYPE, COMM, SIZE, IERROR ) INTEGER INCOUNT, DATATYPE, COMM, SIZE, IERROR

66 92 MPI IN INCOUNT,. IN DATATYPE,. IN COMM,. OUT SIZE,. SIZE,. MPI_PACK MPI_UNPACK, ,,. : program pack include mpif.h c integer maxbuf, len parameter (maxbuf = 200, len = 10) integer myid, p, mycomm, ierr, status(mpi_status_size, 2), & ia(len), count1, count2, i, pos real*8 a(len) character buf(maxbuf) c call mpi_init( ierr ) call mpi_comm_dup( mpi_comm_world, mycomm, ierr) call mpi_comm_rank( mycomm, myid, ierr ) call mpi_comm_size( mycomm, p, ierr ) print *, Process, myid, of, p, is running c if(myid.eq. 0) then do 10 i=1, len ia(i) = i a(i) = dble(i) 10 continue endif call mpi_pack_size(len, mpi_integer, mycomm, count1, ierr) call mpi_pack_size(len, mpi_double_precision, mycomm, count2, i) print *, The pack size of 10 integer and real8 are:, & count1, count2 pos = 1 if(myid.eq. 0.and. p.gt. 1) then call mpi_pack(ia, len, mpi_integer, buf, maxbuf, pos, & mycomm, ierr) call mpi_pack(a, len, mpi_double_precision, buf, maxbuf, pos, & mycomm, ierr) call mpi_send(buf, pos-1, mpi_packed, 1, 1, mycomm, ierr) elseif(myid.eq. 1) then call mpi_recv(buf, maxbuf, mpi_packed, 0, 1, mycomm, & status, ierr) call mpi_unpack(buf, maxbuf, pos, ia, len, mpi_integer, & mycomm, ierr)

67 call mpi_unpack(buf, maxbuf, pos, a, len, mpi_double_precision, & mycomm, ierr) print *, The received values are:, a(1), a(2), a(3) endif call mpi_comm_free(mycomm, ierr) call mpi_finalize(ierr) stop end 5.13 (collective communication) ( 2).,, (barrier synchronization) (global communication functions) (global reduction operations) MPI BARRIER C int MPI_Barrier( MPI_Comm comm ) Fortran MPI_BARRIER( COMM, IERROR ) INTEGER COMM, IERROR IN COMM,. MPI, COMM,,., MPI BCAST C int MPI_Bcast( void* buffer, int count, MPI_Datatype datatype, int root, MPI_Comm comm ) Fortran MPI_BCAST( BUFFER, COUNT, DATATYPE, ROOT, COMM, IERROR ) <type> BUFFER(*) INTEGER COUNT, DATATYPE, ROOT, COMM, IERROR INOUT BUFFER,. IN COUNT,. IN DATATYPE,. IN ROOT,. IN COMM,.

68 94 MPI COMM, ROOT, BUFFER COMM., MPI. C Fortran MPI GATHER int MPI_Gather( void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, int root, MPI_Comm comm ) MPI_GATHER( SENDBUF, SENDCOUNT, SENDTYPE, RECVBUF, RECVCOUNT, RECVTYPE, ROOT, COMM, IERROR ) <type> SENDBUF(*), RECVBUF(*) INTEGER SENDCOUNT, SENDTYPE, RECVCOUNT, RECVTYPE, ROOT, COMM, IERROR IN SENDBUF,. IN SENDCOUNT,. IN SENDTYPE,. OUT RECVBUF,. IN RECVCOUNT,. IN RECVTYPE,. IN ROOT,. IN COMM,. MPI (gather), COMM ( ROOT) ROOT, RECVBUF. ROOT, RECVBUF. C Fortran MPI GATHERV int MPI_Gatherv( void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, int *recvcounts, int *displs, MPI_Datatype recvtype, int root, MPI_Comm comm ) MPI_GATHERV( SENDBUF, SENDCOUNT, SENDTYPE, RECVBUF, RECVCOUNTS, DISPLS, RECVTYPE, ROOT, COMM, IERROR ) <type> SENDBUF(*), RECVBUF(*) INTEGER SENDCOUNT, SENDTYPE, RECVCOUNTS(*), DISPLS(*), RECVTYPE, ROOT, COMM, IERROR

69 IN SENDBUF,. IN SENDCOUNT,. IN SENDTYPE,. OUT RECVBUF,. IN RECVCOUNTS,. IN DISPLS,. IN RECVTYPE,. IN ROOT,. IN COMM,. MPI_GATHER,,, MPI_GATHER. MPI : , ROOT,,. program gather include mpif.h c integer maxbuf, len, mp parameter (maxbuf = 200, len = 10, mp = 5) integer myid, p, mycomm, ierr, root, ia(len), iga(maxbuf), i, & displs(mp), counts(mp) c call mpi_init( ierr ) call mpi_comm_dup( mpi_comm_world, mycomm, ierr) call mpi_comm_rank( mycomm, myid, ierr ) call mpi_comm_size( mycomm, p, ierr ) print *, Process, myid, of, p, is running c do 10 i=1, len ia(i) = i+myid*len 10 continue c do 20 i=1, p displs(i) = 20*i-20 counts(i) = len 20 continue root = 0 c c c call mpi_gather(ia, len, mpi_integer, iga, len, mpi_integer, & root, mycomm, ierr) if(myid.eq. root) then print *, The gather values are:, & iga(1), iga(len+1), iga(2*len+1) endif call mpi_gatherv(ia, len, mpi_integer, iga, counts, displs,

70 96 MPI c c & mpi_integer, root, mycomm, ierr) if(myid.eq. root) then print *, The gatherv values are:, & iga(2), iga(2*len+2), iga(4*len+2) endif call mpi_comm_free(mycomm, ierr) call mpi_finalize(ierr) stop end : 1, 11, 21 2, 12, 22. MPI_GATHER, MPI_GATHERV,, COUNTS, 20. C Fortran MPI SCATTER int MPI_Scatter( void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, int root, MPI_Comm comm ) MPI_SCATTER( SENDBUF, SENDCOUNT, SENDTYPE, RECVBUF, RECVCOUNT, RECVTYPE, ROOT, COMM, IERROR ) <type> SENDBUF(*), RECVBUF(*) INTEGER SENDCOUNT, SENDTYPE, RECVCOUNT, RECVTYPE, ROOT, COMM, IERROR IN SENDBUF,. IN SENDCOUNT,. IN SENDTYPE,. OUT RECVBUF,. IN RECVCOUNT,. IN RECVTYPE,. IN ROOT,. IN COMM,. MPI (scatter), MPI_GATHER. C Fortran MPI SCATTERV int MPI_Scatterv( void* sendbuf, int *sendcounts, int *displs, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, int root, MPI_Comm comm ) MPI_SCATTERV( SENDBUF, SENDCOUNTS, DISPLS, SENDTYPE, RECVBUF, RECVCOUNT, RECVTYPE, ROOT, COMM, IERROR ) <type> SENDBUF(*), RECVBUF(*) INTEGER SENDCOUNTS(*), DISPLS(*), SENDTYPE, RECVCOUNT, RECVTYPE, ROOT, COMM, IERROR

71 IN SENDBUF,. IN SENDCOUNTS,. IN DISPLS,. IN SENDTYPE,. OUT RECVBUF,. IN RECVCOUNT,. IN RECVTYPE,. IN ROOT,. IN COMM, C Fortran MPI ALLGATHER int MPI_Allgather( void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, MPI_Comm comm ) MPI_ALLGATHER( SENDBUF, SENDCOUNT, SENDTYPE, RECVBUF, RECVCOUNT, RECVTYPE, COMM, IERROR ) <type> SENDBUF(*), RECVBUF(*) INTEGER SENDCOUNT, SENDTYPE, RECVCOUNT, RECVTYPE, COMM, IERROR IN SENDBUF,. IN SENDCOUNT,. IN SENDTYPE,. OUT RECVBUF,. IN RECVCOUNT,. IN RECVTYPE,. IN COMM,. C Fortran MPI ALLGATHERV int MPI_Allgatherv( void* sendbuf, int sendcount, MPI_Datatype sendtype, void* recvbuf, int recvcount, MPI_Datatype recvtype, MPI_Comm comm ) MPI_ALLGATHERV( SENDBUF, SENDCOUNT, SENDTYPE, RECVBUF, RECVCOUNT, RECVTYPE, COMM, IERROR ) <type> SENDBUF(*), RECVBUF(*) INTEGER SENDCOUNT, SENDTYPE, RECVCOUNT, RECVTYPE, COMM, IERROR