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Constraints:

• SHADOW_START directive must be executed after SHADOW_GROUP one.

• SHADOW_WAITT directive must be executed after SHADOW_START one.

• Updated values of the shadow edges may be used only after SHADOW_WAIT directive.

• These directives may not be used within a parallel loop.

6.3.1. REMOTE_ACCESS Directive

6.3.3. Asynchronous specification of REMOTE type references

Constraint:

• The identifier, specified in the directive, can be used only in REMOTE_ACCESS, PREFETCH and RESET directives.

Constraints.

• Repeated usage of PREFETCH directive is correct, if the remote reference group characteristics (the loop parameters, distribution of arrays and the values of index expressions in remote references) are not updated.

• PREFETCH directive can be executed for the several loops (several REMOTE_ACCESS directives), if there are no data dependencies between the loops for distributed arrays specified in the REMOTE_ACCESS directives.

6.4. Asynchronous group reduction

Constraints.

• Before executing REDUCTION_START directive, the reduction variables of the group may be used in reduction statements of parallel loops only.

• REDUCTION_START and REDUCTION_WAIT directives must be executed after the completion of the loop (loops) where the values of the reduction variables are calculated. The only statements allowed between these directives are those that are not using the values of the reduction variables.

• REDUCTION_WAIT directive deletes the group of the reduction operations.

7.1. Description of task array

7.2. Mapping tasks on processors. MAP directive

7.4. Distribution of computations. TASK_REGION directive

9. Procedures

Annex2. Code examples

Seven small scientific programs are presented to illustrate Fortran DVM language features. They are intended for solving a system of linear equations:

A x = b

where A - matrix of coefficients,

b - vector of free members,

x - vector of unknowns.

The following basic methods are used for solving this system.

Direct methods. The well-known Gaussian Elimination method is the most commonly used algorithm of this class. The main idea of this algorithm is to reduce the matrix A to upper triangular form and then to use backward substitution to diagonalize the matrix.

Explicit iteration methods. Jacobi Relaxation is the most known algorithm of this class. The algorithm perform the following computation iteratively

x_i,j^new = (x_i-1,j^old + x_i,j-1^old + x_i+1,j^old + x_i,j+1^old ) / 4

Implicit iteration methods. Successive Over Relaxation (SOR) refers to this class. The algorithm performs the following calculation iteratively

x_i,j^new = ( w / 4 ) * (x_i-1,j^new + x_i,j-1^new + x_i+1,j^old + x_i,j+1^old ) + (1-w) * x_i,j^old

By using «red-black» coloring of variables each step of SOR consists of two half Jacobi steps. One processes «red»variables and the other processes «black» variables. Coloring of variables allows to overlap calculation and communication.

Example 1. Gauss elimination algorithm

PROGRAM GAUSS

C Solving linear equation system A x = b

PARAMETER ( N = 100 )

REAL A( N, N+1 ), X( N )

C A : Coefficient matrix with dimension (N,N+1).

C Right hand side vector of linear equations is stored

C into last column (N+1)-th, of matrix A

C X : Unknown vector

C N : Number of linear equations

CDVM$ DISTRIBUTE A ( BLOCK, *)

CDVM$ ALIGN X(I) WITH A(I, N+1)

C

C Initialization

C

*DVM$ PARALLEL ( I ) ON A( I , * )

DO 100 I = 1, N

DO 100 J = 1, N+1

IF (( I .EQ. J ) THEN

A( I, J ) = 2.0

ELSE

IF ( J .EQ. N+1) THEN

A( I, J ) = 0.0

ENDIF

ENDIF

100 CONTINUE

C

C Elimination

C

DO 1 I = 1, N

C the I-th row of array A will be buffered before

C execution of I-th iteration, and references A(I,K), A(I, I)

C will be replaced with corresponding reference to buffer

*DVM$ PARALLEL ( J ) ON A( J, * ) , REMOTE_ACCESS (A ( I, : ))

DO 5 J = I+1, N

DO 5 K = I+1, N+1

A( J, K ) = A( J, K ) - A( J, I ) * A( I, K ) / A( I, I )

5 CONTINUE

1 CONTINUE

C First calculate X(N)

X( N ) = A( N, N+1 ) / A( N, N )

C

C Solve X(N-1), X(N-2), ...,X(1) by backward substitution

C

DO 6 J = N-1, 1, -1

C the (J+1)-th elements of array X will be buffered before

C execution of J-th iteration, and reference X(J+1)

C will be replaced with reference to temporal variable

*DVM$ PARALLEL ( I ) ON A( I , * ) , REMOTE_ACCESS ( X( J+1 ))

DO 7 I = 1, J

A( I, N+1 ) = A( I, N+1 ) - A( I, J+1 ) * X( J+1 )

7 CONTINUE

X( J ) = A( J, N+1 ) / A( J, J)

6 CONTINUE

PRINT *, X

END

Example 2. Jacobi algorithm

PROGRAM JACOB

PARAMETER (K=8, ITMAX=20)

REAL A(K,K), B(K,K), EPS, MAXEPS

CDVM$ DISTRIBUTE A ( BLOCK, BLOCK)

CDVM$ ALIGN B( I, J ) WITH A( I, J )

C arrays A and B with block distribution

PRINT *, '********** TEST_JACOBI **********'

MAXEPS = 0.5E - 7

CDVM$ PARALLEL (J,I) ON A(I, J)

C nest of two parallel loops, iteration (i,j) will be executed on

C processor, which is owner of element A(i,j)

DO 1 J = 1, K

DO 1 I = 1, K

A(I, J) = 0.

IF(I.EQ.1 .OR. J.EQ.1 .OR. I.EQ.K .OR. J.EQ.K) THEN

B(I, J) = 0.

ELSE

B(I, J) = 1. + I + J

ENDIF

1 CONTINUE

DO 2 IT = 1, ITMAX

EPS = 0.

CDVM$ PARALLEL (J, I) ON A(I, J), REDUCTION ( MAX( EPS ))

C variable EPS is used for calculation of maximum value

DO 21 J = 2, K-1

DO 21 I = 2, K-1

EPS = MAX ( EPS, ABS( B( I, J) - A( I, J)))

A(I, J) = B(I, J)

21 CONTINUE

CDVM$ PARALLEL (J, I) ON B(I, J), SHADOW_RENEW (A)

C copying shadow elements of array A from

C neighboring processors before loop execution

DO 22 J = 2, K-1

DO 22 I = 2, K-1

B(I, J) = (A( I-1, J ) + A( I, J-1 ) + A( I+1, J) + A( I, J+1 )) / 4

22 CONTINUE

PRINT *, 'IT = ', IT, ' EPS = ', EPS

IF ( EPS . LT . MAXEPS ) GO TO 3

Характеристики

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