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Figure 10: Time in seconds and speed-up w.r.t 1 processor. This is a problem of moderate size. MGS and COL are the more slowest methods. In addition their speed-ups are very poor. ROC and ROD perform slightly better. They achieve a speed-up of about 8 with 24 processors but they didn’t go further with more processors.

Figure 10 Time in seconds and speed-up w.r.t 1 processor. This is a problem of moderate size. MGS and COL are the more slowest methods. In addition their speed-ups are very poor. ROC and ROD perform slightly better. They achieve a speed-up of about 8 with 24 processors but they didn’t go further with more processors.

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Data distribution. Remember that r is the number of processors, the matrix A,, is parti- tionned into r portions of contiguous rows. If the euclidian division gives N = Nr + then, in
Data distribution. Remember that r is the number of processors, the matrix A,, is parti- tionned into r portions of contiguous rows. If the euclidian division gives N = Nr + then, in
The first thing we do is to roll up the ring in order to «see)) Py, in the tail end. This is done by means of the (local) variable q which keeps trace of the new position of each processor. We obtain: The way the proposed algorithm achieves this is better illustrated throughout an ex- ample. We consider a ring of r = 3 processors and a matrix of m = 10 columns, Am = [a1,.-, 419], % = 3,F =1, km =O and Qny = Hy-++Ayo[y,0]!. We assume that P,,, owns y. At the end of COLDEC we have the following distribution:
The first thing we do is to roll up the ring in order to «see)) Py, in the tail end. This is done by means of the (local) variable q which keeps trace of the new position of each processor. We obtain: The way the proposed algorithm achieves this is better illustrated throughout an ex- ample. We consider a ring of r = 3 processors and a matrix of m = 10 columns, Am = [a1,.-, 419], % = 3,F =1, km =O and Qny = Hy-++Ayo[y,0]!. We assume that P,,, owns y. At the end of COLDEC we have the following distribution:
The problem now comes down to a production-line work. We must manage to let the reflectors reach processor P;,,, in the correct order. The crucial observation to make at this juncture is that, by unrolling the entire communication pattern, we came across a problem consisting of moving data along the rows of an upper triangular grid. To proceed further, we introduce some notations (Table 1).
The problem now comes down to a production-line work. We must manage to let the reflectors reach processor P;,,, in the correct order. The crucial observation to make at this juncture is that, by unrolling the entire communication pattern, we came across a problem consisting of moving data along the rows of an upper triangular grid. To proceed further, we introduce some notations (Table 1).
Data distribution. Contiguous rows as in MGSDEC (§ 3.1, Figure2). The factor R goes to P,_4 upon completion. For ROCVEC which computes the product z = Qy, the vector y € R” must be at the beginning in processor P,_1 but the resulting vector z € R% is split at completion.
Data distribution. Contiguous rows as in MGSDEC (§ 3.1, Figure2). The factor R goes to P,_4 upon completion. For ROCVEC which computes the product z = Qy, the vector y € R” must be at the beginning in processor P,_1 but the resulting vector z € R% is split at completion.
Figure 4: ROCDEC. Factorization in progress.
Figure 4: ROCDEC. Factorization in progress.
jth column the elements below its main diagonal whereas any other processor applies the reflector that annihilates on its jth column the elements below its first row (this is illustrated is Figure 4 when 7 = 2). Then the non null portion of the jth row of processor Py is passed round the ring and on the way, in each processor, this row is used to apply the Givens rotation that zero-out the unique element of the column which was not eliminated by the Householder transformation.
jth column the elements below its main diagonal whereas any other processor applies the reflector that annihilates on its jth column the elements below its first row (this is illustrated is Figure 4 when 7 = 2). Then the non null portion of the jth row of processor Py is passed round the ring and on the way, in each processor, this row is used to apply the Givens rotation that zero-out the unique element of the column which was not eliminated by the Householder transformation.
Figure 5: RODDEC. Factorization in progress. Description.
Figure 5: RODDEC. Factorization in progress. Description.
Figure 7: RODVEC. Rotations are applied in the order \X\.
Figure 7: RODVEC. Rotations are applied in the order \X\.
In this section, for several values of NV, we present the time spent in running the Paragon Fortran code corresponding to the algorithms. The values indicated on the tables represent the time spent in the obtention of the decomposition augmented by the time spent in computing one product Qy. We use the following abbreviations: During our experiments, we have to face lack of memory when attempting to run very large problems on few nodes. As a consequence, the times obtained were perturbed by memory swaps. That is why we decide to consider speed-ups with respect to the number of nodes in which little memory misses occur. More specifically, if n is the smallest number of nodes for which the main matrix A, € RY*™ fits in the distributed-memory without performance overheads, the speed- up with respect to (w.r.t) m processors is T(r)/T(n). Those numbers are indicated by arrows on the tables. Our timing does not take into consideration the time used to load the matrix. In fact, we use dynamic space allocation to reserve exactly the sufficient memory storage needed within each processor for the problem in hand. Then the data were randomly generated.
In this section, for several values of NV, we present the time spent in running the Paragon Fortran code corresponding to the algorithms. The values indicated on the tables represent the time spent in the obtention of the decomposition augmented by the time spent in computing one product Qy. We use the following abbreviations: During our experiments, we have to face lack of memory when attempting to run very large problems on few nodes. As a consequence, the times obtained were perturbed by memory swaps. That is why we decide to consider speed-ups with respect to the number of nodes in which little memory misses occur. More specifically, if n is the smallest number of nodes for which the main matrix A, € RY*™ fits in the distributed-memory without performance overheads, the speed- up with respect to (w.r.t) m processors is T(r)/T(n). Those numbers are indicated by arrows on the tables. Our timing does not take into consideration the time used to load the matrix. In fact, we use dynamic space allocation to reserve exactly the sufficient memory storage needed within each processor for the problem in hand. Then the data were randomly generated.
Figure 11: Time in seconds and speed-up w.r.t 4 processors. Here once again MGS and COL are the slowest. No speedup is achieved for MGS while COL becomes more and more poorer. The behaviors of ROC and ROD are almost identical. After achieving a speedup of about 5.6 for 32 processors, they are about to stagnate. This problem comes from the fact that from there, m is no more large when compared to r. Hence this observation is in accordance to what we were expecting in advance (see the discussions about their potentiel parallelism).
Figure 11: Time in seconds and speed-up w.r.t 4 processors. Here once again MGS and COL are the slowest. No speedup is achieved for MGS while COL becomes more and more poorer. The behaviors of ROC and ROD are almost identical. After achieving a speedup of about 5.6 for 32 processors, they are about to stagnate. This problem comes from the fact that from there, m is no more large when compared to r. Hence this observation is in accordance to what we were expecting in advance (see the discussions about their potentiel parallelism).
per processor. In the following series of examples, we present the time behavior of the algorithms when we fix a constant load for each processor whatever is the total number of processors. If for reference, we consider that the complexity of these algorithms is about O(2N m7), we can clearly see with the very last example that we cross a rate performance of 779 Mflops for 56 nodes, i.e. 13.9 Mflops per processor.
per processor. In the following series of examples, we present the time behavior of the algorithms when we fix a constant load for each processor whatever is the total number of processors. If for reference, we consider that the complexity of these algorithms is about O(2N m7), we can clearly see with the very last example that we cross a rate performance of 779 Mflops for 56 nodes, i.e. 13.9 Mflops per processor.
Figure 12: Time in seconds and speed-up w.r.t 8 processors.
Figure 12: Time in seconds and speed-up w.r.t 8 processors.