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Texas Instruments A Parallel Approach for Matrix Multiplication on the TMS320C4x DSP Application notes
A Parallel Approach for Matrix
Multiplication on the TMS320C4x DSP
Application Report
Rose Marie Piedra
Digital Signal Processing — Semiconductor Group
SPRA107
February 1994
Printed on Recycled Paper
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Introduction
Matrix operations, like matrix multiplication, are commonly used in almost all areas of scientific research.
Matrix multiplication has significant application in the areas of graph theory, numerical algorithms, signal
processing, and digital control.
With today’s applications requiring ever higher computational throughputs, parallel processing is an
effective solution for real-time applications. The TMS320C40 is designed for these kinds of applications.
This application note shows how to achieve higher computational throughput via parallel processing with
the TMS320C40. Although the focus is on parallel solutions for matrix multiplication, the concepts stated
here are relevant to many other applications employing parallel processing.
The algorithms that are presented were implemented on the Parallel Processing Development System
(PPDS), which has four TMS320C40s and both shared- and distributed-memory support. The algorithms
make use of parallel-runtime-support library (PRTS) functions available with the ’C40 C compiler for easy
message passing.
This report is structured in the following way:
Matrix Multiplication
Gives a brief review of matrix multiplication and some common
application areas.
Fundamentals of Parallel Processing
Presents some basic concepts of parallel processing. Partitioning, memory configuration, interconnection topologies, and
performance measurements are some of the issues discussed.
Parallel Matrix Multiplication
Focuses on parallel implementations of matrix multiplication.
Shared- and distributed-memory implementations are considered, as well as TMS320C40 suitability for each.
Results of Matrix Multiplication on a
TMS320C40-Based Parallel System
Presents the results of shared- and distributed-memory imple
mentations of parallel matrix multiplication on the ’C40 PPDS.
Includes analysis of speed-up, efficiency, and load balance.
Conclusion
States conclusions.
Appendices
List the code for parallel matrix multiplication. The programs
have been written in C. For faster execution, a C-callable assembly language routine is also supplied.
Matrix Multiplication
Let A and B be matrices of size n × m and m × l, respectively. The product matrix C = A * B is an n × l matrix,
for which elements are defined as follows [7]:
*
ȍ
+
+
m
c ij
k
1
0
a ik b kj
where
0 ≤ i < n, 0 ≤ j < l
The matrix multiplication requires O(nml) arithmetic operations, with each arithmetic operation requiring
a cumulative multiply-add operation. When l=1, a matrix-vector multiplication exists. Assuming that
n=m=l, matrix multiplication is an O(n3) operation.
1
Matrix-multiplication applications range from systems-of-equations solutions to graph representation.
Also, matrix-vector multiplication can be applied to compute linear convolution. Refer to [2] and [7] for
further information on these techniques.
Fundamentals of Parallel Processing
When applications require throughput rates that are not easily obtained with today’s sequential machines,
parallel processing offers a solution.
Generally stated, parallel processing is based on several processors working together to accomplish a task.
The basic idea is to break down, or partition, the computation into smaller units that are distributed among
the processors. In this way, computation time is reduced by a maximum factor of p, where p is the number
of processors present in the multiprocessor system.
Most parallel algorithms incur two basic cost components[7]:
•
•
computation delay—under which we subsume all related arithmetic/logic operations, and
communication delay—which includes data movement.
In a realistic analysis, both factors should be considered.
This application report presents some basic concepts of parallel processing. Refer to [2], [4], [5], [6], and
[7] for more detailed information.
Partitioning Schemes
From the software point of view, two basic approaches are used to create a parallel application:
•
•
Functional Partitioning: In this case, the task is a single function that has been subdivided
between the processors. Each processor performs its subfunction on the data as it moves from
one processor to the next in an assembly line or pipeline fashion.
Data Partitioning: In this case, the task is partitioned so that each processor performs exactly
the same function, but on different subblocks of the data. This approach requires algorithms with
strong intrinsic parallelism. The parallel matrix multiplication implemented with the
TMS320C40 PPDS applies this data-partitioning approach.
Architectural Aspects
From the hardware point of view, two important issues should be considered:
•
•
2
Memory configuration (shared- versus distributed-memory): In a distributed-memory
system, each processor has only local memory, and information is exchanged as messages
between processors. In contrast, the processors in a shared-memory system share a common
memory. Although data is easily accessible to any processor, memory conflict constitutes the
bottleneck of a shared-memory configuration. Because the PPDS has both shared and
distributed memory, it is an excellent tool for implementing and evaluating different parallel
configurations.
Connectivity network: This issue relates to the way the processors are interconnected with each
other. Fully connected networks (in which all the processors are directly connected to each other)
are the ideal networks from an “ease of use” point of view. However, they are impractical in large
multiprocessor systems because of the associated hardware overhead. Linear arrays, meshes,
hypercubes, trees, and fully-connected networks are among the topologies most commonly
used. Hypercube topologies are widely popular in commercially available multiprocessor
systems because they provide higher connectivity and excellent mapping capabilities. In fact,
it is possible to embed almost any other topology in a hypercube network [4]. Mesh topologies
are also commonly used to make systems modular and easily expandable. When
distributed-memory systems are used, interconnectivity issues play an important role in the
message-passing mechanism.
Performance Measurements
Two measurements apply to the performance of a parallel algorithm—speed-up and efficiency.
•
•
Speed-up of a parallel algorithm is defined as Sp = Ts /Tp, where Ts is the algorithm execution
time when the algorithm is completed sequentially, and Tp is the algorithm execution time using
p processors. Theoretically, the maximum speed-up that can be achieved by a parallel computer
with p identical processors working concurrently on a single problem is p. However, other
important factors (such as the natural concurrence in the problem to be computed, conflicts over
memory access, and communication delay) must be considered. These factors can reduce the
speed-up.
Efficiency, defined as Ep = Sp /p with values between (0,1), is a measure of processor utilization
in terms of cost efficiency. An efficiency close to 1 reveals an efficient algorithm. If the
efficiency is lower than 0.5, it is often better to use fewer processors because using more
processors offers no advantage.
Generally, the communication cost should be minimized by using wiser partitioning schemes and by
overlapping CPU and I/O operations. DMA channels help to alleviate the communication burden.
Parallel Matrix Multiplication
In parallel matrix multiplication, successive vector inner products are computed independently.
Because this application report focuses on multiple instruction multiple data (MIMD) implementations
(shared- and distributed-memory approaches), systolic implementations are not discussed. However,
single instruction multiple data (SIMD) implementations are also feasible with the TMS320C40.
Shared-Memory Implementation
Let n = qp, where n is the number of rows of matrix A, p is the number of processors, and q ≥ 1 is an integer.
Matrices A and B are stored in global memory so that each processor can have access to all the
rows/columns. The basic idea is to allocate a different working set of rows/columns to each processor.
Processor i computes row vectors qi, qi+1, ... , qi+q–1 of product matrix C, where i = 0,1,..., p–1. This is
illustrated in Figure 1 for p = 4 and n = m = l = 8.
3
Figure 1. Shared-Memory Implementation
Matrix A
(8×8)
×
Matrix B
(8×8)
0 1 2 3 4 5 6 7
P0
0
1
2
3
4
5
6
7
P1
P2
P3
Note:
Matrix C
(8×8)
0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
P0
P1
P2
P3
0
1
2
3
4
5
6
7
All processors have full access to the entire matrix B.
Two different approaches can be followed:
1.
Execute operations totally in shared memory (full memory conflict): This implementation
does not require any initial data transfer, but a conflict among memory accesses results (see code
in Appendix A).
2.
Transfer data for execution in on-chip memory of each processor (reduced memory
conflict): This approach reduces the delay caused by memory conflicts, but it requires extra data
transfer. This moving of data can be executed by the CPU or DMA via double-buffering
techniques.
Using double-buffering techniques can minimize the data-transfer delay. For matrix-vector
multiplication, vector B is initially transferred to on-chip RAM. While the CPU is working on
row A(i), the DMA is bringing row A(i+1) to on-chip RAM. If the two buffers are allocated in
different on-chip RAM blocks, no DMA/CPU conflict will be present. If the DMA transfer time
is less than or equal to the CPU computation time, the communication delay will be fully
absorbed. The TMS320C40 has two 4K-byte on-chip RAM blocks that enable it to support
double buffering of up to 1K words. Although this approach is not implemented in this
application report, page 6 shows what kind of performance can be expected.
Distributed-Memory Implementation
Let n = qp, where n is the number of rows of matrix A, p is the number of processors, and q ≥ 1 is an integer.
Matrix A has been partitioned into p regions with each region containing q rows and being assigned to the
local-memory (LM) of each processor. Matrix B is made available to all the processors. The
data-partitioning scheme is similar to the shared-memory approach. The differences are the extra time
required for data distribution/collection via message passing and the fact that all computations are done in
the LM of each processor with no memory-access conflict involved. With the use of double-buffering, this
communication delay can be reduced.
In this implementation, it is assumed that only processor 0 has access to matrix A and B. Processor 0 acts
as a host processor responsible for broadcasting the needed data to each of the other processors and waiting
for the vector results from the other processors. This is illustrated in Figure 2. Data distribution/collection
is system-specific and may not be needed for certain applications.
4
Figure 2. Distributed-Memory Implementation
Step 1: Data Broadcasting (Asynchronous)
P0
q = (n/p)=(8/4)=2
P1
Matrix A: Rows 2, 3
Complete Matrix B
P2
Matrix A: Rows 4, 5
Complete Matrix B
P3
Matrix A: Rows 6, 7
Complete Matrix B
Step 2: Distributed Matrix Multiplication (Processor i)
Partition of
Matrix A
×
0 1 2 3 4 5 6 7
Matrix B
Partition of
Matrix A×B
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
7
Row (i*q)
Row (i*q+1)
Row (i*q)
Row (i*q+1)
Step 3: Data Collection (Asynchronous)
Note:
P1
Rows 2, 3
P2
Rows 4, 5
P3
Rows 6, 7
P0
Rows 0, 1
Asynchronous = using DMA channels.
TMS320C40 Implementation
The TMS320C40 is the first parallel-processing DSP. In addition to a powerful CPU that can execute up
to 11 operations per cycle with a 40- or 50-ns cycle time, it contains 6 communication ports and a
multichannel DMA [3]. The on-chip communication ports allow direct (glueless) processor-to-processor
communication, and the DMA unit provides concurrent I/O by running parallel to the CPU. Also, special
interlocked instructions provide support for shared-memory arbitration. These features make the
TMS320C40 suitable for both distributed- and shared-memory computing systems.
5
Results of Matrix Multiplication on a TMS320C40-Based Parallel System
Parallel matrix multiplication was implemented in the TMS320C40 PPDS. The PPDS is a stand-alone
development board with four fully interconnected TMS320C40s. Each ’C40 has 256K bytes of local
memory (LM) and shares a 512K-byte global memory (GM)[1].
Features of implementing parallel matrix multiplication in the TMS320C40 PPDS:
•
•
•
•
•
•
•
The programs are generic. You can run the programs for different numbers of processors in the
system just by changing the value of P (if you set P=1, you will have a serial program).
Data input is provided in a separate file to preserve the generality of the programs.
A node ID must be allocated to each processor. In this way, each processor will select
automatically the row/column working set allocated to it. In this implementation, a different
node ID is allocated to each processor by using the ’C40 debugger commands to initialize that
variable. It is also possible to allocate a node ID by using the my_id function in the
parallel-runtime-support library (PRTS), which reads a predetermined set node ID value from
a user-specified memory location.
For benchmarking of shared-memory programs, a global start of all the processors is absolutely
necessary; otherwise, the real-memory-access conflict will not be observed. To help with this
process, a C-callable assembly routine is provided in Appendix C (syncount.asm) for debugging
systems without global start capability. Rotating priority for shared memory access should be
selected by setting the PPDS LCSR register to 0x40. On this basis, the total execution time of
the parallel algorithm can be defined as T = max (Ti) , where Ti is the execution time taken by
processor i (see Appendix A, shared.c.: Ti = time between labels t2 and t1).
For benchmarking of distributed-memory programs, I/O-execution time is optional. Data I/O
is system-specific and normally is not considered. In this application report, speed-up/efficiency
figures are given for both cases—including and not including I/O—in order to show the effect
of the communication delay in a real application. In this program (see Appendix B, distrib.c),
when processor 0 is acting as a host, then
Execution time
(I/O included)
= time between labels t1 and t4 in processor 0.
Execution time
(I/O not included)
= time between labels t2 and t3 in the processor with more load, or in any
processor in the case of load balancing.
If a debugger with benchmarking options (runb) is not available, the ’C40 analysis module or
the ’C40 timer can be used. In this application report, the ’C40 timer and the timer routines
provided in the PRTS library have been used for speed-up efficiency measures. Serial program
timings for the speed-up figures were taken with the shared-memory program with P = 1.
When the number of rows of matrix A is not a multiple of the number of processors in the system,
load imbalance occurs. This case has been considered for the shared-memory
(full-memory-conflict) implementation. (See Appendix A, shared.c.)
In parallel processing, speed-up/efficiency figures are more important than cycle counting because
speed-up/efficiency figures show how much performance improves if you make an application parallel.
You can apply the speed-up factors to any known sequential benchmarks to get a rough idea of the
parallel-execution time (assuming that the same memory allocation is used). Appendix D includes a
6
C-callable assembly-language function that executes matrix multiplication in approximately
nrowsa*(5+ncolsb*(6+ncolsa)) cycles in single-processor execution. This assumes use of program and
data in on-chip RAM. It also shows how you can use that function for parallel-processing execution.
Analysis of the Results
The performance of a parallel algorithm depends on the problem size (matrix size in our case) and on the
number of processors in the system. Speed-up and efficiency figures covering those issues can be observed
from Figure 3 to Figure 8 for the parallel algorithms presented. As you can see:
•
•
•
•
•
•
Shared-memory (full-memory conflict) has the lowest speed-up and efficiency. However, the
initial transfer of data to on-chip memory increases the speed-up, and if double-buffering
techniques are used, shared-memory implementation becomes as ideal as the
distributed-memory approach.
Speed-up is proportional to the number of processors. In the shared-memory implementation
(reduced-memory conflict) or in the distributed case (computation only), an optimal speed-up
of p can be reached. See Figure 3 and Figure 5. This result occurs because matrix multiplication
does not require any intermediate communication steps. In Figure 4, when p = 3, there is a
decline in efficiency due to load imbalance.
In general, efficiency is a better measure to analyze a parallel algorithm because it is more
meaningful in processor utilization. For example, compare Figure 3 and Figure 4—the
efficiency figure shows more clearly how increasing the number of processors negatively affects
the performance of the shared-memory (full-memory conflict) implementation.
Speed-up/efficiency increases for larger matrices in all cases, except for the shared-memory
(full-memory conflict) case. In the distributed-memory case (with I/O), speed-up/efficiency
increases because the communication delay (O(n 2 ) operation) becomes negligible against the
computation delay (O(n3) operation) for large n. See Figure 6 and Figure 7.
In the case of load imbalance, efficiency decreases because computation is not evenly distributed
among the processors. This is plotted in Figure 8. As you can see, if P = 4, the worst case occurs
when matrix size = 5, because while processor 0 is calculating the last row (row 4), all the other
processors are idle. The results shown here were taken for the shared-memory implementation
but are applicable for the distributed case.
The shared-memory implementation requires n*m+m*l+n*l words of shared memory (for
matrices A, B, and C, respectively). When this amount of memory is not available in the system,
intermediate-file downloading can be used. Another option is in-place computation ( A * B →
A ) using one intermediate buffer of size n per processor. For the distributed-memory case, the
performance depends on the way you implement your initial data distribution. In the application,
processor 0 requires n*m+m*l+n*l words of local memory. The other processors require
q*m+m*l+q*l of local memory, where q = n/p.
The programs in Appendices A and B have been used to calculate the speed-up/efficiency figures. For the
assembly-language case (Appendix D), the speed-up figures for the computation timing are still valid. For
the total timing (I/O included) using the assembly-language routine, the C implementation of the PRTS
routines lowers the speed-up, but for larger matrices, this is minimized.
7
Figure 3. Speed-Up Vs. Number of Processors (Shared Memory)
Speed-Up
5
4
3
2
1
0
1
2
3
Number of Processors
4
5
Shared-memory implementation with full memory conflict
Shared-memory implementation with double buffering for reduced-memory conflict
Note:
Matrix size = 16×16
Figure 4. Efficiency Vs. Number of Processors (Shared Memory)
Efficiency
110%
100%
90%
80%
70%
60%
50%
40%
30%
0
1
2
3
Number of Processors
Shared-memory implementation with full memory conflict
Shared-memory implementation with double buffering for reduced-memory conflict
Note:
8
Matrix size = 16×16
4
5
Figure 5. Speed-Up Vs. Number of Processors (Distributed Memory)
Speed-Up
5
4
3
2
1
0
1
2
3
4
5
Number of Processors
Distributed-memory implementation with computation and I/O delay
Distributed-memory implementation with computation only
Note:
Matrix size = 16×16
Figure 6. Speed-Up Vs. Matrix Size
Speed-Up
5
4
3
2
1
0
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
Matrix Size
Distributed-memory implementation with computation and I/O delay
Distributed-memory implementation with computation only
Shared-memory implementation with full memory conflict
Shared-memory implementation with double buffering for reduced-memory conflict
Note:
Number of Processors = 4
9
Figure 7. Efficiency Vs. Matrix Size
Efficiency
120%
110%
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
0
4
8
12
16
20
24
28
32
36
Matrix Size
40
44
48
52
56
60
64
68
Distributed-memory implementation with computation and I/O delay
Distributed-memory implementation with computation only
Shared-memory implementation with full memory conflict
Shared-memory implementation with double buffering for reduced-memory conflict
Note:
Number of Processors = 4
Figure 8. Speed-Up Vs. Matrix Size (Load Imbalance for Shared-Memory Program)
Speed-Up
5
4
3
2
1
0
3
4
5
6
7
8
9
10
Matrix Size
11
12
13
Shared-memory implementation with full memory conflict
Shared-memory implementation with double buffering for reduced-memory conflict
Note:
10
Number of Processors = 4
14
15
16
17
Conclusion
This report has presented parallel implementations of matrix multiplication using both shared- and
distributed-memory approaches. Matrix multiplication is an excellent algorithm for parallel processing,
as the speed-up/efficiency figures have shown. To avoid memory conflict when using the shared-memory
approach, it is important to transfer the data to on-chip/local memory for execution. Because interprocessor
communication is required only initially, it does not have a strong effect on the performance of the
distributed-memory approach; but with double-buffering techniques, this can be minimized even more.
Load balancing must also be considered.
11
References
[1] D.C. Chen and R. H. Price. “A Real-Time TMS320C40-Based Parallel System for High Rate Digital
Signal Processing.” ICASSP91 Proceedings, May 1991.
[2] S. G. Akl. The Design and Analysis of Parallel Algorithms. Englewood Cliffs, New Jersey:
Prentice-Hall, 1989, page 171.
[3] TMS320C4x User’s Guide, Texas Instruments, Incorporated, 1991.
[4] D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation, Numerical Methods,
Englewood Cliffs, New Jersey: Prentice-Hall, 1989.
[5] S. Y. Kung. VLSI Array Processors. Englewood Cliffs, New Jersey: Prentice-Hall, 1988.
[6] U. Schendel. Introduction to Numerical Methods for Parallel Computers. England: John Wiley & Sons,
1984.
[7] J. J. Modi. Parallel Algorithms and Matrix Computation. New York: Oxford University Press, 1988.
12
Appendix A: Shared-Memory Implementation
INPUT0.ASM
*****************************************************************
*
*
INPUT0.ASM: Contains matrix A and B input values.
*
*****************************************************************
.global
.global
.global
.global
_MAT_A
_MAT_B
_MAT_AxB
_synch
; counter for synchronization (global start)
.data
_synch
.int
_MAT_A
0
.float
.float
.float
.float
; stored by rows
1.0, 2.0, 3.0, 4.0
5.0, 6.0, 7.0, 8.0
9.0, 10.0, 11.0, 12.0
13.0, 14.0, 15.0, 16.0
.float
.float
.float
.float
; stored by rows
1.0, 2.0, 3.0, 4.0
1.0, 2.0, 3.0, 4.0
1.0, 2.0, 3.0, 4.0
1.0, 2.0, 3.0, 4.0
_MAT_B
_MAT_AxB .space
16
; must produce
;
;
;
;
(by rows):
10,20,30,40
26,52,78,104
42,84,126,168
58,116,174,232
.end
/***********************************************************************
13
SHARED.C
/************************************************************************
SHARED.C : Parallel matrix multiplication (Shared memory version: full memory
conflict)
– All the matrices (A,B,C) are stored by rows.
To run:
cl30 –v40 –g –o2 –as –mr shared.c
asm30 –v40 –s input0.asm
asm30 –v40 –s syncount.asm
lnk30 shared.obj input0.obj shared.cmd
************************************************************************/
#define
NROWSA
4
/* number of rows in mat A
*/
#define
NCOLSA
4
/* number of columns in mat A */
#define
NCOLSB
4
/* number of columns in mat B */
#define
P
4
/* number of processors
*/
extern
extern
extern
float MAT_A[NROWSA][NCOLSA];
float MAT_B[NCOLSA][NCOLSB];
float MAT_AxB[NROWSA][NCOLSB];
extern
extern
int
void
float
*A[NROWSA],*B[NCOLSA],*AxB[NROWSA],temp;
synch;
syncount();
/* synchronization for global start */
int
*synch_p
= &synch,
q = NROWSA/P,
l1 = 0,
my_node, i, j, k,tcomp;
/***********************************************************************/
main()
{
asm(” OR 1800h,st”);
/* cache enable */
/* accesing matrices declared in an external assembly file
for (i=0;i<NROWSA;i++) A[i] = MAT_A[i];
for (i=0;i<NCOLSA;i++) B[i] = MAT_B[i];
for (i=0;i<NROWSA;i++) AxB[i] = MAT_AxB[i];
*/
syncount(synch_p,P);
/* global start:loop until counter=P */
if (((i = NROWSA %P) >0)) {
if (my_node<i) ++q; else l1 =i;
}
l1 += q*my_node;
/* load imbalancing:optional
t1: time_start(0);
/* benchmarking with C40 timer
*/
for (i=l1;i<(l1+q);i++)
for (j=0;j<NCOLSB;j++)
{
temp = 0;
for (k=0;k<NCOLSB;k++)
AxB[i][j] = temp;
}
/* matrix multiplication
*/
14
/* select beginning of row working set
for processor ”my_node”
*/
temp += A[i][k] * B[k][j] ;
t2 : tcomp = time_read(0);
syncount(synch_p,2*P);
} /*main*/
*/
/* shared–memory benchmark
*/
/* optional: if you want all processors
finish at the same time
*/
SHARED.CMD
/************************************************************************
SHARED.CMD: Linker Command File for Shared-Memory Program
************************************************************************/
syncount.obj
–c
–stack 0x0100
–lrts40r.lib
–lprts40r.lib
–m a.map
/* link using C conventions
*/
/* get run–time support
*/
/* SPECIFY THE SYSTEM MEMORY MAP */
MEMORY
{
ROM:
org = 0x00
RAM0: org = 0x0002ff800
RAM1: org = 0x0002ffc00
LM: org = 0x040000000
GM: org = 0x080000000
}
len
len
len
len
len
=
=
=
=
=
0x1000
0x0400
0x0400
0x10000
0x20000
/*
/*
/*
/*
RAM block0
RAM block1
local memory
global memory
*/
*/
*/
*/
/* SPECIFY THE SECTIONS ALLOCATION INTO MEMORY */
SECTIONS
{
.text:
.cinit:
.stack:
.bss :
.data:
}
{}
{}
{}
{}
{}
>
>
>
>
>
RAM0
RAM1
RAM0
RAM1
GM
/*
/*
/*
/*
/*
code
initialization tables
system stack
global & static vars
for input matrix
*/
*/
*/
*/
*/
15
Appendix B: Distributed-Memory Implementation
INPUT.ASM
*****************************************************************
*
*
INPUT.ASM : Input file for processors 1 to (P–1)
*
*****************************************************************
.global
.global
.global
_MAT_A
_MAT_B
_MAT_AxB
.data
_MAT_A
_MAT_B
_MAT_AxB
.space
.space
.space
16
16
16
.end
DISTRIB.C
/*********************************************************************
DISTRIB.C : Parallel matrix multiplication (distributed–memory
implementation)
(no load imbalancing has been considered)
cl30 –v40 –g –mr –as –o2 distrib.c
asm30 –v40 –s input0.asm
(see Input0.asm on page 13 )
asm30 –v40 –s input.asm
lnk30 distrib.obj input0.obj distrib.cmd –o a0.out (For processor 0)
lnk30 distrib.obj input.obj distrib.cmd –o a.out (For processors 1 to (P–1))
**********************************************************************/
#define
NROWSA
4
/* number of rows in mat A
*/
#define
NCOLSA
4
/* number of columns in mat A */
#define
NCOLSB
4
/* number of columns in mat B */
#define
P
4
/* number of processors
*/
extern
extern
extern
float MAT_A[NROWSA][NCOLSA];
float MAT_B[NCOLSA][NCOLSB];
float MAT_AxB[NROWSA][NCOLSB];
float
*A[NROWSA], *B[NCOLSA], *AxB[NROWSA], temp;
int
my_node ,
q = NROWSA/P,
tcomp, ttotal,
i,j,k,l1;
int port[4][4] = {
0,0,4,3,
3,0,0,4,
1,3,0,0,
0,1,3,0 };
/* connectivity matrix: processor i is
connected to processor j thru port[i][j]:
system specific PPDS
*/
/***********************************************************************/
16
main()
{
asm(” OR 1800h,st”);
/* accesing assembly variables
for (i=0;i<NROWSA;i++) A[i] =
for (i=0;i<NCOLSA;i++) B[i] =
for (i=0;i<NROWSA;i++) AxB[i]
*/
MAT_A[i];
MAT_B[i];
= MAT_AxB[i];
t1: time_start(0);
/* Processor 0 distributes data. Other processors receive it */
if (my_node==0)
for(i=1;i<P;++i){
/* asynchronous sending (DMA)
send_msg(port[0][i],&A[i*q][0],(q*NCOLSA),1);
send_msg(port[0][i],&B[0][0],(NCOLSA*NCOLSB),1);
/* autoinitialization can also be used */
}
else {
/* synchronous receiving (CPU)
k = in_msg(port[my_node][0],&A[0][0],1);
k = in_msg(port[my_node][0],&B[0][0],1);
}
*/
*/
t2: tcomp = time_read(0);
for (i=0;i<q;i++)
/* Matrix multiplication
for (j=0;j<NCOLSB;j++)
{
temp = 0;
for (k=0;k<NCOLSB;k++)
temp += A[i][k] * B[k][j];
AxB[i][j] = temp;
}
t3: tcomp = time_read(0) – tcomp;
*/
/* Processors 1–(P–1) send result to proc. 0. Processor 0:ready to receive it */
if (my_node==0)
for(i=1;i<P;++i)receive_msg(port[0][i],&AxB[i*q][0],1); /* asynchronous*/
else send_msg(port[my_node][0],&AxB[0][0],(q*NCOLSB),1);
if (my_node==0)
/* Wait for interprocessor communication to finish
for (i=1;i<P;++i) while (chk_dma(port[0][i]) );
else while (chk_dma(port[my_node][0])) ;
t4: ttotal = time_read(0);
/* this is including: comp + input + output + 2 timer_reads
*/
*/
} /*main*/
17
DISTRIB.CMD
/************************************************************************
DISTRIB.CMD: Linker Command File for Distributed-Memory Program
************************************************************************/
–c
–stack 0x0100
–lrts40r.lib
–lprts40r.lib
–m a.map
/* link using C conventions
/* get run–time support
*/
*/
/* SPECIFY THE SYSTEM MEMORY MAP */
MEMORY
{
ROM:
RAM0:
RAM1:
LM:
GM:
}
org
org
org
org
org
= 0x0
= 0x0002ff800
= 0x0002ffc00
= 0x040000000
= 0x080000000
len
len
len
len
len
=
=
=
=
=
0x1000
0x0400
0x0400
0x10000
0x20000
/*
/*
/*
/*
RAM block0
RAM block1
local memory
global memory
/* SPECIFY THE SECTIONS ALLOCATION INTO MEMORY
*/
SECTIONS
{
.text:
.cinit:
.stack:
.bss :
.data:
}
*/
*/
*/
*/
*/
18
{}
{}
{}
{}
{}
>
>
>
>
>
RAM0
RAM1
RAM0
RAM1
LM
/*
/*
/*
/*
/*
code
initialization tables
system stack
global & static vars
for input matrix
*/
*/
*/
*/
Appendix C: Synchronization Routine for Shared-Memory Implementation
*****************************************************************
*
* syncount.asm :
assembly language synchronization routine to provide a
* global start for all the processors. Initially, a counter in shared
* memory is set to zero. Each processor increments the counter by 1. When
* the counter equals value, the processors exit this routine. Rotating
* priority for shared-memory access should be selected. The processors
* start with a maximum cycle difference of 3 instruction cycles, which for
* practical purposes is acceptable. This routine is C–callable and uses
* registers for parameter passing.
*
* Calling conventions:
*
void syncount((int *)counter,int value)
ar2 , r2
*
* where counter
= synchronization counter in shared memory
*
value
= counter value to be reached.
*
*****************************************************************
.global _syncount
.text
_syncount:
LDII
*AR2,R1
ADDI
1,R1
CMPI
R1,R2
STII
R1,*AR2
BZ
L1
AGAIN LDI
CMPI
BNZ
L1
.end
*AR2,R1
R1,R2
AGAIN
RETS
19
Appendix D: C-Callable Assembly Language Routine for Matrix Multiplication
INPUT0_A.ASM
*****************************************************************
*
* INPUT0_A.ASM: Contains matrix A and B input values. Matrix B is
* stored by columns.
*
*****************************************************************
.global
.global
.global
.global
_MAT_A
_MAT_B
_MAT_AxB
_synch
; counter for synchronization
.data
_synch
.int
0
_MAT_A
.float
.float
.float
.float
; stored by rows
1.0, 2.0, 3.0, 4.0
5.0, 6.0, 7.0, 8.0
9.0, 10.0, 11.0, 12.0
13.0, 14.0, 15.0, 16.0
.float
.float
.float
.float
1.0,
2.0,
3.0,
4.0,
_MAT_B
_MAT_AxB
; stored by columns!!!
.space 16
.end
20
1.0,
2.0,
3.0,
4.0,
1.0,
2.0,
3.0,
4.0,
1.0
2.0
3.0
4.0
;
;
;
;
;
must produce (stored by rows)
10,20,30,40
26,52,78,104
42,84,126,168
58,116,174,232
MMULT.ASM
****************************************************************************
*
*
MMULT.ASM: Matrix multiplication (assembly language C–callable program)
*
* mmult(&C, &A, &B, nrowsa, ncolsa, ncolsb)
*
ar2, r2, r3, rc, rs, re
*
*
– Matrix A (nrowsa×ncolsa): is stored by rows (row–major order)
*
– Matrix B (ncolsa×ncolsb): is stored by columns (column–major order)
*
– Matrix C (nrowsa×ncolsb): is stored by rows (row–major order)
*
– ”ncolsb” must be greater or equal to 2
*
– This routine uses register to pass the parameters (refer to C compiler
*
users’guide for more information)
*
*
5/1/90 : Subra Ganesan
*
10/1/90: Rosemarie Piedra
*
****************************************************************************
.global
.text
_mmult
_mmult
LDI
LDI
R2,AR0
R3,AR1
LDI
LDI
PUSH
PUSH
PUSH
RS,IR0
RE,R10
AR5
AR3
R5
SUBI
SUBI
SUBI
1,RC,AR3
2,RS,R1
1,RE,R9
LDI
R9,AR5
;
;
;
;
;
;
AR0: address of A[0][0]
AR1: address of B[0][0]
AR2: address of C[0][0]
IR0: NCOLSA
R10: NCOLSB
preserve registers
; AR3: NROWSA–1
; R1: NCOLSA–2
; R9: NCOLSB–1
ROWSA
COLSB
||
||
; initialize R2
*AR0++(1),*AR1++(1),R0 ; perform one multiplication
R2,R2,R2
; A(I,1)*B(1,I) –> R0
R1
; repeat the instruction NCOLSA–1 times
*AR0++(1),*AR1++(1),R0
R0,R2,R2
; M(I,J) * V(J) –>R0
; M(I,J–1) * V(J–1) + R2 –> R2
DBD
AR5,COLSB
; loop for NCOLSB times
ADDF
R0,R2
; last accumulate
STF
R2,*AR2++(1)
; result –> C[I][J]
SUBI
IR0,AR0
; set AR0 to point A[0][0]
DBD
AR3,ROWSA
; repeat NROWSA times
ADDI
IR0,AR0
; set AR0 to point A[1][0]
MPYI3 IR0,R10,R5
; R5 : NCOLSB*NROWSB(IR0)
SUBI
R5,AR1
; set AR1 to point B[0][0]
POP
R5
POP
AR3
POP
AR5
; recover register values
RETS
MPYF3
SUBF3
RPTS
MPYF3
ADDF3
21
SHAREDA.C
/***********************************************************************
SHAREDA.C : Parallel Matrix Multiplication (shared memory version: full
memory conflict)
–This program uses an assembly language C–callable routine for matrix
multiplication for faster program execution.
–Matrix A and C are stored by rows. Matrix B is stored by columns to
take better advantage of the assembly language implementation.
To run:
cl30 –v40 –g –o2 –as –mr shareda.c
asm30 –v40 –s input0_a.asm
asm30 –v40 –s syncount.asm
asm30 –v40 –s mmult.asm
lnk30 mmult.obj shareda.obj input0_a.obj shared.cmd
************************************************************************/
#define
NROWSA 4
/* number of rows in mat A
*/
#define
NCOLSA 4
/* number of columns in mat A
*/
#define
NCOLSB 4
/* number of columns in mat B
*/
#define
P
4
/* number of processors
*/
extern
extern
extern
extern
float
float
float
void
MAT_A[NROWSA][NCOLSA];
/* stored by rows
*/
MAT_B[NCOLSB][NCOLSA];
/* stored by columns */
MAT_AxB[NROWSA][NCOLSB];
/* stored by rows
*/
mmult(float*C, float*A, float*B, int nrowsa, int ncolsa,
int ncolsb);
extern
extern
int
void
synch;
syncount();
/* synchronization for benchmarking */
float
*A[NROWSA], *B[NCOLSB], *AxB[NROWSA], temp;
int
*synch_p = &synch,
q = NROWSA/P,
l1 = 0,
my_node, i, j, k, tcomp;
/***********************************************************************/
main()
{
asm(” OR 1800h,st”);
/* cache enable
*/
/* accesing matrices declared in an external assembly file
for (i=0;i<NROWSA;i++) A[i] = MAT_A[i];
for (i=0;i<NCOLSB;i++) B[i] = MAT_B[i];
for (i=0;i<NROWSA;i++) AxB[i] = MAT_AxB[i];
*/
syncount(synch_p,P);
*/
/* global start
if (((i = NROWSA %P) >0)) { /* load imbalancing: optional
*/
if (my_node<i) ++q; else l1 =i;
}
l1 += q*my_node;
/* select beginning of row working set
for processor ”my_node”
*/
t1: time_start(0);
/* benchmarking with C40 timer
*/
mmult (&AxB[l1][0],&A[l1][0],&B[0][0],q,NCOLSA,NCOLSB);/* matrix mult.*/
t2 : tcomp = time_read(0);
syncount(synch_p,2*P);
} /*main*/
22
/* shared–memory benchmark
*/
/* optional: if you want all processors
finish at the same time
*/
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