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Matrix Multiplication [50 points]
1. Naive Matrix Multiplication [10 points]
Implement the naive matrix multiplication to multiply two matrices of dimension
4K × 4K. Report the execution time (in ns) and performance (in F LOP S).
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Programming homework
• Native Matrix Multiply
Implement C = A B (A, B, C are n n matrixes)
=
C(i,j)
A(i,:)
B(:,j)
for i = 1 to n
for j = 1 to n
for k = 1 to n
C(i,j) = C(i,j) + A(i,k) B(k,j)
Figure 1: Naive Matrix Multiplication
2. Block Matrix Multiplication [15 points]
A matrix can be viewed to be constructed by bigger blocks. Assume an n × n
matrix is partitioned into m × m blocks and each block is a b × b matrix ,where
1
b =
n
m
, b is called the block size. As shown in Figure 2, a 4 × 4 (n = 4) matrix
can be viewed as a 2 × 2 (m = 2) block matrix; each block matrix is a 2 × 2
(b = 2) matrix.
13
Programming homework
• Blocked Matrix Multiply
The matrix can be partitioned into four 2×2 blocks
The partitioned matrix can then be written as
Figure 2: Block Matrix
Block matrix multiplication computes the output block by block. Figure 3 depicts the principle of Block Matrix Multiplication. Here, the algorithm has m3
iterations as oppose to n
3
iterations for Naive Matrix Multiplication; the computation of each iteration is based on blocks rather than a single element for Naive
Matrix Multiplication.
Programming homework
• Blocked Matrix Multiply
C = A B, Consider A,B,C to be N-by-N matrices of b-by-b
sub-blocks, where b=n / N is called the block size
=
C’(i,j)
A’(i,:)
B’(:,j)
for i = 1 to m
for j = 1 to m
for k = 1 to m //do a matrix multi. on blocks
C’(i,j) = C’(i,j) + A’(i,k) * B’(k,j)
Figure 3: Block Matrix Multiplication
Use block matrix multiplication to solve the previous problem with block size
b = 4, 8, 16 respectively. Report the execution time (in ns) and performance (in
F LOP S) respectively. Compare the result against the result obtained by using
naive matrix multiplication. Report your observation.
2.1 Examples
• A matrix-vector multiplication example code, “example.c”, which multiplies
a matrix with a vector is provided. To compile “example.c”, type: ‘gcc -O3
-o run example.c’. You can refer to it for some useful functions.
• A matrix-matrix multiplication example code framework “problem1.c”; you
can either insert your codes into it or write the whole program by yourself.
To compile “problem1.c”, type: ‘gcc -O3 -o run problem1.c’.
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• To run the compiled program in hpc, type: ‘srun -n1 ./run’.
• Initialize the matrices: A[i][j] = i, B[i][j] = i + j, C[i][j] = 0, for any 0 ≤
i, j < n. After the computation, print out the value of C[100][100]. Refer
to “problem1.c”.
2.2 Submission
• Two .c/.cpp files: ‘problem1a.c’ for naive matrix multiplication; ‘problem1b.c’ for block matrix multiplication. Make sure your program is runnable.
(10+20 pts)
• Report. Write clearly how to compile and run your code. Screenshot of the
execution time and performance on hpc. (20 pts)
3 Submission Instructions
You may discuss the algorithms. However, the programs have to be written individually. Submit the code and the report via ee451spring2019@gmail.com. Please make
sure to include your name, student ID and the homework number in the PDF, and
name your PDF file lastname firstname hw#.
References
[1] “Command Line Parameter Parsing,”
[2] “Using make and writing Makefiles,”
http://www.cs.swarthmore.edu/~newhall/unixhelp/howto_makefiles.html
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