How to Efficiently Transpose a Matrix in C ?

How to Swiftly Transpose a Matrix in C ?

Problem:

Consider a substantial matrix with elements arranged as:

a b c d e f
g h i j k l
m n o p q r 

The goal is to transpose this matrix, resulting in:

a g m
b h n
c I o
d j p
e k q
f l r

Solution:

To transpose the matrix efficiently, consider the following approaches:

1. Naive Transpose:

void transpose(float *src, float *dst, const int N, const int M) {
    #pragma omp parallel for
    for(int n = 0; n<n n int i="n/N;" j="n%N;" dst src><p>This straightforward method iterates through each element and copies it to the transposed position. However, it may suffer from cache misses due to unpredictable memory access patterns.</p>
<p><strong>2. Transpose for Matrix Multiplication:</strong></p>
<p>When performing matrix multiplication C = A*B, it can be advantageous to transpose B. This approach eliminates cache misses and significantly speeds up the computation.</p>
<pre class="brush:php;toolbar:false">transpose(B);
for(int i=0; i<n i for j="0;" float tmp="0;" l="0;" a c transpose><p><strong>3. Block Transpose Using Loop Blocking:</strong></p>
<p>For large matrices, loop blocking offers exceptional performance. It divides the matrix into smaller blocks and transposes them independently.</p>
<pre class="brush:php;toolbar:false">void transpose_block(float *A, float *B, const int n, const int m, const int lda, const int ldb, const int block_size) {
    #pragma omp parallel for
    for(int i=0; i<n i for j="0;" transpose_scalar_block lda ldb block_size><p><strong>4. Transpose Using SSE Intrinsics:</strong></p>
<p>This advanced technique leverages SSE intrinsics to achieve unparalleled speed. It efficiently transposes 4x4 blocks at a time using a single instruction.</p>
<pre class="brush:php;toolbar:false">void transpose4x4_SSE(float *A, float *B, const int lda, const int ldb) {
    __m128 row1 = _mm_load_ps(&A[0*lda]);
    __m128 row2 = _mm_load_ps(&A[1*lda]);
    __m128 row3 = _mm_load_ps(&A[2*lda]);
    __m128 row4 = _mm_load_ps(&A[3*lda]);
     _MM_TRANSPOSE4_PS(row1, row2, row3, row4);
     _mm_store_ps(&B[0*ldb], row1);
     _mm_store_ps(&B[1*ldb], row2);
     _mm_store_ps(&B[2*ldb], row3);
     _mm_store_ps(&B[3*ldb], row4);
}

5. Loop Blocking with SSE:

Combining loop blocking with SSE intrinsics further enhances performance. This approach processes 4x4 blocks of the matrix efficiently.

void transpose_block_SSE4x4(float *A, float *B, const int n, const int m, const int lda, const int ldb ,const int block_size) {
    #pragma omp parallel for
    for(int i=0; i<n i for j="0;" int max_i2="i+block_size" n block_size : max_j2="j+block_size" m i2="i;" j2="j;" transpose4x4_sse lda ldb></n>

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