This code implements a **single-precision general matrix-vector multiplication (SGEMV)** operation,...

This code implements a single-precision general matrix-vector multiplication (SGEMV) operation, optimized for performance using SIMD (Single Instruction, Multiple Data) intrinsics available in Intel's AVX (Advanced Vector Extensions) and SSE instruction sets.

Here’s what the code does:

Purpose:

The function computes the matrix-vector multiplication:

[ out = weights \cdot x ]

Where:

• weights is a matrix of dimensions rows x cols.
• x is a vector of size cols.
• out is the resulting vector of size rows.

Details:

Input Parameters:

1. float *out:
   This is the output vector where the result of the matrix-vector multiplication is stored.

2. const float *weights:
   The input matrix (rows x cols) stored in row-major order.

3. int rows:
   The number of rows in the weights matrix (and size of the out vector).

4. int cols:
   The number of columns in the weights matrix (the size of the x vector).

5. int col_stride:
   The stride (distance in memory) between consecutive columns of weights.

6. const float *x:
   The input vector of size cols.

Output:

• Computes out[i] for each row i as the dot product of the row i in the matrix weights with the input vector x.

Implementation Overview:

1. SIMD Vectorization: The function uses AVX (256-bit registers) and SSE (128-bit registers) intrinsics for efficient computation. SIMD enables the function to process multiple elements of data in parallel, significantly improving performance for large matrices.

2. Loop Structure: The computation is divided into several loops to handle different numbers of elements in SIMD registers:

   Step 1 (Main Loop with 16-wide Vectorization):

   • Processes 16 rows at a time using AVX instructions (__m256).
   • Two __m256 vectors (vy0 and vy8) accumulate results for the first and second halves of these 16 rows.
   • Each column of x is broadcast to an AVX register (vxj) and multiplied with the corresponding 8 elements of weights in two batches (8 each).
   • Results are stored back into the out vector after processing all columns.

   Step 2 (8-wide Vectorization):

   • Processes 8 rows at a time when the remaining rows are less than 16 but at least 8.
   • Similar to the first stage but without splitting vy0 into two halves.

   Step 3 (4-wide Vectorization):

   • When the remaining rows are less than 8 but at least 4, SSE instructions (__m128) are used instead of AVX.
   • Uses __m128 registers for 4-wide processing.

   Step 4 (Scalar Fallback):

   • For any remaining rows (less than 4), falls back to a simple scalar loop that computes the dot product for each remaining row using standard arithmetic.

3. SIMD Intrinsics Used:

   • _mm256_setzero_ps(): Zero-initializes a 256-bit AVX register.
   • _mm_setzero_ps(): Zero-initializes a 128-bit SSE register.
   • _mm256_broadcast_ss(&x[j]): Broadcasts a single scalar value (x[j]) to all lanes of a 256-bit register.
   • _mm256_loadu_ps(): Loads 8 float values into a 256-bit register from memory (weights).
   • _mm_loadu_ps(): Loads 4 float values into a 128-bit register from memory.
   • _mm256_fmadd_ps() / _mm_fmadd_ps(): Performs fused multiply-add (FMA), computing vy = a * b + vy for improved performance and accuracy.
   • _mm256_storeu_ps() / _mm_storeu_ps(): Stores 8 or 4 float values from the 256-bit or 128-bit register back to memory.

4. Final Result: After the computation, the resulting vector out contains the dot product of each row of the weights matrix with the x vector.

Key Features:

1. Fusion of Multiply and Add (FMA): The function uses FMA instructions (_mm256_fmadd_ps() and _mm_fmadd_ps()) to optimize the computation of dot products. FMA reduces the latency and improves accuracy compared to separate multiply and add instructions.

2. SIMD Optimization:

   • Highly optimized for modern CPUs with AVX or SSE support.
   • Processes multiple rows in parallel via vectorized operations.

3. Handling Edge Cases:

   • The function includes fallback scalar loops for cases where the number of rows is not a multiple of 16, 8, or 4.

4. Column Stride:

   • Accounts for non-contiguous memory layout of the weights matrix using col_stride.

Usage Example:

If weights is a 3x3 matrix and x is a vector:

weights = [1  2  3
           4  5  6
           7  8  9]
x       = [1  1  1]

Then the function computes: [ out = weights \cdot x = [6, 15, 24] ]

Performance:

This function is designed for scenarios where performance is critical, such as deep learning, numerical computing, or handling large data matrices. However, it requires a CPU with AVX and SSE support to realize its full potential.