Principle:Ggml org Ggml BLAS Matrix Multiplication

Field	Value
sources	GGML OpenBLAS Intel MKL Apple Accelerate
domains	Linear_Algebra, Performance
last_updated	2026-02-10

Overview

BLAS Matrix Multiplication is the principle of delegating dense matrix multiplication operations to highly optimized, vendor-tuned Basic Linear Algebra Subprograms (BLAS) libraries rather than computing them with hand-written kernels.

Description

The Basic Linear Algebra Subprograms (BLAS) specification defines a standard set of routines for performing common linear algebra operations. BLAS libraries are organized into three levels:

• Level 1 -- Vector-vector operations (dot products, norms)
• Level 2 -- Matrix-vector operations (matrix-vector multiply)
• Level 3 -- Matrix-matrix operations (general matrix multiply, or GEMM)

In the context of neural network inference, the dominant computational workload is matrix multiplication, which maps directly to the Level 3 BLAS routine GEMM (General Matrix Multiply). The GEMM operation computes:

 C = alpha * A * B + beta * C

where A, B, and C are matrices, and alpha and beta are scalars.

Vendor-provided BLAS libraries such as Apple Accelerate, Intel MKL, OpenBLAS, BLIS, and NVIDIA NVPL apply years of micro-architecture-specific tuning -- including cache blocking, SIMD vectorization, multi-threaded parallelism, and prefetching strategies -- to achieve near-peak floating-point throughput on their target hardware.

The GGML BLAS backend takes advantage of these libraries by:

1. Dequantizing quantized weight tensors (e.g., Q4_0, Q8_0) to float32 in a temporary work buffer
2. Calling cblas_sgemm to perform the actual matrix multiplication in single-precision floating-point
3. Broadcasting across batch dimensions (ne2/ne3) when the source tensors have different batch sizes

Because BLAS libraries only operate on float32 (or float64) data, the backend maintains an intermediate work buffer sized to hold the dequantized representation of the weight matrix. The dequantization itself can be parallelized across threads when OpenMP is available.

Usage

Use the BLAS matrix multiplication principle when:

• The target platform has a highly optimized BLAS library available (Accelerate on macOS, MKL on Intel, OpenBLAS on Linux)
• The model uses weight types that benefit from dequantization to float32 for compute (all quantized types)
• Matrix multiplication is the computational bottleneck and the overhead of dequantization is amortized by GEMM performance
• GPU acceleration is not available or not desired

This approach is not ideal when:

• The overhead of dequantizing large weight matrices exceeds the GEMM speedup
• The hardware already provides dedicated low-precision matrix multiply units (e.g., AMX, Tensor Cores)

Theoretical Basis

The core algorithm for BLAS-accelerated matrix multiplication in GGML follows this pattern:

 Input: Tensor A (weights, possibly quantized), Tensor B (activations, float32)
 Output: Tensor C = A * B (float32)

 1. Determine work buffer size
    If A is not float32:
      work_size = batch_dims * rows_A * cols_A * sizeof(float32)
    Else:
      work_size = 0

 2. For each batch slice (i03, i02):
    a. If A is quantized:
       Dequantize A[i03][i02] into work buffer (parallel across rows using thread pool)
       src_A_ptr = work_buffer pointer
    Else:
       src_A_ptr = A[i03][i02] data pointer

    b. Compute broadcast indices:
       i13 = i03 * broadcast_factor_3
       i12 = i02 * broadcast_factor_2

    c. Call BLAS GEMM:
       cblas_sgemm(
         CblasRowMajor / CblasColMajor,
         CblasNoTrans, CblasTrans,
         M = rows_of_B,   // ne11
         N = rows_of_A,   // ne01
         K = cols_of_A,   // ne00 (shared dimension)
         alpha = 1.0,
         B_ptr = B[i13][i12],  ldb = stride_B,
         A_ptr = src_A_ptr,    lda = stride_A,
         beta = 0.0,
         C_ptr = C[i03][i02],  ldc = stride_C
       )

 3. Return C

The key insight is that the BLAS library handles all the low-level optimizations (cache tiling, SIMD, threading within GEMM) internally, so the GGML backend only needs to manage data layout, dequantization, and batch iteration.

Related Pages

• Implementation:Ggml_org_Ggml_Ggml_blas_backend
• Ggml_org_Ggml_Ggml_blas_backend -- The backend implementation that applies this principle
• Ggml_org_Ggml_Architecture_Specific_SIMD_Quantization -- Quantization formats that must be dequantized before BLAS calls
• Ggml_org_Ggml_CPU_Compute_Engine -- The CPU backend that provides an alternative non-BLAS compute path