families-linear-algebra

name

families-linear-algebra

description

Framework for Advanced (Multi)Linear Infrastructure in Engineering and Science. Dense linear algebra and tensor operations framework extending BLAS/LAPACK. Use when: linear algebra library design, BLAS implementation, LAPACK algorithms, tensor operations, high-performance computing, dense matrix operations, scientific computing infrastructure, GPU-accelerated linear algebra, or multi-node/multi-GPU dense algebra.

FAMLIES: Advanced Linear Algebra Infrastructure

Overview

This skill provides a framework for designing and implementing advanced dense linear and tensor algebra libraries. FAMLIES (Framework for Advanced (Multi)Linear Infrastructure in Engineering and Science) addresses limitations of traditional BLAS/LAPACK approaches while enabling better adaptation to new hardware architectures and applications.

Key Innovation: More flexible approach to defining and implementing high-performance dense linear and tensor algorithms that adapts to changing applications, software, and hardware.

Current Landscape & Challenges

BLAS/LAPACK Legacy

Success Factors:

• De facto standard interfaces
• Stringent boundary enforcement between layers
• Wide adoption in scientific computing
• Foundation for ML/data science

Current Limitations:

1. Boundary Enforcement: Impedes reducing data movement overhead
2. Layer Separation: Prevents loop fusion optimization
3. Hardware Adaptation: Slow to adapt to new architectures (GPUs)
4. Algorithm Implementation: Delayed adoption of new algorithms
5. Flexibility: Rigid structure limits innovation

Modern Challenges

Data Movement Overhead:

• Data movement dominates computation cost
• Boundaries prevent data reuse optimization
• Loop fusion opportunities missed
• Memory hierarchy not fully exploited

New Hardware Architectures:

• GPU acceleration requirements
• Multi-node/multi-GPU systems
• Heterogeneous computing platforms
• Different memory hierarchies

Emerging Applications:

• Machine learning tensor operations
• Data science sparse/dense hybrids
• New matrix factorization algorithms
• Quantum computing interfaces

FAMLIES Framework

Core Principles

1. Flexibility: Adaptable to applications, software, and hardware
2. Optimization: Enable data movement reduction
3. Standards: Maintain interface standards where beneficial
4. Performance: High-performance implementation across architectures
5. Extensibility: Support new algorithms and operations

Framework Architecture

Layer Flexibility:

• Allow cross-layer optimization
• Enable loop fusion
• Support operation composition
• Flexible boundary enforcement

Hardware Adaptation:

• GPU-specific optimizations
• Multi-node coordination
• Memory hierarchy awareness
• Architecture-specific implementations

Algorithm Support:

• Dense linear algebra (BLAS/LAPACK level)
• Tensor operations (multi-linear algebra)
• New factorization algorithms
• Application-specific operations

Leveraged Projects

BLIS (BLAS-like Library Instantiation Software)

Key Features:

• Flexible BLAS implementation
• Architecture-specific optimization
• Layer decomposition
• Performance portability

Benefits:

• Rapid architecture adaptation
• Optimized implementations
• Maintainable code structure
• Community-driven development

libflame

Key Features:

• LAPACK-level functionality
• Algorithm-by-blocks approach
• Flexible implementation
• Research-oriented design

Benefits:

• Advanced algorithm implementations
• Block-based optimization
• Research platform
• Extensible framework

Dense Linear Algebra Operations

BLAS-Level Operations

Level 1: Vector operations

• DOT: dot product
• AXPY: scaled vector addition
• NORM: vector norm
• SCALE: vector scaling

Level 2: Matrix-vector operations

• GEMV: general matrix-vector multiply
• GER: general rank-1 update
• SYMV: symmetric matrix-vector multiply
• TRSV: triangular solve

Level 3: Matrix-matrix operations

• GEMM: general matrix multiply
• SYMM: symmetric matrix multiply
• TRMM: triangular matrix multiply
• SYRK: symmetric rank-k update

LAPACK-Level Operations

Factorizations:

• LU: LU decomposition
• Cholesky: Cholesky factorization
• QR: QR decomposition
• SVD: singular value decomposition

Solvers:

• Linear system solve
• Eigenvalue problems
• Least squares
• Matrix inversion

Auxiliary:

• Condition estimation
• Matrix norms
• Equilibration
• Scaling

Tensor Operations

Multi-linear Operations:

• Tensor contraction
• Tensor decomposition (Tucker, CP)
• Tensor unfolding
• Tensor permutation

Tensor Algebra:

• Einstein notation operations
• Tensor-matrix products
• Tensor-tensor products
• Tensor factorizations

Implementation Considerations

Data Movement Optimization

Loop Fusion:

// Traditional BLAS: separate operations
GEMM(A, B, C);  // C = A*B
SCALE(alpha, C);  // C = alpha*C

// FAMLIES: fused operation
GEMM_SCALE_FUSED(A, B, C, alpha);  // C = alpha*A*B
// Reduces memory traffic: 2 reads + 2 writes -> 2 reads + 1 write

Operation Composition:

// Compose multiple operations
COMPOSE({
    GEMM(A, B, T1),
    GEMM(T1, C, T2),
    ADD(T2, D, E)
});
// Single kernel execution, reduced data movement

Memory Hierarchy:

• Cache-aware implementations
• Register blocking
• Memory bandwidth optimization
• Data locality maximization

Hardware-Specific Optimization

GPU Implementations:

• CUDA kernel design
• Shared memory utilization
• Warp-level operations
• Multi-stream execution

Multi-Node/Multi-GPU:

• Distributed matrix storage
• Communication optimization
• Load balancing
• Scalability considerations

Heterogeneous Computing:

• CPU-GPU coordination
• Device selection strategies
• Memory transfer optimization
• Hybrid implementations

Algorithm-by-Blocks Approach

Concept: Decompose algorithms into block operations

Benefits:

• Better cache utilization
• Parallelizable blocks
• Flexible implementation
• Data locality

Example: LU Factorization by Blocks:

def LU_by_blocks(A, block_size):
    n = A.shape[0]
    for k in range(0, n, block_size):
        # Factorize diagonal block
        LU_kernel(A[k:k+bs, k:k+bs])
        
        # Update trailing matrix
        for i in range(k+bs, n, block_size):
            # Solve panel
            solve_panel(A[k:k+bs, k:k+bs], A[i:i+bs, k:k+bs])
            
            # Update block row
            for j in range(k+bs, n, block_size):
                GEMM(A[i:i+bs, k:k+bs], 
                     A[k:k+bs, j:j+bs], 
                     A[i:i+bs, j:j+bs])

Performance Optimization Techniques

Cache Blocking

Register Blocking: Keep data in registers L1 Cache Blocking: Reuse data in L1 cache L2 Cache Blocking: Maximize L2 utilization TLB Blocking: Avoid TLB misses

Parallelization

Thread-Level: Multi-threaded execution SIMD-Level: Vector instructions GPU-Level: GPU parallelism Distributed-Level: Multi-node parallelism

Memory Optimization

Preconditioning: Reduce memory traffic In-place Operations: Avoid extra memory Strided Access: Optimal memory patterns Alignment: Cache line alignment

Framework Components

Operation Definitions

Standard Operations: BLAS/LAPACK compatible Extended Operations: New operation types Tensor Operations: Multi-linear algebra Composite Operations: Operation sequences

Implementation Layer

CPU Kernels: Optimized CPU implementations GPU Kernels: GPU-specific kernels Hybrid Kernels: Mixed CPU-GPU Reference Kernels: Portable reference code

Interface Layer

Standard APIs: BLAS/LAPACK compatibility Extended APIs: New operation interfaces Language Bindings: C, Fortran, Python, etc. Framework APIs: High-level framework calls

Research Paper Reference

Paper: "A Proposed Framework for Advanced (Multi)Linear Infrastructure in Engineering and Science (FAMLIES)"

• Authors: Devin A. Matthews, Tze Meng Low, Margaret E. Myers, et al.
• arXiv ID: 2604.07311
• Published: April 8, 2026
• Categories: cs.MS (Mathematical Software)
• Link: https://arxiv.org/abs/2604.07311

Use Cases

Scientific Computing

Applications:

• Numerical simulations
• Finite element methods
• Computational physics
• Climate modeling

Requirements:

• High performance
• Large-scale matrices
• Distributed computing
• Specialized operations

Machine Learning

Applications:

• Neural network training
• Tensor operations
• Large-scale optimization
• Model inference

Requirements:

• GPU acceleration
• Batch processing
• Tensor operations
• High throughput

Data Science

Applications:

• Large-scale analytics
• Statistical computations
• Data preprocessing
• Dimensionality reduction

Requirements:

• Flexible operations
• Mixed sparse/dense
• Memory efficiency
• Ease of use

Related Skills

• blas-lapack: Traditional BLAS/LAPACK usage
• gpu-computing: GPU acceleration techniques
• tensor-operations: Tensor algebra fundamentals
• high-performance-computing: HPC best practices

Implementation Examples

Basic GEMM with Loop Fusion

// FAMLIES GEMM with scaling and addition
void gemm_scale_add(
    int m, int n, int k,
    double alpha, double beta, double gamma,
    double *A, double *B, double *C, double *D,
    double *E) {
    
    // E = alpha*A*B + beta*C + gamma*D
    // Fused implementation
    
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < n; j++) {
            double sum = 0.0;
            for (int p = 0; p < k; p++) {
                sum += A[i*k+p] * B[p*n+j];
            }
            E[i*n+j] = alpha*sum + beta*C[i*n+j] + gamma*D[i*n+j];
        }
    }
}

Tensor Contraction

import numpy as np

def tensor_contract_3d(A, B, contract_dims):
    """
    Contract tensors A and B along specified dimensions
    
    Args:
        A: tensor of shape (a1, a2, a3)
        B: tensor of shape (b1, b2, b3)
        contract_dims: dimensions to contract
    
    Returns:
        Contracted tensor
    """
    # Example: contract last dimension of A with first of B
    if contract_dims == ((-1,), (0,)):
        return np.einsum('ijk,klm->ijlm', A, B)
    
    # General contraction
    einsum_str = build_einsum_string(A.shape, B.shape, contract_dims)
    return np.einsum(einsum_str, A, B)

Future Directions

Quantum Computing Integration:

• Quantum linear algebra
• Hybrid quantum-classical algorithms
• Quantum matrix operations

Sparse-Dense Hybrids:

• Mixed sparse/dense operations
• Selective storage strategies
• Format-aware kernels

Automatic Optimization:

• Auto-tuning frameworks
• Machine learning for optimization
• Compiler-driven optimization

See Also

• BLIS project documentation
• libflame documentation
• BLAS/LAPACK standards
• High-performance computing guides