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EXPANSION_PLAN.md

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LinAlgKit Expansion Plan

Goal: Transform LinAlgKit into a comprehensive, high-performance scientific computing library that outperforms existing alternatives.

Executive Summary

This plan outlines the expansion of LinAlgKit from a basic linear algebra toolkit to a full-featured scientific computing library with:

  • 100+ mathematical functions (basic → advanced)
  • Performance optimizations targeting 2-5x speedup over NumPy for key operations
  • Multiple backend support (CPU, GPU, distributed)
  • Memory efficiency rivaling or exceeding competitors

Phase 1: Core Linear Algebra (v0.2.0 - v0.3.0)

1.1 Matrix Decompositions

Function Description Complexity Priority
lu() LU decomposition with partial pivoting O(n³) HIGH
qr() QR decomposition (Householder) O(mn²) HIGH
cholesky() Cholesky for positive-definite O(n³/3) HIGH
svd() Singular Value Decomposition O(min(mn², m²n)) HIGH
schur() Schur decomposition O(n³) MEDIUM
hessenberg() Hessenberg form O(n³) MEDIUM

1.2 Eigenvalue Problems

Function Description Priority
eig() Eigenvalues and eigenvectors HIGH
eigvals() Eigenvalues only (faster) HIGH
eigh() Hermitian/symmetric eigenvalue HIGH
eigsh() Top-k eigenvalues (iterative) MEDIUM

1.3 Linear System Solvers

Function Description Priority
solve() Direct solver (LU-based) HIGH
lstsq() Least-squares solution HIGH
inv() Matrix inverse HIGH
pinv() Moore-Penrose pseudoinverse HIGH
solve_triangular() Fast triangular solver MEDIUM
solve_banded() Banded matrix solver MEDIUM

1.4 Matrix Norms & Conditions

Function Description Priority
norm() Frobenius, 1, 2, inf norms HIGH
cond() Condition number HIGH
rank() Matrix rank HIGH
nullspace() Null space basis MEDIUM
orth() Orthonormal basis MEDIUM

Phase 2: Advanced Linear Algebra (v0.4.0 - v0.5.0)

2.1 Sparse Matrix Support

  • SparseCSR - Compressed Sparse Row format
  • SparseCOO - Coordinate format for construction
  • SparseDiag - Diagonal matrix (O(n) storage)

2.2 Iterative Solvers

  • cg() - Conjugate Gradient (SPD matrices)
  • gmres() - General non-symmetric
  • bicgstab() - Bi-Conjugate Gradient Stabilized
  • minres() - Symmetric indefinite

2.3 Matrix Functions

  • expm() - Matrix exponential
  • logm() - Matrix logarithm
  • sqrtm() - Matrix square root
  • funm() - General matrix function

Phase 3: Calculus & Optimization (v0.6.0 - v0.7.0)

  • Numerical differentiation (gradient, jacobian, hessian)
  • Numerical integration (quad, simps, trapz)
  • Optimization (minimize, least_squares, linprog)
  • Interpolation (interp1d, spline, rbf)

Phase 4: Statistics & Random (v0.8.0)

  • Descriptive statistics
  • Probability distributions
  • Statistical tests
  • Random number generation

Phase 5: Signal Processing (v0.9.0)

  • FFT (1D, 2D, N-D)
  • Digital filtering
  • Spectral analysis

Performance Optimization Strategy

  1. SIMD Vectorization - 4-8x speedup for element-wise operations
  2. Cache-Optimized Algorithms - Blocked matrix multiply
  3. Multi-threading - OpenMP parallel execution
  4. Memory Layout Optimization - Row/column major selection
  5. Lazy Evaluation - Expression templates
  6. GPU Acceleration - CUDA/ROCm backends

Implementation Roadmap

Version Target Key Features
v0.2.0 Q1 2025 LU, QR, Cholesky, solve, inv, norms
v0.3.0 Q2 2025 Eigenvalues, SVD, lstsq, pinv
v0.4.0 Q3 2025 Sparse matrices, iterative solvers
v0.5.0 Q4 2025 C++ core, BLAS integration, benchmarks
v0.6.0 Q1 2026 Calculus, optimization
v0.7.0 Q2 2026 Interpolation, fitting
v0.8.0 Q3 2026 Statistics
v0.9.0 Q4 2026 Signal processing
v1.0.0 2027 GPU, distributed, production ready

Contributing

We welcome contributions! Priority areas:

  1. Implementing functions from Phase 1
  2. Writing comprehensive tests
  3. Performance benchmarking
  4. Documentation improvements

See CONTRIBUTING.md for guidelines.