aip-engine 0.4.1

Algebraic Independence Processor — auto-detection of matrix structure + memory-efficient computation for ultra-large sparse systems

AIP Engine

Algebraic Independence Processor — auto-detection of matrix structure + memory-efficient computation for ultra-large sparse systems.

pip install aip-engine

What it does

AIP Engine solves a real problem: building and solving sparse linear systems that are too large for conventional tools. It does three things:

1. Detects matrix structure automatically (sparse/dense, square/rectangular)
2. Routes to the optimal solver (LSQR, spsolve, LAPACK)
3. Accordion Memory: builds and solves ultra-large systems without running out of RAM

Quick start

import aip

# Auto-detect structure and solve
report = aip.detect_matrix(A)
x = aip.solve(A, b)

Accordion Memory

For systems with millions or billions of entries that don't fit in RAM:

from aip.accordion import PascalIndex, AccordionBuilder, solve_chunks

# 1. Mathematical indexing (replaces dictionary, saves GB of RAM)
index = PascalIndex(num_vars=30, max_degree=10)
idx = index.combo_to_index((3, 7, 12))  # O(k) time, 0 extra memory

# 2. Batch construction (never all in RAM at once)
builder = AccordionBuilder(num_rows=53_000_000)
# ... add entries in batches ...
builder.flush()  # converts to CSR chunk, frees raw data
chunks = builder.finalize()

# 3. Streaming solve (never assembles full matrix)
result = solve_chunks(chunks, b, max_iter=10000)
print(result['residual'], result['size_l2'])

Why Accordion?

	Without Accordion	With Accordion
Monomial index (8.6M entries)	~2 GB dictionary	0 MB (computed mathematically)
Matrix construction (150M entries)	~12 GB Python lists	~2.4 GB array.array per batch
Full matrix (53M x 1.17B)	496,052 TB dense	14.5 GB sparse chunks
Solve	needs full matrix in RAM	streaming over chunks

Real-world results:

Problem	Matrix size	Dense would be	Accordion uses	Compression
PHP n=4 d=8	8.6M x 78M	5.4 PB	1.2 GB	4,640,586x
PHP n=5 d=10	53M x 1.17B	496,052 TB	14.5 GB	34,215,310x

How it works

PascalIndex

Uses the Combinatorial Number System to compute the index of any monomial in O(k) time using a precomputed Pascal table. No dictionary needed.

index = PascalIndex(num_vars=45, max_degree=10)
print(index)  # PascalIndex(vars=45, deg=10, monomials=4,346,814,276, pascal=4048 bytes)
# 4.3 billion monomials indexed with 4 KB of memory

AccordionBuilder

Builds sparse matrices in batches using array.array (C-native, 4-8 bytes/element) instead of Python lists (28 bytes/element). Each batch is converted to CSR immediately and raw arrays are freed.

solve_chunks

LSQR solver that operates on a LinearOperator built from column chunks. The matvec/rmatvec operations iterate over chunks sequentially, never needing the full matrix in memory.

Requirements

• Python >= 3.8
• NumPy >= 1.20
• SciPy >= 1.7

License

MIT License - Carmen Esteban, 2025-2026