blocksolver 0.8.3

Block Quasi-Minimal-Residual sparse linear solver

BlockSolver - Block Quasi-Minimal Residual (BLQMR) Sparse Linear Solver

BlockSolver is a Python package for solving large sparse linear systems using the Block Quasi-Minimal Residual (BLQMR) algorithm. It provides both a high-performance Fortran backend and a pure Python/NumPy implementation for maximum portability.

Features

• Block QMR Algorithm: Efficiently solves multiple right-hand sides simultaneously
• Complex Symmetric Support: Designed for complex symmetric matrices (A = Aᵀ, not A = A†)
• Dual Backend: Fortran extension for speed, Python fallback for portability
• ILU Preconditioning: Built-in incomplete LU preconditioner for faster convergence
• SciPy Integration: Works seamlessly with SciPy sparse matrices
• Optional Numba Acceleration: JIT-compiled kernels for the Python backend

Algorithm

Block Quasi-Minimal Residual (BLQMR)

The BLQMR algorithm is an iterative Krylov subspace method specifically designed for:

1. Complex symmetric systems: Unlike standard methods that assume Hermitian (A = A†) or general matrices, BLQMR exploits complex symmetry (A = Aᵀ) which arises in electromagnetics, acoustics, and diffuse optical tomography.

2. Multiple right-hand sides: Instead of solving each system independently, BLQMR processes all right-hand sides together in a block fashion, sharing Krylov subspace information and reducing total computation.

3. Quasi-minimal residual: The algorithm minimizes a quasi-residual norm at each iteration, providing smooth convergence without the erratic behavior of some Krylov methods.

Key Components

• Quasi-QR Decomposition: A modified Gram-Schmidt process using the quasi inner product ⟨x,y⟩ = Σ xₖyₖ (without conjugation) for complex symmetric systems.

• Three-term Lanczos Recurrence: Builds an orthonormal basis for the Krylov subspace with short recurrences, minimizing memory usage.

• Block Updates: Processes m right-hand sides simultaneously, with typical block sizes of 1-16.

When to Use BLQMR

Use Case	Recommendation
Complex symmetric matrix (A = Aᵀ)	✅ Ideal
Multiple right-hand sides	✅ Ideal
Real symmetric positive definite	Consider CG first
General non-symmetric	Consider GMRES or BiCGSTAB
Very large systems (>10⁶ unknowns)	✅ Good with preconditioning

Installation

From PyPI

pip install blocksolver

From Source

Prerequisites:

• Python ≥ 3.8
• NumPy ≥ 1.20
• SciPy ≥ 1.0
• (Optional) Fortran compiler + UMFPACK for the accelerated backend
• (Optional) Numba for accelerated Python backend

# Ubuntu/Debian
sudo apt install gfortran libsuitesparse-dev libblas-dev liblapack-dev

# macOS
brew install gcc suite-sparse openblas

# Install
cd python
pip install .

Quick Start

import numpy as np
from scipy.sparse import csc_matrix
from blocksolver import blqmr

# Create a sparse matrix
A = csc_matrix([
    [4, 1, 0, 0],
    [1, 4, 1, 0],
    [0, 1, 4, 1],
    [0, 0, 1, 4]
], dtype=float)

b = np.array([1., 2., 3., 4.])

# Solve Ax = b
result = blqmr(A, b, tol=1e-10)

print(f"Solution: {result.x}")
print(f"Converged: {result.converged}")
print(f"Iterations: {result.iter}")
print(f"Relative residual: {result.relres:.2e}")

Usage

Main Interface: blqmr()

The primary function blqmr() automatically selects the best available backend (Fortran if available, otherwise Python).

from blocksolver import blqmr, BLQMR_EXT

# Check which backend is active
print(f"Using Fortran backend: {BLQMR_EXT}")

# Basic usage
result = blqmr(A, b)

# With options
result = blqmr(A, b, 
    tol=1e-8,           # Convergence tolerance
    maxiter=1000,       # Maximum iterations
    use_precond=True,   # Use ILU preconditioning
)

Multiple Right-Hand Sides

BLQMR excels when solving the same system with multiple right-hand sides:

import numpy as np
from blocksolver import blqmr

# 100 different right-hand sides
B = np.random.randn(n, 100)

# Solve all systems at once (much faster than solving individually)
result = blqmr(A, B, tol=1e-8)

# result.x has shape (n, 100)

Complex Symmetric Systems

BLQMR is specifically designed for complex symmetric matrices (common in frequency-domain wave problems):

import numpy as np
from blocksolver import blqmr

# Complex symmetric matrix (A = A.T, NOT A.conj().T)
A = create_helmholtz_matrix(frequency=1000)  # Your application
b = np.complex128(source_term)

result = blqmr(A, b, tol=1e-8)

Custom Preconditioning

For the Python backend, you can provide custom preconditioners:

from blocksolver import blqmr, make_preconditioner

# Create ILU preconditioner
M1 = make_preconditioner(A, 'ilu')

# Or diagonal (Jacobi) preconditioner
M1 = make_preconditioner(A, 'diag')

# Solve with custom preconditioner
result = blqmr(A, b, M1=M1, use_precond=False)

SciPy-Compatible Interface

For drop-in replacement in existing code:

from blocksolver import blqmr_scipy

# Returns (x, flag) like scipy.sparse.linalg solvers
x, flag = blqmr_scipy(A, b, tol=1e-10)

Low-Level CSC Interface

For maximum control, use the CSC component interface:

from blocksolver import blqmr_solve

# CSC format components (0-based indexing)
Ap = np.array([0, 2, 5, 9, 10, 12], dtype=np.int32)  # Column pointers
Ai = np.array([0, 1, 0, 2, 4, 1, 2, 3, 4, 2, 1, 4], dtype=np.int32)  # Row indices
Ax = np.array([2., 3., 3., -1., 4., 4., -3., 1., 2., 2., 6., 1.])  # Values
b = np.array([8., 45., -3., 3., 19.])

result = blqmr_solve(Ap, Ai, Ax, b, 
    tol=1e-8,
    droptol=0.001,      # ILU drop tolerance (Fortran only)
    use_precond=True,
    zero_based=True,    # 0-based indexing (default)
)

API Reference

blqmr(A, B, **kwargs) -> BLQMRResult

Main solver interface.

Parameters:

Parameter	Type	Default	Description
A	sparse matrix or ndarray	required	System matrix (n × n)
B	ndarray	required	Right-hand side (n,) or (n × m)
tol	float	1e-6	Convergence tolerance
maxiter	int	n	Maximum iterations
M1, M2	preconditioner	None	Custom preconditioners (Python backend)
x0	ndarray	None	Initial guess
use_precond	bool	True	Use ILU preconditioning
droptol	float	0.001	ILU drop tolerance (Fortran backend)
residual	bool	False	Use true residual for convergence (Python)
workspace	BLQMRWorkspace	None	Pre-allocated workspace (Python)

Returns: BLQMRResult object with:

Attribute	Type	Description
x	ndarray	Solution vector(s)
flag	int	0=converged, 1=maxiter, 2=precond fail, 3=stagnation
iter	int	Iterations performed
relres	float	Final relative residual
converged	bool	True if flag == 0
resv	ndarray	Residual history (Python backend only)

blqmr_solve(Ap, Ai, Ax, b, **kwargs) -> BLQMRResult

Low-level CSC interface.

blqmr_solve_multi(Ap, Ai, Ax, B, **kwargs) -> BLQMRResult

Multiple right-hand sides with CSC input.

blqmr_scipy(A, b, **kwargs) -> Tuple[ndarray, int]

SciPy-compatible interface returning (x, flag).

make_preconditioner(A, type) -> Preconditioner

Create a preconditioner for the Python backend.

Types: 'diag'/'jacobi', 'ilu'/'ilu0', 'ssor'

Utility Functions

from blocksolver import (
    BLQMR_EXT,        # True if Fortran backend available
    HAS_NUMBA,        # True if Numba acceleration available
    get_backend_info, # Returns dict with backend details
    test,             # Run built-in tests
)

Performance Tips

1. Use the Fortran backend when available (10-100× faster than Python)

2. Enable preconditioning for ill-conditioned systems:

   result = blqmr(A, b, use_precond=True)
   

3. Batch multiple right-hand sides instead of solving one at a time:

   # Fast: single call with all RHS
   result = blqmr(A, B_matrix)
   
   # Slow: multiple calls
   for b in B_columns:
       result = blqmr(A, b)
   

4. Install Numba for faster Python backend:

   pip install numba
   

5. Reuse workspace for repeated solves with the same dimensions:

   from blocksolver import BLQMRWorkspace
   ws = BLQMRWorkspace(n, m)
   for b in many_rhs:
       result = blqmr(A, b, workspace=ws)
   

Examples

Diffuse Optical Tomography

import numpy as np
from scipy.sparse import diags, kron, eye
from blocksolver import blqmr

def create_diffusion_matrix(nx, ny, D=1.0, mu_a=0.01, omega=1e9):
    """Create 2D diffusion matrix for DOT."""
    n = nx * ny
    h = 1.0 / nx
    
    # Laplacian
    Lx = diags([-1, 2, -1], [-1, 0, 1], shape=(nx, nx)) / h**2
    Ly = diags([-1, 2, -1], [-1, 0, 1], shape=(ny, ny)) / h**2
    L = kron(eye(ny), Lx) + kron(Ly, eye(nx))
    
    # Diffusion equation: (-D∇² + μ_a + iω/c) φ = q
    c = 3e10  # speed of light in tissue (cm/s)
    A = -D * L + mu_a * eye(n) + 1j * omega / c * eye(n)
    
    return A.tocsc()

# Setup problem
A = create_diffusion_matrix(100, 100, omega=2*np.pi*100e6)
sources = np.random.randn(10000, 16)  # 16 source positions

# Solve for all sources at once
result = blqmr(A, sources, tol=1e-8)
print(f"Solved {sources.shape[1]} systems in {result.iter} iterations")

Frequency-Domain Acoustics

import numpy as np
from blocksolver import blqmr

# Helmholtz equation: (∇² + k²)p = f
# Results in complex symmetric matrix

def solve_helmholtz(K, M, f, frequencies):
    """Solve Helmholtz at multiple frequencies."""
    solutions = []
    for omega in frequencies:
        # A = K - ω²M (complex symmetric if K, M are symmetric)
        A = K - omega**2 * M
        result = blqmr(A, f, tol=1e-10)
        solutions.append(result.x)
    return np.array(solutions)

Troubleshooting

"No Fortran backend available"

Install the package with Fortran support:

# Install dependencies first
sudo apt install gfortran libsuitesparse-dev  # Linux
brew install gcc suite-sparse                  # macOS

# Reinstall blocksolver
pip install --no-cache-dir blocksolver

Slow convergence

1. Enable preconditioning: use_precond=True
2. Reduce ILU drop tolerance: droptol=1e-4 (Fortran backend)
3. Check matrix conditioning with np.linalg.cond(A.toarray())

Memory issues with large systems

1. Use the Fortran backend (more memory efficient)
2. Reduce block size for multiple RHS
3. Use iterative refinement instead of tighter tolerance

License

BSD-3-Clause / LGPL-3.0+ / GPL-3.0+ (tri-licensed)

Citation

If you use BlockSolver in your research, please cite:

@software{blocksolver,
  author = {Qianqian Fang},
  title = {BlockSolver: Block Quasi-Minimal Residual Sparse Linear Solver},
  url = {https://github.com/fangq/blit},
  year = {2024}
}

See Also

• BLIT - The underlying Fortran library
• SciPy sparse.linalg - Other iterative solvers
• PyAMG - Algebraic multigrid solvers