Numpy Linalg

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name: numpy-linalg description: Linear algebra operations in NumPy, including matrix multiplication, SVD, system solving, and least squares fitting. Triggers: linalg, matrix multiplication, SVD, eigenvalues, matrix decomposition, lstsq, multi_dot.

Overview

NumPy's linalg module provides high-performance implementations for standard linear algebra routines. It leverages optimized backends (like BLAS/LAPACK) for operations such as Singular Value Decomposition (SVD), batch equation solving, and least-squares optimization.

When to Use

• Solving systems of linear equations ($Ax = b$).
• Dimensionality reduction or matrix reconstruction using SVD.
• Optimizing chains of matrix multiplications to reduce FLOPs.
• Finding best-fit solutions for overdetermined systems.

Decision Tree

1. Multiplying two matrices?
   • Use the @ operator (modern standard).
2. Multiplying 3+ matrices?
   • Use np.linalg.multi_dot to optimize multiplication order.
3. Is the matrix square and full-rank?
   • Yes: Use np.linalg.solve.
   • No: Use np.linalg.lstsq for a best-fit solution.

Workflows

1. Solving Multiple Linear Systems in Batch

   • Organize coefficient matrices into a stack of shape (K, M, M).
   • Organize ordinate values into a stack of shape (K, M).
   • Call np.linalg.solve(a, b) to compute all solutions simultaneously.

2. Rank-Reduced Reconstruction via SVD

   • Perform Singular Value Decomposition using np.linalg.svd(a, full_matrices=False).
   • Set small singular values in 's' to zero to perform noise reduction.
   • Reconstruct the matrix using (u * s) @ vh.

3. Least Squares Fitting for Overdetermined Systems

   • Construct the matrix 'A' of variables and vector 'b' of results.
   • Use np.linalg.lstsq(A, b) to find the solution that minimizes the Euclidean 2-norm.
   • Retrieve the coefficients and residuals from the returned tuple.

Non-Obvious Insights

• Modern Operator: The @ operator is preferred over dot() for 2D matrix products for readability and intent.
• Multi-Dot Optimization: multi_dot uses dynamic programming to find the optimal parenthesization of matrix products, which can significantly speed up calculations with varying matrix sizes.
• Batch Processing: Most linalg functions support "stacked" arrays, processing multiple independent problems in leading dimensions automatically.

Evidence

• "The @ operator... is preferable to other methods when computing the matrix product between 2d arrays." Source
• "a must be square and of full-rank... if either is not true, use lstsq for the least-squares best “solution”." Source

Scripts

• scripts/numpy-linalg_tool.py: Demonstrates SVD reconstruction and batch solving.
• scripts/numpy-linalg_tool.js: Simulated vector normalization logic.

Dependencies

• numpy (Python)

References

• references/README.md