How Does MATLAB\'s `mldivide` Operator Solve Linear Systems Efficiently?

Implementing MATLAB's mldivide Operator for Linear System Solutions

MATLAB's backslash operator () serves as a versatile tool for computing solutions of linear systems, encompassing a variety of algorithms to tackle different matrix types effectively.

Implementation Details

For square matrices A:

• Triangular matrices are solved using simple forward or backward substitution.
• Positive-definite symmetric matrices leverage Cholesky decomposition.
• General square matrices employ LU decomposition.

For rectangular matrices, QR decomposition is employed.

Algorithm Selection

The mldivide operator selects an appropriate algorithm based on matrix characteristics to achieve optimal performance and numerical stability. Users can specify options directly or let the operator determine the best strategy.

Alternative Solutions

For rectangular or singular matrices, the pinv function provides a least-squares solution using SVD decomposition.

Considerations for Sparse Matrices

Iterative solvers are typically used for sparse matrices. MATLAB utilizes UMFPACK for direct sparse solvers.

GPU and Distributed Computing

The backslash operator extends its functionality to gpuArray and distributed arrays, utilizing specialized libraries for computations on GPUs and distributed systems.

Implementation Complexity

Replicating all the capabilities of mldivide is a challenging task due to the diverse algorithms and use cases involved.

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