How Does MATLAB\'s Backslash Operator (\\) Solve Linear Equations?

How MATLAB's mldivide Operator Solves Linear Equations

MATLAB's mldivide operator, also known as the backslash operator "", is a versatile tool for computing solutions to linear systems expressed in matrix form. It encompasses various algorithms to handle different types of input matrices to ensure efficiency and numerical stability.

Algorithm Selection for Non-Square Matrices:

For non-square matrices, the backslash operator employs QR decomposition to determine the solution. QR decomposition factorizes the matrix A into orthogonal vectors Q and an upper triangular matrix R. The solution x is then obtained by solving the upper triangular system R (Q' * b).

Algorithm Selection for Square Matrices:

For square matrices, mldivide diagnoses the matrix and selects the appropriate algorithm based on its characteristics:

• Triangular Matrices (Lower or Upper): Forward or backward substitution is performed directly.
• Symmetric Positive Definite Matrices: Cholesky decomposition is used for faster and more stable solution.
• General Square Matrices: LU decomposition is employed for a combination of forward and backward substitution.

Special Cases:

• If A is singular or rectangular, the pseudoinverse (pinv) method can be used for a minimal norm least-squares solution.
• For sparse matrices, MATLAB utilizes iterative solvers and external libraries like UMFPACK.
• For computations on GPUs, mldivide leverages cuBLAS and MAGMA.
• For distributed arrays, ScaLAPACK is employed for solving linear systems in a distributed computing environment.

Implementing mldivide:

Implementing the entire functionality of mldivide in a custom matrix library would be a significant endeavor. However, by understanding the underlying algorithms and employing existing libraries for specific matrix types (e.g., LAPACK for dense matrices, iterative solvers for sparse matrices), it is possible to build a reasonably efficient and versatile solver for linear systems.

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