aBFT

ABFT, which stands for All-But-First, is an algorithm used in the fields of mathematics and computer science. It was developed to efficiently compute the inverse of a matrix, a crucial operation in many linear algebra algorithms. This algorithm is particularly useful for dense matrices, which are matrices that have more elements than the average matrix.

ABFT relies on the use of a cached inversion in the calculation of the inverse of a matrix. By recursively dividing the problem of inverting the entire matrix into smaller sub-problems, ABFT reduces the computational complexity of the inversion operation. This results in a method that requires significantly less arithmetic operations than traditional methods.

The ABFT algorithm has been widely adopted in various linear algebra applications due to its efficiency. It has been implemented in several libraries, including the LINPACK and BLAS libraries, which are widely used in numerical computations. ABFT's use in these libraries has contributed to its widespread adoption in various industries, including finance and data analysis.

The ABFT algorithm has been compared to other matrix inversion methods, such as the Doolittle and Cholesky methods. While it has been shown to be effective in reducing computational overhead, it also has limitations. The algorithm is more effective for dense matrices, and performance.degrades for sparse matrices. As a result, ABFT is typically used for specific tasks where dense matrices are prevalent.

Research into ABFT and its variants has led to improvements in computational efficiency and wider adoption in scientific computing. Its inclusion in popular libraries has made it a standard tool in linear algebra applications, demonstrating its value in efficiently computing the inverse of dense matrices.