GaussAlgorithmus

GaussAlgorithmus, commonly known as Gaussian elimination, is a fundamental algorithm in linear algebra for solving systems of linear equations, computing determinants, and obtaining matrix inverses. The method is named after Carl Friedrich Gauss, though the concept predates him, and it remains a core tool in numerical linear algebra.

The algorithm operates on an n-by-n coefficient matrix A and a right-hand side vector b. It transforms the augmented matrix [A|b] into an equivalent upper triangular form through a sequence of elementary row operations: swapping rows, scaling rows, and adding multiples of one row to another. With partial or complete pivoting, the algorithm selects pivot elements to improve numerical stability. Once the matrix is upper triangular, the solution x is obtained by back-substitution.

In practice, Gaussian elimination is often implemented as a decomposition step: A is factored into A = P L U (with a pivot matrix P), where L is lower triangular and U is upper triangular. This LU (with pivoting) factorization enables efficient solving of multiple right-hand sides by forward and backward substitution.

The method has a typical computational cost of O(n^3) operations for dense systems and O(n^2) storage. It is widely taught due to its simplicity and serves as a basis for more advanced algorithms, including iterative methods for large sparse systems.

Applications include solving linear systems, computing determinants (as the product of the diagonal entries of U, adjusted for row swaps), and deriving matrix inverses.