Direktmetoder

Direktmetoder, or direct methods, are numerical algorithms that aim to solve a problem in a finite sequence of operations, typically giving an exact solution up to machine precision. They contrast with iterative methods, which reach a solution through successive approximations that progressively converge.

In linear systems, direktmetoder include Gaussian elimination with pivoting, LU decomposition (including Crout and Doolittle variants), and Cholesky factorization for symmetric positive definite matrices. These methods factorize the coefficient matrix in a preparatory step, after which the unknown vector can be computed by forward and backward substitutions. They can also yield the matrix inverse or the determinant as byproducts of the factorization.

For least squares problems, direct methods use QR factorization of the design matrix, or the Cholesky factorization of the normal equations. QR is generally preferred for numerical stability, as it avoids squaring the condition number that can occur when forming A^T A.

Direct solvers are also adapted for sparse matrices, using techniques that reduce fill-in during factorization. Examples include specialized sparse LU and Cholesky-based methods, often with reordering strategies to minimize memory usage.

Advantages of direktmetoder include predictable numerical behavior, robustness, and a finite, well-understood workflow. Limitations include high memory and computational cost for large dense systems, and potential inefficiency for very large sparse problems where iterative methods or hybrid approaches may be favored. In practice, direct methods are foundational in engineering and scientific computing for solving linear systems, performing factorization for multiple right-hand sides, and providing stable baselines against which iterative methods are compared.