ABSTRACT
The need to solve systems of linear equations arises often within diverse disciplines of science, engineering, and finance. The expression “direct solution of linear systems” refers generally to computational strategies that can produce solutions to linear systems after a predetermined number of arithmetic operations that depends only on the structure and dimension of the coefficient matrix. The evolution of computers has and continues to influence the development of these strategies as well as fostering particular styles of perturbation analysis suited to illuminating their behavior. Some general themes have become dominant, as a result; others have been pushed aside. For example, Cramer’s Rule may be properly thought of as a direct solution strategy for solving linear systems; however it requires a much larger number of arithmetic operations than Gauss elimination and is generally much more susceptable to the deleterious effects of rounding. Most current approaches for the direct solution of a linear system, A x = b, are patterned after Gauss elimination and favor systematically decoupling the system of equations. Zeros are introduced systematically into the coefficient matrix, transforming it into triangular form; the resulting triangular system is easily solved. The entire process can be viewed in this way:
Find invertible matrices { S i } i = 1 ρ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429138492/cc3be78c-f644-49c2-b40d-db06d778c1a5/content/eq6025.tif"/> such that Sρ … S 2 S 1 A = U is triangular; then
Calculate a modified right-hand side y = Sρ … S 2 S 1 b; and then
Determine the solution set to the triangular system U x = y.
