Direct Methods for Linear Systems
This chapter is devoted to the solution of systems of linear equations of the form Ax = b, (6.1) where A is a nonsingular square matrix with real entries, b is a vector called the “right-hand side,” and x is the unknown vector. For simplicity, we invariably assume that A ∈ ℳn(葷) and b ∈ 葷n. We call a method that allows for computing the solution x within a finite number of operations (in exact arithmetic) a direct method for solving the linear system Ax = b. In this chapter, we shall study some direct methods that are much more efficient than the Cramer formulas in Chapter 5. The first method is the celebrated Gaussian elimination method, which reduces any linear system to a triangular one. The other methods rely on the factorization of the matrix A as a product of two matrices A = BC. The solution of the system Ax = b is then replaced by the solution of two easily invertible systems (the matrices B and C are triangular or orthogonal) By = b, and Cx = y.
KeywordsDiagonal Entry Gaussian Elimination Cholesky Factorization Triangular System Band Matrix
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