Direct Methods for the Solution of Linear Systems

A system of m linear equations in n unknowns consists of a set of algebraic relations of the form 3.1% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca % WGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaadIhadaWgaaWcbaGa % amOAaaqabaGccqGH9aqpcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaai % ilaiaadMgacqGH9aqpcaaIXaGaaiilaiabl+UimjaacYcacaWGTbaa % leaacaWGQbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa!4BA4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\sum\limits_{j = 1}^n {{a_{ij}}{x_j} = {b_i},i = 1, \cdots ,m} $$ where x _j are the unknowns, a_ij are the coefficients of the system and b_i are the components of the right hand side. System (3.1) can be more conveniently written in matrix form as 3.2% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadI % hacqGH9aqpcaWGIbaaaa!39A4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$Ax = b$$ where we have denoted by A = (a_ij) ∈ ℂm×n the coefficient matrix, by b=(b_i) ∈ (ℂm the right side vector and by x=(x_i) ∈ ℂm the unknown vector, respectively. We call a solution of (3.2) any n-tuple of values x_i which satisfies (3.1).