ABSTRACT

We know that the m × n system (5.2) has a solution if and only if r(A) = r(A:b), i.e., b ∊ C(A) . However, in a wide range of applications we encounter problems in which b may not be in C(A) . For such a problem we seek a vector(s) x ^ ∈ R n $ \hat{x} \in R^{n} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315208657/c3a398e7-9861-47a8-b5b5-8b0c24135fcb/content/inline-math21_1.tif"/> so that the error | | A x ^ - b | | 2 $ ||A\hat{x} - b||_{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315208657/c3a398e7-9861-47a8-b5b5-8b0c24135fcb/content/inline-math21_2.tif"/> is as small as possible (minimized), i.e., | | A x ^ - b | | 2 ≤ | | A x - b | | 2 $$ ||A\hat{x} - b||_{2} \le \,||Ax - b||_{2} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315208657/c3a398e7-9861-47a8-b5b5-8b0c24135fcb/content/math21_1.tif"/>