8 2 Error Analysis for Ax=b 设b精确 A|·A‖是关键x+Sx,即 的误差放大因子,称为 的条件数,记为cond(A4), 越大则A越病 难得准确解。 A(x+)+S4(x (A+d4)x+(4+d4)G=b =-4-S4(+&c →(4+d4)a=-4x l ac A4(I+A4)=-64x s‖A‖·G4‖ lx+&x‖ &c=-(+A SA) A SAx 4|-1A31D 4‖ A‖l (要高充分使得 sd#+4<1 is Invertib ‖4t·才田 ‖<‖A-‖‖4‖ A c‖-1-‖A-l·‖4‖ 4| A‖l§2 Error Analysis for Ax b .   = ➢ 设 b 精确，A有误差 ，得到的解为 ，即   A x x   +  A A x x b    ( + )( + ) = A x x A x x b      ( + )+ ( + ) = ( ) 1 x A A x x      = −  + − || || || || || || || || || || || || || || || || 1 1 A A A A A A x x x     =     +  − −    A A x A A x b    ( + ) + ( + ) = A A x Ax    ( + ) = − A I A A x Ax    +   = − − ( ) 1 x I A A A Ax      1 1 1 ( ) − − −  = − + Wait a minute … Who said that ( I + A−1 A ) is invertible? (只要 A充分小，使得 || || || || || || 1 ) 1 1    − − A A A A || || || || 1 || || || || || || || || || || || || 1 || || || || || || || || || || || || 1 1 1 1 A A A A A A A A A A A A x x      −     = −     − − − −   是关键 的误差放大因子，称为 A的条件数，记为cond (A) , 越 则 A 越病态， 难得准确解。 || || || || −1 A  A 大