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Chemical Engineering Hand Written_Notes_Part_46

# Chemical Engineering Hand Written_Notes_Part_46 - 94 3...

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94 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES gives an upper bound on the possible ampli f cation of errors in b while comput- ing the solution. 5.3.2. Case: Perturbation in matrix A. Suppose ,instead of solving for A x = b due to truncation errors we end up solving (5.41) ( A + δA )( x + δ x )= b Then by subtracting A x = b from the above equation we obtain (5.42) x + δA ( x + δ x )= 0 or (5.43) δ x = A 1 δA ( x + δ x ) Taking norm on both the sides, we have || δ x || = || A 1 δA ( x + δ x ) || (5.44) or || δ x || || A 1 || || δA || || x + δ x | (5.45) || δ x || / || x + δ x || ( || A 1 || || A || ) || δA || / || A || (5.46) || δ x || / || x + δ x || / || δA || / || A || C ( A )= || A 1 || || A || (5.47) Again,the condition number gives an upper bound on % change in solution to % error A. In simple terms, condition number of a matrix tells us how serious is the error in solution of A x = b due to the truncation or round o

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Chemical Engineering Hand Written_Notes_Part_46 - 94 3...

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