Chemical Engineering Hand Written_Notes_Part_29

Chemical Engineering Hand Written_Notes_Part_29 - 60 3....

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60 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES Thus, if we add any vector from the null space of A to the solution of (1.5), then we get another solution to equation (1.5). If N ( A ) { ¯ 0 } and a solution exists, i.e. b R ( A ) , then the solution is unique. If N ( A ) 6 = { ¯ 0 } and b R ( A ) , then there are in f nite solutions to equation (1.5). Methods for solving linear algebraic equations can be categorized as (a) direct or Gaussian elimination based schemes and (b) iterative schemes. In the sections that follow, we discuss these techniques in detail. 2. Direct Solution Techniques There are several methods which directly solve equation (1.5). Prominent among these are such as Cramer’s rule, Gaussian elimination, Gauss-Jordan method and LU decomposition. We assume that you have some exposure to these method in earlier courses on engineering mathematics. Let ϕ denote the number of divisions and multiplications required for generating solution by a particular method. We
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Chemical Engineering Hand Written_Notes_Part_29 - 60 3....

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