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Chemical Engineering Hand Written_Notes_Part_43

Chemical Engineering Hand Written_Notes_Part_43 - 88 3...

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88 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES From example 3, we can clearly see that the rate of convergence depends on ρ ( S 1 T ) . From analysis of some simple problems, we can generate the following table [ 5 ] Method Convergence Rate No. of iterations Jacobi O (1 / 2 n 2 ) O (2 n 2 ) Gauss_Seidel O (1 /n 2 ) O ( n 2 ) Relaxation with optimal ω O (2 /n ) O ( n/ 2) 5. Well Conditioned and Ill-Conditioned Problems One of the important issue in computing solutions of large dimensional linear system of equations is the round-o ff errors caused by the computer. Some ma- trices are well conditioned and the computations proceed smoothly while some are inherently ill conditioned, which imposes limitations on how accurately the system of equations can be solved using any computer or solution technique. Before we discuss solution techniques for (1.5), we introduce measures for as- sessing whether a given system of linear algebraic equations is inherently ill conditioned or well conditioned .
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