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# lecture5 - Numerical Methods in Engineering ENGR 391 System...

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Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University System of linear equations

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Linear algebraic equations ±A n e q u a t i o n o f t h e f o r m ax+by=c is called a linear equation in x and y variables. ± ax+by+cz=d is a linear equation in three variables, x, y , and z . ±T h u s , a l i n e a r e q u a t i o n i n n variables is a 1 x 1 +a 2 x 2 + … +a n x n = b ±A s y s t e m o f l i n e a r e q u a t i o n s c o n s i s t s o f n number of these equations and a solution of such system consists n number of real numbers.
Engineering Problems

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Noncomputer methods for solving systems of linear equations ±For sma l l number of equat ions (n ± 3) linear equations can be solved readily by simple techniques such as ²method of elimination.³ ±L inear a
Solving system of linear algebraic Equations For n ± 3 There are many ways to solve a system of linear equations: Graphical method Cramer±s rule Method of elimination Computer methods Gauss elimination LU decomposition Gauss-Seidel

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Graphical Method (n=2; mostly) 2 2 22 1 21 1 2 12 1 11 b x a x a b x a x a ± ± ±This plot method is used for two equations (linear system) ±X 2 is a function of X 1 . ±p lo t X 1 vs. X 2 for each equation ²plot the slope and the intercept³ ±the so lu t ion is the in tersec t ion o f the two l ines ° ° ¯ ° ° ® ­ ± ¸ ¸ ¹ · ¨ ¨ © § ² ² ± ³ ± ¸ ¸ ¹ · ¨ ¨ © § ² ² 22 2 1 22 21 22 1 21 2 2 1 2 12 1 1 12 11 12 1 11 1 2 intercept (slope) a b x a a a x a
±P l o t x 2 vs. x 1 on rectilinear paper, the intersection of the lines present the solution. Fig. 9.1 Chapra and Canale The graphical method can be used for n=3 (3 equations), but beyond, it will be very complex to determine the solution.

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Figure 9.2 Chapra and Canale Singular Ill-defined No solution Infinite solution However, this technique is very useful to visualize the properties of the solutions: -No so lut ion
Determinants and Cramer±s Rule ²A s y s t e m o f l i n e a r e q u a t i o n s c a n be written in matrix form: ² Where [A] is the coefficient matrix: >@ ^ ` ^ ` B x A >@ » » » ¼ º « « « ¬ ª 33 32 31 23 22 21 13 12 11 a a a a a a a a a A >@ » ¼ º «

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lecture5 - Numerical Methods in Engineering ENGR 391 System...

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