Lecture26 - Algorithms in Systems Engineering IE170 Lecture...

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Unformatted text preview: Algorithms in Systems Engineering IE170 Lecture 26 Dr. Ted Ralphs IE170 Lecture 26 1 References for Today’s Lecture • Required reading – CLRS Chapter 28 IE170 Lecture 26 2 Systems of Equations • In some applications, we must determine values for a given set of unknowns , or variables , that satisfy one or more equations . • Example : IE170 Lecture 26 3 Linear Equations • A linear equation in n variables x 1 , . . . , x n is an equation of the form a 1 x 1 + a 2 x 2 + ··· + a n x n = b where a 1 , a 2 , . . . , a n and b are constants. • A solution to the equation is an assignment of values to the variables such that the equation is satisfied. • Suppose we interpret the constants a 1 , a 2 , . . . a n as the entries of an n-dimensional vector a . • Let’s also make a vector x out of the variables x 1 , x 2 , . . . , x n . • Then we can rewire the above equation as simply a T x = b . IE170 Lecture 26 4 Systems of Linear Equations • Suppose we are given a set of n variables whose values must satisfy more than one equation. • In this case, we have a system of equations , such as a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = b 1 (1) a 21 x 1 + a 22 x 2 + ··· + a 2 n x n = b 2 (2) . . . . . . (3) a m 1 x 1 + a m 2 x 2 + ··· + a mn x n = b m (4) where a ij is a constant for all 1 ≤ i ≤ m and 1 ≤ j ≤ n and b 1 , . . . , b m are constants....
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This note was uploaded on 02/29/2008 for the course IE 170 taught by Professor Ralphs during the Spring '07 term at Lehigh University .

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Lecture26 - Algorithms in Systems Engineering IE170 Lecture...

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