matrix - CGN 3421 Computer Methods Spr 1998 Solution of...

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CGN 3421 matrices page 1 of 10 7/1/98 CGN 3421 Computer Methods Spr 1998 Solution of Simultaneous Linear Equations Matrices can be used as an organized way of presenting a set of coupled equations. Generally speaking, we seek a single unique solution that satisfies all the equations at the same time. Consider the equations below: These are referred to as coupled linear equations . Coupled because each equation has one or more terms in common with the others, , so that a change in one of these variables will affect more than one equation. Linear because each equation contains only first order terms of . For example, there are no terms like , or , or , or , etc.Using the rules of matrix multiplication, we can represent the above equations in matrix form: We’ll refer to the coefficient matrix as , the vector of unknowns as , and the solution vector as . Note that we are seeking the unknowns vector , not the solution vector , which is known information. There are several ways to solve for the values of the unknown vector . Each method involves some manipulations to the coefficient matrix using algebraic rules, such that we create a new and equivalent problem in a more easily solvable form. These manipulations involve the addition of multiples of one row to another. Since each row 3 X 1 5 X 2 2 X 3 ++ 8 = 2 X 1 3 X 2 1 X 3 +1 = 1 X 1 2 X 2 –3 X 3 –1 = X 1 X 2 X 3 ,, X 1 X 2 X 3 X 1 2 X 2 X 3 (29 log 1 X 1 X 2 coefficient matrix A unknowns vector X solution vector B 35 2 23 1 12 X 1 X 2 X 3 8 1 1 = AX BX B X
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CGN 3421 matrices page 2 of 10 7/1/98 represents an equation, where the left hand side (l.h.s.) is equal to the right hand side (r.h.s.), adding one row to another results in an equivalent equation. For example, starting with the two equations: (1) (2) their addition (3) will not change the equality, or the solution to . This addition does not add any new information either, but it does present a new form of the old information, i.e. .( 4 ) We will use this technique of adding the equations together to recast the original problem into a form that is easier to solve. Gaussian Elimination (method #1): Let’s consider the three coupled linear equations given on the previous page. The original form looks like this: . (5) But what if we could recast the same problem to look like this? (6) This makes life easier, since there is less coupling between equations. In fact, can be solved immediately using the bottom equation ==> . Now the result can be used to write the middle equation as to get ==> . Finally, the known X 1 5 X 2 +3 = 2 X 1 3 X 2 –5 = X 1 5 X 2 2 X 1 3 X 2 5 + = X 1 X 2 , 1 X 1 –2 X 2 +8 = 35 2 23 1 12 –3 X 1 X 2 X 3 8 1 1 = 35 2 0 11 3 ----- 11 3 00 2 X 1 X 2 X 3 8 11 3 4 = X 3 X 3 2 = 11 3 X 2 11 3 ----- 2 (29 11 3 = X 2 1 =
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CGN 3421 matrices page 3 of 10 7/1/98 values are used to solve for in the first equation to get ==> . This easy
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matrix - CGN 3421 Computer Methods Spr 1998 Solution of...

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