Simultaneous equations 11 Row echelon form

Transcription

1 1 Simultaneous equations 11 Row echelon form J A Rossiter For a neat organisation of all videos and resources

2 Introduction 2 The previous videos demonstrated that both a matrix inverse and Cramer s rule can be used to solve linear simultaneous equations. However, both of these methods are very inefficient for large numbers of unknowns. Consequently, we need to consider more numerically efficient methods that can deal with 10s and 100s of unknowns. This video introduces the row-echelon form which is one tool used by efficient solvers.

3 Two linear simultaneous equations When solving 2 equations in 2 unknowns, the basic technique is to eliminate one variable. x + 2y = 3 Add row 1 to row 2 to eliminate x. x + 3y = 1 5y = 4 Write one of the original equations alongside the new equation and then put in matrix form. x + 2y = 3 0x + 5y = 4 This matrix is, in effect, triangular due to the zero x y = 3 4 We can now solve for y and then use this to find x. 3

4 Solving 3 equations The typical technique is to use the equations to eliminate one variable at a time, and then back substitute. x + 2y + z = 3 5x + 3y z = 1 2x + 2y 4z = 5 4x + 5y = 4 6x + 10y = 17 Add row 1 to row 2 to eliminate z. Add 4 times row 1 to row 3 to eliminate z. Subtract 2 times row 1 from row 2 to eliminate y. 4 14x = 9 We can now solve for x, use this to find y and then use both to find z.

5 3 unknowns final form Let us look at the equations we ended up with having done the variable elimination and rearrange 3 of these into matrix form. 5 x + 2y + z = 3 5x + 3y z = 1 2x + 2y 4z = 5 4x + 5y = 4 6x + 10y = 17 This matrix is triangular! x y z = x = 9

6 Triangular forms 6 Consider a set of equations represented as follows. a b c 0 e f 0 0 j x y z = k m n This matrix is upper triangular. It is now straightforward to solve for all the unknowns using a sequence. Use the 3 rd row to solve for z using: j z =n Use the 2 nd row to solve for y using: ey = m- fz Use the 1 st row to solve for x using: ax = k- by - cz

7 Proposed methodology If we can represent our linear simultaneous equations using the form AX=B where the matrix A is upper (or lower) triangular, then the equations can be solved very efficiently x y z w = Use the 4 th solve for w row to Use the 3rd row to solve for z using w. 7 Use the 2 nd row to solve for y using w, z. Use the 1 st row to solve for x using w, z y.

8 Summary observation If we can represent our linear simultaneous equations using the form AX=B where the matrix A is upper (or lower) triangular, then the equations can be solved very efficiently This is known as row-echelon form. x y z w = Efficient solvers first aim to modify the equations into row-echelon form.

9 Summary 9 This brief resource introduced the row-echelon form. If the linear simultaneous equations, in matrix form, have a triangular matrix, then the unknowns can be solved very efficiently. A popular numerical technique is to force the A matrix into triangular form; this is introduced in the next video.

10 Anthony Rossiter Department of Automatic Control and Systems Engineering University of Sheffield For a neat organisation of all videos and resources /indexwebbook.html 2018 University of Sheffield This work is licensed under the Creative Commons Attribution 2.0 UK: England & Wales Licence. To view a copy of this licence, visit or send a letter to: Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA. It should be noted that some of the materials contained within this resource are subject to third party rights and any copyright notices must remain with these materials in the event of reuse or repurposing. If there are third party images within the resource please do not remove or alter any of the copyright notices or website details shown below the image. (Please list details of the third party rights contained within this work. If you include your institutions logo on the cover please include reference to the fact that it is a trade mark and all copyright in that image is reserved.)