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Lecture_14_E7_L14__linear_algebraic_equations_F07

Lecture_14_E7_L14__linear_algebraic_equations_F07 - E7 L14...

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Unformatted text preview: E7 L14 1 E7: INTRODUCTION TO COMPUTER E7: INTRODUCTION TO COMPUTER PROGRAMMING FOR SCIENTISTS AND PROGRAMMING FOR SCIENTISTS AND ENGINEERS ENGINEERS Lecture Outline (pages 358-376) 1. Linear algebraic equations 2. Basic matrix algebra 3. Solving linear algebraic equations with MATLAB (# of unknowns = # of equations) Copyright 2007, Horowitz, Packard. This work is licensed under the Creative Commons Attribution-Share Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. E7 L14 2 Linear algebraic equations Linear algebraic equations Many engineering and science problems reduce to the simultaneous solution of several linear algebraic equations. Some typical examples: • Curve-fitting of data • Governing physical phenomena such as • deformation of solids, flow of liquids, conduction of heat and electricity, etc. • Design of engineering systems – Example: simple truss structures • Consist of beams • Frictionless “pin” joints E7 L14 3 Linear algebraic equations Linear algebraic equations Consider n LINEAR equations and m unknowns. m unknowns n equations 11 12 1 1 21 22 2 2 1 1 2 2 2 1 1 2 m m n n m m n m m n A A A b A A A b A A A b x x x x x x x x x + + + = + + + = + + + = L L M L E7 L14 4 Linear algebraic equations Linear algebraic equations Consider n LINEAR equations and m unknowns. Think of the A ij ’s and b i ’s as known numbers j 11 12 1 1 21 22 2 2 1 1 2 2 2 1 1 2 m m n n m m n m m n A A A b A A A b A A A b x x x x x x x x x + + + = + + + = + + + = L L M L E7 L14 5 Example: intersection of two lines on a plane Example: intersection of two lines on a plane 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 intersection E7 L14 6 Example: intersection of two lines on a plane Example: intersection of two lines on a plane • The intersection of the two lines is the solution of: 2 equations 2 unknowns: x 1 and x 2 Gauss elimination is an efficient method for solving linear algebraic equations when the number of equations n is equal to the number of unknowns m E7 L14 7 Gauss elimination method Gauss elimination method n = 2 equations m = 2 unknowns • Multiply one equation by a factor and add to the other equation to eliminate one variable E7 L14 8 Solution using Gauss elimination method Solution using Gauss elimination method E7 L14 9 Solution using Gauss elimination method Solution using Gauss elimination method pivot add first equation to second equation result E7 L14...
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