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# Mechanical and Aerospace Engineering Laboratory MAE 315

Syracuse

GPA 3.51

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This 20 page Class Notes was uploaded by Vinnie Hansen V on Wednesday October 21, 2015. The Class Notes belongs to MAE 315 at Syracuse University taught by Mark Glauser in Fall. Since its upload, it has received 70 views. For similar materials see /class/225608/mae-315-syracuse-university in Engineering Mechanical & Aero at Syracuse University.

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Date Created: 10/21/15

Underdetermined and Overdetermined Linear Algebraic Systems ESlOO March 1 1999 TS Whitten Objectives 0 De ne underdetermined systems 0 De ne overdetermined systems 0 Least Squares Examples Review SOON 2Fx20 500 FBCsin45020 sz 0 FBC cos4SO FBA 0 sin45O O FBC 500 005450 1 FBA O V coe icz ents variables Review cont sin45O O FgC 500 cos45O l 5amp4 O x V J E coe icienis variables The system of matrices above is of the form AX b and can be solved using MATLAB left division thus x Ab results in a lgtlt2 matrix of values for F BC and F B A Review Summary 0 A system of two Equations and two unknowns may yield a unique solution 0 The exception is when the determinant of A is equal to zero Then the system is said to be singular 0 The left division operator will solve the linear system in one step by combining two matrix operations OAB is equivalent to A391B Graphical Representation of Unique vs Singular Systems I unique solution I y singular system Underdetermined Systems 0 A system of linear equations is may be undetermined if 1 The determinant of A is equal to zero A 0 2 The matrix A is not square ie the are more unknowns than there are equations X3y2Z22 1 3 2 x 2 xyZ4 1 1 li39y 4 Z Overdetermined Systems 0 The converse of an underdetermined system is an overdetermined system Where there are more equations than there are variables 0 This situation arises frequently in engineering For example suppose a linear relationship is expected between x and y and there are multiple data points Data Distribution of Linear Phenomena A experimental data Data Distribution of Linear Phenomena A experimental data ymxb The line y mx b that best describes this data is Obtained by the method ofleast squares Method of Least Squares xpyl The line that results in the minimum value of J is the least squares linear t to the data 25 20 15 Y 10 Example 0 data linear t 5 10 15 LOOJNOUO lFOONAIH O data 300 316 566 815 1184 1385 1488 1767 1653 2165 The Overdetermined System 0 Once the curve t is obtained a y value may be interpolated for any xvalue Within the x data range sometimes extrapolation is possible 0 In the following example fmins is used to minimize the sum of the squared residuals with respect to the slope and intercept Flowchart for leastrn load datatxt J calculate line assign columns 4 plot i line call fmins l assign output plot data points 4 and line generate X vector Run leastm Solving The Overdetermined System Method 11 0 Solving the overdetermined system is carried out the same way as the other linear algebra solutions using the left division method m x AB b Method 11 cont d 0 If the system is not overdetermined the method will not work Overdetermined system cont 1mb300 data 1 300 5mb1184 2 316 3 566 10mb2165 4 815 5 1184 6 1385 1 300 2 1312 m 9 1653 5 1 b 1184 10 2165 10 1 2165 A X B Program least2m Load the data Build matrix A Build matrix b Print slope intercept Generate X y pairs using fit Left diVide parametersAB Plot data and titgenerated Xy set Run leasth

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