Underdetermined and Overdetermined Systems and Least Squares
Underdetermined and Overdetermined Systems and Least Squares
Underdetermined and Overdetermined Systems and Least Squares
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<strong>Underdetermined</strong><strong>and</strong><strong>Overdetermined</strong>Linear Algebraic <strong>Systems</strong>ES100March 1, 1999T.S. Whitten
Objectives● Define underdetermined systems● Define overdetermined systems● <strong>Least</strong> <strong>Squares</strong> Examples
ReviewB500 N∑F x= 0 ; 500 − F sin 45°=BC0F BC∑Fy=0 ; F cos 45° − F =BCBA0F BA⎡−sin 45°⎢⎣ cos 45°142 4coefficients0⎤⎥ ⋅⎦143 4⎡F⎤ BC⎢ ⎥123 ⎣FBA⎦variables=⎡−⎢⎣500⎤⎥0 ⎦
Review cont.⎡−sin 45°⎢⎣ cos 45°142 4coefficients0⎤⎥ ⋅⎦143 4⎡F⎤ BC⎢ ⎥123 ⎣FBA⎦variables=⎡−⎢⎣500⎤⎥0 ⎦The system of matrices above is of the form:Ax = b<strong>and</strong> can be solved using MATLAB left division thus, x = A\bresults in a 1×2 matrix of values for F BC <strong>and</strong> F BA
Review Summary● A system of two Equations <strong>and</strong> twounknowns may yield a unique solution.● The exception is when the determinant of A isequal to zero. Then the system is said to besingular.● The left division operator will solve thelinear system in one step by combining twomatrix operations●A\B is equivalent to A -1 *B
Graphical Representation ofUnique vs. Singular <strong>Systems</strong>unique solutionyxsingular system
<strong>Underdetermined</strong> <strong>Systems</strong>● A system of linear equations is may beundetermined if;1 The determinant of A is equal to zeroA = 02 The matrix A is not square, i.e. the are moreunknowns than there are equationsx + 3 y + 2z=x+ y + z= 42⎡1⎢⎣131⎡x⎤2⎤⎢ ⎥ ⎡ ⎤⎥ ⋅ 2⎢y⎥=1⎢ ⎥⎦⎢ ⎥⎣4⎦⎣z⎦
<strong>Overdetermined</strong> <strong>Systems</strong>● The converse of an underdetermined systemis an overdetermined system where there aremore equations than there are variables● This situation arises frequently inengineering. For example: suppose a linearrelationship is expected between x <strong>and</strong> y <strong>and</strong>there are multiple data points.
Data Distribution of LinearPhenomenayexperimental datax
Data Distribution of LinearPhenomenayexperimental datay = mx + bxThe line y = mx + b, that best describes thisdata is obtained by the method of least squares
Method of <strong>Least</strong> <strong>Squares</strong>}(x 1 ,y 1 )(x 2 ,y 2 )J = ∑( mx )i+ b−y iJ {i=12}J 1J 3(x 3 ,y 3 )nThe line that results in theminimum value of J isthe least squares linear fitto the data.2
Exampley2520151050datalinear fit0 5 10 15t data1 3.002 3.163 5.664 8.155 11.846 13.857 14.888 17.679 16.5310 21.65t
The <strong>Overdetermined</strong> System● Once the curve fit is obtained, a y-value maybe interpolated for any x-value within the x-data range (sometimes extrapolation ispossible).● In the following example, fmins is used tominimize the sum of the squared residualswith respect to the slope <strong>and</strong> intercept.
Flowchart for least.mload data.txtassign columnscall fminsassign outputcalculatelineplotlineplot data points<strong>and</strong> linegenerate xvector
Run least.m
Solving The <strong>Overdetermined</strong>System, Method II● Solving the overdetermined system iscarried out the same way as the other linearalgebra solutions using the left divisionmethod⎡m⎤x = ⎢ = A \b⎥⎣ ⎦B
Method II (cont’d)● If the system is not overdetermined, themethod will not work
<strong>Overdetermined</strong> system cont.⎡ 1⎢⎢5⎢⎣101m+ b5m+ b10m+ b111⎤⎥⎥⎥⎦⋅⎡⎢⎣mb=3.00= 11.84= 21.65⎤⎥⎦=⎡ 3.00 ⎤⎢ ⎥⎢11.84⎥⎢⎣21.65⎥⎦A x Bt data1 3.002 3.163 5.664 8.155 11.846 13.857 14.888 17.679 16.5310 21.65
Program least2.mStartLoad the dataBuild matrix APrint slope,interceptGenerate x, y pairsusing fitBuild matrix bLeft divide(parameters=A\B)Plot data<strong>and</strong>fit-generated x,y setStop
Run least2.m