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12.6: Over-Determined Systems of Equations

  • Page ID
    85172
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    Consistent Equations

    Simple examples of consistent over-determined system of equations:

    • One equation is a multiple of another equation
    • One equation is the of 2 other equations

    The rank of a consistent set of equations is equal to the number of unknowns.

    Example \(\PageIndex{1}\) "Overdetermined" consistent system: 3 equations, 2 unknowns

    x = -1 : 0.2 : 4;
    y1 = 5 - 1*x;
    y2 = -1 + 2*x;
    y3 = 2 + 0.5*x;

    The 3 lines intersect in 1 point, so these equations are consistent.

    Graphical intersection at (2, 3)

    Intersection_of_3_Lines.png
    %% Matrix reformulation:

    % Put the variables on the left and the constants on the right.
    % 1*x + y = 5
    % -2*x + y = -1
    % -0.5*x + y = 2

    % A*xy = B
    A = [ 1 1
    -2 1
    -0.5 1]
    B = [ 5
    -1
    2]
    rank(A) % 2
    xy = A\B % Matrix algebra solution = (2, 3)

    Solution

    Add example text here.

    .

    Exercise \(\PageIndex{1}\) Over-Determined Consistent System of Equations HW1

    Over-Determined consistent system of equations of 4 equations in 3 unknowns.

    A3 = [3 2 5
    4 5 -2
    1 1 1
    2 -4 -7]
    % X3 = [x y z]'
    B3 = [22
    8
    6
    -27]

    Use Left matrix-division to solve this system of equations for X3
    Check your solution by computing this:
    X3ck = A3*X3

    The result should be close to B3.

    Answer

    Add texts here. Do not delete this text first.

    .

    Inconsistent Equations

    Example \(\PageIndex{2}\) "Overdetermined" inconsistent system: 3 equations, 2 unknowns

    x = -1 : 0.2 : 4;
    y1 = 5 - 1*x;
    y2 = -1 + 2*x;
    y3 = 1.6 + 0.5*x;

    % The 3rd equation has been modified, so the 3 lines do not intersect in a single point.

    Measurement Errors were added to the right-hand-side values of the consistent version of this exercise. Extra measure ments are often taken to help average-out measurement errors.

    These these equations are inconsistent. Rather, there are 3 intersection points of each pair of lines.
    Inconsistent_3_Lines.png

    %% We can still use left-division (Gaussian elimination)
    % to find a good approximate solution.
    % Matrix formulation
    % 1*x + y = 5
    % -2*x + y = -1
    % -0.5*x + y = 1.6]
    % xy = [x
    % y]
    % A*xy = B]
    A = [ 1 1
    -2 1
    -0.5 1]
    B = [ 5
    -1
    1.6]
    rank(A) % 2

    xy = A\B % Solution = (2, 2.87)

    Solution

    Add example text here.

    .

    Exercise \(\PageIndex{2}\) Over-Determined Inconsistent System of Equations HW2

    Over-Determined inconsistent system of equations of 4 equations in 3 unknowns.

    Measurement Errors were added to the right-hand-side values of the consistent version of this exercise. Extra measure ments are often taken to help average-out measurement errors.

    A4 = [3 2 5
    4 5 -2
    1 1 1
    2 -4 -7]

    % X4 = [x y z]'
    B4 = [22+1
    8-0.5
    6+1
    -27-1]

    Use Left matrix-division to solve this system of equations for X4
    Check your solution by computing this:
    X4ck = A4*X4

    The result should be somewhat close to B4, but will not match because these equations are inconsistent.

    Answer

    Add texts here. Do not delete this text first.

    .


    This page titled 12.6: Over-Determined Systems of Equations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Carey Smith.