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Example: Least Squares for Polynomials

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Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Polynomial Interpolation

3 Activities

Wize University Linear Algebra Textbook > Least Squares Approximation

Least Squares Approximation

4 Activities

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Example: Least Squares for Polynomials

Find the equation of a quadratic polynomial that comes as close as possible to passing through the points: (1,2),(1,3),(2,3)(1,2),(1,3),(2,3).

More Polynomial Interpolation Questions:
Find an interpolating polynomial of degree at most 2 of the data points (1,2)\left(1,2\right), (2,0)\left(2,0\right), and (3,12)\left(3,12\right).
Applications
Example: Polynomial Interpolation
Find an interpolating polynomial for the points A(1,1),  B(2,2),  C(0,−2)A(1,1),\ \ B(2,2),\ \ C(0,-2).
Find an interpolating polynomial of degree at most 2 of the data points (0,2)\left(0,2\right), (1,−1)\left(1,-1\right), and (2,1)\left(2,1\right).
Polynomial of least degree
Find the interpolating polynomial of least degree for the data points (0,−3)\left(0,-3\right), (1,−1)\left(1,-1\right), and (−2,5)\left(-2,5\right).
Quadratic equations and matrices
Find a quadratic equation (of the form ax2+bx+c=0ax^2+bx+c=0) that passes through the points (0,-3), (1,-1) and (2,3).
Practice Question: Least Squares for Polynomials
Find the cubic polynomial which best fits the data points (−2,−1),(−1,1),(0,0),(1,−2),(2,3)(-2,-1),(-1,1),(0,0),(1,-2),(2,3) in the least squares sense.
Example: Least Squares for Linear Functions
Example: Least Squares for Linear Functions
Find the equation of the line that comes as close as possible to passing through the points: (0,1),(2,3),(3,4),(4,7)(0,1),(2,3),(3,4),(4,7).
Example: Least Squares for Linear Functions
Example: Least Squares for Linear Functions
Find the equation of the line that comes as close as possible, in the least squares sense, to passing through the points: (0,1),(2,3),(3,4),(4,7)(0,1),(2,3),(3,4),(4,7) .
Solution:
More Least Squares Approximation Questions:
Practice: Least Squares Approximation
Find the cubic polynomial which best fits the data points (−2,−1),(−1,1),(0,0),(1,−2),(2,3)(-2,-1),(-1,1),(0,0),(1,-2),(2,3).
19.4F_133_8.2_Mock_F1_$\tkco{eg2}$_$\key{Final}$_Builder_$\tkcth{8.2.}\tkcf{2}$_
Show that, in the case that the matrix A‾\A is invertible, the least squares solution corresponding to the system:
A‾ x⃗  =  b⃗, \A \, \vx \; = \; \vb,
is the same as the solution to the system itself.
Least Squares for Mix of Quadratic and Linear Functions
Find a least squares approximation of form a+bx+cx2+dya+bx+cx^2+dy for the following data:
x‾y‾result‾11112−121−3222011\begin{array}{r|r|r}\underline{x}&\underline{y}&\underline{result}\\1&1&1\\1&2&-1\\2&1&-3\\2&2&2\\0&1&1\end{array}
Least Squares
Find a function of the form a0+a1x+a3x3a_0+a_1x+a_3x^3 (note there is no squared term) which is a best approximation in the sense of least squares to the following data points: (0,1),(−1,1),(1,2),(−2,1)(0,1),(-1,1),(1,2),(-2,1) .
Practice Question: Least Squares for Polynomials
Find the cubic polynomial which best fits the data points (−2,−1),(−1,1),(0,0),(1,−2),(2,3)(-2,-1),(-1,1),(0,0),(1,-2),(2,3) in the least squares sense.
Example: Least Squares for non-Polynomials
Example: Least Squares for non-Polynomials
Find the function of form c1sin⁡(x)+c2cos⁡(x)c_1\sin(x)+c_2\cos(x) which best fits the data points (−π,−3),(0,−4),(π/2,−5)(-\pi,-3),(0,-4),(\pi/2,-5) in the least squares sense.
Solution:
Example: Least Squares for Linear Functions
Example: Least Squares for Linear Functions
Find the equation of the line that comes as close as possible to passing through the points: (0,1),(2,3),(3,4),(4,7)(0,1),(2,3),(3,4),(4,7).
Example: Least Squares for Linear Functions
Example: Least Squares for Linear Functions
Find the equation of the line that comes as close as possible to passing through the points: (0,1),(2,3),(3,4),(4,7)(0,1),(2,3),(3,4),(4,7).
Example: Least Squares for Linear Functions
Example: Least Squares for Linear Functions
Find the equation of the line that comes as close as possible, in the least squares sense, to passing through the points: (0,1),(2,3),(3,4),(4,7)(0,1),(2,3),(3,4),(4,7) .
Solution: