Question

Use least squares method to find a quadratic polynomial that fits the following data:  (-2, 15.7), (-1, 6.7), (0, 2.7), (1, 3.7), (2, 9.7).

06 Feb 2024
Answer :
Word Count : 811

To find a quadratic polynomial that fits the given data using the least squares method, we want to minimize the sum of the squared differences between the actual y-values and the y-values predicted by the quadratic polynomial. 

Let's assume the quadratic polynomial has the form:

\[ y = ax^2 + bx + c \]

We want to find the coefficients \( a \), \( b \), and \( c \) that minimize the sum of the squared errors. We'll set up the equations using the given data points:

1. For the point \((-2, 15.7)\):
\[ 15.7 = a(-2)^2 + b(-2) + c \]

2. For the point \((-1, 6.7)\):
\[ 6.7 = a(-1)^2 + b(-1) + c \]

3. For the point \((0, 2.7)\):
\[ 2.7 = a(0)^2 + b(0) + c \]

4. For the point \((1, 3.7)\):
\[ 3.7 = a(1)^2 + b(1) + c \]

5. For the point \((2, 9.7)\):
\[ 9.7 = a(2)^2 + b(2) + c \]

Now, let's solve this system of equations to find \( a \), \( b \), and \( c \).

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Related Questions

Let

A = \begin{bmatrix} 2 & 2 & 1\\ -1 & -1 & 2\\ 0 & 0 & -2 \end{bmatrix}

Find a unitary matrix U such that U^*AU is upper triangular.

Use least squares method to find a quadratic polynomial that fits the following data:  (-2, 15.7), (-1, 6.7), (0, 2.7), (1, 3.7), (2, 9.7).

Find the SVD of the following matrices:

i). \begin{bmatrix} -1 & 1 & 1\\ 1 & 1 & 0\\ \end{bmatrix}

ii). \begin{bmatrix} -1 & 1 \\ 1 & 1 \\ 1 & 2 \end{bmatrix}

Which of the following statements are true and which are false? Give reasons for your answer

i) If V is a finite dimensional vector space and T:V \to V is a diagonalisable linear operator, then there is a basis, unique up to order of the elements, with respect to which the matrix of T
is diagonal.
ii) Up to similarity, there is a unique 3 \times 3 matrix with minimal polynomial (x-1)^2(x-2).
iii) If \lambda is the eigenvalue of a matrix

Let T : C^2 \to C^2 : T\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x + 2y - iz \\ 2y + iz \\ ix + z - 2z \end{bmatrix}.  FInd [T]B ,  [T]B' and P where

B = \left \{ \begin{bmatrix} 0\\ i\\ 0 \end{bmatrix}, \begin{bmatrix} i\\ 1\\ -1 \end{bmatrix} , \begin{bmatrix} 0\\ 0\\ 2 \end{bmatrix} \right \}, B' = \left \{ \begin{bmatrix} 1\\ -i\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} , \begin{bmatrix} 1\\ i\\ 0 \end{bmatrix} \right \},

[T]_{B'} = P^{-1}[T]_BP​​​​​​​

Find the Jordan canonical form J for

B = \begin{bmatrix} -1 &0 & -2 & -4 \\ 2 & 1 & 2 & 4 \\ -4 & 2 & -1 & -4\\ 2 & -1 & 1 & 3 \end{bmatrix}

Also, find a matrix P such that J = P-1BP

Check which of the following matrices is positive definite and which is positive semi-definite:

A = \begin{bmatrix} 1 & 1 & 0\\ 1 & 2 & 1\\ 0 & 1 & 1 \end{bmatrix}, B \begin{bmatrix} 2 & 0 & 1\\ 0 & 2 & -1\\ 1 & -1 & 3 \end{bmatrix}

Also, find the square root of the positive definite matrix.

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