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Sonbull [250]
3 years ago
14

Formulate the following problem as least squares problems. For each problem, give a matrixA and a vector b such that the problem

can be expressed asargmin)‖Ax − b‖..(you are not asked to solve the problems. Just state define matrix A and vector b)a. minimize x/. + 2x.. + 3x3. + (x/ − x. + x3 − 1). + (−x/ − 4x. + 2).;b. minimize x8x + ||Bx − d||., where the p × n matrix B and the p-vector d are givenc. minimize ||Bx − d||.+2||Fx − g||.. The p × n matrix B, the ???? × n matrix F, the p-vector d and the ????-vector g are given.
Mathematics
1 answer:
hammer [34]3 years ago
6 0

Answer:

a) A=\left[\begin{array}{ccc}1&2&3\\1&-1&1\end{array}\right]

b=\left[\begin{array}{ccc}0\\1\end{array}\right]

b) ||Ax-b||^{2} =(-bx_{2}+4)^{2}  (-4x_{1} +3x_{2} -1)^{2} +(x_{1} +8x_{2} -3)^{2}

c) A=\left[\begin{array}{ccc}0&6\sqrt{2} &0\\\sqrt{3} &3\sqrt{3} &0\\2&-16&0\end{array}\right]

x=\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right]

b=\left[\begin{array}{ccc}-\sqrt{2} \\\sqrt{3} \\6\end{array}\right]

Step-by-step explanation:

a) considering the equation:

Minimize x_{1}^{2}  +2x_{2}x^{2}  +3x_{3}^{2}+(x_{1}   -x_{2} +x_{3} -1)^{2} +(-x_{1} -4x_{2} +2)^{2}

A=\left[\begin{array}{ccc}1&2&3\\1&-1&1\end{array}\right] (matrix A)

vector b

b=\left[\begin{array}{ccc}0\\1\end{array}\right]

b) If Pxn is matrix B and p-vector d, we have:

minimize (-6x_{2}+4)^{2}  +(-4x_{1} +3x_{2} -1)+(x_{1} +8x_{2} -3)^{2}

Ax=\left[\begin{array}{ccc}0&-6&0\\-4&3&0\\1&8&0\end{array}\right]

\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right]

b=\left[\begin{array}{ccc}-4\\1\\3\end{array}\right]

Ax-b=\left[\begin{array}{ccc}-bx_{2}+4 \\-4x_{1}+3x_{2}-1  \\x_{1}+8x_{2}-3  \end{array}\right] =1

||Ax-b||^{2} =(-bx_{2}+4)^{2}  (-4x_{1} +3x_{2} -1)^{2} +(x_{1} +8x_{2} -3)^{2}

c) minimize 2(-bx_{2}+4)^{2}  +3(-4x_{1} +3x_{2} -1)^{2} +4(x_{1} -x_{2} -3)^{2} -(6\sqrt{2}x_{2}  +4\sqrt{2} )^{2} +(-4\sqrt{3} x_{1} +3\sqrt{3}x_{2}  -\sqrt{3})^{2}  +(2x_{1} -16x_{2} -6)^{2}

in matrix:

A=\left[\begin{array}{ccc}0&6\sqrt{2} &0\\\sqrt{3} &3\sqrt{3} &0\\2&-16&0\end{array}\right]

x=\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right]

b=\left[\begin{array}{ccc}-\sqrt{2} \\\sqrt{3} \\6\end{array}\right]

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