Contents

Module 1 Programming Assignment

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**Problem 1 ** Plot the curve of function

$\(f(x) = x^3 - x,~~ x \in[-1,1].\)$

# write your code for solving probelm 1 in this cell

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**Problem 2 **

(1) Find the two roots of the function

\[f(x) = 2x^2 - x - 1,~~ x \in[-1,2].\]

(2) Plot the curve of the function \(f(x)\) defined in (1) and mark all the roots on the curve.

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**Problem 3 **. Given \(P=\begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix}\), investigate the two different multiplications \(P*P\) and \(torch.mm(P,P)\).

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**Problem 4 ** Given

$\(A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, ~~ b=\begin{bmatrix} 1 \\ 2 \\ 3\end{bmatrix}.\)$

Solve \(A^2 x = b.\)

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**Problem 5 ** Given \(f(x,y) = x^2+y^2+(xy)^3\), compute \(\frac{\partial f(x,y)}{\partial y}\) at \(x=1,y=2\).

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**Problem 6 ** Define a function to find the minimum of three numbers \(a,~b,~c\). Test your code and print the minimum of the three numbers, where \(a=\sqrt{2},~ b=\frac{4}{3},~ c=0.5e\).

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**Problem 7 ** Define a function to find the maximum and minimum of a sequence with n numbers.

Hint: x = np.random.randint(a,b,size=n) can randomly generate n numbers (saved in a row vector x ) and each number is between a and b.

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**Problem 8 ** Define a function to sort a sequence with n numbers in ascending order.

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Problem 9 Given a function

$\(f(x,y)=(x-2)^2 + y^2 + 2(y-4)^2 + 2x^2 + 2xy + 4y -16x +1.\)$

Please write a code to apply gradient descent method to find the minimum of \(f(x,y)\) with initial value \(x=y=0\).