Find the gradient
(a)
(b)
(c)
(a)
To calculate: The gradient vector and Hessian matrix for the function
Answer to Problem 5P
Solution:
The gradient vector for the function
The Hessian matrix for the function
Explanation of Solution
Given:
The function
Formula used:
The gradient vector for the function
The Hessian matrix for the function
Calculation:
Consider the function,
Partial differentiate the above function with respect to x,
Again, partial differentiate the above equation with x,
Partial differentiate the
Now, partial differentiate the function
Again, partial differentiate the above equation with y,
Partial differentiate the
Therefore, the gradient vector for the function is,
And, the Hessian matrix for the function is,
(b)
To calculate: The gradient vector and Hessian matrix for the function
Answer to Problem 5P
Solution:
The gradient vector for the function
The Hessian matrix for the function
Explanation of Solution
Given:
The function
Formula used:
The gradient vector for the function
The Hessian matrix for the function
Calculation:
Consider the function,
Partial differentiate the above function with respect to x,
Again, partial differentiate the above equation with x,
Partial differentiate the
Partial differentiate the
Now, partial differentiate the function
Again, partial differentiate the above equation with y,
Partial differentiate the
Partial differentiate the
Now, partial differentiate the function
Again, partial differentiate the above equation with z,
Partial differentiate the
Partial differentiate the
Therefore, the gradient vector for the function is,
And, the Hessian matrix for the function is,
(c)
To calculate: The gradient vector and Hessian matrix for the function
Answer to Problem 5P
Solution:
The gradient vector for the function
The Hessian matrix for the function
Explanation of Solution
Given:
The function
Formula used:
The gradient vector for the function
The Hessian matrix for the function
Calculation:
Consider the function,
Partial differentiate the above function with respect to x,
Again, partial differentiate the above equation with x,
Simplify furthermore,
Partial differentiate the
Simplify furthermore,
Now, partial differentiate the function
Again, partial differentiate the above equation with y,
Simplify furthermore,
Partial differentiate the
Simplify furthermore,
Therefore, the gradient vector for the function is,
And, the Hessian matrix for the function is,
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