
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,
Want to see more full solutions like this?
Chapter 14 Solutions
Numerical Methods for Engineers
Additional Math Textbook Solutions
Pathways To Math Literacy (looseleaf)
Precalculus
Elementary Statistics ( 3rd International Edition ) Isbn:9781260092561
Intermediate Algebra (13th Edition)
Probability And Statistical Inference (10th Edition)
APPLIED STAT.IN BUS.+ECONOMICS
- Recall the RSA encryption/decryption system. The following questions are based on RSA. Suppose n (=15) is the product of the two prime numbers 3 and 5.1. Find an encryption key e for for the pair (e, n)2. Find a decryption key d for for the pair (d, n)3. Given the plaintext message x = 3, find the ciphertext y = x^(e) (where x^e is the message x encoded with encryption key e)4. Given the ciphertext message y (which you found in previous part), Show that the original message x = 3 can be recovered using (d, n)arrow_forwardTheorem 1: A number n ∈ N is divisible by 3 if and only if when n is writtenin base 10 the sum of its digits is divisible by 3. As an example, 132 is divisible by 3 and 1 + 3 + 2 is divisible by 3.1. Prove Theorem 1 2. Using Theorem 1 construct an NFA over the alphabet Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}which recognizes the language {w ∈ Σ^(∗)| w = 3k, k ∈ N}.arrow_forwardRecall the RSA encryption/decryption system. The following questions are based on RSA. Suppose n (=15) is the product of the two prime numbers 3 and 5.1. Find an encryption key e for for the pair (e, n)2. Find a decryption key d for for the pair (d, n)3. Given the plaintext message x = 3, find the ciphertext y = x^(e) (where x^e is the message x encoded with encryption key e)4. Given the ciphertext message y (which you found in previous part), Show that the original message x = 3 can be recovered using (d, n)arrow_forward
- Find the sum of products expansion of the function F(x, y, z) = ¯x · y + x · z in two ways: (i) using a table; and (ii) using Boolean identities.arrow_forwardGive both a machine-level description (i.e., step-by-step description in words) and a state-diagram for a Turing machine that accepts all words over the alphabet {a, b} where the number of a’s is greater than or equal to the number of b’s.arrow_forwardCompute (7^ (25)) mod 11 via the algorithm for modular exponentiation.arrow_forward
- Prove that the sum of the degrees in the interior angles of any convex polygon with n ≥ 3 sides is (n − 2) · 180. For the base case, you must prove that a triangle has angles summing to 180 degrees. You are permitted to use thefact when two parallel lines are cut by a transversal that corresponding angles are equal.arrow_forwardAnswer the following questions about rational and irrational numbers.1. Prove or disprove: If a and b are rational numbers then a^b is rational.2. Prove or disprove: If a and b are irrational numbers then a^b is irrational.arrow_forwardProve the following using structural induction: For any rooted binary tree T the number of vertices |T| in T satisfies the inequality |T| ≤ (2^ (height(T)+1)) − 1.arrow_forward
- (a) Prove that if p is a prime number and p|k^2 for some integer k then p|k.(b) Using Part (a), prove or disprove: √3 ∈ Q.arrow_forwardProvide a context-free grammar for the language {a^ (i) b^ (j) c^ (k) | i, j, k ∈ N, i = j or i = k}. Briefly explain (no formal proof needed) why your context-free grammar is correct and show that it produces the word aaabbccc.arrow_forwardDo College Students With Part-Time Jobs Sleep Less? College students were surveyed about the number of hours they sleep each night.Group A = With part-time jobs | Group B = Without jobs Group A: 6, 5, 7, 6, 5Group B: 8, 7, 9, 8, 7 Instructions: State your hypothesis and perform a two-sample t-test with all formulas. Create histograms for each group. Label axes and add titles. Comment on the distribution shape (e.g., normal, skewed, etc.).Solve on pen and paperarrow_forward
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY
- Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,





