![Introduction to Algorithms](https://www.bartleby.com/isbn_cover_images/9780262033848/9780262033848_largeCoverImage.gif)
To find the LU decomposition of Matrix A given as:
![Check Mark](/static/check-mark.png)
Explanation of Solution
LU Decomposition of A: The LU Decomposition of A means to decompose a matrix A into a product of lower triangular matrix and upper triangular matrix.
In Decomposition of A the input is lower triangular matrix and output is upper triangular matrix obtain by applying following row operation on A.
Here,
and
And lower triangular matrix is given by
Consider a matrix,
Initially calculate the value of all entries other than 0 and 1 of equation (1).
Apply the row operation to the matrix A to find the upper triangular matrix U
Calculate the value of
(2)
Calculate the value of ,
(3)
Calculate the value of
(4)
Calculate the value of
(5)
Apply row transformation to find upper triangular matrix.
Similarly calculate the value of,
…… (6)
Calculate the value of,
…… (7)
…… (8)
Apply row transformation to find upper triangular matrix.
Calculate the value of,
(9)
Calculate the value of,
(10)
Apply row transformation to find upper triangular matrix.
The upper triangular matrix is-
The lower triangular matrix is-
Hence, the LU decomposition of A is-
To solve the equation by using forward and backward substitution.
![Check Mark](/static/check-mark.png)
Explanation of Solution
Consider an expression-
To solve this, the following steps are followed-
Express in the
decomposition form-
Let for any
• LUP Decomposition
• Forward Substitution
Let
After solving, the following value of is obtained-
• Backward Substitution
Solving for
To find the inverse of A.
![Check Mark](/static/check-mark.png)
Explanation of Solution
Inverse of a matrix is a matrix
such that
where
is an identity matrix.
To solve this, the following steps are followed-
• Forward Substitution
Let
After solving, the following value of is obtained-
Similarly, other columns can also be obtained in the following way-
Inverse of the matrix can be formed by using the five results which comprises the columns of the inverse matrix.
• Backward Substitution
Now find from
Solving for
To show how, for any symmetric positive-definite, tridiagonal matrix
and any n-vector
to solve the equation
![Check Mark](/static/check-mark.png)
Explanation of Solution
Inverse of a matrix is a matrix
such that
where
is an identity matrix.
To solve this, the following steps are followed-
• Forward Substitution
Let
After solving, the following value of is obtained-
Similarly, other columns can also be obtained in the following way-
Inverse of the matrix can be formed by using the five results which comprises the columns of the inverse matrix.
• Backward Substitution
Now find from
Solving for
To show that, for any non-singular, tridiagonal matrix.
![Check Mark](/static/check-mark.png)
Explanation of Solution
Forward and Backward substitution take the same amount of operations as in the previous case. The only different case is the decomposition step. In this case, the decomposition is done into unit lower triangular and upper triangular matrix, and. The pivoting is done by the swapping rows such that the highest element of a column comes to the diagonal place. The decomposition looks like the following:
Here, loses the tri-diagonal property of matrix
but it is still a banded matrix. The essence to solve this is similar to the tri-diagonal case. The time complexity now depends on the width of the band or the distance between the farthest diagonals of the resultant matrix. Matrix becomes wider than the upper triangle of the initial matrix. The final complexity becomes
Hence the time complexity of LUP decomposition
Want to see more full solutions like this?
- STEP 1: The skeleton Let's start by creating a skeleton for some of the classes you will need. • Write a class called Tile. You can think of a tile as a square on the board on which the game will be played. We will come back to this class later. For the moment you can leave it empty while you work on creating classes that represents characters in the game. • Write an abstract class Fighter which has the following private fields: - A Tile field named position, representing the fighter's position in the game. A double field named health, representing the fighter's health points (HP). An int field named weaponType, representing the type of weapon the fighter is using. This value is used to rank different weapon types: higher values indicate higher weapon ranks. -An int field named attackDamage, representing the fighter's attack power. The class must also have the following public methods: 3 A constructor that takes as input a Tile indicating the position of the fighter, a double…arrow_forwardA company database needs to store information about employees (identified by SIN, with salary and phone as attributes), departments (identified by DID, with dname and budget as attributes), and children of employees (with name and age as attributes). Employees work in departments; each department is managed by an employee; a child must be identified uniquely by name when the parent (who is an employee; assume that only one parent works for the company) is known. We are not interested in information about a child once the parent leaves the company. Draw an ER diagram using Crows Foot notation that captures this information. Important: Must submit both ER Diagram and Relational Schema images in your solution here.arrow_forwardGiven the dependency diagram of attributes C1,C2,C3,C4,C5 in a table shown in the following figure, the primary key attributes are underlined. Make a database with multiple tables from attributes as shown above that are in 3NF, showing PK, non-key attributes, and FK for each table? Assume the tables are already in 1NF. Hint: 3 tables will result after deducing 1NF -> 2NF -> 3NF]arrow_forward
- 1. Using one of the method described in class and/or textbook (Section 9.1) convert the following regular expression into a state transition diagram: (0+ 10*1)* (01 + 10) Indicate in your answer how did you arrive at the result as follows: Write down all the state transition diagrams that you constructed for all the subexpressions and clearly indicate which diagram corresponds to which expression. Do not simplify any state transition diagram. 2. Consider the following state transition diagram over Σ = {a,b}: b A a a C b B a a b D За a Using the method described in class and in the textbook (Section 9.2) convert the diagram into an equivalent regular expression. Include all the intermediate steps in your answer. 3. Are the languages L1, L2, and L3 below over the alphabet Σ = {a, b, c} regular or non-regular? Justify your answer carefully. (a) L₁ = {a¹b2jc²i : i ≥ 0, j > 2} (b) L₂ = L₁n {akbm c³p: k,m,p≥ 0} (c) L3 = {a²ib²j+1 : i,j ≥ 0}^{akbm c³p : k,m,p ≥ 0}arrow_forward(1 point) By dragging statements from the left column to the right column below, give a proof by induction of the following statement: an = = 9" - 1 is a solution to the recurrence relation an = 9an-18 with ao = : 0. The correct proof will use 8 of the statements below. Statements to choose from: Note that a₁ = 9a0 + 8. Now assume that P(n) is true for all n ≥ 0. Your Proof: Put chosen statements in order in this column and press the Submit Answers button. Let P(n) be the predicate, "a = 9″ – 1". απ = 90 − 1 = Note that Let P(n) be the predicate, "an 9" - 1 is a solution to the recurrence relation an = 9an-1 +8 with ao = 0." - Now assume that P(k + 1) is true. Thus P(k) is true for all k. Thus P(k+1) is true. Then ak+1 = 9ak +8, so P(k + 1) is true. = 1 − 1 = 0, as required. Then = 9k — 1. ak Now assume that P(k) is true for an arbitrary integer k ≥ 1. By the recurrence relation, we have ak+1 = ak+1 = = 9ak + 8 = 9(9k − 1) + 8 This simplifies to 9k+19+8 = 9k+1 − 1 Then 9k+1 − 1 = 9(9*…arrow_forwardGiven the following request to an AI chatbot, which response is more accurate? (You should use Google to verify that the factual claims in each response are accurate.) Prompt Is this true? The first recorded use of the phrase "when pigs fly" in print is in a 1616 play by Ben Jonson called "The Devil is an Ass," in which a character says, "When pigs fly, good-night!!" Response Response A Response B Yes, that's true! The phrase "when pigs fly" indeed appears in Ben Jonson's 1616 play The Devil is an Ass and was said during the second act in a conversation between Fitzdottrel and Pug. The phrase was first used to express skepticism or disbelief, suggesting that something is as unlikely to happen as the fanciful idea of pigs flying. This usage is often cited as one of the earliest instances of the phrase in print. The claim that the phrase "when pigs fly" was first recorded in Ben Jonson's 1616 play "The Devil is an Ass" is not accurate. There is no evidence from the…arrow_forward
- This is for my Computer Organization & Assembly Language Classarrow_forwardPlease answer the homework scenario below and make a JAVA OOP code. You have been hired by GMU to create and manage their course registration portal. Your first task is to develop a program that will create and track different courses in the portal. Each course has the following properties: • a course number ex. IT 106, IT 206, • A course description, ex. Intro to Programming • Total credit hour ex. 3.0, and • current enrollment ex. 30 Each course must have at least a course number and credit hours. The maximum enrollment for each course is 40 students. The current enrollment should be no greater than the maximum enrollment. A course can have a maximum of 4 credit hour. The DDC should calculate the number of seats remaining for the course. Design an object-oriented solution to create a data definition class for the course object. The course class must define all the constructors, mutators with proper validation, accessors, and special purpose methods. The DDC should calculate the…arrow_forwardFor this case study, students will analyze the ethical considerations surrounding artificial intelligence and big data in healthcare, as explored in the case study found in the textbook (pages 34-36) and in the extended version available here There will also be additional articles in this weeks learning module to show both sides of the coin. https://www.delftdesignforvalues.nl/wp-content/uploads/2018/03/Saving-the-life-of-medical-ethics-in-the-age-of-AI-and-Big-Data.pdf Students should refer to the syllabus for specific guidelines regarding length, format, and content requirements. Reflection Questions to Consider: What are the key ethical dilemmas presented in the case? How does AI challenge traditional medical ethics principles such as autonomy, beneficence, and confidentiality? In what ways can responsible innovation help address moral overload in healthcare decision-making? What are the potential risks and benefits of integrating AI-driven decision-making into patient care?…arrow_forward
- Can you please solve this. Thanksarrow_forwardcan you solve this pleasearrow_forwardIn the previous homework scenario problem below: You have been hired by TechCo to create and manage their employee training portal. Your first task is to develop a program that will create and track different training sessions in the portal. Each training session has the following properties: • A session ID (e.g., "TECH101", "TECH205") • A session title (e.g., "Machine learning", "Advanced Java Programming") • A total duration in hours (e.g., 5.0, 8.0) • Current number of participants (e.g., 25) Each session must have at least a session ID and a total duration and must met the following requirements: • The maximum participant for each session is 30. • The total duration of a session must not exceed 10 hours. • The current number of participants should never exceed the maximum number of participants. Design an object-oriented solution to create a data definition class(DDC) and an implementation class for the session object. In the DDC, a session class must include: • Constructors to…arrow_forward
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks ColeC++ for Engineers and ScientistsComputer ScienceISBN:9781133187844Author:Bronson, Gary J.Publisher:Course Technology Ptr
![Text book image](https://www.bartleby.com/isbn_cover_images/9780534380588/9780534380588_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781133187844/9781133187844_smallCoverImage.gif)