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Author Archives: Academic Wizard

June 24, 2025
June 24, 2025

DNA vs. Physical Evidence

Τhis  week I would like each student to post a question. Use Compose for your  topic so that each of you start a new thread. Note the topic in the  subject line. Your topic can be about anything covered in the course.  For full credit you need to post one question and respond to one of your  classmates. If you do not understand these instructions, email me.

DNA vs. Physical Evidence

An example would be:  “A young woman is missing. After more than two weeks, her body was  found. Investigators believed she has been sexually assaulted and killed  the same day since she was gone. Do you think it is possible to find  sperm in her vaginal vault? If not, why?”

OR

DNA vs. Physical Evidence

“What are your thoughts/opinions on the collection of DNA from  arrestees for submission into CODIS in accordance with the  Katie Sepich Enhanced DNA Collection Act of 2010, also known as Katie’s  Law?”

AND

Type of Evidence

In an alleged rape case, the victim comes forward 10 days after  the commission of the crime. The victim has showered several times  washing away any potential DNA evidence.  The clothes she wore were  discarded and long gone.  You are on a jury that identifies the  potential rape suspect based on the collection of a fingerprint and a  shoe impression obtained from the scene of the crime?  How  strongly would you feel about this identification in comparison to an  identification made based solely off of DNA evidence in this scenario?   What are the differences or similarities?

DNA vs. Physical Evidence

  1. What kind of question is each student required to post?,

  2. What are possible question examples for full credit?,

  3. What is Katie’s Law, and what is its purpose?,

  4. How reliable is evidence like fingerprints and shoe impressions compared to DNA?,

  5. What are the differences between physical and DNA evidence in sexual assault cases?

June 24, 2025
June 24, 2025

Felony Investigation Documentation

· In your discussion, please identify the requirements for documenting felony investigations involving people.

· Please discuss the difference in the types of information that might be required and compare that information to the information that might be required for felony investigations involving property

· Finally, please discuss the role of effective note taking and accurate, clear writing when documenting felony investigations regarding people as compared to felony investigations and documentation involving property.

Felony Investigation Documentation

Felony Investigation Documentation

  1. What are the requirements for documenting felony investigations involving people?,

  2. What types of information are required in felony investigations involving people?,

  3. What types of information are required in felony investigations involving property?,

  4. How does information differ between people-focused and property-focused felony investigations?,

  5. What is the role of effective note-taking and writing in documenting both types of felony investigations?


Comprehensive General Response:

1. Requirements for Documenting Felony Investigations Involving People:

When documenting felony investigations that involve individuals (victims, suspects, witnesses), investigators must ensure thorough and legally sound reporting. Key requirements include:

  • Identification details (names, dates of birth, addresses, physical descriptions).

  • Statements from suspects, victims, and witnesses, recorded verbatim or summarized with clear attribution.

  • Behavioral observations (e.g., emotional state, demeanor during interviews).

  • Chronology of events leading up to, during, and following the incident.

  • Evidence collected, particularly linking individuals to the crime (e.g., DNA, fingerprints, surveillance footage).

  • Miranda warnings documentation when applicable.

  • Use of force or arrest details if they occurred.

This information must be detailed, accurate, and capable of standing up to scrutiny in court.


2. Types of Information in People-Focused Investigations:

Investigations involving people emphasize:

  • Personal interactions and relationships.

  • Felony Investigation Documentation
  • Motive, intent, and behavior.

  • Psychological or emotional states.

  • Injuries sustained and medical reports.

  • Witness reliability and credibility assessments.

This type of documentation often requires more narrative explanation and subjective assessment than property crimes.


3. Types of Information in Property-Focused Investigations:

Property crime documentation focuses on:

  • Detailed property description (type, value, serial numbers).

  • Condition before and after the crime (e.g., damage, theft).

  • Point of entry/exit if a break-in occurred.

  • Security measures in place (e.g., alarms, locks).

  • Ownership documentation (receipts, titles).

  • Scene sketches/photos showing property layout and disturbances.

Property-related documentation is typically more technical and objective, with emphasis on physical evidence and valuation.


4. Comparison Between People and Property Investigations:

  • Nature of Evidence: People-related cases rely heavily on testimonial and behavioral evidence, while property cases focus on physical and forensic evidence.

  • Complexity of Narratives: People cases involve interpersonal dynamics and often require in-depth interviews. Property cases often follow a more linear, factual account.

  • Emotional Sensitivity: Crimes against individuals often involve trauma and require more careful, sensitive documentation.

  • Legal Considerations: Documentation involving people often includes constitutional considerations (e.g., rights advisement), which are less prevalent in property crimes.


5. Role of Effective Note-Taking and Writing:

Clear and accurate documentation is vital in all felony investigations, but its importance differs slightly between the two types:

  • In People-Related Investigations:

    • Notes must capture nuances of statements and behaviors.

    • Precise wording is critical to avoid misinterpretation in court.

    • Emotional and psychological contexts must be respectfully and accurately conveyed.

  • In Property-Related Investigations:

    • Accuracy in details (e.g., serial numbers, dimensions) ensures evidence can be traced and verified.

    • Organized writing supports reconstruction of events like burglaries or vandalism.

    • Clear diagrams, inventory lists, and technical descriptions strengthen the case.

In both cases, well-kept notes ensure that investigators can later write thorough, credible reports and testify confidently in court.

June 23, 2025
June 23, 2025

To prepare:

  • Review what you have learned about evaluating research articles in terms of their relationship to your discipline and to your proposed topic.
  • Review the articles in your Research Articles Summaries and apply the evaluation questions to each one. You may need to reopen the article to do this.
  • Review the video resources on finding the gap in the research literature related to your topic.

Research Gap Evaluation

Post your response to the following:

  • Last week, you described the steps to identifying a research gap in literature. Describe how that process went as you reviewed previous studies and any new ones you have identified for this week’s Assignment. For example, what challenges in evaluating a research study are you encountering?

Note: Your posts should be substantial (500 words minimum), supported with scholarly evidence from your research and/or the Learning Resources, and properly cited using APA style.

Research Gap Evaluation

  1. What steps did you follow to identify the research gap in the literature?

  2. How did reviewing previous studies help you in that process?

  3. Did you find any new articles this week to support your topic?

  4. What challenges are you encountering while evaluating research studies?

  5. How are you applying evaluation questions to each article?

Identifying a research gap within forensic psychology literature has been both enlightening and complex. The process required careful reading, comparative analysis, and critical thinking. My topic of focus involves the psychological assessment of criminal offenders and how forensic evaluations influence court decisions, particularly surrounding issues of competency and risk assessment. To find a relevant gap, I had to review various empirical studies, meta-analyses, and literature reviews, while applying key evaluation criteria—such as relevance to forensic psychology, clarity of research objectives, methodology, and findings.

The first step in identifying the research gap was to compile a comprehensive list of peer-reviewed articles related to my topic. I began by reviewing the sources used in my earlier Research Article Summaries and expanded my search to include additional databases such as PsycINFO, Academic Search Complete, and Criminal Justice Abstracts. I also used specific search terms like “forensic psychological assessment,” “competency to stand trial,” “risk assessment tools,” and “courtroom decision-making.”

As I reviewed these studies, I applied questions such as:

  • What was the purpose of the research?,

  • Is the study relevant to forensic psychology?,

  • What methodology was used, and is it sound?,

  • Are there limitations or biases in the study?,

  • What are the implications and suggestions for futu

Research Gap Evaluation

June 23, 2025
June 23, 2025

Nursing Inspiration Board

1. Inspiration (45 points/30%)

1. Creates a one-page vision board that reflects own inspiration and motivation for becoming a nurse. The format may be a PowerPoint slide, word document, infographic etc.

Nursing Inspiration Board

2. Includes pictures, a collage, a series of quotes, etc. Choose words and images that are meaningful to you. This should be a page that you can post somewhere, perhaps where you study, to motivate yourself as you move forward on your journey to become a nurse.

3. Links the inspiration to your ePortfolio’s “About Me” section.

2. Presentation (65 points/43%)

1. Presents the inspiration to the class.

2. Describes the inspiration and the relationship of the inspiration to being a nurse.

3. Answers the following questions during presentation:,

i. What does your inspiration mean?,

ii. What concept challenged you during this course?,

iii. What has been the greatest lesson you have learned?,

iv. How will what you have learned help you be successful at Chamberlain University?

Nursing Inspiration Board

The vision board includes a collage of meaningful images and quotes. Key elements:

  • Images: A stethoscope, heart monitor, hands holding, Florence Nightingale, a hospital bedside, and a diploma—symbolizing compassion, service, healing, achievement, and the path ahead.

  • Quotes:

    • “Nursing is an art: and if it is to be made an art, it requires an exclusive devotion.” – Florence Nightingale

    • “The best way to find yourself is to lose yourself in the service of others.” – Mahatma Gandhi

  • Nursing Inspiration Board

  • Personal Words: “Empathy,” “Dedication,” “Healing,” and “Hope.”

This board is designed to hang near the study area as a daily motivator and is linked directly to the “About Me” section of the ePortfolio, describing a personal story of why nursing was chosen.


Part 2: Presentation Overview

What does your inspiration mean?
My inspiration reflects a deep desire to make a difference in people’s lives through compassion and care. It is a reminder of why I chose this path—to be a source of hope and healing for others, especially in their most vulnerable moments.

What concept challenged you during this course?
The most challenging concept was time management and balancing academic responsibilities with personal life. Nursing is demanding, and this course pushed me to grow in discipline and organization—skills vital in clinical practice.

What has been the greatest lesson you have learned?
I’ve learned that nursing is more than technical skill—it’s emotional intelligence, communication, and resilience. The importance of being present with patients, even during difficult times, has been my biggest takeaway.

How will what you have learned help you be successful at Chamberlain University?
This course has given me the foundational tools and mindset needed to thrive—self-awareness, the ability to reflect, and a commitment to lifelong learning. I now understand how to use both inspiration and discipline to persist through challenges at Chamberlain and beyond.

How is your inspiration connected to becoming a nurse?
Each image and quote on my vision board connects to a personal story—whether it’s a family member’s health battle or the admiration I have for nurses I’ve encountered. They all serve as emotional fuel to continue pursuing nursing with purpose and heart.

June 23, 2025
June 23, 2025

Federal Data Analytics Plan

Freedom is not the permission to do what you like; it’s the power to do what you ought.” – Os Guinness

“I have traveled the length and breadth of this country and talked with the best people, and I can assure you that data processing is a fad that won’t last out the year.” –Editor in Charge 1957 Business books for Prentice Hall

 

Federal Data Analytics Plan

Prepare a 20-minute PowerPoint presentation with speaker’s notes for senior leadership of a Federal Government agency of your choice that outlines a strategic plan for implementing a data analytics solution to a problem or challenge that you have identified.  You will have time for a maximum of 20 slides to present the plan. Use the notes section to guide your presenter through the presentation. Begin by clearly stating the challenge that you propose to resolve.  Make your case for adoption of the project.  Follow with a clear annotation of the life cycle of the project to manage the expectations of the leadership.

  • Federal Data Analytics Plan
  • Business case evaluation
  • Data identification
  • Data acquisition and filtering
  • Data extraction
  • Data validation and cleansing
  • Data aggregation and representation
  • Data analysis
  • Data visualization
  • Utilization of analysis results

Your presentation should report new and emerging technologies in TWO of the critical areas listed above. Describe the technologies and their proposed applications. Consider the implications of these developments for leadership policy and planning. Please be sure to explain how these technologies will enhance corporate operations on a multi-national scale. What must leadership do now to prepare for the technological innovations you describe? Offer concrete recommendations for action. Your speaker’s notes should incorporate resources and references.

The course objective for this project includes your ability to demonstrate appreciation for the potential impact of technology on leadership thought and action. Provide sufficient reasoned masters-level information for senior leadership to make a decision regarding both your proposed action and its associated impacts.

Make sure you provide:

  • A presentation that shows understanding and application of technological concerns and benefits of data analytics.
  • A clear and reasonable display of leadership, application of technology, and innovation stewardship of Government resources.
  • A clear, reasonable, and complete discussion of critical the leadership may expect upon successful completion of the project.
  • Clear, reasonable, and complete set of speaker’s notes with appropriate references.
  • Clear, reasonable, and complete set of recommendations ending with a call for action by the organization’s senior leadership based on acceptance of the presentation
  • Insights as to how to prepare leaders and their organizations for future technological developments

In addition to your slide deck, provide a one-page executive summary of your presentation to provide the head of the department with the essence of your proposal

Federal Data Analytics Plan

  • What is the specific challenge being addressed in the federal agency?,

  • What is the strategic plan for implementing data analytics?,

  • How will the project lifecycle be managed from business case to analysis utilization?,

  • What new technologies are emerging in two lifecycle phases?,

  • What actions should leadership take to prepare for these innovations?

June 23, 2025
June 23, 2025

Remote Access Domain Security

· Identify common risks, threats, and vulnerabilities found in the LAN-to-WAN Domain that require proper security controls for mitigation

· Identify network and security policies needed to properly secure the LAN-to-WAN portion of the network infrastructure

Remote Access Domain Security

Write a 2-3 page APA-formatted essay that identifies network and security policies needed to properly secure the LAN-to-WAN portion of the network infrastructure

Written Essay Assi

Remote Access Domain Securitygnment 3-2:

· Identify common risks, threats, and vulnerabilities found in the Remote Access Domain that require proper security controls for mitigation

· Identify network and security policies needed to properly secure the Remote Access Domain portion of the network infrastructure

· Write a 2-3 page APA-formatted essay recommending Remote Access Domain hardening solutions by implementing proper security controls at the Internet ingress/egress point within an IT infrastructure

 

  1. What are common risks threats, and vulnerabilities in the LAN-to-WAN domain?,

  2. What network and security policies help secure the LAN-to-WAN domain?,

  3. What are common risks threats and vulnerabilities in the Remote Access domain?,

  4. What security controls should be implemented at the Internet ingress/egress point?,

  5. How can the Remote Access domain be hardened for improved security?

Remote Access Domain Security


Written Essay Assignment 3-1

LAN-to-WAN Domain Security

Introduction

The LAN-to-WAN domain forms a critical part of the IT infrastructure where internal networks interface with external networks like the internet. This boundary is a common target for cyberattacks and requires strong security controls and policies to prevent unauthorized access, data exfiltration, and service disruptions.

Common Risks, Threats, and Vulnerabilities

The LAN-to-WAN domain is exposed to a range of threats including:

  • Malware and viruses entering from external sources.

  • Denial of Service (DoS) or Distributed Denial of Service (DDoS) attacks that flood network resources.

  • Unauthorized access via poorly secured firewalls or open ports.

  • Man-in-the-Middle (MitM) attacks, which intercept data between LAN and WAN.

  • IP spoofing and routing attacks, leading to misdirection of traffic.

  • Data leakage due to insufficient outbound controls.

These threats exploit vulnerabilities such as weak firewall configurations, outdated firmware, lack of intrusion detection systems (IDS), and improper network segmentation.

Network and Security Policies

To mitigate these risks, organizations must establish and enforce strict policies:

  1. Firewall Policy

    • All incoming and outgoing traffic must pass through a configured stateful firewall.

    • Only necessary ports and protocols should be allowed.

  2. Intrusion Detection/Prevention System (IDS/IPS) Policy

    • Deploy IDS/IPS at the LAN-WAN edge to monitor and block malicious traffic.

  3. Access Control Policy

    • Implement Role-Based Access Control (RBAC) and Multi-Factor Authentication (MFA) for administrative interfaces.

    • Block anonymous or unauthorized external access attempts.

  4. Patch Management Policy

    • Regular updates for firewall and router firmware.

    • Security patches must be applied promptly.

  5. Network Segmentation Policy

    • Use DMZ (demilitarized zones) for public-facing servers.

    • Separate critical systems from general user networks.

  6. VPN and Encryption Policy

    • Enforce encrypted communication between LAN and WAN using secure VPN protocols (e.g., IPSec).

    • Apply TLS/SSL for web traffic.

Conclusion

The LAN-to-WAN domain is the gateway between private networks and the internet, making it essential to implement layered security policies. Combining firewalls, IDS/IPS, encryption, and strict access controls ensures resilient and secure network operations.

June 23, 2025
June 23, 2025

Challenging Group Topics

Imagine the group you have been designing has had the opportunity to meet for at least two sessions. Over the two group sessions, some topics have surfaced that are challenging for group members. Choose three of the topics from the following list that have surfaced in your group. o    Sexual abuse o    Homosexuality o    Religious preferences o    Abortion o    Use of illegal drugs o    Drinking and driving

Challenging Group Topics

Write a 1,000-1,500-word summary that includes the following: o    A description of the selected topics o    Details about how the topics were manifested in the group sessions. For example, someone states he/she is uncomfortable discussing the topic and/or being a part of the group. o    Details about how you would handle the situations o    A minimum of three scholarly sources in addition to the course textbook to support the findings. Scholarly sources include peer reviewed articles, textbooks, the Bible, etc.

Challenging Group Topics

Prepare this assignment according to the guidelines found in the APA Style Guide, located in the Student Success Center.

This assignment uses a rubric. Please review the rubric prior to beginning the assignment to become familiar with the expectations for successful completion.

You are required to submit this assignment to LopesWrite. A link to the LopesWrite technical support articles is located in Class Resources if you need assistance.

Challenging Group Topics

  1. What are the three sensitive topics discussed in the group?,

  2. How did each topic emerge or affect group dynamics?,

  3. What were some member reactions to these topics?,

  4. How should a group leader handle such discomfort or conflict?,

  5. What scholarly support guides best practices in managing sensitive discussions?


📝 Comprehensive General Response:


Navigating Challenging Topics in Group Counseling

Introduction

Group counseling often creates a dynamic space where participants can explore difficult emotions, share personal experiences, and confront complex social or psychological issues. As the group evolves, sensitive topics often surface, challenging both members and facilitators to maintain safety, respect, and therapeutic progress. In this summary, we examine three controversial topics—sexual abuse, homosexuality, and abortion—that arose during two group sessions. Each topic elicited strong reactions, impacted group cohesion, and tested the facilitator’s skill in managing dialogue. The following analysis discusses how these issues manifested, how they were addressed, and what best practices should be used to support the group ethically and compassionately.


Topic 1: Sexual Abuse

Description

Sexual abuse refers to any non-consensual sexual act or behavior inflicted on a person, often leaving deep psychological and emotional trauma. It is a highly sensitive topic due to the stigma, shame, and pain survivors may experience.

Manifestation in Group Sessions

During the second session, one group member shared a traumatic experience involving childhood sexual abuse. While this disclosure was brave and important, it led to discomfort among other members. One participant appeared visibly shaken and asked for a short break. Another expressed concern over being “triggered” by the conversation, mentioning past experiences they were not ready to discuss.

Handling the Situation

As a facilitator, it is critical to validate the courage it takes to share such experiences while also protecting the emotional safety of the group. I would thank the participant for their vulnerability and acknowledge the complexity of the emotions present. I would then offer a brief grounding exercise for the group, followed by a check-in to assess the emotional states of all members.

Moving forward, I would revisit the group’s ground rules, emphasizing confidentiality, the right to pass, and trauma-informed language. In individual follow-ups, I would encourage those struggling to seek one-on-one support, if necessary, and reaffirm the importance of pacing disclosures. The goal is to allow space for healing while preserving emotional safety for all members.

June 23, 2025
June 23, 2025

Employee Benefits & Retention

Please read the following for a review of employee benefits:

Doyle, A. (2019).  Types of employee benefits and perks . Retrieved from https://www.thebalancecareers.com/types-of-employee-benefits-and-perks-2060433. (Be sure to click on the links within this source to gain depth about specific benefits.)

Glassdoor Team. (2018).  11 Companies with seriously impressive benefits . Retrieved from https://www.glassdoor.com/blog/companies-seriously-impressive-benefits/

Employee Benefits & Retention

Hamill, L. (2019). Workplace culture: The invisible perk your employees actually need: Good benefits can create hype, but they fail to address employees’ greatest need: Care.  Employee Benefit News, 33(5), 33. Available in the Trident Online Library.

The Georgia Society of CPAs. (2020, January 17).  Your employee benefits & retention  [Video file]. Retrieved from https://www.youtube.com/watch?v=MW-zlOT4IMM

Case Assignment

You have been asked to give a presentation at a college Society for Human Resource Management student chapter meeting on the following topic:

The relationship among employee benefits, employee retention, and employee turnover.

Create a presentation of 10–12 slides with audio voiceover and provide detailed speaker notes under each slide. Be sure to utilize at least 3 references from the Trident Online Library.

Assignment Expectations

Employee Benefits & Retention

Your submission will be evaluated using the criteria as stated in the Case rubric. The following is a review of the rubric criteria:

· Assignment-Driven: Does the submission fully address all aspects of the assignment? Is the assignment addressed accurately and precisely using sound logic? Does it meet minimum length requirements?

· Critical Thinking: Does the submission demonstrate graduate-level analysis, in which information derived from multiple sources, expert opinions, and assumptions has been critically evaluated and synthesized in the formulation of a logical set of conclusions? Does it address the topic with sufficient depth of discussion and analysis?

· Business Writing: Is the submission logical, well organized and well written? Are the grammar, spelling, and vocabulary appropriate for graduate-level work? Are section headings included? Are paraphrasing and synthesis of concepts the primary means of responding, or is justification/support instead conveyed through excessive use of direct quotations?

· Effective Use of Information: Does the submission demonstrate that the student has read, understood, and can apply the background materials for the module? If required, has the student demonstrated effective research, as evidenced by student’s use of relevant and quality (library?) sources? Do additional sources used provide strong support for conclusions drawn, and do they help in shaping the overall paper?

· Citing Sources: Does the student demonstrate understanding of APA Style of referencing, by inclusion of proper citations (for paraphrased text and direct quotations) as appropriate? Have all sources (e.g., references used from the Background page, the assignment readings, and outside research) been included, and are these properly cited? Have all sources cited in the paper been included on the References page?

· Timeliness: Has the assignment been submitted to TLC (Trident’s learning management system) on or before the module’s due date?

Employee Benefits & Retention

  1. What are common types of employee benefits?,

  2. How do benefits influence employee retention?,

  3. What happens when benefits are poor or lacking?,

  4. What are current trends in employee benefits?,

  5. How can HR use benefits strategically?

June 23, 2025
June 23, 2025
  • Discussion Prompt: Now that you’ve explored emotional intelligence (Head & Heart), AI, and strategic simplicity (Business Made Simple), it’s time to bring it all together. Reflect on how you will integrate emotional intelligence, AI-driven insights, and simple business strategies to ensure the long-term success of your business or process improvement. How will each of these elements play a role in creating a sustainable, values-driven business? Consider what data you will continue to collect and analyze to support ongoing decision-making and improvement.
  • EQ, AI, and Strategy Integration

EQ, AI, and Strategy Integration

  • Response Prompt: Provide feedback on how a peer can further integrate emotional intelligence, AI, and data to strengthen their business or process improvement plan. How can they ensure that all three elements work together harmoniously for long-term success?
  • EQ, AI, and Strategy Integration
  1. How will you integrate emotional intelligence into your business or process improvement?,

  2. What role will AI-driven insights play in your strategy?,

  3. How will simple business strategies contribute to long-term success?,

  4. What data will you continue to collect and analyze?,

  5. How will these elements support ongoing decision-making and improvement?

June 23, 2025
June 23, 2025

Linear Algebra Home Exam

General Instructions 1. Submission of a solution to this exam constitutes a statement that you followed these instructions.

If you failed to follow some of the instructions for whatever reason, explain at the top of the exams which instructions you failed to follow, how you solved the relevant parts of the exam, and why.

Linear Algebra Home Exam

  • How can we efficiently and stably factor a real anti-symmetric matrix into A=LTLTA = LTL^T?,

  • How many operations does the anti-symmetric factorization perform compared to LU?,

  • What happens when we apply partial pivoting to the anti-symmetric factorization?,

  • How do we benchmark and compare solvers for Laplacian systems?,

  • How can we multiply sparse matrices efficiently in CSC format?

2. The exam is individual. You may not discuss the exam, your solutions or other solutions, or anything having to do with the algorithms and computations that the exam focuses on with any person except Sivan Toledo. You must follow this rule before, during, and after you solve the exam (including in the periods before you start the exam and after you submit your solution, since different students may solve it at different times).

3. You may only discuss this exam with persons other than Sivan Toledo after the grades are published.

4. You may use any publicly-accessible non-human resource to solve the exam, but you must cite every resource that you use. For books, articles, web sites, and so on, cite the title, author, and URL, and explain how you used the resource. For AI tools and other responsive resources, give the name of the tool (and URL if not well known) and the prompts. If the interaction was complex, just summarize the interaction. In both cases, explain how you used the results.

5. Include all the code that you used, and mark clearly code that you wrote, code from other sources (static or responsive/AI). If you fixed or corrected existing or generated code, indicate your changes or fixes clearly.

6. All code must be clearly documented using comments.

7. Use diagrams and graphs where appropriate to clarify and/or enhance your answers.

1 Efficient and Stable Factorization of Anti-Symmetric Matrices (40 points)

We can solve a linear system of equation with an <-by-< symmetric positive-definite matrix � using the Cholesky factorization using 1

3< 3 + =(<3) arithmetic operations. The algorithm is backward stable and it is

about twice as efficient as an !* factorization with partial pivoting, which is also usually backward stable, but does not exploit symmetry. In some vague way it makes sense that Cholesky would be twice as efficient,

Linear Algebra Home Exam

 

ראינו בשיעור 2

 

because due to the symmetry of the matrix, we can represent it using its upper and lower half, so we do not need to update the entire trailing submatrix after every elimination step.

In this problem we want to factor in a backward-stable a real matrix that is not symmetric, but satis- fies the constraint �) = −�. In other words, its elements satisfy �7 8 = −�87. We will call such matrices anti symmetric. Such matrices are fully defined by about <2/2 real values, not <2, so we hope to achieve computational savings similar to the savings in Cholesky.

The main building block in our new algorithm will be to zero most of the first column by subtracting a multiple of the second row of � from rows 3 : <. We will then zero most of the first row by subtracting the same multiples of the second column from columns 3 : <. We continue with the trailing submatrix in the same way, until we reduce � to a tridiagonal matrix) , in only sub- and super-diagonal elements)7+7,7,)7,7+1 are nonzero, and which is also anti symmetric.

1. Implement this algorithm and test it on random matrices with Gaussian elements. The algorithm should return twomatrices, a lower triangularmatrix !with 1s on the diagonal and an anti symmetric matrix ) , such that � satisfies � = !)!) (but a weaker condition in floating point). For simplicity, operate on all the elements of � and the trailing submatrices, not only on one triangle. Try to make the code as efficient as possible (while operating on both triangles). Test that it works correctly and explain how you performed the tests. Hint: As in !* with Gaussian elimination, if you eliminate column 8 with using a matrix !8 with ones on the diagonal and the negation of the multipliers below the diagonal in one column (which one?), thematrix ! should have themultipliers in the same position, but not negated.

2. Roughly how many arithmetic operation does your code perform? The answer should be given as an explicit multiple of <3; you can ignore low order terms. Please relate your answer to the number of operations that the !* factorization performs. Explain the result.

3. If you were to modify the algorithm and the code from Part 1 so that it would only operate on one triangle, say the lower one, howmany operations would the improved algorithm perform? How does this relate to the number of operations performed by the !* factorization?

4. The main motivation for this algorithm is that we can incorporate into it partial pivoting. Show that the algorithm, as presented up to now, can fail, and explain why it might be unstable even in cases where it does not fail.

5. We can avoid these problems by exchanging before we eliminate column 8 and row 8 two rows and the same two columns so that the element in position 8+1, 8 in the trailing submatrix has the largest mag- nitude among elements 8 + 1 : <, 8. We can express the overall factorization as %�%) = !)!) ,where % is a permutation matrix. Explain how these exchanges affect the matrix !. How do you think we can characterize the trailing submatrices and) if we perform these exchanges, both in the worst case and in practice? You only need to provide a reasonable characterization and to justify it from what we know about !* with partial pivoting; you do not need to provide a formal analysis.

6. We can make this algorithm even more efficient by noticing that the matrix � = )!) is upper Hes- senberg (elements �7 8 with 7 > 8 + 1 are zeros). Prove that this is indeed the case and explain how elements of ) relate to elements of � (so that we can construct ) from �).

7. Modify your code from Part 1 so that it computes ! and � . You can again update the entire trailing submatrix.

Linear Algebra Home Exam

8. Explain how this modification reduces the number of arithmetic operations and give an estimate for the number of operations that the algorithm performs if it exploits the anti-symmetry (the leading term only).

9. To solve a system �F = %)!)!)%F = 1, we need to solve linear systems with a anti-symmetric tridi- agonal matrix) . Explain howwe can do that in a backward-stable way in$(<) arithmetic operations and storage.

2 Benchmarking Linear Solvers (30 Points) The aim of this problem is to benchmark a number of exiting solvers of linear equations. The focus is on producing valid and robust results and on presenting them well. For this problem, it is important that the solvers be efficient. Matlab is considered the gold standard for this problem. All its solvers are good enough (if used correctly); you can use some other programming environment (e.g., Python), but if one or more of its solvers is inefficient, you may lose points.

1. Implement a function lap2d that creates the Laplacian of an <-by-< two dimensional grid. Theweight of all edges should be 1, and the row-sums of all rows other than the first should be 0. The sum of the elements in the first row should be 1. The row/column ordering of the matrix should be a row- by-row ordering of the grid vertices (the so-called natural order). Thematrixmust be represented efficiently as a sparsematrix (inmatlab, the form S=sparse(i,j,a) is a goodway to construct the matrix from a set of nonzero values 07 8) Plot the nonzero structure of the matrix (spy) for a 10-by-10 grid.

2. Implement a similar function lap3d for 3-dimensional grids. Plot it for some convenient size for which the structure of the matrix is clear from the plot.

3. Write a code that measures the performance of several linear solvers for �F = 1, where � is a Lapla- cian from Part 1 and 1 is a random vector. You The solvers that you need to implement are:

(a) A dense Cholesky factorization. In Matlab, A=full(S) creates a dense representation of a sparse matrix.

(b) A sparse Cholesky factorization, with a row/column permutation to enhance sparsity in the factor. In Matlab, help chol (or the more detailed doc chol) will tell you how to invoke the Cholesky factorization so that it can permute the rows and columns of a sparse matrix.

(c) Conjugate gradients without a preconditioner (pcg in Matlab).

(d) Conjugate gradients with an incomplete-Cholesky factorization with no fill (ichol in Matlab).

4. Demonstrate that each of the solvers is correct. For the iterative solvers, explain their accuracy of the solution that they return and compare to the solution of the direct (Cholesky factorization) solvers. For the sparse solver, show the nonzero structure of the factor, to demonstrate that it is indeed sparse.

5. Compare the performance of the 4 solvers by finding out the size of the largest grid they can handle (in both 2D and 3D) in 10s (only the linear solver should take 10s; exclude the time to construct the test problem). You will need to write code that finds these problem sizes. You are free to also present the performance of the 4 solvers in other ways. Make sure your results are statistically robust; the

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actual running times are not deterministic even if the algorithm is. You need not perform a statistical analysis, but you need to account for the running-times variability and to explain how do you did it. In this part of the problem, do your best to ensure that the solver is using only 1 CPU core (in Matlab, maxNumCompThreads(1) should work); explain how you restricted the solvers to 1 core and how you checked that they are indeed using only one.

6. Repeat Part 5, but with at as many cores as you can. Try to the speedups.

3 Efficient Multiplication of Sparse Matrices (20 Points) This problem investigates the efficient multiplication of sparse matrices represented using a compressed sparse column (CSC) representations. In this representation, the nonzero values of elements each column are stored consecutively in an array, and the row indices of these elements are stored in the same order in an integer array. These arrays need not be sorted in increasing row order. A third array points to the two arrays representing each column; this array represents the columns in increasing index order.

1. Given the CSC representation of two sparse matrices � ∈ R;×< and � ∈ R<×9, show how to create the CSC representation of the product �� in $(; + 9 + >) operations, where > is the number of arithmetic operations on nonzeros required to multiply the two matrices. Make sure you explain how to determine the size of each column in the product (that is, the size of the arrays that represent that column).

2. Given the CSC representation a sparse matrix � ∈ R;×<, show how to create the CSC representation of the product �) �. Try tomake this as efficient as possible and analyze the total number of operations that the multiplication requires in terms of ;, <, and >.

3. Suggest a mechanism to improve the efficiency of the solution of Part 1 in cases in which ; � >. That is, try to find a way to remove the dependence on ; from the running time.

4 A Tradeoff in the LLL Basis Reduction Algorithm (10 points) The !!! algorithm ensures that in the output matrix ‘ satisfies��’7,8�� ≤ 1

2 ��’7,7��

‘27,7 ≥ H’27−1,7−1 − ’27−1,7 .

for some 1/4 < H < 1. That is, the client code that calls LLL sets H to some value in this range. Explain the tradeoff: what happens when we set H just above 1/4? What happens when we set H just below 1?

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