Numerical Analysis

Foundations and Exercises

TANAKA, Kazuaki (田中 一成)
k.tanaka@aoni.waseda.jp

Global Center for Science and Engineering, Waseda University

Before We Start

📥
Please clone (or pull) the course materials repository right now.
First time
git clone https://github.com/waseda-num-analysis-2026/materials
Already cloned
cd materials
git pull

💡 Not sure how? Just ask your AI — it can run these commands for you.

Today's materials are inside 030/:
3rd-handout.qmd 📘 main material — full content of this lecture
3rd-handout.html rendered handout — viewable in browser
3rd.qmd slides (this file) — exercises & instructions only
3rd.html rendered slides

Pair Work — Ex 2.3

TODAY'S CLASS ACTIVITY (~90 min)
Watch your partner's Ex 2.3 video, exchange feedback, and submit a sheet.
  1. Form pairs — pairing method announced in class  (10 min)
  2. Exchange video URLs with each other  (5 min)
    Just share the URL you submitted to Ex 2.3.
  3. Watch your partner's video  (10 min)
  4. Submit feedback — the Before Discussion section  (15 min)
  5. Discuss based on the feedback sheet  (30 min)
    Share three positive points first. Then "what would make it even more perfect".
  6. Submit your self-evaluation — the After Discussion section  (15 min)
📻 Please bring your earphones. Feedback sheet template: Will Be Announced.

Exercise 3.0 — Evolve the Handout with AI

Schedule update (May 3)

All Session 3 assignment deadlines have been extended by one week: May 14 (Thu), 23:59 JST.

This was originally May 7. Also, the pair work for Exercise 3.2 will be held in Session 5 or later, not in the next class.

Repository: same as Ex 3.1   |   Submit: 3rd-handout.qmd (your edited copy)   |   Deadline: May 14 (Thu), 23:59 (extended from May 7)

While studying 3rd-handout.qmd, insert at least 3 Q&A blocks of your own — questions you actually had while reading, answered with the help of GitHub Copilot (or another AI) following the protocol in AI_TUTOR.md.

How to submit

  1. Copy materials/030/3rd-handout.qmd into the root of your Ex 3 repository
  2. Open it in VS Code, then for each question:
    • First prime the chat once with AI_TUTOR.md
    • Highlight the line you don’t understand
    • Press ⌘L / Ctrl+L, then ask the AI to add a Q&A block
    • Verify the AI’s answer, then commit the inserted Q&A block
  3. Repeat for ≥ 3 distinct questions, then git push

💡 The Q&A block format and an English example are at the top of 3rd-handout.qmd.

Exercise 3.1 — Normalized Floating-Point Numbers

Repository: Ex 3 — GitHub Classroom link   |   Edit: ex3-1.qmd   |   Deadline: May 14 (Thu), 23:59 (extended from May 7)

For two toy floating-point systems, find normalizations and the smallest / largest positive normalized numbers.

  1. With \(\beta = 10,\ p = 4,\ -2 \leq e \leq 3\):
    • Normalize \(-0.123\).
    • Find the minimum and maximum positive normalized numbers.
  2. With \(\beta = 2,\ p = 5,\ -6 \leq e \leq 7\):
    • Normalize \(0.25\) and \(0.1\) (use round-to-nearest if not exact).
    • Find the minimum and maximum positive normalized numbers.

Reminder. Normalization means writing a number with the leading digit \(d_0 \neq 0\) (in binary, that means \(d_0 = 1\)). See the handout § Floating-Point Numbers in Computers.

Exercise 3.2 — \(1 + \eta - 1\) in float64

Repository: same as Ex 3.1   |   Edit: ex3-2.qmd   |   Deadline: May 14 (Thu), 23:59 (extended from May 7)

🎥 Video still required. Record a 5-minute video explaining your answer. The pair work using this video is rescheduled to Session 5 or later.

For IEEE 754 binary64 (float64), compute and explain

\[ 1 + \eta - 1 \quad\text{for}\quad \eta = 2^{-52},\ 2^{-53},\ 2^{-54}. \]

You should explain why the answer is what it is in each case — not just print the number.

Hint. Recall the handout § Machine Epsilon. What is the gap between \(1\) and the next representable number? What does round-to-nearest do when \(1 + \eta\) is closer to \(1\) than to \(1 + \varepsilon\)?

Exercise 3.3 — Where the Integer Grid Breaks

Repository: same as Ex 3.1   |   Edit: ex3-3.qmd   |   Deadline: May 14 (Thu), 23:59 (extended from May 7)

Among the positive integers representable in float64, find the smallest pair of consecutive integers whose float64 representations differ by 2 instead of 1.

⚠️ The answer must be derived theoretically from the definition of float64. Code may be used as a verification tool only.

Hint. When does the spacing between consecutive float64 values exceed \(1\)? Express the gap in terms of \(p\) and \(e\).

Assignments and Preparation for Next Lecture

1

Accept the Ex 3 GitHub Classroom link once — one repository for all four parts

  • Edit ex3-1.qmd, ex3-2.qmd, ex3-3.qmd + your copy of 3rd-handout.qmd for Ex 3.0 (all .qmd)
  • All four parts: May 14 (Thu), 23:59 (extended from May 7)
  • Submit by git push — the last commit before the deadline is graded
2

Record a 5-minute explanatory video for Ex 3.2 and paste its URL into ex3-2.qmd

  • YouTube unlisted (限定公開) recommended; otherwise any link-shareable service (Google Drive, Dropbox, OneDrive, …)
  • Use Zoom, QuickTime, or OBS to record
  • Do not commit the MP4 file directly to the repository — paste a URL instead
3

Session 5 or later: pair work for Ex 3.2

  • You will be assigned to pairs (announced later)
  • Each person will watch the partner's video
  • Discuss strengths and areas for improvement together
  • Both members submit separate feedback sheets
  • 📻 Please bring your earphones when the pair work is announced
Evaluated on 5 axes
  1. Accuracy & Depth of Content
  2. Clarity of Explanation
  3. Structure & Time Management
  4. Audience Engagement
  5. Effective Use of Visual Aids