# 6-23-25 ## Meeting Start - Meeting started at 01:15 ## Course Introduction - Introduction to MATH307: Scientific Computing - Overview of course content and assessments - Course materials have been published on Canvas - Standard syllabus available on course web space - Web space can be found by searching 'Scott Strong Math' ## Website Access - Add 'math' to search to avoid confusion with other Scott Strongs - Direct link to website provided if DNS server issues occur ## Course Content Management - Final grades calculation method to be discussed - Changes to deliverables for summer session due to smaller class size - Link to general course description to be removed from top level - Focus on organizing content for Introduction to Scientific Computing for Summer 2025 ## Instructor Information - Instructor: Scott Strong, flexible with name usage - Biographical information to be shared in slides ## Course Description - Course designed for STEM audience of applied scientists and engineers - Scientific computing as an offshoot of applied mathematics ## Course Schedule - Course meeting times: Monday to Thursday, 01:15 to 3PM - Location: M Z 026 - Two fifty-minute lectures with a five-minute break ## Lecture and Office Hours Details - Transition between computers and boards during lectures - Two types of office hours: open and by appointment - Open office hours location: Stratton Hall 102 - Open office hours on Monday, Tuesday, and Thursday - No open office hours on Wednesdays - Exceptions: July 8 and July 29, open hours will be rescheduled - Additional office hours from 3PM to 4PM, Monday to Thursday, in Stratton 102 - Additional office hours by appointment to be held in the computer lab - Hyperlink provided for booking individual appointments in 15-minute blocks ## Scheduling Issues - Previous scheduling issue: course changed to MWF 50-minute sessions, reducing contact time ## Grading System - Introduction of specifications grading: tasks need to be completed to a specified level ## Use of Technology in Course - Discussion on the use of Generative AI in academic work - Use of school's large language model interface for educational purposes - Interface provides hints but does not require coding from scratch - Emphasis on working within C language - Safeguards on generative AI have been adjusted for course use - Emphasis on visual aids like pictures and graphs in computational mathematics - Loosened restrictions on educational generative AI interface for quicker task completion - Generative AI interface includes pre-class objectives and interactive activities ## Classroom Practices - Pictures of the board are taken for students who are absent - Emphasis on attending classes as board work is hard to interpret later - Post-class assignment: interact with board content, due within a couple of days ## Learning Strategies - Emphasis on multiple interactions with content for better understanding - Pre-homework activities to encourage pre-thinking about homework topics ## Generative AI Interactions - Students expected to complete 75% of interactions with generative AI to pass - Interactions designed to be brief and not time-consuming ## Homework and Grading Details - Grades are broken down into letter grades for participation - Six homework assignments planned, each limited to a single front side of a page ## Homework Structure and Grading - Homework sets consist of five problems each - Grading: 2 means complete, 1 indicates issues to address - Generative AI can assist when students hit a wall in the decoding process ## Homework Completion Criteria - Zero given for incomplete homework with no revision opportunity - Completion of 80% of total problems required for homework - Minimal software engineering knowledge required for computational mathematics - Grade variations: 75-80% completion for C-, 80-85% for C+, >85% for B- ## Problem Revision Policy - Students may revise level one problems to achieve completeness ## Additional Grading Components - Micro projects introduced for higher grades (B and A) ## Presentation and Homework Impact - PowerPoint slides will be used to discuss point of view, worth 1-2 points - Completion of B or C level homework affects grading, treated as 80% completion ## Project Grading Criteria - Ten two-point projects will result in a 100% grade - Five two-point projects will result in a 90% grade, treated as an A- - A grades start at 95% - 10 to 12 different one to two point projects are being developed to apply scientific computing in practical scenarios ## Use of Generative AI in Projects - Completing more than 85% of homework grants two additional points, starting at 82% for B-level work - Generative AI can be used to enhance codes for scientific computing projects ## Interaction and Project Development - Interaction with generative AI is encouraged for homework and micro projects - Micro projects aim to build a comprehensive code library ## Instructor's Insights - Encouragement to have fun with coding and utilize modern tools - Instructor has been teaching since 2001, witnessing the growth of the Internet ## Adapting to Information Availability - Educators need to adapt to the availability of information online, similar to the impact of Wikipedia ## Course Introduction - Course is an introduction to scientific computing - Instructor: Scott Strong, grew up in North Pole, Alaska ## Instructor's Background - Instructor was born in Enid, Oklahoma, and moved to North Pole at age one - Moved to Anchorage, Alaska, at age seven - Worked at Arlist library in Anchorage for two summers - Worked as a grocery store clerk at Fred Meyer in the Pacific Northwest for two summers ## Teaching Experience - Started teaching C++ in 2001 using transparency machines ## Academic Background - Holds a Master's in Science in Applied and Computational Mathematics - PhD in Applied Physics, focusing on mathematical physics of one-dimensional ambient space - Expertise in vector and tensor analysis, differential geometry, and nonlinear PDEs ## Personal Anecdotes - Instructor shared a story about driver's license picture and headshots - School started bringing in people for headshots before it became common practice ## Personality Insights - Instructor consistently scores as INTP on Myers Briggs type indicator - Instructor took an alignment test: 92% true neutral, 69% chaotic good, 69% neutral good ## Teaching Anecdotes - Instructor no longer does practical jokes on April Fools due to a past incident with a fake electronic quiz - Shared an anecdote about a prank involving a Futurama splash screen in class - Last practical joke involved a prank that made a student so anxious they vomited ## Personal Preferences - Prefers dogs over cats - Would like to be an owl in the air, a turtle in the sea, obsidian as a rock, tulips as a flower, and orange as a color - Believes comedy is better than tragedy ## Interactive Teaching Methods - Occasionally uses Mentimeter system for interactive sessions ## Interactive Teaching Methods - Instructor plans to start another Mentimeter session - Instructor felt they had a rough go earlier in the year - Instructor is using a word cloud visualization in class ## Group Dynamics - Discussed the presence of a mathematician in the group - Noted a significant presence of mechanical engineering students in the course - Instructor found a new icebreaker question for interactive sessions - Plans to use scaled questions and information polling in future sessions ## Engineering Relevance - Discussed debates about parking spots and control time ## Course Content - Course will cover implementing algorithms and procedures - Focus on encoding and approximating solutions - Discussed the importance of approximations in computing and selecting appropriate algorithms for specific problems - Emphasized understanding the margin of error in numerical approximations - Focus on solving algebraic equations using techniques for systems with many variables - Representation of discrete data from digitized aspects of the real world will be a key feature - Students should have completed a calculus sequence and a course in ODEs concurrently or prior ## Sound and Mechanical Engineering - Instructor plans to discuss sound generation and its relevance to mechanical engineering - Discussed storing sounds digitally and breaking them into components - Discussed the conversion of mechanical energy to electrical information in the ear - Instructor has videos from last year that may still be relevant for current discussions ## Numerical Path Representation - Discussed numerical representation of paths and curves in space - Discussed solving equations to determine when algebraic terms representing paths are equal ## Computer Graphics Discussion - Discussed the visual representation of water in computer graphics - Mentioned issues with ray tracing affecting visual quality - Ray tracing significantly enhances the visual representation of water in computer graphics - New chips released last summer may have contributed to improvements in ray tracing - Photons in ray tracing follow straight line paths and interact with virtual objects - Discussed computational problem of tracing rays and computing light intensity through integrals - Discussed the difference between raw data and expected visual output in computer graphics ## Animation Techniques - Discussed interpolating frames to create smooth motion in animations - Generative AI was used last year for interpolating frames in animations - Stitching frames together improves with closely aligned frames, reflecting the state of generative AI last year - Generative AI in animations can humanize robotic movements ## Robotics Discussion - Robots exhibit human-like errors, adding a relatable aspect to their operation - Boston Dynamics released a new robot model with enhanced rotational capabilities ## Computational Crash Tests - Discussed the representation of solid objects in computational crash tests ## Future Plans - Discussed potential projects and problems to tackle this summer ## Programming Languages Discussion - MATLAB will be used as the standard environment for code development - Generative AI facilitates quick porting of code from MATLAB to Python or R - Students have the option to use Python, MATLAB, or R for their projects ## Language Preferences - Last summer, the course infrastructure was heavily tied to MATLAB, limiting language choice - 41% of students prefer MATLAB for their projects - MATLAB is commonly used in industry, but many firms are transitioning to Python due to cost and capabilities - MATLAB usage in companies can restrict personal projects or side gigs, pushing individuals towards Python for more flexibility - Preference towards Python due to its open-source nature and flexibility - Generative AI can be used to port code between MATLAB and Python - Students encouraged to explore multiple programming languages - R is acknowledged as more of a statistics program but capable of similar tasks ## AI and Machine Learning Integration - Introduction of the high TA system, a user interface leveraging OpenAI and ML anthropic APIs - The system is connected to course knowledge and designed to be educational in its responses - The high TA system prioritizes education over solution provision ## Critical Thinking and AI Usage - Emphasized the importance of critical thinking when using advanced AI systems - Encouraged clarity on personal development versus tool utility in course projects ## Hi TA System Enhancements - Hi TA system will incorporate worksheets, transcriptions, and summarized lectures into its knowledge base - Content in the Hi TA system will be anonymized to improve understanding and response tailoring ## Hi TA System Interaction Guidelines - Discussion on the movie 'Hidden Figures' and its relevance to understanding ordinary differential equations in space missions - High TA link will be available on Canvas for students - Access to Hi TA system requires authentication through Mynd's user accounts - High TA interactions have due dates on Canvas, with objectives due before the course and reflections due two days later at 11:59 PM - Due dates for High TA interactions are considered soft deadlines ## Lecture Objectives - LO1 has been completed and posted to the website - Interface allows side-by-side view on the website, embedded into Canvas - Lectures will focus on identifying how linear transformations affect standard basis vectors - Interaction with the Hi TA system is designed to be brief, ideally between 10 to 20 minutes - Conversations with the Hi TA system can be exported as a markdown file from the website ## Hi TA System Submission and Feedback - Submit markdown file of Hi TA interactions to Canvas as part of the process - Interactions should be spaced to enhance learning through neuron activation - Hi TA system provides feedback on task completion status - Recommended interaction time with Hi TA is 10 to 20 minutes - Encouragement to work with the Hi TA system and provide feedback for adjustments ## Source Material Access - Source material for problems is publicly available through the website ## Additional Resources - Homework assignments are available on scottastrong.org - High TA guides available showing examples of interactions and capabilities - Lecture objectives and reflections available on boards ## Hi TA System Usage Tips - Check-in with Hi TA to ensure all points are covered before submission - Look for the word 'complete' in Hi TA feedback before submitting - Ongoing efforts to link Canvas and Hi TA system - No required textbook; information will be provided in lectures ## Mathematica Usage - Lecture duration is approximately 50 minutes - Mathematica prefers exact symbolic calculations over double precision - Expertise in using Mathematica for mathematical tasks - Option to use Python or MATLAB for majors requiring experience in these languages ## Attendance and Participation - Attendee will miss Thursday, Monday, and Tuesday due to a funeral - Plan to take pictures of the boards and upload to Hi TA - Plan to discuss a better strategy for capturing board information on Wednesday ## Hi TA System Navigation - Access recent conversations on Hi TA webpage through embedded view # 6-24-25 # MATH307 Scientific Computing - Day 2 Meeting Notes ## June 24, 2025 ## Meeting Overview **Topic:** Matrix Multiplication Implementation and Linear Transformations **Duration:** ~105 minutes (two 50-minute sessions with break) **Location:** M Z 026 --- ## Course Administration Updates ### Website Resources - **Homework assignments** now embedded on course website with AI-generated concepts/vocabulary - **Hi TA guides** available showing interaction examples and capabilities - **Learning objectives (LOs)** loaded into Hi TA system with 3 bullet points each - **Lecture boards** section updated with Monday's boards (will expand to 3 pages: boards, audio summaries, notes) - **Projects section** populated with micro-projects worth 1-2 points each for B/A grades ### Hi TA System Changes - **Lecture reflections** simplified from 2 per class to 1 per day (24 total for course) - **Export process**: Access markdown export via 3-dots menu when in conversation (not activity view) - **Unknown checkbox** in activities - instructor unsure of function, likely for future Canvas integration - **Completion verification**: Ask the bot if work is complete rather than relying on checkbox --- ## Technical Implementation: Matrix Multiplication ### Mathematical Foundation **Matrix multiplication formula:** For A ∈ ℝᵐˣⁿ and B ∈ ℝᵖˣᵠ where n = p: ``` C[i,j] = Σ(k=1 to n) A[i,k] × B[k,j] ``` ### Algorithm Structure **Three nested loops required:** 1. **Outer loop:** i = 1 to M (output rows) 2. **Middle loop:** j = 1 to Q (output columns) 3. **Inner loop:** k = 1 to N (accumulation) **Computational cost:** O(N³) for square matrices ### MATLAB Implementation #### Matrix Definition and Syntax - **Matrix creation:** Use semicolons for new rows, commas for new columns - **Example:** `A = [1, 2; 3, 4]` creates 2×2 matrix - **Dimension access:** `size(A,1)` for rows, `size(A,2)` for columns - **Matrix initialization:** `zeros(M,Q)` creates M×Q zero matrix #### Live Coding Session ```matlab % Define input matrices A = [1, 2; 3, 4]; % 2x2 matrix X = [1; 0]; % 2x1 vector % Get dimensions M = size(A,1); N = size(A,2); P = size(X,1); Q = size(X,2); % Initialize output matrix C = zeros(M,Q); % Triple nested loop implementation for i = 1:M for j = 1:Q for k = 1:N C(i,j) = C(i,j) + A(i,k) * X(k,j); end end end ``` #### Error Handling Discussion - **Dimension compatibility:** Need to verify n = p before multiplication - **Conditional structure:** `if n ~= p` then break/exit - **MATLAB syntax:** `~=` for "not equal", `==` for equality comparison - **Program termination:** `exit` terminates entire MATLAB (too aggressive), `break` only works in loops --- ## Linear Algebra Theory ### Identity Matrix **Definition:** I ∈ ℝⁿˣⁿ where I[i,j] = δᵢⱼ (Kronecker delta) - δᵢⱼ = 1 if i = j (diagonal elements) - δᵢⱼ = 0 if i ≠ j (off-diagonal elements) **Property:** A × I = I × A = A (multiplicative identity) ### Matrix Inverse For A ∈ ℝ²ˣ² with det(A) ≠ 0: ``` A⁻¹ = (1/det(A)) × [d, -b; -c, a] ``` where A = [a, b; c, d] and det(A) = ad - bc **Key requirement:** det(A) ≠ 0 (otherwise inverse doesn't exist) --- ## Linear Transformations ### Fundamental Properties **Transformation T: ℝ² → ℝ²** defined by T(x) = Ax #### Property 1: Linearity T(c₁x₁ + c₂x₂) = c₁T(x₁) + c₂T(x₂) - Maps linear combinations of inputs to linear combinations of outputs #### Property 2: Origin Preservation T(0) = 0 (origin maps to origin) #### Property 3: Parallel Line Preservation Parallel lines in input space remain parallel in output space ### Standard Basis Transformation Analysis #### Example 1: Reflection Matrix ``` A = [0, 1; 1, 0] ``` - **I-hat transformation:** [1,0] → [0,1] - **J-hat transformation:** [0,1] → [1,0] - **Effect:** Swaps x and y coordinates (reflection across y=x line) - **Determinant:** det(A) = 1 (no orientation flip) #### Example 2: Rotation Matrix ``` A = [0, 1; -1, 0] ``` - **I-hat transformation:** [1,0] → [0,-1] - **J-hat transformation:** [0,1] → [1,0] - **Effect:** 90° clockwise rotation - **Determinant:** det(A) = -1 (orientation flip) ### Geometric Interpretation - **Positive determinant:** Preserves orientation (rotation, scaling) - **Negative determinant:** Reverses orientation (reflection component) - **Unit determinant:** Preserves area - **Parametric lines:** T(P + tV) = T(P) + tT(V) (parallel lines stay parallel) --- ## Computational Visualization ### MATLAB Visualization Tools - **Basic mesh plotting:** Shows before/after grid transformations - **AI-generated enhancement:** Animated visualization showing continuous transformation - **Animation feature:** Parameter sweeping to show transformation process dynamically ### Key Observations from Animations 1. **Reflection transformation:** Grid appears to "flip" during animation 2. **Rotation transformation:** Grid rotates continuously 3. **Parallel preservation:** Grid lines maintain parallelism throughout 4. **Origin fixation:** Center point remains stationary --- ## Programming Environment Notes ### MATLAB vs Python Discussion - **MATLAB:** Primary environment for course (Matrix Laboratory) - **Python option:** Available for students preferring open-source - **Code portability:** Generative AI can assist with language translation - **Industry trends:** Companies moving from MATLAB to Python for cost/flexibility ### Error Debugging - **Common issues:** Case sensitivity (`zeros` vs `Zeros`) - **Semicolon behavior:** Suppresses output display but code still executes - **File execution:** Returns filename when semicolons used (normal behavior) --- ## Upcoming Work ### Homework Assignment Progress - **Problem 1:** Should be completable after Day 1 content - **Problem 2:** Should be completable after Day 2 content - **Problem 3:** Will be covered in upcoming sessions ### Next Session Topics 1. **Error handling** implementation in matrix multiplication 2. **Eigenvalue/eigenvector** introduction 3. **Advanced transformation** analysis 4. **Mesh visualization** code distribution --- ## Action Items ### For Students - [ ] Complete Hi TA interactions for Day 2 learning objectives - [ ] Practice matrix multiplication implementation in chosen language - [ ] Begin Problem 2 of homework assignment - [ ] Export and submit Hi TA conversations to Canvas ### For Instructor - [ ] Research proper MATLAB program termination commands - [ ] Publish mesh visualization codes to website - [ ] Create Python equivalent of matrix multiplication implementation - [ ] Update course website with Day 2 materials # 6-25-25 ## Lecture Updates - Lecture pictures have been posted. - Learning reflections encourage reviewing lecture pictures and asking questions - Learning objectives 5 and 6 are not yet available as they cover Taylor series, which hasn't been covered yet - Learning objectives 5 and 6 will be available on Canvas tomorrow - Learning objectives 7 and 8 have been unpublished to maintain consistency - Lectures are planned to cover two learning objectives per day, but this may vary ## Matrix Discussions - Discussion on matrices and coding with matrices is ongoing - Aim to build intuition on matrix transformations without a full linear algebra class - Upcoming discussion on eigen problems and their applications to regular stochastic processes - Explanation of matrix jargon and its connection to prior topics - All matrix multiplication code is available on high TA, including MATLAB and Python modules - MATLAB code has been ported to Python for convenience - Ports to R or Mathematica can be made upon request - Discussion topic took longer than expected, causing some frustration - Computational costs related to scientific computing were discussed, though considered outside the current scope - Plan to improve matrix multiplication code by addressing the triple loop structure - New project to compare performance of matrix multiplication code across different platforms - Project on matrix multiplication runtime to be used as a basis for comparison in class discussions ## Error Handling in MATLAB - Discussion on error handling in MATLAB, including the use of the 'error' statement - The 'error' statement halts execution and displays a message when run within an if statement - The 'return' command hands off control when encountered - The 'return' command in MATLAB sends control back to the command line without a display message ## Error Handling in Python - In Python, error messages can be raised using 'raise Exception', which allows for a custom message ## MATLAB Code Precautions - Be cautious with MATLAB code as it contains an 'exit' command that will close MATLAB if an error is encountered - Modify the MATLAB code to remove the 'exit' command before use - Plan to send the procedure to a function that takes two matrices as inputs ## Matrix Multiplication Project - Plan to create a project for matrix multiplication outputting their product - Include error handling with if statement for dimensionality issues - Discussion to continue tomorrow on integrating this into a function ## New Projects - New project to compare code against computer's multiplication algorithm for speedup analysis - Project will be a collaborative effort to track progress in class - Draft for the new project is ready for release - Internal routines handle system memory uniquely to find speedup - Explore the use of sparse routines for matrices with many zeros to improve efficiency - Information on sparse routines will be released on the website ## Code Base Updates - Edits to the code base will be made today, with updates to be shared tomorrow - Discussion on computational run times in relation to matrix transformations ## Linear Transformations - Matrices map the origin to the origin and maintain linear combinations and parallel lines in transformations ## Matrix Transformations - Explore other parts of linear algebra for potential projects - Matrix A transforms the standard basis vectors - A maps input vector to output vector after transformation - Discussion on color coding in the xy-plane: horizontal as red (i hat) and vertical as blue (j hat) - I hat remains unchanged after transformation by matrix A - J hat transforms to two units right and one unit up after transformation by matrix A - Discussed benefits of using standard basis vectors for transformations - Discussion on the unit square and its transformation under matrix A ## Vector Manipulation - Attempt to create a vector from the tip of I hat parallel to a non-straight line - Use red to take its tip to the tail of blue and go parallel - Transformation of unit square into a parallelogram using matrix A - Transformation described as a shearing, similar to pushing a table to feel forces - Cartesian grid described as malleable, bending into a parallelogram ## Parallelogram Area Calculation - Area of the parallelogram is calculated as base times height - Base of the parallelogram is the I hat vector with length one - Height is determined by the shift of J hat vector, two units to the right - Unit square transforms into a parallelogram with unit area - No area changes observed, determinant of matrix is one - Reference to multivariate calculus: transformation from x's and y's to r's and thetas involves Jacobian matrix determinant - Embedded in the determinant calculation ## Shear Transformation Details - Shear transformation acting on I hat and J hat - Encouragement to run calculations for better understanding - Preparation for visual representation with axes for picture - Mathematicians often include humor in linear algebra books - Waiting for thirty more seconds before proceeding - Preparing to transfer unit square for transformation analysis ## Matrix Transformation Analysis - Matrix transformation applied to I hat results in a vector with components (1, 1) - Matrix transformation applied to J hat results in a vector with components (0, 1) - I hat mapped to vector (1, 1) in the output plane, labeled as red - J hat mapped to vector (0, 1) in the output plane, labeled as blue - Horizontal transformation results in negative shift instead of positive - Reference to 'box trick' for visualization - Tail to tip, tip to tail method discussed for vector manipulation - Familiarity with geometric concepts emphasized ## Rectangular Shape Analysis - Discussion on the rectangular shape formed by red and blue vectors at a 90-degree angle - Calculation of the area of the rectangular shape - Width of red vector is 1, hypotenuse calculated as square root of 2 - Area of rectangular shape calculated as 2, indicating dilation of space ## Rotation and Scaling Effects - Rotation and scaling of square increases area - Determinant of matrix A calculated as two, indicating area increase ## Transformation Properties - Rotations and shear transformations can occur simultaneously - Compression occurs when determinant is between zero and one - Negative determinant indicates a flip - Two flips result in no change, similar to looking in a mirror twice - If determinant is greater than one, it enlarges the area or volume ## Transition to Computer Analysis - Transition to computer-based analysis initiated - Code generated by machines includes animation feature for visualizing underlying space - First matrix inputted for analysis, expected to be uninteresting - Animation shows frame by frame transformation with a parameter to shear - Humorous reference to 'sheared sheep' in linear algebra context - Second matrix (1, -1, 1, 1) applied, resulting in rotation and enlargement - Animation shows rotation and scaling, multiplying area of cells from ones to twos - Discussion on matrix with determinant of zero from homework - Discussion on matrix with determinant of zero indicating loss of dimension ## Dimensional Reduction Discussion - Condensing two-dimensional plane to one-dimensional line subpart discussed - Determinant indicates loss of dimension, but not the extent of loss - In three-dimensional space, reduction can be to two or one dimension - Discussion on determinant vanishing indicating complete loss of area, potentially reducing to a point ## Eigenvalue Analysis - Examination of dependency relations within columns - Counting zero eigenvalues to determine significance ## Simplifying Transformations - Discussion on simplifying complex transformations by extracting simpler structures - Discussion on eigenvalues and eigenvectors in transformation ## Shear Transformation Example - Mona Lisa example used to illustrate shear transformation - Top part of Mona Lisa pushed to the right, demonstrating vector effect - Discussion on vector transformation from domain to range - Discussion on the concept of vector having direction and magnitude - Discussion on changes in direction and magnitude of vectors ## Space Manipulation Discussion - Discussion on manipulating space like Play Doh, pushing and pulling to grow the space - Attempt to identify a blue vector that maintains its direction during transformation - Discussion on subdomains or subspaces where vectors maintain their direction during transformation - Discussion on vectors whose directions remain unchanged during transformation, despite stretching or compressing - Discussion on maintaining vector direction while allowing changes in length - Goal to find lambda and x that satisfies the equation for unchanged direction ## Eigenvector and Eigenvalue Discussion - Discussion on eigenvectors being special to a transformation as their direction doesn't change - Lambda is identified as the corresponding eigenvalue to eigenvector x - Issue identified: Cannot subtract a scalar from a matrix, need a matrix to maintain x vector unchanged - Discussion on homogeneous system with zero solution, where both sides equal zero when x is zero ## Homogeneous Problem Discussion - Discussion on neglecting the origin point in favor of a subspace line - Zero is a solution to the homogeneous problem, but seeking non-zero solutions - Setting matrix to zero to explore non-zero solutions - Requirement for matrix: determinant must be zero for non-trivial solutions - Determinant of (A - lambda * I) = 0 for non-trivial x - Prioritize solving the determinant equation first as Lambda is the only unknown - Address eigenvectors after solving the determinant equation - Instructor plans to work through an example, but notes a change in the expected progression ## Matrix Introduction - Introduction of a matrix: [0, 1; 1, 0] as a segue into the next topic ## Animation and Transformation Discussion - Animation used to demonstrate transformation of points on a grid - Discussion on finding one-dimensional invariant subspaces for the matrix [0, 1; 1, 0] ## Eigenvalues and Eigenvectors Importance - Eigenvalues and eigenvectors provide fundamental information about a matrix - Expression for characteristic polynomial: Lambda squared minus the trace times Lambda plus the determinant - Emphasis on using computers for calculations, but understanding the concepts is crucial ## Matrix Properties Discussion - Trace of the matrix is the sum of the diagonal elements, which is zero - Determinant of the matrix is negative one - Lambda one is identified as one and lambda two as negative one ## Technical Setup - Instructor successfully navigated technical setup with document camera - Instructor plans to solve only one eigenproblem by hand, rest will be computed using software ## Matrix Line Representation - Calculation of matrix after subtracting Lambda one from identity matrix results in [-1, 1; 1, -1] ## Equation Solving Reminder - Reminder of solving equations: negative one times negative plus y equals zero and ex minus y equals zero - Rows are the same line due to determinant set to zero - Eigenvector for this eigenvalue is any vector on that line with non-zero length ## Eigenvector Choices - Eigenvector choices for lambda one include (1,1), (2,2), (-π,-π), but not (0,0) - Running routine for lambda two equals negative one ## Eigenstructure Overview - For a 1x1 matrix times [x, y] equals [0, 0], y equals negative x - When y becomes negative one, x becomes one - Eigenvalue one has eigenvector (1,1) - Second eigenvector is (1,-1) with eigenvalue negative one - Animation shows transformation around the line y equals x - Animation illustrates the flipping effect of eigenvalue negative one - Discussion on eigen lines and their visibility in matrix eigenstructure ## Matrix Review and Ion Lines - Review of matrices and discussion on potential eigenvalues - Ion lines remain unchanged after transformation ## Recent Achievements - Successful implementation of multiplication code ## Computational Techniques - Discussion on copying and pasting techniques in mathematical computations ## Notation and Data Management - Discussion on matrix size reaching 5.6 megabytes - Use of underlines for vectors instead of over arrows for clarity ## Document Management Issues - Folder error encountered in document management - Suggested solution: recreate the missing folder or choose a different working directory - MATLAB will rewire your path when trying to run specific class files - Errors encountered when using x and y as variables - Discussion on checking process against a vector and handling x and y as zero or one ## Eigenvalues and Eigenvectors - Plan to modify the system to accept dot n b input ## Meeting Observations - No one took the five minute break - Working at the command line in MATLAB to define matrices instead of using scripts - Use of IGE command in MATLAB to strip out matrix elements - Discussion on eigenvalues and eigenvectors in relation to matrix A - Use of Igg A command in MATLAB to process a 2x2 matrix and extract eigenvalues and eigenvectors ## Large Matrix Computation - Instant computation for large matrices (e.g., million by million) to identify important features - Discussion on the structure of eigenvectors in matrices and their symmetry - Discussion on matrix D as a 2x2 matrix with eigenvalues on the main diagonal and zeros elsewhere - Explanation of eigenvectors in column form with lambda equal to negative one - Discussion on eigenvectors corresponding to lambda equal to one and negative one - Discussion on matrix A with elements 0, 1, -1, 0 and its role in rotation - Animation grids appear identical, indicating a rotation - New matrix A defined and used to sort eigenvectors and store eigenvalues ## Matrix Transformations - Plane rotation observed, drawing on Diffie Q intuition for eigenvalues - Execution of command line operations on matrix A for eigenvalue analysis - Observation of complex eigenvalues: one is imaginary I, another is negative imaginary I - Discussion on spirals and complex eigenvalues in relation to Diffie Q knowledge - Observation of unit square rotation and scaling, increasing area by a factor of two - Matrix A defined with ones in upper right, lower left, and lower right - Transformation involves rotation and change in size - Discussion on the role of real parts in stretching and changing size during rotation - Discussion on shearing effect on matrix and its impact on eigenvectors - Eigenvalues are both one, indicating no change in area - Eigenvector identified as I hat, leaving the horizontal line unchanged ## Linear Algebra Conclusion - Last topic in the linear algebra section discussed ## Anecdotes - Humorous anecdote shared about a string and a bartender ## Stochastic Matrices - Transition to discussion on regular stochastic matrices - Discussion on left stochastic matrices and their properties - Stochastic matrix is square with non-negative real entries and columns summing to one ## Deterministic Systems - Stochastic processes are random processes, opposite of deterministic systems - Columns summing to one represent a probability state, indicating total probability in the column is one - Regular stochastic matrix has all entries related to probabilistic settings - Discussion on matrix powers and their positivity in linear algebra context - Regular stochastic matrix has positive entries in some power, important for theoretical applications - Matrix A acts on vector x0 to update it one step forward in time - Subscript in x0 can be interpreted as time - Transition from x0 to x2 requires updating x - X1 is the update of x0, leading to A squared times x0 for the nth state - Updates by matrix multiplication relate to x0, involving repeated calculations - Inefficient calculation coded up and repeated n times for regular stochastic matrix - Limit of x subscript n as n approaches infinity is called p vector - Stationary vector defined as the point where successive matrix multiplication updates result in no change - Regular stochastic matrix with non-negative entries can have zero entries - Elevating a regular stochastic matrix to a power where all entries are positive indicates a stationary vector ## Chessboard Problem Exploration - Mention of a PBS video exploring knight's moves on a chessboard ## Markov Chain Theory - Markov chain consists of a state space and a probability transition function - Example of Markov chain: radio station playing K-pop and Ska as states - Probability transition: two-thirds chance of playing the same genre next - One-third probability of switching genres in Markov chain - Introduction of stationary distribution in Markov chain - Stationary distribution assigns a number to each state based on long-term behavior - Stationary distribution example: equal probability distribution between K-pop and Ska ## Algebraic Problem Solving - Discussion on mathematical results involving one half and one third - Mention of linear algebra and eigenvectors in relation to these results - Discussion on addressing algebraic problems in different contexts ## Matrix Label Redefinition - Diagram with nodes A and B on a graph, allowing transitions A to B, B to A, B to B, and A to A - Discussion on interpreting drawings from a probability and Markov chain perspective - Example given with two-thirds probability on one part and one-third on another - Redefinition of matrix labels from A and B to K and S for clarity - K represents K-pop and S represents Ska in the matrix - Filling matrix with data encoded in the graph, focusing on k and a rows and columns - Element in k row k column represents probability of k transitioning to k - Probability of A transitioning to K is two-thirds - Clarification needed on matrix labels, should be rewritten as S's instead of A's - Transition probability from Ska to K-pop is one-third - Transition probability from K-pop to Ska is one-third, indicating symmetry - Transition probability from Ska to Ska is two-thirds - Columns of the matrix sum to one, confirming probability distribution - All entries are positive for a matrix raised to the power of one, confirming it as a regular stochastic matrix - Limit can be attained by repeatedly applying the matrix to initial distributions between K-pop and Ska - x0 is considered an initial probability vector - Symmetry in the system may simplify calculations - Define SK matrix labels consistently, either as row-column or column-row - K-pop and Ska transition probabilities are one-third, interchangeable without impact - Personal preference leans towards defining Ska to K-pop transition - Probability vectors are columns in stochastic matrices - Important defining characteristic of a probability vector discussed - Elements of the matrix sum to one, confirming it as a probability matrix - Question raised about the limit of the matrix as n approaches infinity - Expectation of a 50-50 distribution between K-pop and Ska after applying matrix n times ## Eigenvector Independence - Theory A states that if lambda one and lambda two are different, then the eigenvectors are linearly independent - Colloquial definition of linear independence: two column vectors are linearly independent if they point in different directions in the plane - Linear independence in R2 involves two different directions - In higher dimensions, more directions are compared - Example: East and West as opposite directions on a map - South is considered the negative direction of North, and they are linearly independent - East and its negative, North and its negative, are orthogonal directions - Orthogonal directions allow for complete navigation on a map - Directions in terms of North and Northeast can still allow full navigation on a map - Having two different directions in the plane allows for complete directional control - Any vector in the plane can be made using eigenvectors v1 and v2 - In three dimensions, more permutations need to be considered - Constants c1 and c2 are used to build vectors - Linear combinations involve multiplying vectors by constants and adding them together - Constants c1 and c2 are used in linear combinations to express x vector as c1 v1 plus c2 v2 - Working in R2 involves two directions: v1 and v2 - Directions v1 and v2 may not align with cardinal directions like North or East - Theoretical ability to determine distance in direction v1 and v2 to reach x0 - If matrix A is raised to the nth power and acts on x0, it is equivalent to A^n acting on c1v1 + c2v2 ## Linear Transformations and Combinations - Linear transformation acts on a linear combination - Editorial article mentioned linear transformations - Common math class phrases: derivative of sum is sum of derivatives, integral of sum is sum of integrations - Linear transformation of a linear combination is the linear combination of the corresponding linear transformations - Plan to take a board picture and write out details later - Current focus is on linear transformations, with intention to elaborate - Matrix n is a linear transformation with many multiplications - Distribution over linear combination is possible - Matrix A raised to nth power acts on eigenvector v1 - v1 is an eigenvector with an origin story - Eigenvector v1 remains in the same direction when acted upon by matrix A - Eigenvector v1 can be scaled by its corresponding eigenvalue, Lambda - Lambda one is a scalar and remains unaffected by matrix A - Matrix A acts on eigenvector v1, producing another Lambda one - Repeated application of matrix A on v1 results in consistent Lambda one output - Repeated occurrence of eigenvector v1 results in n many Lambda ones - Expression: c1 times n many Lambda ones, v1, plus c2 times n many Lambda twos, v2 - Inquiry about the mathematical nature of Lambda - Discussion on the computational complexity of raising a matrix to a high power - Scalars, such as lambdas, can be easily raised to high powers using calculators - Discussion on using computers to determine eigenvalues (Lambdas) and eigenvectors (V's) for large n - Encouragement to apply matrix repeatedly to vectors to observe behavior ## Demographic Transitions and Migration Patterns - Discussion on demographic transitions between cities like Denver and LA - Analogy of K-pop and Ska used to illustrate migration patterns - Data from U-Haul and moving companies used to track demographic transitions - Graph and matrix can represent comings and goings between different buckets - Examples of buckets: states, cities, radio stations - Matrix values mentioned: one-third, two-thirds - Discussion on eigenvalues and eigenvectors in relation to matrix power - Lambda one identified as the first column of matrix D, value is one - Computers report eigenvectors as unit vectors by default, dividing by their own length - Unit vectors may have gross entries due to normalization - As n approaches infinity, raising one-third to higher powers results in it getting smaller - In the limit, as n goes to infinity, the probability approaches zero - Probability transition matrix applied to any initial state results in the first term becoming negligible - Raising one to the nth power remains one, resulting in c two times one - Inquiry about the distribution of people between K-pop and Ska - c two is a constant affecting the distribution outcome - Discussion on vector entries not summing to one, questioning its validity as a probability vector - Freedom in choosing constant c2 to achieve desired vector effect - Construction of vector in terms of a probability vector with flexibility in c2 - Long-term distribution between two radio stations or cities expected to be fifty-fifty - Flowchart symmetries support this distribution - Plan to interpret the flowchart further - Initial state can be set to 100% K-pop - Initial state can be set to 0% Ska, 100% K-pop - Dot product used to calculate transition probabilities - Two-thirds probability for K-pop to K-pop transition - One-third probability for K-pop to Ska transition - Problem formulation can be approached through rights or lefts - Discussion on left formulation and its agreement - Mention of the need for probabilities to sum to one - Intuitive approach suggested for averaging after hearing a 'k' ## Upcoming Topics and Clarifications - Change in values mentioned, further clarification needed - Plan to discuss left versus right formulation - Taylor series to be covered in the next session - Matrices topic concluded for now # 6-26-25 ## Project Updates - Meeting has started - Two projects posted: matrix multiplication and image compression through eigen information - Projects available on the website with a deliverable checklist - Projects aim to build a larger code base with generative AI to solve real-world problems - Projects include a one-page reflection on accomplishments and learnings - A five to seven minute video is required to demonstrate the code base and explain its components ## Deliverable Considerations - Concerns about the ease of submitting work to generative AI without understanding the process - Emphasis on ensuring students gain knowledge from the projects - Video format is considered the easiest way to demonstrate understanding - Consideration of different media formats for project deliverables - Discussion on traditional exam methods like blue books - Companies producing blue books are seeing increased purchases as traditional exam methods are revisited - Students are encouraged to explore both projects but focus on matrix multiplication ## Future Discussions - Codes are posted on high TA for reference - Discussion planned for Monday on deliverables and capabilities ## Technical Enhancements - Python ports have been made for all projects to ensure compatibility ## Submission Guidelines - Submission zone will accept multiple submissions including handwritten and code materials - Option to upload handwritten responses and built codes - Submission zone will accept zip files for convenience ## Taylor Series Discussion - Files should be named in order with a simple structure like '01.name', '02.name' for easy reference - Question 5 on the current homework assignment is designated for group work - Introduction to Taylor series will begin today - Two dimensional Taylor series might be briefly touched on today - Helper materials for two dimensional Taylor series will be provided by Monday - Continuation of Taylor series discussion planned for today ## Probability Transition Matrix - Discussion on probability transition matrix and its symmetric properties - Discussion on regular stochastic matrix problem setup - A matrix acting on an initial probability gives rise to a new state in a discrete time process - A matrix is used to represent transitions between states, with rows and columns labeled for each state - Vector representation includes percentage on k and percentage on s - Suggestion to visualize the vector representation with a clock analogy - Quick back of the envelope check suggested for matrix setup - Decision to discuss probability vector in detail - Task assigned to populate the matrix based on the given statement - Transition from state j to state i is the typical formulation - Additional 30 seconds allocated for discussion ## Matrix Value Placement - Discussion on placing values in the probability transition matrix, specifically for state k to state k - Emphasis on avoiding common mistakes like flipping values - Discussion on flipping diagonal elements for left vs right stochastic matrices - Placement of one half and one fourth in the matrix for state transitions - Clarification on transition probability: first row, second column should be transition to state k from state s, with a value of one half - Columns in the left stochastic matrix add up to one - Transition from state k to state s should have a value of one fourth - Basis vectors used for interpretation of matrix representation - Matrix columns add up to one, indicating a regular stochastic matrix - All matrix entries are non-negative - Application of matrix to standard basis I hat discussed ## Initial State and Transition Predictions - Initial state vector (I hat) represents the initial configuration of radio listening - Current configuration: All listening to k pop, none to Scott - Application of probability transition matrix predicts future state - Calculation: 3/4 times 1 plus 1/2 times 0 equals 3/4 - Calculation: 1/4 times 1 plus 1/2 times 0 equals 1/4 - Current state: All listening to K pop, none to Ska - Transition to next state: 3/4 listening to K pop, 1/4 to Ska - J hat represents an even split: 1/2 listening to K pop and 1/2 listening to Ska - Red notes indicate potential errors in transition probability matrix setup - Right stochastic matrix discussed, where rows sum to one - Application of right stochastic matrix to initial state of all listening to K pop results in 3/4 in the first row - Discussion on potential errors in transition probability matrix setup, indicated by red notes - Verification of matrix setup by checking if a equals three fourths ## Eigenvalues and Eigenvectors Discussion - Eigenvalues and eigenvectors are being calculated from the a matrix - Eigenvectors are stored in the x matrix, eigenvalues in the d matrix - MATLAB function 'eig' is used to compute eigenvalues and eigenvectors - One eigenvalue identified as one - Discussion on the effect of repeatedly applying the matrix to the probability vector, specifically the one quarter eigenvalue approaching zero - Need to normalize the probability vector for accurate representation - Probability vector formulation resulted in one third and two thirds ## Linear Algebra in Scientific Computing - Discussion on the relevance of eigenstructure in linear algebra for scientific computing ## Code Development - Discussion on building a code from scratch and commenting for clarity - Educational code with comments available on HiTA and Python port for replication ## Polynomial Functions - Discussion on scalar functions of scalar variables in mathematics - Simple functions refer to basic mathematical functions often introduced in high school - Introduction to polynomials as linear combinations of powers - Focus on two simple polynomial functions for analysis ## Polynomial Visualization - Visualization of x squared and x cubed polynomials: x squared goes up, x cubed has one side going up and one going down - Polynomials exhibit bumpy features when more terms are added - As x approaches infinity, polynomials diverge in positive or negative directions - Polynomials diverge as magnitude goes to infinity - Derivative of a polynomial results in another polynomial - Derivative of a cubic polynomial results in a quadratic polynomial - Polynomials differentiate to lower order polynomials until they vanish ## Oscillatory Functions - Discussion on oscillatory functions and their non-perpetual oscillation - Oscillations eventually diverge at the sides, not perpetual - Hyperbolas considered in relation to oscillations ## Functions with Asymptotes - Discussion on functions with finite vertical asymptotes and their behavior - Functions associated with staining petri dishes and cell division - Mention of exponential growth in relation to these functions ## Limitations and Extensions of Polynomial Models - Discussion on limitations of polynomials in modeling certain STEM phenomena - Inquiry on methods to incorporate other behaviors into polynomial models ## Polynomial Examples - Computers cannot represent transcendental numbers like pi, which go on forever ## Taylor Polynomial Approximation - Discussion on approximation by Taylor Polynomial to match functions locally - Example given: f(x) = x^n, discussing low degree polynomials - Discussion on nth degree Taylor Polynomial centered at x₀ = 0 - Explanation of nth degree polynomial with highest order x power being n - Discussion on nth order polynomial with terms up to aₙ(x-x₀)ⁿ - Key statement: Polynomial at x₀ equals the result of substituting x₀ into the polynomial - Agreement at x₀: Evaluate polynomial at x₀, a₀ should match the function value - Agreement at x₀: a₀ should be e^(x₀) for polynomial and its derivatives ## Derivatives of Polynomial Functions - Derivative of pₙ at x₀: Constant term becomes zero, derivative of (x-x₀) yields a₁ - Further derivatives: Coefficient a₂ emerges, subsequent terms follow pattern - Application of the power rule in derivatives, with x₀ substitution leading terms to zero - Derivative process: n comes out front, reducing power by one - Discussion on derivative process: Only a₁ remains after substitution, aligning with f'(x₀) - Forcing the derivative of polynomial to agree with target function's derivative at x₀ - Observation of pattern formation with pₙ'' at x₀ - Second derivative: a₁ term is eliminated, resulting in 2a₂(x-x₀) - Matching polynomial's second derivative at x₀ with the function's second derivative - Higher order terms in derivatives reduce sequentially: n, n-1, n-2, etc. - Substitution of x = x₀ results in terms becoming zero ## Third Derivative Discussion - Introduction of third derivative in discussion, aiming for clarity in notes - Plan to write a statement about the third derivative for documentation - Taking three derivatives results in 2 * 3 * a₃ * (x-x₀), with only a₃ term remaining - Demand for polynomial to match third derivative of function at x₀ - Choice of e^x simplifies calculation of derivatives - Coefficients for Taylor Polynomial: a₀ = e^(x₀), a₁ = e^(x₀), a₂ = e^(x₀)/2, a₃ = e^(x₀)/(3*2) - Observation of factorial pattern emerging in Taylor series construction - Expression for pₙ(x): Sum from k=0 to n of aₖ(x-x₀)ᵏ - Repeated appearance of e^(x₀) in all coefficients due to chosen function - Connection between denominators and subscript of coefficient: 6 * 5 * 3 - Coefficient aₖ in Taylor Polynomial: Denominator is k factorial, with (x-x₀)ᵏ term - Centering at x₀ = 0 simplifies calculations, symbolic x₀ used for generality - McLaurin representation is the name for Taylor polynomial centered at zero - Aim to build a polynomial that matches the function e^x - Key point: First derivative provides geometric interpretation of function behavior - Higher order derivatives provide geometric information: concave up, concave down, etc. - Aim to align polynomial with function at specific point, capturing geometric hierarchy - Collect geometric information of f at x₀ to build polynomial - Taylor polynomials and series aim to extract essential features of a function to match data in the polynomial ## Taylor Series and Infinity - As n approaches infinity, the Taylor series becomes exact - Mention of series and sequences in relation to Taylor series - Humorous remark about not admitting to letting polynomials become infinitely long - Taylor series becomes identical to the function on its domain as n approaches infinity - Question raised: How good is the approximation when using computers? ## Error Estimation in Polynomial Approximations - Importance of estimating error in polynomial approximations, similar to measuring liquid in a graduated cylinder - Maintaining awareness of information lost in polynomial representation - Mention of Taylor's theorem as a reference for understanding timing in teaching ## Intermediate Value Theorem - Taylor Lembrough's theorem: Assume f is an n+1 times differentiable function on interval I - Introduction of typical notation used in class for x₀ ## Greek Letters and Notation - Discussion on Greek letter c and its pronunciation in Greek and American English - Anecdote about math classes with multiple squiggles - Visual description of drawing Greek letter xi: resembles Charlie Brown or Homer Simpson with a tuft of hair and a tail - Existence of a sea between x and x₀ ensuring certain conditions hold ## Taylor Polynomial and Remainder - Taylor polynomial equals nth degree polynomial plus Taylor remainder - Various formulations for Taylor remainder, including Lagrange's formulation ## Taylor Series Formulation - Formula: f^(n+1)(c)/(n!) * (x - x₀)^(n+1) where x is the independent variable and x₀ is the point of expansion - Centering the polynomial representation at x₀ - Discussion on the notation of f function with superscript in parentheses indicating n+1 derivative - The function f is exactly the Taylor polynomial plus an error term for some c ## Error Identification in Polynomial Approximation - The theorem captures what is lost in polynomial approximation, but the exact size of the error is uncertain - The location of c is within the interval, but its exact position is unknown - Example calculation of Taylor polynomial and estimation of Taylor remainder - Discussion on identifying the amount of information lost in polynomial approximation ## Coursework and Assignments - Learning objectives (LOs) will involve interaction with Hatiya - Handwritten work will be required for homeworks and projects - Scratch work should be organized, linearized, and submitted with code ## Submission Details - Drafts can be reviewed before final submission - Canvas will be opened for submissions at the end of next week, with rolling submissions thereafter ## Project Requirements - Projects may require typesetting for reflections - Projects will have rolling due dates, feedback will be solicited before finalizing dates ## Lecture Pacing and Planning - Discussion on the pacing of Taylor series topics, with some topics being too advanced - Clarification on f(x) as the exact equation and P sub n as the polynomial approximation - Discussion on taking the limit as a function approaches zero, leaving the polynomial as the function when n approaches infinity - Estimation of the worst-case scenario for polynomial approximation error ## Taylor Series Error Analysis - Discussion on the potential divergence of Taylor series if derivatives are ill-behaved - Importance of the c value and derivative behavior in error term analysis - Example replication in coding base, using MATLAB for Taylor polynomial - Discussion on the remainder term in Taylor's theorem and its exactness - Clarification on the remainder term being the only component on the right-hand side in Taylor's theorem - Discussion on the magnitude of error between the function and its approximation ## Uncertainties in Derivative Behavior - Uncertainty about the behavior of the n+1 derivative evaluated at some c, where c is between x and x naught - Suggestion to bound the error term using a placeholder 'm' for the boxed term, multiplied by (x - x naught)^(n+1) over n factorial - Clarification on 'm' as the maximum value of the n+1 derivative on the domain - Estimation of an upper bound for the error in Taylor's theorem without knowing the exact c value ## Building Taylor Polynomials - Example started with f(x) = x log x to find a third derivative - Taylor polynomial centered at x naught equal to one to approximate f(2) - Discussion on estimating 'm' for the remainder term in Taylor's theorem - Discussion on using Taylor polynomial to approximate x log x at x = 2 - Requirements for building a Taylor polynomial to third order: constant term, first derivative term, second derivative term, and third derivative term - Need three derivatives for the polynomial and one more for the remainder term - Calculation of the derivative of x log x: f'(x) = log x + 1 - Calculation of the second derivative of x log x: f''(x) = 1/x - Calculation of the third derivative of x log x: f'''(x) = -1/x^2 ## Derivative Calculations - Mention of the quadruple derivative for further analysis - Calculation of f'(1) for x log x: f'(1) = 1 - Calculation of f''(1) for x log x: f''(1) = 0 - Calculation of f'''(1) for x log x: f'''(1) = -1 - Quadruple derivative not centered at x = 1, used for remainder estimate - Explanation of third order Taylor polynomial: P_3 = f(1) + f'(1)/1! * (x - x_0)^2, with known x_0 values - Explanation of centering the Taylor polynomial at x naught = 1 - Calculation of f(1) and f'(1) for centering, with f(1) = 0 and f'(1) = 1 - Cubic approximation to x log x: P_3 = (x - 1) + (1/2!)(x - 1)^2 - (1/3!)(x - 1)^3 - Explanation of centering at x naught = 1 as the point where the polynomial and function match exactly - Discussion on the effectiveness of the cubic approximation slightly away from x naught = 1 - Calculation of f(2) using a pocket calculator instead of plugging into x log x - Use of third order Taylor polynomial P_3 to approximate f(2) by plugging in x = 2 - Clarification on the sign of the third derivative term in the Taylor series, confirming it is plus f'''(x) - Question raised on how to bound the error in the approximation of f at x = 2 - Discussion on the application of Taylor series in scientific computing and error bounding in approximation schemes - Instructor preparing to use the projector for further explanations ## Lecture Anecdotes - Anecdote about a chaotic lecture with overhead projectors and rapid slide changes - Mention of a bilingual class with dialogue in Chinese - Discussion on the challenges of bilingual classes and rapid slide changes ## Error Estimation in Taylor Polynomials - Error estimate bounded by (x - x naught) with x naught = 1 and x = 2 - n value in the approximation is 3, confirming the use of a third order Taylor polynomial ## Error Bound Estimation - Attempt to make an informed decision on error bound estimation using MATLAB - Inquiry about the fourth derivative term, initially thought to be two ## MATLAB Function Definitions - Correction on the fourth derivative term: 2/x^3 - Discussion on plotting features in MATLAB for visualization - Explanation of defining a function with four derivatives in MATLAB using anonymous functions ## MATLAB Environment Discussion - Discussion on MATLAB's default setting to work with matrices and vectors - Visualization of functions as parabolas in MATLAB environment ## MATLAB Plotting Discussion - Reminiscing about algebra class and the method of plotting points on graph paper ## MATLAB Domain Specification - Use of linspace in MATLAB to create a linear spacing of x values for domain specification - Specification of domain in MATLAB from 1 to 2 with 100 evenly spaced points - Discussion on plotting input variable x and output f in MATLAB ## MATLAB Taylor Series Discussion - Issues with Taylor series implementation in MATLAB, specifically with the fourth derivative - Discussion on creating a table of points for plotting in MATLAB - Clarification that x is a vector, consisting of 100 numbers, in MATLAB - Discussion on the misconception of cubing vectors in MATLAB - Explanation of element-wise operations in MATLAB using the dot operator for cubing vectors ## MATLAB Error Reports - Error encountered during plotting in MATLAB related to pre-class activities - Error due to matrix dimensions not agreeing during plotting in MATLAB ## MATLAB Taylor Series Limitations - Discussion on the role of the fourth derivative in the Taylor remainder theorem - Discussion on the limitations of Taylor series in identifying the exact value of c between 1 and 2 - Consideration of the worst-case scenario for the fourth derivative on the specified domain - Inference from graph about the largest value of m, which is 1 - Fourth derivative value is 2 when m equals 1 - Known values: m = 2, x = 2, egg naught = 1 - Discussion on bounding the error in MATLAB Taylor series - Discussion on using graphical methods to develop bounds for the fourth derivative in MATLAB ## MATLAB Function Behavior - Observation that as x decreases towards 1, the function grows, indicating the largest value is at x = 1 - Mention of using tools to determine the remainder R3 of x at 2 - Clarification on the use of the fourth derivative in Taylor series, specifically with n = 3 and the expression involving factorial and powers ## MATLAB Taylor Series Calculations - Magnitude of the fourth derivative is less than or equal to 2 on the domain from 1 to 2 - Calculation involves (2 - 1)^4 over 3 factorial - Calculation confirmed: 2 divided by 3 for the fourth derivative term ## MATLAB Remainder Concerns - Concern about the presence of a four factorial in the denominator in notes - Mention of Taylor remainder and Lagrange remainder formation ## MATLAB Correction Notes - Correction made: n plus one factorial should be used instead of n factorial - Calculation detail: 1 raised to the fourth power, 2 over 4 factorial equals 1/12 ## General Suggestions - Suggestion to make a note for future reference if an issue does not occur again - Discussion on notation for Taylor series: 'a' as the centering point (x₀) and 'x' as the estimation point ## MATLAB Visualization Plans - Plan to leverage computer power to visualize completed Taylor work ## MATLAB Function Definition - Defining the original function in MATLAB without using dot products - Generating a list of input values for x and applying log x to create a new list ## MATLAB Vector Operations - Accepting vectors as inputs without needing dots on pluses and minuses - Element-wise operations are default for pluses and minuses in vectors - Products require special attention when dealing with vector inputs in MATLAB ## Taylor Series Details - Third derivative referred to as f triple p, fourth derivative as f quadruple p - Centering point x₀ is equal to one for Taylor series - First degree Taylor polynomial is also known as the tangent line approximation - Second degree Taylor polynomial referred to as parabolic approximation - Cubic approximation follows the parabolic approximation in Taylor series - At the centering point, Taylor series approximations are exact - Outputs include the function itself represented as a column vector ## Plotting and Approximation Details - Plotting from zero to five with 100 points for approximation - Capital X represents the approximation point, x = 2 - As the radius increases, the approximation near the center improves - Convergence of Taylor series can be poor depending on the function - Plan to plot inputs with corresponding outputs for visualization - Use 'hold on' in MATLAB for additional graphical work - Plot includes function and polynomial approximations up to cubic order - Increase line width for better visibility - Use marker size 30 for plotting specific points ## Plotting Enhancements - Plot will include a point on the f curve at x = 2 to indicate the target - Output from the polynomial will be plotted to compare with the target - Use 'hold off' after plotting to stop adding to the current graph - Add a legend to the plot for clarity ## Error Analysis and Aesthetic Improvements - Utilize generative AI for enhancing plot aesthetics, including legend and titling - Print absolute error and absolute percent error for approximation - Print bound on Taylor error for analysis - First order Taylor polynomial is flat, representing the tangent line approximation - Point of tangency for the first order Taylor polynomial is at x = 1 ## Approximation Observations - Marker size is considered too large, appearing as a blob - Second order approximation is not quite accurate - Third order approximation improves curvature, described as cubic style - Blue point represents the target value, 1.38 - Purple point represents the output value, 1.33 - Quantifiable error between third order Taylor polynomial and target function is 0.052961 - Absolute error is calculated as the absolute value of f(2) minus p - Absolute percent error calculated as 3.30.8% - Error less than the bound put down by Taylor ## Homework Updates - Code with extensive comments will be posted for those who have questions - Concise version of the code will also be posted for those who prefer it - Homework due date moved to next Friday - Second homework due date set for Wednesday, a week after the first homework submission ## Miscellaneous - Casual conversation about personal plans, no relevant meeting details ## Course Material Access - Assignments will remain accessible after the course closes - Exporting conversation in .m format results in a PDF on Canvas - Images in the PDF are rendered as text, not from markdown - Obsidian is an open-source markdown viewer that allows PDF export ## Advanced Approximation Discussion - Discussion on multivariate Taylor or quadratic approximation - Possibility of cubic approximation with higher dimension tensor - Discussion on third order derivatives from multivariate functions using tensorial notation - Multi-index Taylor notation is convoluted but effective ## Arbitrary Approximation Goals - Discussion on the goal of achieving arbitrary approximation - Discussion on the value of code for cubic multivariate polynomial approximation ## Academic Publication Discussion - Discussion on the effectiveness of quadratic and cubic terms in polynomial approximation - Discussion on the novelty required for academic publication in mathematics ## Computational Efficiency and Modeling - Discussion on efficiency improvements in computational sciences as a publishable topic - Mention of a published article on modeling a zipline, potentially relevant to current discussions - Discussion on the importance of applying mathematical models to real-world scenarios for computational efficiency ## Domain Extension in Approximation - Discussion on extending the domain of approximation by adding more terms or re-centering Taylor series ## Slope Fields and Tangent Lines in Approximation - Discussion on using slope fields and tangent lines in approximation, similar to techniques in computer graphics ## Visual Techniques in Approximation - Mention of using squinting as a technique in approximation, potentially related to visual perception in modeling - Mention of a gambling matrix in relation to visual techniques ## Relevance to Other Disciplines - Mention of capturing words on the backboard during discussions ## Feasibility of Mathematical Models - Discussion on the feasibility of certain mathematical models being achievable due to their simplicity - Discussion on the feasibility of certain mathematical models being achievable due to their simplicity ## Goal Determination in Mathematical Models - Discussion on the goal of determining relationships between two points - Question raised about the requirement of this topic for physics # 6-30-25 ## Topics - Course logistics and administration updates - Homework 1 adjustments and project allocation - Introduction to Taylor series (concept, coding implementation) - Finite difference approximations via Taylor expansion ## Key Notes ### Course Logistics - Issue persists with ‘567 not loading with AI’; instructor plans workflow streamlining to reduce manual switches. - Homework 1 Problem 5 is being moved to a **project** due to its length (15–17 pages when processed as a project). - Homework due date clarified: **Wednesday by 11:59 PM**, Canvas will accept multiple uploads, PDFs, and zipped code. - Instructor is transitioning away from Canvas “activities” in favor of direct instructions embedded in lecture pictures to streamline work across courses. ### Project Updates - Six draft projects are being prepared for proofing today; Homework 2 will drop tomorrow. - Projects will integrate multivariable Taylor series, eigenvalue/eigenvector computations, and scientific visualization. ### LOs/LRs Grading Workflow - Generative AI assessed recent LOs/LRs based on prompt counts, word counts, and completion detection. - Errors in automated grading are being surveyed; students should report mismatches via the survey linked in announcements. - First week’s LOs/LRs will be marked as complete for submissions regardless of automated detection issues. --- ## Lecture Content ### Taylor Series: Conceptual Introduction - Defined Taylor series as an infinite sum of polynomial terms representing a function. - **Transcendental nature**: True Taylor series require the infinite sum; Taylor polynomials truncate at finite $n$. - Introduced remainder term as the difference between the polynomial approximation and true function value. ### Coding Implementation (MATLAB) - **Example:** Exponential function $f(x)=e^x$, centered at $x_0=0$ (Maclaurin series) - Developed code to: - Define $x$ as a vector input using `linspace`. - Initialize $y$ as a zero vector matching the size of $x$. - Loop from $k=0$ to `Taylor_upper` adding each term $\frac{x^k}{k!}$. - Discussed use of dot operators (`.*`, `.^`) for vectorized computation. - Introduced anonymous function handles for Taylor coefficients for modularity. ### Sin(x) Series Considerations - Discussed odd-only summation for $\sin(x)$ Maclaurin series: - Required zeroing coefficients for even indices using logical checks with `mod(k,2)`. - Implemented sign alternation via $(-1)^{(k-1)/2}$ for odd indices. --- ### Numerical Differentiation via Taylor Expansion - Introduced $\Delta x = h$ as step size. - Derived forward difference approximation: f′(x0)≈f(x0+h)−f(x0)hf'(x_0) \approx \frac{f(x_0 + h) - f(x_0)}{h} - Noted: - First-order accurate (truncates after linear term; error is $O(h)$). - Approximations trade off between **truncation error** (from series truncation) and **round-off error** (from machine precision). - Students questioned: - How H affects accuracy: smaller $h$ reduces truncation error but may increase round-off error. - Differences between finite difference and symbolic derivatives: finite differences approximate using function values; symbolic differentiation yields exact expressions. --- ## Implementation Reflection - Emphasis on automating Taylor expansion approximations in code rather than hard-coding known series. - Previewed transition to constructing finite difference formulae for numerical derivatives in upcoming sessions. --- # 7-1-25 ## Project Updates - Two more projects have been posted - Review of project formats is planned - Bullet points will be added to describe project contents - Canvas will be used for project submissions starting next Monday - Feedback and completion status will be provided based on submissions ## Feedback Process - Audio transcriptions will be used for quicker feedback ## Homework Updates - Homework number one is due tomorrow - Problem five moved to a project due to length - Homework two will be posted tomorrow ## Lecture Updates - Lecture objectives are slightly behind schedule - Lecture objectives will be synced with Canvas - Efforts are being made to catch up on lecture objectives ## Current Lecture Topics - Discussion on finite difference approximations to derivatives - Upcoming topic: truncation versus round-off errors - Introduction to calculus earlier in the course than usual - Introduction to Taylor series in the context of scientific computing - Discussion on constructing derivatives in a computational environment - Further exploration of Taylor series for additional insights ## Lecture Materials - Discussion on constructing Taylor series centered at x₀ = 1/2 instead of 0 - Markdown file in lecture picture category contains instructions ## Lecture Activities - Centering Taylor series at points other than zero or negative numbers is recommended - Developers updated the site to make the green check box functional ## Teaching Assistant Tools - Use high TA to learn material, green check mark testing is optional - Feedback on green check mark testing will be provided to administrators ## Upcoming Meetings - Last meeting scheduled for Thursday, 07/11 ## Additional Insights - Detailed breakdown of Taylor series using order notation - Explanation of Taylor series terms stopping at order x to the fourth without using summation - Highest power index in Taylor series indicates the limit of terms discussed - Taylor series relies on derivatives and plugging them into the equation ## Historical Context - Historical context: Early radar development involved solving complex partial differential equations - Exotic functions emerged from manual solutions to these equations ## Computational Methods - Truncated Taylor series used as an alternative to exotic functions in early computational algorithms - Taylor series allowed calculations with limited computational power, such as with pocket calculators ## Taylor Series Expansion - Isolating derivatives to create formulas for approximate derivatives using the original function - Expression involves f(x₀) + f'(x₀)h ± f''(x₀)/2! h², indicating the use of Taylor series expansion - Inclusion of f'''(x₀)/3! h³ term in Taylor series, with alternating signs for ±h - Big O notation indicates error term of order h⁴ - f'(x₀) = (f(x₀ + h) - f(x₀))/h + O(h) - Plus h for the first term, minus h for the second term, subtract the two, divide by 2h, resulting in Big O(h²) - Discussion on higher order derivatives and their importance in life applications - Explanation of truncation step in obtaining approximations from Taylor series ## Finite Difference Formulas - Finite difference formulas are used with truncation order to approximate derivatives - First order finite difference formula involves truncating at first order in h (delta x, step size) - Second order finite difference formula is used for the first derivative ## Advanced Derivative Concepts - Discussion on third derivatives and their role in Taylor approximation for any function - Emphasis on calculating derivatives without prior knowledge of the function's derivative structure - Algebraic methods to isolate f'(x₀) and f''(x₀) using truncation ## Truncation Errors - Discussion on first and second order truncation errors in Taylor series - Different truncation methods based on the extent of Taylor series used - h is described as Delta x, providing a standard way to look away from x₀ ## Variable Usage in Computations - Discussion on using h as a variable for limits, avoiding delta h's or delta x's in computations - Clarification on third order division by 3h or 2h² ## Manipulation of Taylor Series - Discussion on different powers of h in the denominator based on Taylor series manipulation ## Visual Aids - Instructor plans to show a partial picture related to the topic - Instructor attempts to find and zoom into a specific region of a picture for better illustration - Instructor discusses the importance of delta x being small and its relation to truncation - Discussion on scaling of first and second order truncation errors with decreasing delta x or h ## Error Reduction and Visual Challenges - Higher order truncation decreases error faster with smaller mesh sizes - Instructor prepared a graph to show in two parts but is unable to find the file ## Error Analysis - Statement of error: difference between derivative and finite difference approximation - Explanation of log-log plots with logarithmic scaling on both axes - Explanation of how power functions appear as lines on log-log plots, with slopes indicating the power (e.g., x² appears as a line with slope 2) ## Truncation Error Comparison - Second order truncation error is more accurate for a given h value compared to first order - As h decreases, second order truncation error decreases faster than first order ## Higher Order Terms in Series - Retained terms up to big O(h^4) or O(h^3) in the series - Discussion on how h^3 line should appear below blue line on graph, indicating steeper descent ## Formula Validity Concerns - Instructor mentions that making h too large can invalidate the formulas ## Graphical Analysis - Making h small is generally beneficial for accuracy - On log-log plots, different methods may converge at a common origin - As h approaches zero, everything should become exact, but this is not reflected on the current graph ## Additional Error Types - Discussion on other types of errors not currently visible on the graph ## Finite Difference Approximation - Finding a second order finite difference approximation for s''(x₀) ## Instructional Techniques - Instructor suggests mentally and physically juggling terms for better understanding in upcoming examples - Two-minute pause for students to think about term derivation ## Data Integrity Issues - Instructor mentions potential data loss during analysis ## Second Order Approximations - Instructor is searching for a second order finite difference approximation for f''(x) - Solving for f''(x₀) with a second order truncation error involves dividing by h² - Dividing h to the fourth by h squared results in a big O of h squared - Truncation error is determined by the power of h in the big O term - Subtracting known formulas at plus h and minus h cancels even terms, isolating f prime - Adding terms for plus h and minus h shows changes in the first term f(x₀) ## Combining Terms in Approximations - Retaining two f(x₀) terms for calculation - f' terms cancel out when adding plus h and minus h - f'' terms have the same sign and add up to two when combined - f''' terms cancel out when adding plus h and minus h variants - Solving for f''(x₀) involves subtracting 2f(x₀) from f(x₀ + h) and f(x₀ - h) - Dividing by h squared reduces the power of h to h squared, affecting truncation ## Taylor Series and Approximations - Taylor series approximation leads to a second order finite difference approximation for f''(x₀) - Taylor series is centered at x₀ - Symmetrical points around x₀ are considered: plus h and minus h - Forward difference involves looking ahead of x₀ by adding h - Instructor discusses the structure of finite difference approximation ## Third Order Approximations - Instructor plans to derive a third order approximation next - Extracting the third derivative is the current focus - Acknowledgment that similar work has been done by others - Instructor aims to combine elements to derive f''' or f'''' - Process involves careful combination of terms to achieve desired derivative ## Extracting Derivatives from Taylor Series - Taylor series contains all necessary derivatives for approximations - Focus is on extracting specific derivatives rather than all information ## Instructor's Plans - Instructor plans to keep certain pieces visible for future reference ## Developing New Formulas - Developing formulas for f(x₀) with plus or minus two h's - Incorporating plus or minus two h's into f(x₀) affects the structure of the series - The squaring of terms affects the plus or minus structure in the series - Squaring terms in the Taylor formula affects the plus or minus structure, resulting in h squared and a factor of four - Squaring leads to the inclusion of f'''(x₀) in the formula - Cubing terms in the Taylor formula results in a factor of eight - Exhausted possibilities from manipulating x₀ plus and minus h ## Higher Order Derivatives - Considering order, adding h to the fourth for constructing other derivatives - Start adding further h contribution after using up terms - Gathering more data by looking one more click away in h ## Coefficient Combinations - Instructor proposes using a combination of coefficients (a, b, c, d, e) with f(-2), f(-1), f(0), f(1), f(2) - Placeholder notation for f(x₀) minus two h's is being used - Discussion on the number of h's in f(x₀) minus two h's, minus one h, plus one h, and plus two h's - Using capital letters A, B, C, D, E for coefficient selection to achieve desired triple derivative - Shorthand notation allows for efficient term selection in f(x₀) representation - f(x₀) remains consistent across different formulas when adjusting by plus or minus h - Factoring off f(x₀) results in different terms: a, b, c, d, e - Changes expected in the structure, increasing complexity ## Interpretation of Notations - Discussion on the meaning of f'₀ in the current context - f'₀ is interpreted as f'(x₀) - Grouping all f' terms results in negative two times a, leading to negative two h - All f' terms should have an h multiplier at the end - f(-1) results in a negative b when considering the formula - f₀ is interpreted as f(f(x₀)), resulting in zero contribution - No 'c' term involved in f₀, only in specific terms - f₁ contributes a 'd' term - f₂ contributes a term - Plus two e and then all terms are multiplied by h ## Contributions of f₀'' - f₀'' contributes h squared to the formula - Evenness of terms results in symmetry, making negative and positive contributions equivalent - Calculation includes a factor of four over two factorial, simplifying to two a - Factoring off f''(x₀) and h squared leaves behind 1/2 b - No f(x₀) involved in f''(x₀) terms ## Simplifications and Factorials - D over two and two two e are part of the formulation - f₀''' contributes h cubed to the formula - Simplification of 4f₋₂ results in negative four thirds, associated with letter A - Next term involves negative one over three factorial - Times b, minimal c, plus d over three factorial, plus four thirds e - Searching for f triple prime, requires dividing everything by three factorial ## Truncation Errors and Orders - Dividing by h cubed results in first order truncation error - Writing out one more step to catch fourth terms - Big O notation for h to the fifth for second order ## Fourth Derivative Discussion - Discussion on the fourth derivative and its independence from plus-minus structure - Raising 2h to the fourth power for fourth derivatives results in h to the fourth - h to the fourth power and two to the fourth due to the 2h part, divided by four factorial - f₋₁ does not require consideration of plus or minus in even order - h raised to the fourth power without plus or minus consideration ## Additional Simplifications - Minimal c, d over four factorial, plus two to the fourth over four factorial - Big O notation for h to the fifth ## Problem Solving Approach - Discussion on the approach to solving the problem using algebra - Choose a, b, d, and e so that the parenthesied quantity is zero - Set the parenthesied quantity to zero, then to one, and back to zero - Five unknowns: a, b, c, d, e - Five equations available to solve for the unknowns - a, b, c, d, and e are raised to the power of one and added together ## Coefficient Matrix Details - Suggestion to consider the powers of a, b, c, d, and e in the formulation - Set up a coefficient matrix for a, b, c, d, and e - First equation needs to be set to a specific value - Second, third, and fifth equations need to be set to zero - Fourth equation needs to be set to one - Coefficient on a, b, c, and e is one - Next equation: negative two, negative, zero, two - Third equation: two, one half, zero, one half, two - Next row: negative one third, one third, one zero, and one third and four thirds - Last row simplifications: two to the fourth over four factorial, one over four factorial, zero - Discussion on factorials and their representation in matrix problems ## Matrix Scaling and Methods - Transition from two by two to five by five matrix - Consider scaling due to square matrix and vector with five elements - Method of undetermined coefficients applied to finite difference - Five by five matrix due to five data points and five coefficient equations ## Truncation Error and Accuracy - Fifth equation necessary for solving batch due to truncation error - Truncation error requires extending to fifth order - Two methods for achieving first order accuracy and beyond ## Data Requirements for Accuracy - Minimum data required for third order accuracy discussed ## MATLAB and Code Utilization - MATLAB file named 'f d matrix Helper' is used for solving matrix problems - FD finite difference matrix helper dot p y code is ready for execution - First row of the matrix consists of ones ## Solving Equations and Derivatives - Solving the equation a times x equals b to find x - Determining values of a, b, c, d, and e for third derivative - In MATLAB, the x vector is equal to a, with orientation from upper left to lower right b - Calculating the determinant of a five by five matrix is costly on a computer - Determinant of the matrix is one, indicating a unique solution exists - Solution from matrix calculation: a = -1/2, b = 1, c = 0, d = -1, e = 1/2 - Coefficient for f triple prime at x naught is one, divided by h cubed ## Derivative Calculations - Zero derivative calculation involves a = -1/2 and b = 1 for f of x naught minus two h's and f of x naught minus h - Plus no minus f of x naught plus h, plus one half f of x naught plus two h's, all over h cubed - Plus order h to the fifth, divided by h cubed, resulting in h squared - More data points needed to capture desired derivative accurately ## Linear Algebra for Coefficient Calculation - Use linear algebra to write coefficients as a linear system for triple derivative - Remove unwanted terms to isolate desired term in the system ## Meeting Adjournment - Meeting adjourned for a short break ## Task Assignments - Discussion on whether additional tasks will be assigned ## Miscellaneous - Discussion on naming suggestions for a new dog - Suggestion to name the new dog 'Truncation Error' ## Software Tools - Discussion on the use of MATLAB for handling imaginary numbers and rotations - Unrelated conversation, no relevant details to add ## File Management - Discussion on file management and organization ## Documentation - Mention of grouping all sightings for documentation ## Mathematical Discussion - Explanation of terms: a = -2h, b = -1h, d = +1h, e = +2h, c = f(x₀) - Discussion on the role of f(x₀) in equations and its relation to other terms ## Miscellaneous - Mention of gambling without money involved ## Live Demonstrations - Attempt to apply finite differences in a live demonstration - Attempt to construct Taylor series approximations for an arbitrary function using finite difference formulas - Discussion on using MATLAB for numerical formulas with sine x - Discussion on differentiating functions using finite difference formulas - Mention of central difference setup for first derivative at second order ## Function Analysis - Explanation of encapsulating f(x-h) in parentheses and dividing by 2h - Description of the anonymous function handle 'd f' requiring a function, a point, and h size ## Derivative Derivations - Derivation of the second derivative: f(x + h) - 2f(x) + f(x - h) ## Expression Simplification - Mention of overuse of parentheses in live settings - Introduction of third derivative concept - Discussion on simplifying expressions by moving constants into the denominator - Simplification by adding 2f(x-h) - 2f(x+h) + f(x+2h) and dividing by 2 ## Expression Concerns - Mention of h cube and tube in the denominator - Concerns about parenthesis work and autocompletion ## Calculation Parameters - Set n equal to 200 for calculations - X axis defined from negative pi to pi - Linearly space from -π to π in MATLAB and slice into 200 points - Set x₀ to 0 as the centering point - Set h to 0.1 ## Finite Difference Approximations - Formulas are second order accurate finite difference approximations to derivatives - Three second order accurate finite difference approximations available - Largest Taylor polynomial constructible is third degree - Domain for Taylor polynomial defined by x variable - x₀ represents the centering point for Taylor polynomial - Building Taylor polynomial with y = f(x₀) + (x - x₀)^1 - Discussion on language for f prime in Taylor polynomial - DF is the derivative of f, centered around x₀ - Importance of sending x₀ and h for derivative calculations - x₀ is a scalar and the centering point of zero - In MATLAB, x is treated as a vector for element-wise computation - MATLAB performs element-wise subtraction of x₀ from each element in the vector - Scalar represents the derivative of f at x₀ with step size h ## Taylor Series Application - Scalar should be applied to x - x₀, which is a vector, using dot multiplication - Add 0.5 times the second derivative to the expression, using f(x₀) and h - Square each element in x - x₀ using dot caret for element-wise operation in MATLAB - Add third term to expression: 1/3! (one divided by six) ## Third Derivative Application - Third derivative denoted as d three f, applied with f, x₀, and h - Dot multiplication applied to x - x₀, raised to the power of three ## Numerical Differentiation and Taylor Series - Numerical differentiation formulas are key components - Constructing Taylor series through numerical differentiation - Plotting the original function using x values and f(x) - Retaining more terms in the Taylor series improves truncation error ## Numerical Approximations - Higher order Taylor series terms are coupled with finite difference approximation - Parenthesis error encountered during execution - No legends in plots, but cubic and sine functions are distinguishable ## Domain and Code Verification - Consider expanding the domain to observe sine function oscillations - Verify if third order Taylor polynomial codes are on the lecture codes section ## Finite Difference Methods - Discussion on transitioning to the next topic or section - Python and MATLAB have built-in functions for finite difference approximation of a certain order - Central difference method involves symmetric selection of neighboring points - High order derivatives can be approximated even with limited domain data - Back differences and center differences discussed for various derivative orders - Language used: 'blank order accurate approximation to the blank derivative' - Transition from symbolic derivatives to numerical approximations using known formulas ## Experimental Calculations - Attempting to use exponential of the sine function in calculations - Discussion on the relationship between derivatives and their applications - Increasing degrees of freedom requires expanding data range from center point ## Finite Difference Tutorial - Issues with vector-related code causing unexpected behavior - Consider using built-in functions to avoid these issues - Brief pause in discussion to adjust line 12 in code - First order derivative being tested, unsure of impact - Finite Difference Tutorial: Truncation versus Roundoff available on the website under intro to scientific computing - Derivative formulas available in both MATLAB and Python for porting purposes ## Graph Replication and Accuracy - Key graph replication possible from code windows, not shared in code base due to narrative context - Second order accurate finite differences expected with step size h ## Error Analysis and Expectations - Expectation of less absolute percent error as step size decreases - Higher order accurate finite differences expected to outperform first order - Avoid analyzing data to the left of a certain point in the formula ## Truncation Error and Accuracy - As h approaches zero, the approximation becomes 100% accurate and truncation error goes to zero - Third derivative representation becomes exact as h decreases - Improvement in accuracy observed as h decreases, ignoring data to the left of a certain point - Achieved absolute percent error of 10^-10 with h = 10^-2, indicating high precision - As h becomes very small, f(x₀ - h) and f(x₀ + h) become approximately equal, affecting subtraction results ## Precision Challenges - Subtractions approaching zero due to finite precision, leading to potential catastrophic loss of precision - Division by increasingly smaller h values causing calculations to appear larger - Making h very small can cause formulas to become ill-behaved due to finite precision issues - Subtracting nearly equal numbers in MATLAB or Python can lead to loss of significance due to differences in decimal places - Round off error caused by subtraction of nearly equal numbers and division by numbers close to zero - Long-term solution isn't just reducing h indefinitely for better accuracy due to truncation limits - Round off error increases when working on calculators as h becomes very small - Computers use a machine number line due to limited bits, unlike the infinite number line in our minds - Round off errors occur due to gaps in the computer's number line, noticeable during subtraction of similar numbers - Floating point arithmetic is prone to round off errors, to be discussed next time ## IEEE Standards in Programming - Preexisting command models available: iv, id, g, with matrix input - Discussion on eigenvectors and eigenvalues in relation to matrix outputs ## Second Order Accuracy Issues - Discussion on correspondence for accuracy improvement - Round off errors appear sooner with higher order methods due to smaller step sizes - Discussion on parameters affecting higher order derivative accuracy: age and truncation origin ## Derivative Analysis - Mapping through h on the graph to analyze first derivative for different truncations - Investigating the role of higher order derivatives in relation to truncations - Third derivative accuracy is influenced by truncation level and order of accuracy - Second order accuracy issues observed when h is too low, leading to instability and unreliable data ## Optimal h Determination - Formal mathematics available online for optimal h determination - Observations show improvements in results as h is adjusted - Sudden changes in results observed after certain adjustments - Python and MATLAB implementations adhere to IEEE standards for arithmetic operations - Libraries exist for infinite precision, but they slow down computations when more decimal places are requested - Internal programming can store additional decimal places, but it affects performance ## Machine Number Line and Precision - Oscillations observed on the round off error side, potentially due to finite precision issues - Symbolic computation aims for 100% precision, but requires significant RAM for high decimal places - Built-in libraries prioritize accuracy over speed - Discussion on finite precision arithmetic in relation to decimal approximation and Taylor series - Taylor series are fundamental for deriving functions, especially with pocket calculators - Taylor series concepts will recur in various settings, emphasizing the importance of remainders ## Software Implementation - Decision to remove unnecessary visual aids from the presentation - Porting code between Python and MATLAB to maintain dual compatibility - Discussion on code base management and decision-making processes - Discussion on typical practices in scientific computing - Discussion on routines and structures in scientific computing - Investigating context and field for further insights - Investigating context and field for further insights ## Team Feedback - Positive feedback on internal team morale - Discussion on the impact of linear approximations on internal processes - Discussion on the impact of linear approximations on internal processes - Concerns raised about efficiency costs in current processes ## Sequence Analysis - Task assignment for first and second room completion - Ensured first derivative calculations were accurate - Discussion on the sequence of numbers and their implications - Discussion on the transition from five to six equations in the analysis ## Handling Unknowns - Discussion on methods to handle unknowns in equations - Discussion on the transition from five to six equations in the analysis - Discussion on zero, one, two, three, and five in sequence analysis - Discussion on the transition from six equations to eleven, three in sequence analysis ## Sequence Complexity - Discussion on potential complexities in sequence analysis - Discussion on the relationship between accuracy and the number of terms in sequence analysis - Discussion on isolating grade in the context of five equations - Discussion on the implications of five equations in current analysis ## Traffic Management - Discussion on strategies for managing traffic effectively # 7-2-25 ## Grading Breakdown - Homework contributes 80% to the final grade - A- starts at 90 and goes up to 100 - Projects have two parts: initial build and additional extensions - Two-part projects have a total of 20 points, with each part worth 2 points - A category starts at 93 ## Homework Deadlines - Homework 1 is due tonight - Homework 2 is posted on the website and is due next Wednesday ## Project Updates - More projects were released yesterday, with additional ones expected soon ## Class Meetings - Homework problems were trimmed down for clarity - Lecture objectives are available in the lecture pictures file - Lecture pictures have the same instructions as reflection activities - Lecture objectives were updated last night - Due dates will be set right before the class to identify waived assignments - LO (Learning Objectives) instructions will be streamlined - Today is the seventh meeting of the class ## Numerical Methods - Discussion on rounding off versus truncation methods - Discussion on modern programming languages for handling matrices and vectors - Introduction to Taylor series as a key topic - Developing numerical calculus using Taylor series - Using Taylor series to create approximate definitions for derivatives on computers - Plan to explore the next topic after derivatives soon - Second order accurate approximation to the first derivative discussed - Second order accurate approximation to the second derivative discussed ## Coding Experiences - Encountered an issue with code running inconsistently, resolved without clear changes - Experience shared about MATLAB code behaving unexpectedly - Cleared MATLAB memory resolved graph inconsistency issue due to cross-wired memory spaces - Discussion on nth order accurate finite difference approximation ## Higher Order Approximations - Second order accurate approximation to the third derivative discussed - Clarification on language used in discussing approximations - Explanation of finite difference as subtractions approximating derivatives - Discussion on nth derivative and its relevance in language - Introduction to the concept of order accuracy in finite difference approximation ## Finite Difference Approximations - Discussion on the role of the big O term in finite difference approximations - Explanation of how the h in the big O term is raised to the power of p - Description of pth order accurate finite difference approximation as a truncated formula - Discussion on achieving pth order accuracy in finite difference approximations by manipulating terms - Clarification on pth order formula truncated at h to the p for nth derivative at x naught - Explanation of using k as an integer in finite difference approximations ## Taylor Series Applications - Discussion on isolating f soup n using Taylor series and zeroing out equations ## Equation Control - Need to gather more points using plus or minus k for accessing derivatives in Taylor series - Process involves zeroing out unnecessary terms in Taylor series - Need more points for controlling equations by looking at f of x not and points away from it ## Current Session Discussions - Discussion on the V-shaped graph from last session and its implications - Mention of upcoming homework due today and its potential impact on student engagement - Discussion on splitting up files in computer systems - Plan to rewrite code to separate anonymous functions into their own files for better accessibility - Agreement to spend 15 minutes on controlling the Taylor series calculations - Discussion on using code from yesterday to calculate Taylor series through finite difference formulas - Performing the third degree Taylor polynomial in a separate folder for organization - Saving work in 'Taylor live' and creating a subfolder 'seven two' for separation ## File Management and Code Documentation - Opening a new file to explore formulas outside the current program context - Copying existing content into the new file and converting it into comments - Using command or control to turn lines into comments in Matlab - Plan to implement function for first derivative using notation DF - Discussion on the function 'diff' and its role in returning the first derivative - Saving progress in MATLAB even when no changes are made - MATLAB attempts to automatically name files upon saving - Clarification on defining DF in terms of the first derivative using second order finite difference - DF will calculate the second order accurate finite difference approximation to the first derivative - Plan to create two more files to replicate the behavior of DF - Opening another file to continue work on finite difference approximations ## Function Naming and Requirements - Naming convention for second derivative function as 'd two f' to maintain consistency - 'd two f' function requires knowledge of the incoming function, the point of interest, and the h spacing for Taylor series derivation - Anonymous function in MATLAB used to replicate 'd two f' functionality - Saved the function as 'd two f' for consistency - Discussion on naming convention for third derivative function as 'd three f' for consistency - 'd three f' function requires inputs of f, x, and h for calculation - Saved third derivative function as 'd three f' in MATLAB, confirming it as a function - Three m files created, each containing code to replicate anonymous functions ## Code Execution and Habits - Discussion on defining code pieces within MATLAB and Python, and the importance of following calculations - Preference for viewing code chunks before understanding their function - Habit of frequently using the run button in MATLAB GUI ## File Locations and Execution - Confirmation that the function is defined but needs to be called for execution - Files related to TaylorSeries saved in 'seven two' directory - Uncertainty about the location of other files, possibly in 'machine' directory ## Script Execution and Environment Management - Running script in 'machine' directory, focusing on anonymous functions - Commenting out certain calculations for clarity - Clearing all to reset the environment before execution ## Anonymous Functions in MATLAB and Python - Driver file contains finite difference formulas for use in other programs - Discussion on anonymous functions in MATLAB and Python, emphasizing their creation within code and conversion to m files for broader use ## Transition to New Files - Transitioning to a different set of files and initiating 'lekker' ## Browser Navigation - Navigating to a personal site using Firefox ## Finite Difference and Code Snippets - Discussion on finite difference truncation versus round-off errors in web space - Code snippets available for educational purposes to recreate the discussed picture - Key feature of the dialogue is reaching the plot near the error analysis section ## Error Analysis and Accuracy Control - Larger p value allows for faster control of accuracy lost in truncation - Graph shows first order accurate approximation to the first derivative - Error decreases with known function calculation - Second order accuracy improves error faster as step size decreases - Blue line outperforms red in terms of absolute percent error by six orders of magnitude - Smaller step sizes initially reduce error but eventually lead to error growth for both lines ## Number Line Details - Increasing detail to the number line and defining numbers on it - Hiccups occur when defining numbers on the number line ## Paradigms in Arithmetic - Real number line is a continuum of values, emphasizing the continuum aspect - Real number line requires infinite precision, which computers lack - Approximating the real number line in a computational environment is challenging due to its vast data - Precision is crucial when selecting a number on the real number line for STEM work - Broad scale understanding is necessary for effective number line utilization - Orders of magnitude are crucial for understanding scales from atomic nuclei (~10^-10 meters) to intergalactic distances (~10^20 meters) - Need for dynamic range to replicate the continuum of the real number line over wide scales - Clear quantifiable precision is essential in encoding machine numbers ## Floating Point Representation - Floating point number representation: x = ±(1 + f) * 2^e - Little f in floating point representation lives between zero and one, known as the significand - Mantissa is the logarithm of the significand, providing precision in floating point representation - Exponent e in floating point representation ranges from -1022 to 1023 - OnePlus f provides precision in floating point representation - Exponent e provides scale in floating point representation - As f approaches one, adding it to one approaches two, leading to an increment on the two scale - MATLAB uses double precision format at 64 bits - 52 bits are used for f and 11 bits for e in MATLAB's representation - Clarification on spelling: 'precision' is spelled with one 's' - Finite precision achieved by allocating bits to f and e - Double precision uses 64 bits, with one bit for the sign - Consequence of finite precision in floating point representation discussed - Finite precision cannot encode all digits of irrational numbers like pi - Finite precision results in gaps, leaving some real numbers inaccessible - Infinite precision libraries in MATLAB can consume significant memory resources ## Infinite Precision and Computer Algebra Systems - Irrational numbers have infinitely long decimal expansions without patterns - Mathematica is a computer algebra system that prioritizes infinite precision before approximations - Infinite precision allows for dictating precision level but increases memory and computation costs - Finite precision results in gaps, leaving some real numbers inaccessible ## Gaps and Scale in Number Representation - Discussed the concept of gaps in number representation and their implications - Introduced the concept of scale using powers of two: 2^0 = 1, 2^2 = 4 - Machine numbers are placed between scales defined by powers of two, with finite precision due to 52 bits for f - Number of machine numbers depends on the number of bits available - One plus f where f is restricted and has 52 bits - Machine numbers exist between orders of magnitude, influenced by available bits for finite precision - Separation in terms of order by two exponents using the same number of bits for one plus f - Total number of representable values remains constant across lines separated by scaling factors - Discussed powers of two: 2^3 = 8, 2^5 = 32 - Explored the width of chunks on the machine number line - Density of machine numbers varies across chunks of the machine number line - Gaps in the machine number line are not uniform - Larger gaps occur when fitting the same amount of machine numbers in a wider space - As the exponent increases, the width of the space for machine numbers increases, leading to larger gaps - Density of machine numbers is uniform and more dense around zero - Precision discussion involves zero, one, three, and four as reference points - Widths increase at larger scales while maintaining the same number of points - Lecture notes used two steps in scale, maintaining point count - Uniformity is a key point in machine number representation - Challenges arise when implementing number representation on computers due to non-uniformity - Computers approximate mathematical numbers to the nearest representable value due to non-uniform gaps - Mathematical numbers are approximated to the nearest representable machine number - Computers map outputs to the nearest machine number due to finite precision - Round off error occurs when landing in a gap and being forced onto a machine number - Round off error is a result of non-uniform scaling in machine number representation ## Real-World Implications of Number Representation - Discussion on thought processes occurring in real-world scenarios - Searching for a link using email and the keyword 'Mario' ## Bit Allocation in Floating Point Representation - Posting lecture notes on machine representation of numbers to the web - Consideration of bits, unsigned bits, and unsigned shorts in machine representation - Discussion on floats and double precision in machine representation - Discussion on the allocation of bits for signed bit, mantissa, and exponent in floating point representation - Density of machine numbers decreases as you move away from zero - Gaps in machine numbers can be large, affecting calculations like Mario's jump in a game - Video discussion resumed, ensuring no repetition of previous points ## Trade-offs in Scale Selection - Discussion on space games exploring interstellar phenomena and encountering gaps in representation - Variability in gaps is due to maintaining scale while preserving decimal precision - Choosing a certain scale allows for reclaiming bits, offering more precision per scale - Trade-offs exist when selecting scales in number representation ## Gapping in Physics Simulations - Demonstration of ragdoll physics with polygon bodies near and away from the origin - Comparison between single precision and double precision in relation to gapping - Close to the origin, the machine number line is dense with minimal gapping - At 10 kilometers from the origin, objects experience less normal bouncing due to gapping - At 50 kilometers, different parts of objects experience different regions of gap ## Precision and Representation Adjustments - Discussion on abnormal twitching due to lack of programmed feelings - Increase to double precision for more bits in representation - Use of double precision for simulations further from the origin to ensure proper simulation - Typical problems in number representation discussed - Typical problems in number representation reiterated - Consideration of graphs in relation to number representation - Confirmation of dialogues covered in previous discussions - Submission of dialogues with the zip file of the code is considered complete ## Scale and Precision Challenges - Anna asked about a previously erased bullet point related to a question from her - Consequences of non-uniform gaps in machine number line when working with finite precision arithmetic - Gaps between machine numbers increase with scale, affecting precision - Double precision helps maintain good precision despite scale - Code used to identify issues with Mario's jump at scale 64 ## Round Off Errors in Number Representation - Round off error occurs when hitting a number that must be represented in a limited space - Discussion on the representation of the number one tenth and its implications - Discussion on binary representation and memory requirements for repeating numbers - Consideration of cutting off repeating bits to manage memory constraints ## Errors in Series and Representation - Truncation error is associated with stopping the Taylor series before completion ## Managing Repeating Numbers and Errors - Discussion on truncating repeating numbers without using 'truncation' to avoid confusion with Taylor series truncation error - Round off error and its implications discussed with examples ## Equation Scaling and Recognition - Discussion on scaling equations and recognizing equivalent equations through scaling - Discussion on lines in space and their points of intersection ## Matrix and Vector Operations - Introduction of matrix A with values (17, 5) and (1.7, 0.5) - Discussion on formatting numbers for better readability in outputs - Explanation of solving the equation Ax = B in MATLAB using built-in commands - Discussion on non-unique points of intersection and verification of points on the line - Discussion on the system having infinitely many solutions not indicated by MATLAB - Discussion on round off error leading to error in solution and its impact on line recognition ## Machine Number Line Limitations - Discussion on machine number line gaps and limitations in representation ## Computational Environment Concerns - Advice on when to be concerned about computational environment issues ## Precision and Subtraction Issues - Discussion on subtraction of nearly equal numbers leading to zeroing out due to precision limitations ## Numerical Interpretation Issues - Discussion on significant zeros zeroing out leading to machine interpreting the number as zero - Discussion on adding and subtracting numbers on very different scales - Analogy of Mario jumping over a gap illustrating machine number rounding issues - Division by very small numbers can lead to overflow and machine interpreting it as infinity ## General Advice - Reminder to be cautious of three key issues in computations ## Taylor Series and Derivatives - Definition of the derivative as represented by a Taylor series - Truncating the Taylor series to get an approximation - Error arises from truncating the Taylor series, affecting accuracy - For the Taylor series to approximate a derivative, h must be small - Issues arise when x naught is large and a small number is added, leading to potential precision problems ## Numerical Derivative Issues - When h is small, f(x naught + small) and f(x naught) may appear similar, causing precision issues - Division by a very small h can lead to numerical derivative issues - Problem two for small h representation discussed - Problem three for small h representation discussed ## Machine Number Line and Precision Issues - Finite difference issues arise when h is too small, leading to interaction of three problematic issues - Round off error as another source of error when using small h due to machine number line structure - Machine number line cannot handle real number line, affecting decimal precision over many magnitudes - Discussion on predicting round off error occurrence - Lowering h improves error from truncation of Taylor series until it becomes too low, causing glitches - Ill-tempered results occur when unexpected outcomes arise despite previous stability - Introduction of new notation for f(x + h) in numerical derivative context - Mathematical results must land on a machine number, resulting in a value of one plus or minus Epsilon - Discussion on epsilon as the smallest number such that one plus epsilon is not equal to one in machine precision - Epsilon is the smallest number such that adding it to one does not result in one due to round off error - Explanation of twiddles as the difference computed due to epsilons landing on a machine number - Discussion on symbolic calculation of f(x + h) and f(x) in numerical derivatives - Both f(x + h) and f(x) can catch epsilon, affecting the calculation - Assumption made to simplify working with epsilon in numerical derivatives ## Control Steps in Numerical Calculations - Control step discussion regarding handling plus or minus issues in calculations - Discussion on handling inequalities by taking absolute values of plus-minus structures to maximize magnitude - Taking absolute values of plus-minus structures to prevent subtraction from reducing magnitude - Assumption that |f(x)| > |f(x + h)| to simplify inequality handling - If |f(x)| > |f(x + h)|, then |f(x + h) - f(x)| ≤ 2ε|f(x)| - Discussion on upper bounds in numerical calculations with additions and subtractions - Assume f(x) is larger in magnitude than f(x + h) for simplification - Bound established for |f(x + h) - f(x)| to be smaller than twice the magnitude of f(x) ## Precision Representation - Delta f tilde represents what the computer sees, while delta f is the exact mathematical symbol - Tilde indicates the difference between infinite precision and computer representation - Discussion on dividing Delta f by h in numerical calculations - Delta f tilde divided by h must be less than delta f exact divided by h plus two epsilon over h ## Magnitude Effects on Derivatives - As h approaches zero, the magnitude of the effect on the derivative is significant - Assumption of first-order accuracy in numerical calculations - f tilde is less than the expression but equal to f prime plus big O of h to the one for first order accuracy - Plus two epsilon over h in the expression for first order accuracy - Discussion on moving f prime to the left side of the equation - Delta f tilde represents the computer's view of the distance in outputs, while delta f is the exact mathematical distance - Discussion on building a derivative as rise over run, with tilde representing the computer's view - Contrast between f prime of x as the exact mathematical derivative and the computer's derivative - Discussion on error in derivative approximation as mentioned by Xavier - Loss of equality in mathematical argument for the point due to big O h plus two epsilon over h magnitude f - Big O notation discussed as a way to describe the order of magnitude without specifying the constant multiplier - Discussion on the power of h in the expression with two epsilon over h times magnitude f - Big O notation indicates truncation error in Taylor series as a source of error - Discussion on the second term contributing to overall error, including truncation and epsilon introduction - Epsilon is the smallest number such that adding it to one results in a number different from one, due to rounding error on the machine number line - Error in computation arises from truncating Taylor series and finite precision floating point arithmetic - c sub eight or c h becomes approximately two epsilon magnitude f of x over h ## Error Transition Analysis - Discussion on the transition point where truncation error and rounding error are equal in magnitude - Analysis of the graph to determine when truncation error is dominant and when rounding error takes over - Confirmation on understanding the transition point where truncation error equals rounding error - Discussion on the approximation of h squared as two epsilon magnitude f over c ## Symbolic Computation and Order Notation - Discussion on symbolic computation and order notation in first order accurate finite difference - Agreement on considering the magnitude as order Epsilon without detailed computation - Discussion on h being approximately on the order of the square root of epsilon - Epsilon is the smallest number such that adding it to one results in a number different from one, due to rounding error on the machine number line - Discussion on the impact of epsilon on the choice of h, where h is on the order of the square root of epsilon - Estimation of case transition from truncation error to round-off error based on epsilon and method knowledge - Estimation of how small h can be made before the trade-off between truncation error and rounding error becomes significant - Discussion on using inequalities to maintain mathematical relationships in equations - Mention of practical approach to estimating where round-off error begins to dominate in calculations ## Epsilon Impact on Calculations - Epsilon is expected to be very small, potentially around 2 x 10^-10 - Focus is on the order of magnitude rather than precise values - Epsilon drives the calculations, but exact values are not typically sought - Aim is to gauge how small values can go before causing computational glitches - Subtraction of nearly equal numbers, large scale arithmetic, and division by zero can lead to computational glitches - Recommendation to avoid making values too small to prevent round-off error - Exact numerical analysis is not typically performed unless necessary for deep analysis - Importance of stating the accuracy level in numerical methods