Oscar Veliz2,500 Subscribers!!! This video covers five ways to help the channel including an updated process for submitting topic requests and a community funding opportunity through @GitHub. Thank you for your feedback and support on these videos. Here's to the next 2,500!
Subscriber Milestone - 5 Ways to Help the ChannelOscar Veliz2019-12-01 | 2,500 Subscribers!!! This video covers five ways to help the channel including an updated process for submitting topic requests and a community funding opportunity through @GitHub. Thank you for your feedback and support on these videos. Here's to the next 2,500!
Chapters: 00:00 Intro 00:36 Factoring with monomial 01:13 Factoring with quadratic 01:40 Synthetic Division 1 variable 02:16 Synthetic Division 2 variables 03:22 Solving for roots 03:50 Different u and v 04:24 Describing Notation 05:22 Solving Nonlinear System 07:01 Another Synthetic Division 07:45 Updating u & v 08:27 Bairstow Iteration Example 08:56 Bairstow Iteration Example 2 09:22 Note on Quadratic Equation 09:42 Bairstow Full Algorithm 10:51 Complex Roots Example 11:33 Bairstow Fractals 12:23 Picking u & v 12:46 Henrici Starting Values 13:08 Bairstow's Original Problem 14:40 Oscar's Notes 15:24 Outro
Background music "Drifting at 432 Hz" by @UnicornHeads
#SoME2 #NumericalAnalysis #BairstowsMethodGraeffes MethodOscar Veliz2022-06-06 | Graeffe's Root-Squaring Method (also called Graeffe-Dandelin-Lobachevskiĭ or Dandelin–Lobachesky–Graeffe method) for finding roots of polynomials. The method solves for all of the roots of a polynomial by only using the coefficients and does not require derivatives nor an interation function. This lesson provides a history of the method, motivates "why" the method works, and walks through an example of root-squaring as well as solving for the roots using logarithms. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters: 00:00 Intro 00:45 History 01:10 Expanding & Reversing 02:21 Bracket Notation 03:53 Bracket Example 04:16 Solving for a 05:46 Solving for b 06:11 Solving for c 06:30 How does this help??? 06:55 Root Squaring Example 07:56 Repeated Root Squaring 08:36 Stopping Criteria 08:58 On Programming Graeffe's Method 09:30 Further Reading 09:52 Oscar's Notes 10:33 Outro
Background music "Drifting at 432 Hz" by @UnicornHeads
#NumericalAnalysis #rootfinding #polynomialsGeneralized Bisection Method for Systems of Nonlinear EquationsOscar Veliz2022-04-18 | Generalization of the Bisection Method for solving systems of equations. This lesson explains the algorithm for a 2 dimension example based on Harvey-Stenger's approach using bisecting triangles. It includes a visualization of the method in action on an example nonlinear system. Other methods for solving in 3 dimensions and for larger systems are also discussed as well as hybrid approaches. Example code hosted on GitHub github.com/osveliz/numerical-veliz written in Python and using numpy and matplotlib.
Reference links: "A two-dimensional analogue to the method of bisections for solving nonlinear equations" by Charles Harvey and Frank Stenger doi.org/10.1090/S0025-5718-1979-0521286-6 "A three-dimensional analogue to the method of bisections for solving nonlinear equations" by Krzysztof Sikorski doi.org/10.1090/S0025-5718-1979-0521286-6 "A bisection method for systems of nonlinear equations" by Eiger et. al. doi.org/10.1145/2701.2705 "An efficient degree-computation method for a generalized method of bisection" by Baker Kearfott doi.org/10.1007/BF01404868 "Abstract generalized bisection and a cost bound" by Baker Kearfott doi.org/10.1090/S0025-5718-1987-0890261-9 "Solving systems of nonlinear equations using the nonzero value of the topological degree" by Michael N. Vrahatis doi.org/10.1145/50063.214384
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#NumericalAnalysis #BisectionMethod #NonlinearSystemGeneralized False Position & Alternative Secant MethodsOscar Veliz2022-01-31 | False Position Method for Nonlinear Systems (aka Generalized Regula Falsi) along with two Alternative Secant Methods. Includes discussion of history and primary sources along with numeric examples and visualizations. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters: 0:00 Scaffolding 0:25 Korganoff 1:02 Robinson 1:32 Some History 1:50 Robinson Continued 2:51 Robinson versus Secant 3:27 Nonlinear System Example 4:16 On Notation 4:37 Efficient Orthogonal Matrix 4:53 Generalized False Position Method 5:18 Oscar's Notes 8:38 Outro
Background music "Drifting at 432 Hz" by @UnicornHeads #NumericalAnalysis #FalsePosition #SecantMethodGlobal Newtons Method - It Always ConvergesOscar Veliz2021-11-26 | Globally convergent modification of Newton's Method that uses backtracking whenever a test point would not cause the function iterations to shrink in absolute value based on the Armijo's Search. Lesson also covers fractals using Global Newton Method as well as solving systems of nonlinear equations. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters: 0:00 Intro 0:30 Literature 1:03 Damped Newton Method 2:28 Armijo's Approach 2:57 Global Newton Method Algorithm 3:50 Numeric Example 4:17 Global Newton Fractals - Single Variable 5:30 On Extra Function Calls 6:11 Nonlinear System Example 6:55 Global Newton Fractals - Nonlinear System 8:11 Oscar's Notes 8:38 Outro
Background music "Drifting at 432 Hz" by @UnicornHeads #NumericalAnalysis #NewtonMethod #NewtonFractalHalleys Method for Systems of Nonlinear EquationsOscar Veliz2021-08-22 | Halley's Method for Solving Systems of Nonlinear Equations. Submission for The Summer of Math Exposition. Lesson includes motivation & explanation of notation, description of the method, numerical example, discussion of order, and comparison with the Method of Tangent Hyperbolas. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters: 0:00 Wikipedia 0:44 Intro 0:54 Recommended Viewing 1:04 Recap 1:42 Generalized Halley's Method 2:55 Nonlinear System Example 3:21 Order 3:45 Method of Tangent Hyperbolas 4:10 Side-by-side 4:36 Higher Order 4:54 Oscar's Notes 5:15 Outro
Background music "The Golden Present" by @JesseGallagher
#NumericalAnalysis #some1 #HalleysMethodBroydens MethodOscar Veliz2021-06-28 | Broyden's Method for solving systems of nonlinear equations. Lesson covers motivation, history, examples, discussion, and order of this Quasi-Newton Method. It also explains the "Good" and "Bad", as well as the third version of the method. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters: 0:00 Intro 0:22 Newton's Method According to Broyden 1:08 Nonlinear System Example 1:18 Newton's Method Example 1:28 Analyzing Newton's Behavior 2:07 Solving for J 2:51 Broyden's Approach 3:25 Almost Broyden's Method 4:13 Solving for the Inverse 4:46 Broyden's "Good" Method 5:11 More Options 5:35 Broyden's "Bad" Method 6:07 Generalizing Root-Finding 6:32 The Case for Secant Method 6:52 The Case for Newton's Method 7:17 Broyden's Take 8:11 The question of Order 8:31 Oscar's Notes 9:03 Outro
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#BroydensMethod #NumericalAnalysis #NonlinearSystemApproximating the Jacobian: Finite Difference Method for Systems of Nonlinear EquationsOscar Veliz2021-05-24 | Generalized Finite Difference Method for Simultaneous Nonlinear Systems by approximating the Jacobian using the limit of partial derivatives with the forward finite difference. Example code on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:13 Prerequisites 0:32 Refresher 0:43 What is the Jacobian? 2:06 Approximating the Jacobian 3:00 Finite Differences 3:21 Note on Notation 4:23 Visualization 6:17 Improving Accuracy 6:42 Note on Notation 2 7:45 Oscar's Notes 8:24 Mathemaniac 8:34 Thank You
Background music "The Golden Present" by @JesseGallagher
#FiniteDifferenceMethod #NumericalAnalysis #NonlinearSystemSteffensens Method for Systems of Nonlinear EquationsOscar Veliz2021-04-05 | Generalized Steffensen's Method for Simultaneous Nonlinear Systems originally credited to J. F. Traub. Video shows how to solve nonlinear systems by approximating the Jacobian. Example code on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Prerequisites 0:20 Intro 0:40 Traub 1:24 Soleymani et al 1:58 Explaining Notation 2:32 1D Example 3:06 Two Methods - Same Method 3:20 System of Equations 3:27 Nonlinear System Example 4:42 Closer = Better ~J 5:03 Oscar's Notes 5:46 Thank You
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#Steffensen'sMethod #NumericalAnalysis #NonlinearSystemSecant Method for Systems of Nonlinear EquationsOscar Veliz2021-03-01 | Generalized Secant Method for Simultaneous Nonlinear Systems originally credited to Wolfe and Bittner. Lesson shows how to solve nonlinear systems without the Jacobian, nor the need to approximate it, in a straightforward and visual manner. Example code on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:15 Prerequisites 0:25 Secant Method Recap 0:45 Literature 1:00 Secant Method Alternative 2:12 Two Methods - Same Method 2:27 Nonlinear System + Example 2:49 Generalized Secant Method Visualized 4:12 Numeric Example 4:28 Order! 4:50 "A Class of Secant Methods" 5:31 Oscar's Notes 6:05 Thank You
References "The Secant method for simultaneous nonlinear equations" by Phillip Wolfe doi.org/10.1145/368518.368542 "Eine Verallgemeinerung des Sekantenverfahrens (regula falsi) zur naherungsweisen Berechnung der Nullstellen eines nichtlinearen Gleichungssystems" by Von Leonhard Bittner borrowed from Technische Universität Dresden "Convergence of Multipoint Iterative Methods" by Leonard Tornheim doi.org/10.1145/321217.321224 "An Algorithm for Solving Non-Linear Equations Based on the Secant Method" by J. G. P. Barnes doi.org/10.1093/comjnl/8.1.66 "Some Efficient Algorithms for Solving Systems of Nonlinear Equations" by Richard P. Brent doi.org/10.1137/0710031 "The Computational Complexity of Iterative Methods for Systems of Nonlinear Equations" by Richard Brent doi.org/10.1007/978-1-4684-2001-2_7
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#SecantMethod #NumericalAnalysis #NonlinearSystemNewtons Method for Systems of Nonlinear EquationsOscar Veliz2021-01-04 | Generalized Newton's method for systems of nonlinear equations. Lesson goes over numerically solving multivariable nonlinear equations step-by-step with visual examples and explanation of the Jacobian, the backslash operator, and the inverse Jacobian. Example code in MATLAB / GNU Octave on GitHub: http://github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:12 Prerequisites 0:32 Background 0:58 Setup 1:54 Jacobian 2:55 Historical Context 5:11 Newton's Method Example Step-by-Step 6:57 End Condition 7:12 Numerical Example in Table 7:33 Newton's Method with Backslash 7:53 Newton's Method with Inverse Jacobian 8:16 MATLAB / GNU Octave 9:01 Newton Fractals 10:51 3D Fractal 11:37 Historical Optimization Newton's Method 12:18 Oscar's Notes 12:59 Thank You
Background music "The Golden Present" by @JesseGallagher #NewtonsMethod #NumericalAnalysis #NonlinearSystemVideo Mistakes II: The SequelOscar Veliz2020-11-30 | This video corrects mistakes in my videos on Taylor Series Origin, Ternary Search, Dichotomous Search, Fixed Point Iteration for Fixed Point Iteration System of Equations with Banach, and Wegstein's Method. Thanks to commenters who pointed these errors out. If you find other mistakes feel free to comment or post in the GitHub Issues Forum for the code repository github.com/osveliz/numerical-veliz
#NumericalAnalysisBrents Minimization MethodOscar Veliz2020-11-05 | Hybrid minimization algorithm combining Golden-section Search and Successive Parabolic Interpolation (Jarratt's Method) that is guaranteed to locate minima with superlinear convergence order. Example code github.com/osveliz/numerical-veliz
Chapters: 0:00 Intro 0:16 Scaffolding 0:31 Motivation 1:17 Parabolic Interpolation Review 1:48 Renaming Variables 2:40 Brent's Method Algorithm 3:19 SPI Behaving? 4:08 Note on Updating 4:38 Brent's Method Visualization 6:02 Numerical Example 6:29 Note on Steps 6:43 MATLAB fminbnd 7:12 Minimum Strategy - Derivative 7:49 Note on Convergence Order 8:04 Oscar's Notes 8:39 Outro
Background music "The Golden Present" by @JesseGallagher
#GoldenSectionSearch #SuccessiveParabolicInterpolation #NumericalAnalysisSuccessive Parabolic Interpolation - Jarratts MethodOscar Veliz2020-09-29 | Optimization method for finding extrema of functions using three points to create a parabola that is then used to find the next approximation to the solution. This lesson visualizes the behavior of the method with numeric examples as well as its convergence through fractals. Based off the paper "An iterative method for locating turning points" by P. Jarratt. Example code github.com/osveliz/numerical-veliz
Chapters: 0:00 Intro 0:21 Scaffolding 0:42 Richard P. Brent 1:01 An Iterative Method for Locating Turning Points 1:33 Graphing 1:46 Create a Quadratic 1:58 Finding the Next Point 2:35 The Next Iteration 2:22 Derivative is Zero 2:56 Avoid Calculating L_2 3:32 Jarratt's Method 3:47 Example 4:10 Fractal Scaffolding 4:20 Complex Plane Discussion 5:47 Jarratt Fractal z^4/4 - z 6:33 Jarratt Fractal -cos(z) 6:55 Jarratt Fractal z^9/9 + 3z^5 - 16z 7:52 Jarratt's Notes 8:32 Oscar's Notes 9:00 Thank You
References: Jarratt's paper doi.org/10.1093/comjnl/10.1.82 Brent's Book https://maths-people.anu.edu.au/~brent/pub/pub011.html
Background music "The Golden Present" by @JesseGallagher
#SuccessiveParabolicInterpolation #NumericalAnalysisSublinear Convergence #MegaFavNumbersOscar Veliz2020-08-31 | Lesson discusses the convergence rate of the Gregory-Leibniz series for computing pi and the mega number of terms needed for 100 decimal digits of accuracy. Example code on github.com/osveliz/numerical-veliz
Chapters: 0:00 Intro 0:18 Scaffolding 1:03 Gregory-Leibniz Series 1:36 Numerical Iterations 2:24 Finding n Setup 3:09 Finding n Bad Example 3:34 Finding n Good Example 4:43 Oscar's Notes 5:17 Thank You
Thank you to @singingbanana for putting together the playlist for #MegaFavNumbers and to my GitHub Sponsor community for requesting this video.
Background music "The Golden Present" by @Jesse Gallagher
#ConvergenceOrder #NumericalAnalysisGolden-section SearchOscar Veliz2020-07-22 | Golden-section Search is a minimization algorithm that expands on the Fibonacci Search scheme described by J. Kiefer and S. M. Johnson. This interval-based numerical method improves on Ternary Search and Dichotomous Search be reusing interval points based on the golden ratio (phi). Code can be found on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:23 Algorithms for Minimization without Derivatives 0:43 Optimum Seeking Methods 1:34 Ternary Recap 2:04 Reusing Points 2:22 Finding c 2:45 Fixed Constant Ratio 3:28 Computing c 3:58 Golden-section Search Algorithm 4:47 GSS Visualized 5:31 Numerical Example 5:55 Comparing Methods 6:18 Search Space Shrinkage 7:01 Golden Ratio Extra History 7:28 Properties of φ 8:13 Oscar's Notes 8:38 Mathemaniac 8:49 Thank You
References: Algorithms for Minimization without Derivatives by Richard P. Brent https://maths-people.anu.edu.au/~brent/pub/pub011.html Optimum Seeking Methods by Douglass Wilde archive.org/details/optimumseekingme00wild Sequential Minimax Search for a Maximum by J. Kiefer jstor.org/stable/2032161 Best Exploration for Maximum is Fibonaccian by S. M. Johnson https://apps.dtic.mil/dtic/tr/fulltext/u2/224385.pdf
Background music "The Golden Present" by @JesseGallagher
#GoldenSectionSearch #NumericalAnalysisFibonacci SearchOscar Veliz2020-06-24 | Fibonacci search scheme for finding the minimum of a function discovered by J. Kiefer and S. M. Johnson. This interval-based numerical method improves on Ternary Search and Dichotomous Search be reusing interval points based on ratios from the Fibonacci Sequence. Code can be found on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:12 Recap 0:23 Optimum Seeking Method 0:41 Sequential Minimax Search for a Maximum 1:06 Best Exploration for Maximum is Fibonaccian 1:23 Kiefer's Ratios 1:33 Kiefer's Ratios Example 1:50 Kiefer's Ratios Visualized 2:49 Fibonacci Search Visualized 3:58 Advantage of Fibonacci 4:16 Stopping Condition 4:47 Finding n 5:12 Johnson's Remarks on n 5:27 Ending Interval Length 5:48 Fibonacci Search Algorithm 6:56 Fibonacci Search Numerical Example 7:26 Finding n from the Example 7:49 Kiefer's Constant Ratio 8:03 Johnson's Golden-section 8:19 Oscar's Notes 8:42 Thank You
References: Optimum Seeking Methods by Douglass Wilde archive.org/details/optimumseekingme00wild Sequential Minimax Search for a Maximum by J. Kiefer www.jstor.org/stable/2032161 Best Exploration for Maximum is Fibonaccian by S. M. Johnson https://apps.dtic.mil/dtic/tr/fulltext/u2/224385.pdf
#FibonacciSearch #NumericalAnalysisDichotomous SearchOscar Veliz2020-05-31 | Dichotomous Search is an improved version of Ternary Search. This video describes the motivation and algorithm followed by a visualized example. Code can be found on GitHub github.com/osveliz/numerical-veliz
*Correction* The numerical example used epsilon = 10^(-6) not 10^(-7) see Video Mistakes II: The Sequel youtu.be/YEUbzqkJBf0
Chapters 0:00 Intro 0:19 Ternary Recap 0:32 Moving Test Points 0:59 Computing c and d 1:19 A Tale of Halving 1:40 Optimum Seeking Methods 2:17 End Condition 3:04 Dichotomous Search Algorithm 3:50 Numerical Example 4:03 Oscar's Notes 4:36 Thank You
#DichotomousSearch #NumericalAnalysisTernary SearchOscar Veliz2020-05-31 | Ternary Search is an interval-based divide-and-conquer algorithm for finding the minimum of a unimodal function. This video describes how to find a minimum when the derivative is know, defines unimodal, presents interval-based approaches for minimum finding, and visualizes the algorithm. Example code available on GitHub github.com/osveliz/numerical-veliz
*Correction* The numerical example used epsilon = 10^(-6) not 10^(-7) see Video Mistakes II: The Sequel youtu.be/YEUbzqkJBf0
Reference: Algorithms for Minimization without Derivatives by Richard P. Brent https://maths-people.anu.edu.au/~brent/pub/pub011.html
#TernarySearch #NumericalAnalysisComputing π: Machin-like formulaOscar Veliz2020-03-14 | Machin-like formulae are used to find many decimal places of pi, although why they work can seem confusing. This lesson shows how the Taylor Series of arctangent is used to compute π, and the power of combining it with Machin's approach, as well as covering the history of Gregory, Leibniz, Euler, and Machin. Example code: github.com/osveliz/numerical-veliz/tree/master/src/series
Chapters 00:00 - Intro 00:11 - Taylor Series 00:20 - Unit Circle & Solve for π 00:57 - Gregory 01:07 - Gregory-Leibniz Series 01:33 - Leibniz 02:27 - arctan(1) 02:48 - Moving a 03:40 - Increasing n 04:11 - Getting Close 04:44 - Nearer to zero 05:07 - Adding arctangents example 05:28 - Arctangent Trick 05:51 - Euler 06:06 - Euler's Equation for π 06:26 - Deriving Euler's Equation 07:00 - Machin intro 07:15 - Machin-like formula 08:00 - John Machin 08:22 - Demo Code & Story Time 09:40 - 10,000 digits of π 10:08 - Beckmann's thoughts on higher digits 10:27 - Oscar's Notes 11:00 - Thank You
#pi #PiDay #πOrigin of Taylor SeriesOscar Veliz2020-02-24 | The history of Taylor Series and Maclaurin Series including the works of de Lagny, Halley, Gregory, and Madhava using primary sources whenever possible. Lesson also presents the Taylor Theorem along with visualizations of James Gregory's equations. Finally the video discusses the time period and context during the battle over calculus.
Chapters 00:00 Intro 00:20 Solving Cube Roots 00:53 de Lagny's Conditions 01:26 Halley's Equations 03:46 Taylor's Letter 04:04 Taylor's Treatise 04:25 Two Mathematical Camps 04:51 Quotes About Taylor 05:29 Methodus 06:34 Going Back in Time 06:47 James Gregory 07:13 Gregory's Letter 07:47 Gregory's Other Series 08:32 Certain Mathematical Achievements 08:59 Taylor Series 09:31 Taylor Series Example 10:27 Colin Maclaurin 11:10 Nilakantha and Madhava 11:28 Oscar's Notes 11:58 Thank You
**Corrections** The second value of b at 2:22 is actually negative. James Gregory was 36 years old, not 37, when he died. The numerator at 9:18 should be f^(k)(a)(x-a)^k not f^(k)(x-a)^k. See Video Mistakes II: The Sequel youtu.be/YEUbzqkJBf0
#TaylorSeries #NumericalAnalysisWhat is Order of Convergence?Oscar Veliz2020-01-05 | Converge order and error reduction can be confusing but this video breaks it down and provides examples showing how order relates to speed and runtime. It also explains how order of convergence relates to Big O. Watching the other videos on this channel can be helpful but is not necessary. Example root finding method code for all of the above methods can be found on github github.com/osveliz/numerical-veliz
Chapters 00:00 - Intro 00:19 - Order Montage 00:54 - Error Definition 01:11 - Introduction of α 01:35 - α equation 01:41 - α example 1 Bisection 02:09 - Solving for M 02:36 - α example 2 False Position 03:13 - α example 3 Newton 03:41 - On Function Calls 04:19 - α with iterations and runtime 05:02 - Note on previous example 05:23 - Generalized operation count 06:28 - How fast is linear? 07:29 - How fast is quadratic? 09:33 - Digits of accuracy 10:15 - Distance impacts α 10:52 - Big O brief intro 11:37 - Big O of Bisection 12:09 - Big O of Newton and Halley 13:02 - Oscar's Notes 13:47 - Thank You
The other methods referenced in this video include: Fixed Point Iteration Method, Bisection Method, False Position Method - Regula Falsi, Newton's Method, Steffensen's Method, Wegstein's Method, Muller's Method, Durand-Kerner Method, Secant Method, Householder's Method, and Halley's Method. All of these method can be found in this root finding playlist youtube.com/playlist?list=PLb0Tx2oJWuYIpNE23qYHGQD42TIR3ThNz
#ConvergenceOrder #NumericalAnalysisVideo Mistakes and How to Fix ThemOscar Veliz2019-12-16 | This video corrects mistakes in my videos on Newton's Method, Newton Fractals, Newton-Bisection Hybrid, Halley's Method, and Cubic Splines. Thanks to commenters who pointed these errors out. If you find other mistakes feel free to comment or post in the GitHub Issues Forum for the code repository github.com/osveliz/numerical-veliz
#NumericalAnalysisHouseholders MethodOscar Veliz2019-10-01 | Householder's Method for finding roots of equations including history, derivation, examples, and fractals. Example code is available on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:25 Derivation 1:58 History 2:34 Householder's Method 4:07 Householder's Method Example 4:41 Higher Order Householder's Method Examples 5:30 Principles of Numerical Analysis 6:03 Householder Fractals 8:10 Summary 8:41 Thank You
#HouseholdersMethod #NumericalAnalysisHalleys MethodOscar Veliz2019-08-25 | Halley's Method (the method of tangent hyperbolas) for finding roots including history, derivation, examples, and fractals. Also discusses Taylor's Theorem relating to Halley's Method as well as Halley's Comet. Sample code and images available on GitHub github.com/osveliz/numerical-veliz
Chapters 00:00 Intro 00:36 History 03:48 Derivation Setup 05:20 Derivation Tangent Hyperbolas 06:00 Derivation Taylor Series 07:21 Example 07:53 Example 2 (see correction) 08:24 Newton versus Halley 08:45 Example 3 09:29 Halley Fractals 11:27 Summary 11:52 Thank You
Corrections: The example at 8:08 had an error with the programming of f'(x). The bug caused the numbers in the table and the plots for that example to be incorrect. There is a separate issue in that example where 2 was written as the input instead of -2. Also there is typo in the name Christopher Wren at 0:46. A video covering these corrections can be found here youtu.be/4jw0cjddmB8
#HalleysMethod #NumericalAnalysisLaguerres MethodOscar Veliz2019-07-14 | Laguerre's method for finding real and complex roots of polynomials. Includes history, derivation, examples, and discussion of the order of convergence as well as visualizations of convergence behavior. Example code available on github github.com/osveliz/numerical-veliz
Chapters 00:00 Intro 00:20 Sources 00:44 Derivation 03:08 Laguerre's Method 03:52 K. A. Redish 04:13 Visuzliation 05:24 Forman S. Acton 06:05 Example 07:04 Example 2 07:43 Example 3 08:01 Convergence 8:35 Laguerre Fractal 10:05 Summary 10:34 Thank You
Chapters 0:00 Intro 0:19 History 0:41 Methodology 0:59 Starting Points 1:11 Starting Points Visualized 1:33 Newton Fractal 2:22 A Modified Newton Method 2:35 Ehrlich's Derivation 4:43 Example 5:09 Durand-Kerner versus Aberth-Ehrlich 6:22 Behavior of Aberth-Ehrlich 6:33 Notes on Aberth-Ehrlich 7:01 Thank You
#DurandKerner #AberthEhrlich #NumericalAnalysisDurand-Kerner MethodOscar Veliz2019-05-29 | The Durand-Kerner Method for solving all roots of a polynomial simultaneously including complex solutions. Example code github: http://github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:24 History 1:04 Methodology & Derivation 2:28 Example 1 real numbers 2:55 Example 2 need for complex numbers 3:19 Complex Roots 3:39 Complex Numbers Visualized 4:16 Example 2 with complex numbers 5:00 Formal Definition 5:28 Starting Points 6:14 Example 3 Visualized 6:57 Behavior of Durand-Kerner 7:33 Computational Order 7:54 Notes on Durand-Kerner 8:23 Thank You
#DurandKerner #AberthEhrlich #NumericalAnalysisGeneralized Aitken-Steffensen MethodOscar Veliz2019-03-14 | Generalized Aitken's delta-squared method and Generalized Steffensen's Method applying Fixed Point Iteration to Systems of Nonlinear Equations. Video goes step-by-step to derive Generalized Aitken-Steffensen and discusses induced and accelerated convergence behavior as well as quadratic order. Example code github: http://github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:40 Motivation 2:32 Solve for X* 3:10 Generalized Aitken 3:44 Generalized Aitken Example 4:10 Generalized Aitken-Steffensen Method 4:40 Generalized Aitken-Steffensen Method Example 1 5:02 Generalized Aitken-Steffensen Method Example 2 5:26 Henrici 5:59 On Order & Proving Convergence 6:31 Proof Intuition 6:55 Notes 7:40 Thank You
#AitkensDeltaSquaredMethod #SteffensensMethod #NumericalAnalysisFixed Point Iteration System of Equations with BanachOscar Veliz2019-03-14 | Fixed Point Iteration Method to solve Systems of Nonlinear Equations with discussion of Banach Fixed Point Theorem, finding the Jacobian, convergence, and order. Example code on GitHub: http://github.com/osveliz/numerical-veliz
Chapters: 00:00 Intro 00:25 Systems of Equations 00:33 Solving Nonlinear Systems 01:03 Fixed Point Iteration 01:26 Rewriting Equations 02:03 Example 1 02:23 Visualized Example 03:12 Measuring Distance and Norm 03:45 End Conditions 04:09 Different Combinations of Rewrites 04:45 When Does it Converge? 05:10 Banach Fixed Point Theorem 05:56 The Jacobian 06:48 Contraction Mapping Test 07:24 Contraction Mapping Test Examples 08:20 Notes on the Contraction Mapping Test 09:06 Order of Convergence 09:41 Exact Order 10:31 Summary 10:49 Thank You
Reference Link: "Elements of Numerical Analysis" by Peter Henrici archive.org/details/ElementsOfNumericalAnalysis/page/n57 #FixedPointIteration #NumericalAnalysisNewton FractalsOscar Veliz2019-01-01 | Using Newton's Method to create Fractals by plotting convergence behavior on the complex plane. Functions used in this video include arctan(z), z^3-1, sin(z), z^8-15z^4+16. Example code and images available at github.com/osveliz/numerical-veliz
Correction: The derivative of arctan(x) should be 1/(1+x^2). This error is only impacts the slides, not the numerical examples nor fractals which used the correct derivative. A video covering this mistake can be found here youtu.be/4jw0cjddmB8
Chapters 0:00 Intro 0:16 Convergence Interval Recap 0:42 Imaginary Numbers 1:05 Newton's Method in Complex Plane 1:29 Basin of Convergence 1:38 Arctangent Fractal 1:50 Newton Fractal 2:10 Why Fractals Emerge 2:54 Example z^3-1 4:07 Example sin(z) 4:33 Example z^8+15z^4-16 4:50 Generalized Newton Fractal 5:05 Generalized Newton Fractal Examples 6:16 Summary 6:42 Thank You
#Fractal #NewtonsMethod #NumericalAnalysisNewton Bisection Hybrid (Newt-Safe)Oscar Veliz2018-12-09 | Newton Bisection Hybrid Method for root finding. Example code available at github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:26 Viewer Request 0:49 Numerical Recipes 1:12 Numerical Methods That Work 1:54 Motivation Examples 3:04 Problems with Newton Recap 3:17 Newt-Safe Basic Algorithm 4:00 Newt-Safe Basic Examples 4:53 Additional Conditions 5:39 ΔX Condition Explanation 6:33 ΔX Example 7:18 Newt-Safe without Interval 7:55 Other Hybrid Methods 8:33 Notes and Summary 8:57 Thank You
Correction: The bracket size test at 5:26 used DeltaX which should be DeltaX_old. A video covering this mistake can be found here youtu.be/4jw0cjddmB8
Reference Links: Based of off NewtSafe from Marc Spiegelman https://www.ldeo.columbia.edu/~mspieg/e4300/BlankPDFs/Lecture06_blank.pdf rtsafe from Numerical Recipes http://numerical.recipes/ Numerical Methods That Work by Forman S. Acton books.google.com/books?id=cGnSMGSE5Y4C&lpg=PA41&pg=PA51#v=onepage&q&f=false
#BisectionMethod #NewtonsMethod #NumericalAnalysisFixed Point Iteration Q&AOscar Veliz2018-11-09 | Fixed Point Iteration Method followup video answering your frequently asked questions like "How do you pick a starting point?" and "How do you use the convergence test without the root?" Example code can be found at github.com/osveliz/numerical-veliz specifically in the programs for Steffensen's and Wegstein's Methods.
Chapters 0:00 Intro 0:14 Why not x = x^2 -1? 1:08 Where did 1.618 come from? 1:29 How do you pick a starting point? 2:04 How do you find the other root? 3:24 Use the Convergence Test without the root? Which root do you use in the test? 4:22 On the Matter of Convergence 5:04 Thank You
#FixedPointIteration #NumericalAnalysisWegsteins MethodOscar Veliz2018-10-09 | Wegstein Method for finding roots, accelerating fixed point iteration, and inducing convergence in fixed point iteration. Explained examples and discussion of order as well as how to compute q. Example code: github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:22 Wegstein's Method Sources 0:52 Wegstein's Methodology 1:56 Wegstein's Method Examples 3:24 Computing q 4:55 Computing q example 5:24 Updating q 5:43 Updating q example 6:03 Computational Order 6:36 Oscar's Notes 7:13 Thank You
See Video Mistakes II: The Sequel youtu.be/YEUbzqkJBf0 for a small error and correction.
References: Wegstein's original paper: dl.acm.org/citation.cfm?id=368871 Gutzler's thesis: https://ir.library.oregonstate.edu/downloads/2r36v1962
#WegsteinsMethod #NumericalAnalysisPower Method with Inverse & RayleighOscar Veliz2018-09-02 | Discussion of Eigenvalues & Eigenvectors, Power Method, Inverse Power Method, and the Rayleigh Quotient with brief overview of Rayleigh Quotient Iteration. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Title Card 0:12 Terminology 0:37 Eigenvalue Example 1:04 Power Method 1:29 Power Method Example 2:33 Notes on Power Method 2:50 Inverse Power Method 3:21 Inverse Power Method Example 4:12 Rayleigh Quotient 4:54 Solve using Determinant 5:15 Inverse Power Method with Shift 5:34 Inverse Power Method with Shift Example 6:12 Rayleigh Quotient Iteration 6:37 Summary 7:01 Thank You
#PowerMethod #RayleighQuotient #NumericalAnalysisHorners MethodOscar Veliz2018-08-01 | Horner's Method (Ruffini-Horner Scheme) for evaluating polynomials including a brief history, examples, Ruffini's Rule with derivatives, and root finding using Newton-Horner. Example code on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:11 - History 1:33 - TLDR 1:47 - Function vs Polynomial 2:23 - Horner's Method 2:50 - Horner's Method Examples 3:36 - Synthetic Division 4:32 - Ruffini's Rule Main Idea 4:58 - Ruffini's Rule 5:34 - Derivative with Ruffini's Rule 6:00 - Derivative Example 6:27 - Polynomial Root Finding 6:36 - Algebraic Root Finding 6:59 - Rational Root Theorem 7:35 - Newton-Horner Method 8:23 - Newton-Horner Example 9:11 - Summary 9:36 - Thank You
References: Horner's paper jstor.org/stable/107508 Ruffini doi.org/10.1090/S0002-9904-1911-02072-9 Chemla's paper doi.org/10.1017/S0957423900001235 Qin Jiushao http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.454.4986&rep=rep1&type=pdf#page=169 Sharaf al-Dīn al-Tūsī doi.org/10.1016/0315-0860(89)90099-2 Jia Xian https://www.math.vt.edu/people/brown/doc/fibo_number.pdf Yong's paper https://sms.math.nus.edu.sg/smsmedley/Vol-14-1/The%20development%20of%20polynomial%20equations%20in%20traditional%20China(Lam%20Lay%20Yong).pdf
#HornersMethod #NumericalAnalysisSteffensens Method with Aitkens Δ²Oscar Veliz2018-07-01 | Discussion of Steffensen's Method and Aitken's Delta-Squared Method with their relation to Fixed Point Iteration including examples, convergence acceleration, order, and code. GitHub: github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:08 Aitken's Δ² Method History 0:23 Derivation with Example 1:01 Aitken's Δ² Method 1:21 Solve for r 2:16 Δ² Notation 2:39 Aitken's Δ² Example 3:12 Steffensen's Method History 3:40 Steffensen's Methodology 4:05 Steffensen's Method Example 4:43 Steffensen's Method 2.0 4:55 One Method, Two Versions 6:30 Steffensen's Method 2.0 Continued 6:55 Order 7:26 Summary 8:02 Thank You
#AitkensDeltaSquaredMethod #SteffensensMethod #NumericalAnalysisBrents MethodOscar Veliz2018-06-01 | Dekker's Method, Inverse Quadratic Interpolation, and Brent's Method including example, code, and discussion of order. GitHub github.com/osveliz/numerical-veliz
Chapters 00:00 Intro 00:12 Secant Method Recap 00:37 Bisection Method Recap 00:54 Dekker's Method History 01:35 Dekker's Method Big Idea 01:50 Dekker's Method 02:28 Dekker's Method Visual Example 02:50 Dekker's Method Update Step 03:48 Dekker's Method Visual Example Continued 04:19 Dekker's Method Example 05:04 Tolerance 05:20 "Remarks on the Paper by Dekker" 05:58 George E. Forsythe 06:30 Brent's Method 06:59 Brent's Method Big Idea 07:19 Inverse Quadratic Interpolation 07:30 Create a Quadratic 07:50 Lagrange Polynomial 08:01 Inverse Quadratic 08:32 Inverse Quadratic Interpolation Methodology 08:54 Inverse Quadratic Simplified 09:23 Trouble with IQI 09:55 Brent's Method - Round 2 11:06 Ill Behaved Functions 11:58 Comparison 12:32 Computational Order 13:06 Summary and Notes 13:40 Thank You
Thank you Adrian and Les for helping and Micheal for the suggestion.
#BrentsMethod #NumericalAnalysisNewtons Method Interval of ConvergenceOscar Veliz2018-05-08 | How to find the Interval of Convergence for Newton-type methods such as Newton's Method, Secant Method, and Finite Difference Method including discussion on Damped Newton's Method and widening the convergence interval. Example code in R hosted on Github: github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:08 Methods with Guaranteed Convergence 0:26 Newton-type Methods 0:42 Convergence vs Divergence Example 1:02 Example Visualized 1:31 Convergence Interval Visual 2:02 Finding Convergence Interval 2:20 Converge Interval Numerical Example 3:25 Widening the Interval 3:55 Widened Interval Visualized 4:31 Summary and Notes 5:14 Thank You
#NewtonsMethod #NumericalAnalysisMullers MethodOscar Veliz2018-04-16 | Muller's Method for finding roots including simple examples, discussion of order, and biography of David Eugene Muller. Example code in Python hosted on GitHub: github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:12 David Muller Bio 0:43 Muller's Method History 1:07 Secant Method with a Parabola 1:38 Visualized Parabola 1:47 Creating a Quadratic 2:09 Finding X4 2:49 Putting It All Together 3:08 Muller's Method 3:30 Muller's Method Examples 4:15 Order 4:47 Notes and Summary 5:04 Special Thanks 5:15 Thank You
#MullersMethod #NumericalAnalysisFalse Position Method - Regula FalsiOscar Veliz2018-03-29 | False Position Method (Regula Falsi) for finding roots of functions. Includes comparison against Bisection and discussion of order. Sample code in C available on GitHub github.com/osveliz/numerical-veliz
Chapters 0:00 Intro 0:21 Regula Falsi Family Tree 0:33 False Position Method 1:01 Bisection Visualized 1:10 False Position Visualized 1:27 Computing c 1:45 You have c now what? 2:10 Bisection vs False Position Examples 2:52 A closer look 3:18 Notes and Summary 3:46 Thank You
#FalsePositionMethod #RegulaFalsi #NumericalAnalysisFinite Difference MethodOscar Veliz2012-12-12 | Finite Difference Method for finding roots of functions including an example and visual representation. Also includes discussions of Forward, Backward, and Central Finite Difference as well as overview of higher order versions of Finite Difference.
#FiniteDifferenceMethod #NumericalAnalysisCubic SplinesOscar Veliz2011-04-24 | Function approximation using Cubic Splines and Natural Cubic Splines including discussion about figuring out if two sets of equations are splines. Important note: around 1:10 the functions also need to go through the points at the ends ax1^3 + bx1^2 + cx1 + d = y1 ex3^3 + fx3^2 + gx3 + h = y3 A video covering this correction can be found here youtu.be/4jw0cjddmB8
Chapters 0:00 Intro 0:05 Ways to Approximate a Function 0:23 Splines 1:06 Calculating Cubic Splines 1:40 Example 1 2:26 Example 2 3:24 Thanks For Watching
#LagrangePolynomials #NumericalAnalysisFixed Point IterationOscar Veliz2011-03-27 | Fixed Point Iteration method for finding roots of functions. Frequently Asked Questions: Where did 1.618 come from? If you keep iterating the example will eventually converge on 1.61803398875... which is (1+sqrt(5))/2.
Why not use x = x^2 -1? Generally you try to reduce the degree of the polynomial you're trying to find the root for.
How did you pick x1? Your starting point should be an educated guess, a point in the neighborhood of your root.
How can you use the convergence test without the root? Think of the convergence test as more of "will this function converge to this root?" When you don't know the root, try iterating a few times to see if the function is converging, bouncing around in a loop, or going to infinity. It will become apparent very quickly.
What happens if a function fails the convergence test? Failing the test means that the function is not guaranteed to converge. It might still converge but it makes no promises. Take the function which I showed fail in the example. If you iterate starting from the root that we found, the function might converge to the same value depending on your calculator's accuracy.
Doesn't this function have two roots? Is there a way to find the second one? Indeed this function has two roots (1+sqrt(5))/2 and (1-sqrt(5))/2 which are the numbers φ (phi) and ψ (psi). I showed how the first example converged to phi and that the other did not for simplicity. You can use the second equation to converge on psi if you start close enough, like -1 for example.
Is there any way to use x = +/- sqrt(x + 1)? In this case you can use x = sqrt(x+1) which will converge to 1.618 as long as the value inside the square root is positive. If you try to take the square root of a negative number you will have to use imaginary and complex numbers.
Is there a way to speed up Fixed Point Iteration? Yes, check out my video on Steffensen's Method with Aitken's Δ² youtu.be/BTYTj0r5PZE and my video on Wegstein's Method youtu.be/T_6mR6rJXQQ
How can I force Fixed Point Iteration to converge? There is a very simple change you can make to induce convergence called Wegstein's Method youtu.be/T_6mR6rJXQQ
Can you make a video that answers these questions? Absolutely check out Fixed Point Iteration Q&A youtu.be/FyCviw2ZA2o
Chapters 0:00 Intro 0:06 Fixed Point Iteration 0:39 Fixed Point Iteration Example 2:12 Convergence Test 2:41 Convergence Test Example 3:18 Order 4:03 Thanks For Watching
#FixedPointIteration #NumericalAnalysisSecant MethodOscar Veliz2011-03-21 | Secant Method for finding roots of functions including examples and discussion about the order.
Chapters 0:00 Intro 0:11 Drawback of Newton's Method 1:05 Secant Method Visualized 1:53 Secant Method Example 2:42 Order 3:05 Order Discussion 3:48 Thanks For Watching
#SecantMethod #NumericalAnalysisNewtons MethodOscar Veliz2011-03-20 | Newton's Method for finding roots of functions including finding a square root example and discussion of the order (newton's method is also known as Newton-Raphson method). Small correction around 2:26 the sign is incorrect it should be x_(n+1) = (1/2)(x_n + a/x_n). A video covering this mistake can be found here youtu.be/4jw0cjddmB8
Chapters 0:00 Intro 0:12 Newton's Method 0:53 Newton's Method Visualized 1:47 Finding Square Root (see correction) 2:30 Example 3:43 Order 4:26 Thanks For Watching
#NewtonsMethodBisection MethodOscar Veliz2011-03-18 | Bisection Method for finding roots of functions including simple examples and an explanation of the order.
Chapters 0:00 Intro 0:14 Bisection Method 1:06 Visual Example 1:49 Difficult Functions 2:14 Order 3:16 Finding Order Example 3:38 Maximum Number of Iterations 4:28 Thanks For Watching