For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations.
The point of departure is mathematical but the exposition strives to maintain a balance between theoretical, algorithmic and applied aspects of the subject. In detail, topics covered include numerical solution of ordinary differential equations by multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; a variety of algorithms to solve large, sparse algebraic systems; methods for parabolic and hyperbolic differential equations and techniques of their analysis.
The book is accompanied by an appendix that presents brief back-up in a number of mathematical topics. Dr Iserles concentrates on fundamentals: deriving methods from first principles, analysing them with a variety of mathematical techniques and occasionally discussing questions of implementation and applications. By doing so, he is able to lead the reader to theoretical understanding of the subject without neglecting its practical aspects.
The outcome is a textbook that is mathematically honest and rigorous and provides its target audience with a wide range of skills in both ordinary and partial differential equations. Apparently written in Btw, the third formula on p. Don't anyone lose your sleep over it. If you start to really dig into the underpinnings of the topic, you find fairly complex things like partial derivatives for gradient descent optimization, for example , but you don't really have to understand it much to take it, apply it, and verify that the results make sense.
On the other hand, I've been around long enough to have learned the hard way that applying something you don't really, fully, from-the-ground-up comprehend can bite you in surprising ways. I'd suggest learning about partial derivatives and in particular, the gradient.
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With an appropriate book, one evening should be enough. Intuitively the negative of the gradient is the direction to ski fastest downhill, and that's why it is heavily involved in optimization problems. Then there are close connections with convexity -- intuitively the inside of most kitchen cereal bowls is convex.
The gradient is crucial in uses of Lagrange multipliers in the non-linear cases; in non-linear optimization with constraints, the gradient is central to the Kuhh-Tucker-Karush necessary conditions for optimality. If are in a deep, long, narrow river valley, want to get to the bottom downstream of the river, and ski downhill using the gradient, then will keep crossing the river over and over, traveling many feet across the river for each foot going downstream.
So can approximate the valley with an ellipse and ski in much better direction along the long axis of the ellipse. People figured this out long ago -- it's called conjugate gradients. If the river wanders, then that's still more difficult. In the best fitting in ML, may be in such river valleys, and some notes on ML recognize this and warn about using just the gradient. A lot more with gradient is known and at times useful -- Newton iteration, quasi-Newton, etc. Vigorously seconded. There is a lot of applications of convex analysis, convex duality, KKT conditions and game theory in ML if one looks at it right.
In fact the thats at the very foundation of techniques such as support vector machines large margin separators in a Hilbert space , regret minimization algorithms, etc etc. Of course one can choose to ignore all that and only focus on stochastic gradient descent. That will carry one for some non-trivial distance. Imagine the Smoky Mountains of east Tennessee, that is, smooth, rolling hills. Then at a point x, y the partial derivative of f x,y with respect to x is just the slope as in ordinary derivative of the mountain at point x,y in the direction of changing x. So, if the X axis runs east and west, the partial derivative of f x,y with respect to x is the slope of the mountain at x,y in the east-west direction.
So the partial derivative is just like the derivative of a function of one variable, that is, a slope, except is for just one variable, say, x, with the other variable s y held constant.
Look at Arnold's book on Diff, it is harder then others, but has absolutely different outlook. That book is totally mindblowing, and obliterates artificial boundaries between physics and mathematics. It also in the older Dover editions had a cover where the phase portrait on the front looked like two angry eyes glaring at you that you hadn't learned enough math yet.
Koshkin on Sept 6, Well, modern theoretical physics seems to be nothing but mathematics mostly advanced differential geometry and group theory. And this is a good thing, as it is the sign of how far along the subject has gone in its evolution. I can only find the first edition on Amazon, but there's apparently a third edition?
On a related note, I can recommend Stanley J. Farlow's "Partial Differential Equations for Scientists and Engineers", which I bought around to better understand a fluid simulation paper I was trying to implement. My Maths course ended with linear algebra and ODEs. Very good read with nice physics examples. The usage of terms 'initial value' and 'boundary value' is a massive failure of mathematics education. There is no notion of time in mathematics.
Catalog Record: Ordinary differential equations: a first course | HathiTrust Digital Library
There is only a notion of space, due to geometry. I had the longest time coming to grips with the question, "mathematically what is the difference between initial value and boundary value? It's a relic of the past when differential equations were studied under physics, where time and space are a huge part of the conceptual foundation. Sometimes I wonder how much progress we would make in education if we didn't confuse the heck of our students in the name of convention and historical baggage. ASipos on Sept 6, Their behavior is quite different.
I vaguely remember Dr. I think we used one of his other books in a class, or maybe he taught one of my classes. This brings back memories. This was my favorite course in college. Super beneficial materials. On a side note, one of the reasons I loved this course was because it was online and only had 2 tests with no other assignments. The professor allowed you to schedule office hours any time you needed, but the course setup was sweet for self-studiers like me.
Here's the book, Chapters are on the midterm, Chapters are on the final. No homework busy work, no other tests. Just 2 exams. When I was in college I supplemented my learning with Notes on Diffyqs, which I thought explained things much clearer and concisely than my assigned textbook. TomMckenny on Sept 5, On a related note, how do you take notes for math subjects with their multiline integrals and sumation and subscripts, division etc?
I'm very much attached to recording every thing in simple text editors. Is there a "Notepad" or "TextEdit" for mathematical notation? I could not imagine trying to take notes with a computer in a math heavy subject. I had to use paper for everything during my undergrad in Civil Engineering. However, if you already have a computer with you, record the lecture and review it later.
Just as a partial counterpoint to this and overall agreement with everyone else who responded, I've had success taking realtime notes on my laptop in a math class. It took a bit of practice and I'm sure I can't do it anymore, but it's not insurmountable by any means. I used latex and made liberal use of keyboard and software macros to do it, and one of the tricks was to realize that if I needed a quick-to-type way to typeset new thing X, I should just pretend I had such an implementation and make up its command on the spot. At my leisure, I could write up a conforming latex command that worked with all the notes I'd taken in realtime.
That said, I've since come to realize that math notes don't help me as much as they seem to help others. I have greater success primarily listening during class and leaning on the textbook as well as online resources outside of class. I do second the use of emacs to handle the latex, but I don't think that realtime rendering is particularly important in a notes setting. Excellent counterpoint. Keep in mind that I was a Civil Engineering student, not computer science. My abilities to use Latex were little-to-none at the time. I didn't get any exposure to it until I was in grad school and we used it to format journal article submissions.
And then learn to read the text book. I'm easily distracted, particularly from internal digressions.
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Taking notes helps me hold on to the thread. Do not deprive yourself of the delight of browsing through your handwritten notes some years in the future! I use mathstackexchange for that purpose and do all my work in the little window where they want you to type up your question. Mathjax will render all your stuff beautifully.
If I want to save my work, I just take a screenshot. There's TeXmacs, which lets you enter mathematics either via point-and-click palettes or using TeX notation. I like it a lot. Downsides: 1. Development seems more or less dead. It's not always been perfectly stable. It's kinda sluggish, especially on older slower machines. For example MathJax, or emacs's org-preview-latex-toggle to show the rendered equations in the same buffer you write in.
Luc on Sept 5, Lately I have been looking for a practical not too much theory book on how to model problems with differential equations, with lots of examples. I imagine I'd plug them into Wolfram Alpha to get a solution. Does anyone have recommendations? Almost all textbooks named "Mathematical Modeling" should have chapters on modelling continuous change with differential equations, hopefully also a chapter on how to model systems of interacting variables with a system of differential equations.
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Then also textbooks with names like "Mathematical Models in Biology" or "Mathematical Biology" should have chapters on population growth, a two-species predator-prey model, diseases and epidemiology, and sometimes also a chapter on chemical kinetics these are all modelled with differential equations. Thank you. Knowing the right search terms will help a lot. You can also look into "system dynamics" software, which allows for the modeling of complex systems with very little user-facing mathematics.
Dana Meadows wrote a book called "Thinking in Systems," which is a great intro to this subject.
A First Course in Ordinary Differential Equations
Luc on Sept 6, Thanks, that looks very interesting! Looks worthwhile. Hacker News new past comments ask show jobs submit. Koshkin on Sept 6, Well, modern theoretical physics seems to be nothing but mathematics mostly advanced differential geometry and group theory. TomMckenny on Sept 5, On a related note, how do you take notes for math subjects with their multiline integrals and sumation and subscripts, division etc?
Koshkin on Sept 6, Do not deprive yourself of the delight of browsing through your handwritten notes some years in the future! Luc on Sept 5, Lately I have been looking for a practical not too much theory book on how to model problems with differential equations, with lots of examples. Luc on Sept 5, Thank you.