Question
Asked 29th Mar, 2014
  • Alberta Research Council (ARC)

What kind of problem solutions do you rate higher: analytical or numerical?

More problems can be solved numerically, using computers. But some of the same problems can be solved analytically. What would your preference be?

Most recent answer

16th May, 2014
Helmut Ziegfeld Baumert
IAMARIS GbR - Research and Consulting, Ludwigslust, Germany
Ljubomir, yes, that's strange and worth to follow the proposal of Chris. I leave that to those who know more about the internals of ResearchGate, which are somewhat a mystery for me as a beginner. It's not bad a network (in my view better than LinkedIn and Xing) but the rating and scores system is somewhat strange and possibly open for manipulation. Anyhow, my answer to our common and important questions is as follows: Analytical theories are more general and form the frameworks of what we then code to do numerical simulations for practical or test purposes. Both are needed!
2 Recommendations

Popular Answers (1)

30th Mar, 2014
Mohammad Firoz Khan
Jamia Millia Islamia
It is really shameful to downvote responses to a serious post that requires knowledge and research aptitude. The downvoter appears to lack both. This button is now more frequently is misused.
Alexander,
A researcher would like to solve it analytically so that it is clear what are premises, assumptions and mathematical rules behind the problem. As such problem is clearly understood. Numerical solution using computers give solution, not the understanding of the problem. It is quite blind. However, in emergency one may resort to this option.
16 Recommendations

All Answers (70)

29th Mar, 2014
Hanno Krieger
I´m not really a specialist. But I prefer the analytical method because there is a logical thinking, a model, in it.
2 Recommendations
29th Mar, 2014
Nageswara Rao Posinasetti
University of Northern Iowa
Alexander
Personally I prefer analytical solution. With the availability of numerical solutions such as finite element and Runge Kutta, it is practically possible to solve many of them easily through them. It has almost like a short cut. But it is always going to be approximate and an error involved. Analytical solution when possible gives the exact value.
2 Recommendations
30th Mar, 2014
Krzysztof Rogowski
Bialystok University of Technology
I prefer analytical solutions. This kind of solutions gives you more. Based on such solution you may infer about the nature of the phenomena that you consider. You may also use this solution in further considerations.
Numerical sollutions can be good confirmation or proof of the analytical solutions.
Unfortunately many problems are to difficult to solve them analitycally, so we have to be satisfied with analytical solutions.
5 Recommendations
30th Mar, 2014
H Chris Ransford
Karlsruhe Institute of Technology
I remember an engineering case a few years ago, when 2 separate teams were modeling a set of rolling mills with an open number of rolling stands in a row.
The mills were meant for a specific, narrow application with newly-developed materials, and were themselves made of a new alloy.
The 2 teams tackled the problem very differently.
One team went the analytical route, and modeled everything - relative & absolute material elasticities, surface roughness statistics (direction cosines, sizes, shapes, interplay), role and distribution of the chemical impurities which could affect the process, etc.
The other team did not bother - they just built several mock-ups of different sizes, built in hundreds of sensors everywhere, assessed the sensor readout numbers numerically and came up with a design.
Both designs wound up within 10% of each other. One team had learnt much about what impacted the process and contributed to an understanding of certain processes & properties (such as dislocation creep under stress within the alloy, etc.) which could later be used elsewhere, and which at the very least heightened understanding.
The other team had learnt nothing and contributed nothing to the furtherance of knowledge. They'd just applied well-known numerical methodologies to content-free numbers they'd read off their sensors.
3 Recommendations
30th Mar, 2014
Vijay K Jindal
Panjab University
Analytical solutions provide far greater insight and thus possibility to modify, alter and design a solution based upon some variables rather easily. The numerical solutions are good only for an accurate result based on assumed model where for designing you need the calculations afresh and getting enormous data. Therefore analytical solutions are laways best option- it is only that these are not easy to obtain in complicated situations. As far preference, analytical would be preferred and when required they can yield quick simple analytic overview under some approximations. Once you have conceived, go to exact solutions numerically.
5 Recommendations
30th Mar, 2014
Fermin De La Fuente Calvo
DeLaFuente Genealogy
Analytical
2 Recommendations
30th Mar, 2014
Muqaddas Tariq
University of Engineering and Technology, Lahore Pakistan.
I think mathematical solutions also based somewhat on any analytical or practical based result so both are important in context of any desired solution, both should be supportive to each other in finding any solution. Further depends on our nature of work, nature of data, nature of required out-put etc.
4 Recommendations
30th Mar, 2014
Demetris Christopoulos
National and Kapodistrian University of Athens
The medium route: Numerical output of a well formulated analytical solution, because our given data has a noisy shape and we cannot apply directly the analytical formulas.
6 Recommendations
30th Mar, 2014
S . Shukla
Indian Institute of Technology Delhi
In my opinion both solution kind of solution are equally rating by a person who know the physic behind both solution although i seen mostly people rates numerically solutions high as compare to analytical but both have equal importance.
3 Recommendations
30th Mar, 2014
Hatem Maraqah
Hebron University
I prefer analytical solution
3 Recommendations
30th Mar, 2014
H Chris Ransford
Karlsruhe Institute of Technology
I'd be quite interested in hearing why my answer above got one negative vote?
2 Recommendations
30th Mar, 2014
Tobias M. Ntuli
Mangosuthu University of Technology
Each and/or every problem possibly and/or probably has its best solution! The method to arrive at the best solution will vary!
Period!
1 Recommendation
30th Mar, 2014
Demetris Christopoulos
National and Kapodistrian University of Athens
@H Chris, if you see above all answers were down-voted including the original question!
I have the same curiosity with you...
4 Recommendations
30th Mar, 2014
Kai Fauth
University of Wuerzburg
Dear Reyhaneh Mehrvar, you write that "numerical solutions are easier and more people can understand them". I find that statement stunning. I find numerical solutions hard to understand. There are only numers, no parameters which have physical significance. When I obtain an "understanding" from numerical solutions, then this is usually the case after studying many of them and identifying the dependence of the numerical solutions on input parameters.
On the other hand, analytical solutions oftentime exist only for cases which cannot cover the complexity of the phenomena involved (i.e. spectral functions (= photoemission line shape) of correlated electrons). Numerical solutions of appropriate models are then absolutely required.
8 Recommendations
30th Mar, 2014
Hanno Krieger
Dear Kai,
impressive declaration. Not the appliance of numerical method is the problem, the problem is to apply numerical techniques without basic understanding and models.
4 Recommendations
30th Mar, 2014
Dejenie Alemayehu Lakew
Hampton University
Although they may be put as alternative methods, but sometimes one is used because of lack of alternative ( in a sense we use numerical methods because we do not have analytical solutions at hand ). But I usually like analytical solutions for mere simplicity and cleanness while numerical solutions are imperative for approximate solutions of problems, in which most applicable problems have approximate solutions - realities are always approximates.
3 Recommendations
30th Mar, 2014
Kai Fauth
University of Wuerzburg
someone seems closely watching and downvoting. Come on. Show your face and discuss what you don't like!
9 Recommendations
30th Mar, 2014
Mohammad Firoz Khan
Jamia Millia Islamia
It is really shameful to downvote responses to a serious post that requires knowledge and research aptitude. The downvoter appears to lack both. This button is now more frequently is misused.
Alexander,
A researcher would like to solve it analytically so that it is clear what are premises, assumptions and mathematical rules behind the problem. As such problem is clearly understood. Numerical solution using computers give solution, not the understanding of the problem. It is quite blind. However, in emergency one may resort to this option.
16 Recommendations
30th Mar, 2014
Alexander J. Babchin
Alberta Research Council (ARC)
I completey agree with you, Mohammad.
4 Recommendations
30th Mar, 2014
Hanno Krieger
Dear all, it´s so easy to kick against others legs, instead of telling the reasons of missing acknowledgement.
So please go on downvoting, it´s so helpful and breeding. Thank you!
5 Recommendations
30th Mar, 2014
Oscar Chavoya-Aceves
Glendale Community College, AZ USA
It depends on the problem I guess. Furthermore, sometimes, the "analytical solution" is an illusion, because the nature of the problem forces us to resort to a numerical procedure, by hand or by computer, to make our solution useful: How do we get the values of trascendental functions if not using numerical methods? Then, of course, as it has already been observed (in a way) by Pavel Komarov , sometimes we just cannot get an analytical solution: like in the case of the three body gravitational problem, then we have to use computers, and face the possibility that no numerical procedure will be stable! http://www.lana.lt/journal/42/NA16306.pdf In those cases we can still use numerical methods to obtain qualitative knowledge on the behavior of a system.
When solving some problems involving integer numbers, using a computer to make our task easier would be advisable.
6 Recommendations
30th Mar, 2014
Herbert H H Homeier
Universität Regensburg
I opt for a hermeneutical circle of both approaches:
Quite often, you can learn about the possibility of an analytical solution from a numerical solution and vice versa.
Further remarks:
Whether or not an analytical solution is possible quite often depends on the complexity of the problem. For real world problems, one cannot really expect complete analytical solutions in most cases. However, sometimes partial insight is possible using general analytical methods.
On the other hand, doing large scale simulations, one often needs some theory to help to interpret the data. Such theories can be of an analytical nature.
In a sense, the analytical vs computational distinction is reflected in the different approaches of mathematicians on the one hand, and natural scientists on the other. They should collaborate in my opinion. That, however, requires to learn the language of the "other" field.
This type of discussion is quite old. The famous slogan
"The purpose of computing is insight, not numbers" of Hamming was - I think by himself - commented by "the purpose of computing numbers is not yet in sight", cf. http://www-history.mcs.st-and.ac.uk/Extras/HammingReviews.html
3 Recommendations
Deleted profile
There are many good answers and great insight for many knowledgeable researchers. I specifically work on developing numerical method for solving problems that, while one can derive analytic solutions, cannot be parallelized easily/make efficient use of computational resources in a straightforward manner unless using numerical methods, especially if ones wants to be able to use everything from small clusters of desktop computers to massively parallel computer architectures such as the K computer or blue waters without having to write different codes for the various situations.
If one can derive analytic solutions then one should, even if it's only for some special cases as these can often give insight into larger problems (e.g., the molecule H4, four hydrogen atoms, in a square-planar arrangement, is a particularly challenging problem to describe accurately quantum mechanically even though, for chemists, a four electron problem is rarely studied in real world applications, but it highlights areas/situations for which one needs to watch out, which otherwise would not be obvious or easy to identify). There are cases, however, where defining an analytic solution is not possible, such as the free compliment method by Hiroshi Nakatsuji which uses monte carlo sampling to generate the integrals for the Hamiltonian matrix, where defining something like an analytic gradient would be somewhat challenging, though it solves the Schrodinger equation exactly for many fermion systems.
I think some people place a little too much emphasis on the analytical solution in the analytic vs. numeric debate, as each can be very useful and complimentary, and insight can be gained from both, as others have mentioned and, as in my case, using numerical methods makes more sense, at least for now (and yes, I have worked out the analytic solutions to the problems I work on because, as others have mentioned, it can be very useful).
P.S. I thought it was J. von Neumann that originally said "The purpose of computing is insight not numbers." I know Hamming uses it as the slogan for his textbook, but does anyone know to who it should be credited?
2 Recommendations
31st Mar, 2014
Jose Alcala
Barry University
Analytical is preferred because it reflects and reveals real-life situations. Numerical is artificial and does not reflect contingencies that might exist in natural situations.
Deleted profile
Jose, your view is a bit underdeveloped, I'm not trying to be offensive, since at some level of the computing process you have to result to numerical techniques (e.g., how does a computer compute the sine, log, etc.). Numerical methods are simply a way to approximate, sometime rather well, the analytic expression for what you want. Granted, there are also cases where the numerical approximation fails and fails catastrophically. But at the same time, even an analytic expression can have singularities or other faults due to approximations made at some level up the food chain (i.e., even the highest level theories are approximations to real life and never truly contain all the information needed for one to be able to say their approach is exact) and those pitfalls are not always apparent even to some of the most genius of women/men (it took several hundred years to develop an integral that worked for all functions, not just continuous or even mostly continuous, and some of the best minds worked on that problem, and even so, there are not closed expressions for the integrals of all functions).
The long and the short is that given certain caveats your statement holds some merit, but to apply it so broadly to everything is naive at best.
1 Recommendation
31st Mar, 2014
Filippo Fratini
Erste bank
Analytical solutions are elegant, fascinating, writable and one normally learns a lot of physics in finding them. No doubts that when doable, one should go for them. However, being doable here include lots of factors, one of which is time: there are processes for which a week or a month is required to find the analytical solution. We researchers often face shortage of time. Dedicating one month for finding an analytical solution might look like a bad investment of time. In that case, numerical methods are appealing and probably the best solution.
4 Recommendations
31st Mar, 2014
Tobias M. Ntuli
Mangosuthu University of Technology
The downvoter is perturbed by the simplicity of the question hence answers! Each situation requires careful consideration to arrive at the best approach to the solution!
2 Recommendations
31st Mar, 2014
C. Martín Saravia
National Scientific and Technical Research Council
I think that it is an error to compare such a different worlds... Analytical solutions are often preferred, if they exist... When they do not exist (almost always) numerical solutions are the ones that are going to help you. I do not think it is a matter or taste or time... it is simply a matter of existence. Nobody is going to use a numerical solutions for predicting the buckling load of a straight isotropic Bernoulli column, use the analytical solution and it is done. But if you are simulating turbulent flow past an obstacle, then there is no choice to make. At last, I think that these worlds are so different that they do not even compete.
5 Recommendations
31st Mar, 2014
Hanno Krieger
@ C. Martin, good comment and examples.
1 Recommendation
1st Apr, 2014
Lin Zhang
Shaanxi Normal University
I agree with most of the answers but I do not think the analytical solutions are actually there! Numerical method is a way to "see" directly the solution of an equation by avoiding to give an explicit form! But what are the analytical solutions? I think they are just mathematical denotations, such as Cos(x), Exp(x) , Bessel function J(n,x) or many other special functions, i.e., we only use specific labels to express the numerical solutions. What important thing for the analytical solutions, I think, is not the solutions themselves but the relations between them. Right?
1 Recommendation
Deleted profile
Here's a slightly different take on it, for those advocating analytic solutions, which are useful when they can be derived, but one needs to take a broader view and keep in mind what they're actually doing.
As an example, I think we can all agree that the N-body problem is important to many fields (physics, chemistry, biology, economics, business, weather, mathematical modeling, risk analysis, etc. etc. etc.) and the basis for most of the work researches do as we rarely deal with just one- or two-body problems. So I would like to make a modest proposal to those advocating analytic solutions, saying you need them to understand the physics or other properties:
Please provide the citation for the analytic, closed form, solution of not the N-body problem, but just the 3-body problem, a toy problem that rarely has broader uses or applications when compared to other N-body problems (N > 3).
1 Recommendation
1st Apr, 2014
Cj Nev
Northwestern University
Analytical solutions can only lead to precise numerical methods (e.g., from https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/: "Interactions that were previously calculated with mathematical formulas thousands of terms long [Feynman diagrams] can now be described by computing the volume of the corresponding jewel-like 'amplituhedron,' which yields an equivalent one-term expression"), or presently to best approximations (due to shortcomings in analysis) of the actual.
1 Recommendation
1st Apr, 2014
Cj Nev
Northwestern University
Okay Jared (and where in the polar vortices have you been hibernating all winter?), you have laid down the gauntlet. Your requested citation for the analytic, closed form solution of the 3- (and n-) body problem is simultaneously anticipated in the citation I provided above as well as the reference contained in (primary author, Princeton) Nima Arkani-Hamed's "Scattering Amplitudes and the Positive Grassmannian" (http://arxiv.org/abs/1212.5605): [82] C.-N. Yang, \Some EXACT [emphasis mine] Results for the Many Body Problems in One Dimension with Repulsive Delta Function Interaction," Phys. Rev. Lett. 19 (1967) 1312{1314.
Analytically, Nima Arkani-Hamed et al has derived much more efficient numerical solutions with the positive Grassmannian "where local space-time physics and unitary evolution in Hilbert space [probabilities] do not play a central role, but arise as emergent phenomena from more primitive principles." Furthermore, "Using this curvature bound [from the generalized positive Grassmannian, or Amplituhedron] it is possible to give a more PRECISE [emphasis mine] picture of the long-time behavior of the flow" (from http://scgp.stonybrook.edu/archives/8698).
1 Recommendation
1st Apr, 2014
Herbert H H Homeier
Universität Regensburg
I think that Jared was addressing the problem of a closed-form solution of the N-body problem in classical mechanics. The latter example shows that a hope for analytical solutions can not always be be satisfied even for few-particle systems with well-defined interactions (at the theoretical level).
2 Recommendations
Deleted profile
While the papers you cite CJ provide "exact" solutions, they are not analytic in the true sense. They are solving an eigenvalue problem, which is an iterative process (i.e., numerical). One can come up with numerically exact solutions for these problems, but the point is, they are not analytic. Any time you have to iterate you do not have a closed form solution and therefore in the strict sense of the word, it is not an analytic solution (numerical analysis exists for a reason).
If it were a true analytic solution to the N-body problem they would have been awarded the Fields medal and possibly the Nobel Prize (physics obviously), among other things, seeing as it is an unsolvable problem in closed analytic form (thus, "a modest proposal").
1 Recommendation
Deleted profile
By the way, seriously, who is the downvoter and what do they have against words?
1 Recommendation
1st Apr, 2014
George Markou
University of Pretoria
If you consider the current trends in any scientific field it is inevitable to have both so as to provide generalized solutions for different kind of problems ("IF" you can provide a solution in some cases, analytical or numerical).
How can we perform numerical simulations without having the analytical solution of the problem or at least analytical solutions of the different parts of the problem at hand? They are both explicitly connected through this scope. On the other hand there is no practical meaning of developing analytical solutions to solve each and every problem that immerses and eventually stick to this approach and expecting to derive results related to large-scale problems that involve multidisciplinary physical phenomena. This is where numerical analysis is very practical and provides the means to approximately derive answers.
“Computers are useless. They can only give you answers.” Pablo Picasso
“Computers have enabled people to make more mistakes faster than almost any invention in history, with the possible exception of tequila and hand guns.” Mitch Ratcliffe
“A computer will do what you tell it to do, but that may be much different from what you had in mind.” Joseph Weizenbaum
and on the other hand:
“I do not fear computers. I fear the lack of them.” Isaac Asimov
“Computing is not about computers any more. It is about living.” Nicholas Negroponte
and last but not least:
“Computers shape the theory.” John Argyris
Finally, it is very interesting to note here that all of us use a kind of "numerical analysis" to express our "analytical analysis" related to this issue just by using our computers...
6 Recommendations
2nd Apr, 2014
Cj Nev
Northwestern University
Dear Jared and Herbert,
Thank you, that these citations are not necessarily analytical solutions is well taken. However, the citations are current works in progress, that I am confident can only derive an analytical solution (I have had long in my mind, under publication, chomping at the bit, it's the year of the horse) from true so far the numerical iterative (positive Grassmannian; amplitudes v. open n-bodies/particles tied to local Hilbert spacetime probabilities). Yet also, THIS iterative of amplitudes (again v. n-bodies) into an "amplituhedron" form is BOUNDED, as well as is self-similar to ONE form (thus abstractly/analytically/non-numerically in that sense a "closed form"). Can not therefore this "bounded," self-similar iterative form be considered -- analytically before numerically as Nima Arkani-Hamed emphasizes in his presentations -- to constitute your "closed" form solution requirement? Please do not allow though a possible Fields Medal and/or Nobel Prize to get in the way of such a consideration (those numerics also follow the analysis in most every case). ; )
Deleted profile
The fact that it's bounded lends to the exactness of the solution but not analyticity (e.g., the variational principle, which is exact and also provides an upper bound to the solution, does not guarantee the solution is analytic). I think there's a little murkiness in what each of us means for a solution to be analytic. I can write a system of linear equations (or nonlinear, which I've linearized, Ax = b, and use the Jacobi method to solve them, which is guaranteed to converge, but it's not an analytic solution. Or, in other words, being able to write an equation (which one can argue is analytic) does not equate to an analytic solution. I hope that makes sense and this is what I mean by analytic, you have a closed form of the solution which you get immediately, no iterative process is needed.
There are a couple of other methods (less sophisticated than the one you cite above CJ, but still worth knowing about) which, for various fermionic quantum systems, provide exact solutions. References are Acc. Chem. Res. 45, 1480 (2012) http://dx.doi.org/10.1021/ar200340j and Physics of Atomic Nuclei 68, 1227 (2005) http://link.springer.com/article/10.1134%2F1.1992578. They provide exact solutions, which can be expressed by a simple equation, but they are not analytic solutions as they require one to iterate to convergence. This is the point I was trying to drive home with the three-body problem. If the solution is analytic then there is no need for iterations, you have the solution. Also, another way to include thinking about it is, I can write the exact equations for the finite difference differentiation of a function, but it is not analytic (and in this case, does not require iterations either).
Again, I think this is from some different definitions/conceptions each of us have of a solution being "analytic" (Wittgenstein is probably very pleased with himself right now).
Finding exact solutions is paramount in any field, regardless of if they are obtained numerically or analytically, as insight can be gained from both approaches about new, or even old, phenomena and they should both be pursued. Having a numerical equation (closed form of a solution, possibly iterative, if you'd like), which one uses to obtain the numerically exact solution, even though it is a numerical solution, can often be used to tease out the underlying physics and is quite useful, even though it is not analytic.
At the end of the day, both forms are needed, and in my not-so-humble opinion, it cannot be definitively said that one is better than the other since both approaches have shortcomings and pitfalls as well as advantages. Being aware of what those are is more important as one tries to understand and make use of them, especially if one would like to improve a model/method/approximation/approach by including previously ignored interactions and/or speeding up the time to a (possibly exact) solution.
2 Recommendations
2nd Apr, 2014
Lin Zhang
Shaanxi Normal University
Actually, this question has a very clear answer: analytical solution! But, now, it is blurred!
2 Recommendations
2nd Apr, 2014
George Markou
University of Pretoria
@Lin. Why did it become "blurred" according to your understanding?
3rd Apr, 2014
Lin Zhang
Shaanxi Normal University
@George. We can say definitely that the analytical solution is better than numerical solution! Just only one example can show this conclusion: the famous Goldbach Conjecture! You can use the numerical solution to verify it again and again but you can never give a positive answer which an analystical solution can definitely give!
2 Recommendations
Deleted profile
Consider, then, a direct proof or proof by construction vs. a proof by induction. You don't get an analytic expression from both, though they both prove/disprove the veracity of a mathematical statement and if someone only proves a theorem by induction, rather than using a direct proof, it is still accepted as just that, a proof. One can draw similar parallels between analytic and numeric solutions.
It does not require an analytic solution to prove the Goldbach Conjecture, in other words, and every time you prove it numerically, you are only strengthening the cause. If one were o numerically come up with an example where it did not hold, this would be a proof by contradiction and would be accepted as valid.
3rd Apr, 2014
George Markou
University of Pretoria
@Lin. Yes analytical solution can be "better" than the numerical solutions in terms of accuracy. I will not argue regarding that. The main problem though with analytical analysis is that in most cases (especially when dealing with multidisciplinary problems) there is no analytical solution. So I guess the answer to the question would have been analytical analysis if we took into account the accuracy.
A few colleagues wrote before that through numerical analyses you can get only answers and this type of analyses do not help you in understanding the mechanics behind the behavior of the problem. This is actually not entirely true. If you are a "user" of a software through which you perform numerical analysis then yes this can be very true. If you are a programmer though, then the exact opposite applies. First of all you need to understand the formulation and then program it. Then you need to run the algorithm and check the results in relation to the analytical. From my experience this is the ultimate learning procedure in understanding how a formula works and behaves numerically. Running the analysis by debugging the algorithm, one may understand different aspects of the problem that cannot be understood when reading formulas.
Another aspect of this issue is the case of studying problems that do not have analytical solutions but can be solved approximately through numerical analysis (as Alexander's question says: More problems can be solved numerically). For example, Fluid-Structure Interaction problem for modeling the case of a full-scale airplane flying with a speed of 1 Mach and its wings are oscillating due to induced vibrations. Or the case of limit state analysis of reinforced concrete structures by accounting for cracking, 3D rebar geometry, rebar yielding and prestress tendon's or limit state analysis of bridges with standard reinforced elastomeric bearings (you cannot even study parts of the prementioned problems by applying analytical analysis). If you have an advanced knowhow on the numerical behavior of the methods that you use during any analysis and their limitations in capturing different physical/mechanical phenomena, then you can benefit qualitatively or even quantitatively in terms of understanding the mechanics behind the behavior of a physically/mechanically complicated system.
In the case you are an Engineer, then providing answers (even approximative onces) is of great significance. Numerical methods gave Engineers the ability to design and built pretty much anything. Analytical analysis was the base and numerical analysis is translated into practical implementation and design. Being an Engineer, I prefer methodologies that provide answers. Mathematicians might like more accurate results given the nature of their science but at the end of the day, as I mentioned before, you cannot have one with out the other.
2 Recommendations
3rd Apr, 2014
Cj Nev
Northwestern University
Dear Jared, In returning to my original citations ("Scattering Amplitudes and the Positive Grassmannian") as a preferable analytical v. numerical solution approach, I should have included these following introductory and concluding remarks from https://www.perimeterinstitute.ca/news/new-face-feynman-diagrams/deeper-dive-shell-and-shell, I believe to be in stronger agreement with your appreciated distinction between analytical and numerical solutions above:
"Feynman's virtual particles ['off-shell' scattering amplitudes that do NOT satisfy (analytically) Special Relativity's energy and momentum relationship]...aren't physical -- they're just representations..in the calculation.
"...Why is that [not] better [preferable]? ...Cachazo stresses the broader implications: 'The advantage of doing things on-shell is that you don’t have to introduce the redundancies that were needed when you work with off-shell [Feynman] physics. The reason Feynman diagrams are complicated to use is that they were designed to make locality manifest, introducing redundancies [unbounded iterations] as the price. [Analytically] Get rid of the off-shell particles and the [unfavorably cumbersome] calculations become strikingly simple [analysis' mere afterthought]. The price to pay is manifest locality. This price is, in fact, not too high as general principles [ANALYTICAL solutions] imply that locality is not a property of nature...'”
Perhaps where our ships cross is the need for further distinction between unbounded iterations, clearly overwhelmingly "unwieldly" numerical approximations, and bounded precise, actual physical iterations as predominately analytical, having priority over subsequent numerical solutions. I guess I just don't see why bounded iterations cannot be regarded as closed in this sense?
2 Recommendations
5th Apr, 2014
Lin Zhang
Shaanxi Normal University
Hi, dear George, an important problem now is: if there is an analytical solution for a system with a large degree of freedom? This question can be refined by considering a 3-body problem! If there exist an analytical solution for 3-body system? Or should we use an infinite number of numerical solutions (just as iterations Cj Nev mentioned above) to give a total picture of this system. I think this is a good topic!
1 Recommendation
7th Apr, 2014
Carlo Gualtieri
University of Naples Federico II
In many problems, the analytical solution could be used as benchmark for the numerical solution. So I think that both the methodology are useful and can be applied to study a problem.
3 Recommendations
7th Apr, 2014
Ljubomir Jacić
Technical College Požarevac
@Alexander, we do need both! In certain control systems I do use Partial Differential Equations! They can be solved on both ways!
2 Recommendations
7th Apr, 2014
Ljubomir Jacić
Technical College Požarevac
Why so many downvotes?!!! This is not fair, neither to the autor of the thread, nor to the contributors!
5 Recommendations
7th Apr, 2014
Tobias M. Ntuli
Mangosuthu University of Technology
Hallellujah!
Because it apparent and/or clear to the uninitiated that both are required and their preference will vary with the problem!
Period!
1 Recommendation
7th Apr, 2014
Nageswara Rao Posinasetti
University of Northern Iowa
Anonymous down vote is bad, since they get away without explaining their stand. It is unfair, and the person should have the courage to explain why he is down voting.
3 Recommendations
7th Apr, 2014
Hanno Krieger
Dear Ljubomir,
then use the faster technique, if you can rely on both.
1 Recommendation
7th Apr, 2014
George Markou
University of Pretoria
Down voting is welcome given a proper reasoning behind it.Nevertheless for those that get frustrated by seeing their comments down voted, I got one also before but don't care to be honest with you, just take it as sign of weakness of the down voter. I guess being anonymous is a trend in the "modern era" that we live in. It is always easier to press a button facing down than provide a solid opinion.
1 Recommendation
7th Apr, 2014
Kai Fauth
University of Wuerzburg
I may be wrong but it seems to me that Tobias Ntuli is "the downvoter" and he has indicated the kind of reasoning behind these downvotes. If this is true, Tobias, I can partly understand your point. The question was formulated in a very broad way and so were some of the answers. But I do think there have been good, general points in the discussion e.g. such as what kind of solution is "better" suited for generating an understanding. And when you state "... that both are required and their preference will vary with the problem!", then this is not a very useful statement to me unless you lay out to us under what kind of circumstances you'd prefer one over the other for being more adequate. Some postings have presented viewpoints in this respect or have given examples I consider worthwhile pondering. If you disagree, please enter the discussion. You'll be welcome.
8th Apr, 2014
Vijay K Jindal
Panjab University
@Kai Fauth, as stated earlier, analytical solutions give us an overall comprehensive assessment of a given problem and its solution, allowing us to vary which parameter to reach a desired goal. The problem is that you cant obtain simple analytical solutions except for a limited case. However, for each limiting case, apart from the confidence these provide to assess the accuracy of a numerical solution, they have always a role in designing or modifying. A numerical solution has got acceptance now a days as it is easly and brute force jump at getting a correct solution without knowing too much about the intermediaries.
However, why so much of downvoting is perplexing- not only a good question but also most answers are getting down vote.
3 Recommendations
8th Apr, 2014
H Chris Ransford
Karlsruhe Institute of Technology
The serial downvoter should be reported to RG and expelled from the community
1 Recommendation
4th May, 2014
Helmut Ziegfeld Baumert
IAMARIS GbR - Research and Consulting, Ludwigslust, Germany
Hi Alexander, I think that stuff we traditionally call science, i.e. mind experiments, qualitative discoveries and/or asymptotic extrapolations, is best expressed in analytical language. The solution techniques (numerics, analogue computers of any kind) are tech tools to handle complex problems in practice. Numerics is a machine science...?
The whole issue reminds of the relation between natural sciences and engineering sciences/technology. My answer: these methods are different but just needed because of these differences?
2 Recommendations
4th May, 2014
Ljubomir Jacić
Technical College Požarevac
This picture summarize the answer!
3 Recommendations
4th May, 2014
Helmut Ziegfeld Baumert
IAMARIS GbR - Research and Consulting, Ludwigslust, Germany
@Ljubomir: Great summary! For me as a physicist remains the question between theory and analytical modelling. Is the latter the more general case? And do we need "Large scale" in front of "Numerical simulation"? I even guess that Ljubomir's trio might represent a general picture of research technology. Well, some people say that numerical modelling belongs into the "Experiment" box. Interesting question!
2 Recommendations
4th May, 2014
Martin Bohle
Ronin Institute
It's a debate that is as old as numerical methods (even before they were executed by non-human computers). A particular twist of the problem is for matters having a mathematical description that assumes continuity (e.g. hydrodynamics) or uses analytical descriptions (formula) for discrete processes (e.g. growth of plankton populations - one might re-think describing that dynamic as "recursive maps"). However, during the last decades the interplay of experiments / observations, theory-building and numerical modelling as shown that the latter are a valuable (and trustworthy) work-horse. For the more philosophical inclined colleagues that may (and should be) less than satisfying. Beyond that one may wonder about "risk" being stuck (main-stream) with one school of "mathematical description".
5 Recommendations
13th May, 2014
Helmut Ziegfeld Baumert
IAMARIS GbR - Research and Consulting, Ludwigslust, Germany
Hi Martin, how are you?
Yes, actually stoneold a question. E.g. geometrical construction as kinda analogue calculations etc. I fully agree about the workhorse picture. However some exciting things have been done still more or less manually (sputnik, moonlanding, yes, also our monsterweapons so that we can now kill people not only once but twice and many more often!!...Halleluohja? Sort of oversaturation with everything?).
In some aspect computing is also kinda mode, it's a must today. In the present historical situation I see a danger that abstract thinking gets lost because the focus of education is on machines and their handling. I realized that in another forum where people discuss the need of the notion of infinity. I insist that we need it because it is the very heart of any theoretical thinking. I made my own experiene with the need for infinity as I learned to understand the essence of turbulence in the context of a project named CARTUM, you should have heard of it :-) . I made a discovery and it was solely based on practical experience ("gut feeling"), some intuition, few calculation, and the concept of infinity. It was the greatest adventure I ever have met!
I hope you are well and have fun in what you are doing. A reasonably science politics is more important than ever. Best wishes, Helmut
1 Recommendation
16th May, 2014
Ljubomir Jacić
Technical College Požarevac
@Helmut, I have forgotten to give a link of web page where the picture from my previous answer comes from. It is about engineering dynamics!
4 Recommendations
16th May, 2014
Ljubomir Jacić
Technical College Požarevac
Dear Alexander, again many downvotes with no reasons! I think all answers have at least one downvote! :( It is not done by scientist!!!
1 Recommendation

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