- Traden4Alpha
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QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnAn interesting follow-on is that having found an elegant closed solution you may want to solve it numerically, for example by using function from a library. This could be expensiveAnother example is the logistic function that has an exact solution but maybe computationally more efficient to solve it as an ODE?Quite true. And one form or the other might be easier to compute in parallel (either in terms of using parallel resources to evaluate the function at a single point or in evaluating multiple values in SIMD architectures in which one wants to avoid iterative algorithms.)P.S. How does the KISS principle affect things. If the user thinks that the logistic function library call uses the exact solution math, might a different implementation violate the user's expectations for how the library call behaves (e.g, computation time, precision, handling of extreme or corner case values, etc.)Some formulae in the past were not used because they were computationally too intensive. But new hardware/software technologies may cause them to be interesting again. People tend to use things they are used to and forget to look back.Very true! What was scarce yesterday (and drove the selection of solution approaches) is not scarce today or tomorrow. Big O analysis ignores both the Moore's law gains in computational power and the even larger improvements algorithm efficiency.By the same token, symbol manipulation technologies (e.g., Mathematica, Maple, etc.) let one compute exact solutions that would have been intractable in the days of pencil and paper.Good remark. The Collector in his book(page 319 for a 2-factor) has a series solution for non-path-dependent options using Bernoulli paths. Looking at the formula you would think it would be slower than a binomial method but maybe not because the loops can be parallelized and you can use precomputed lookup tables for factorials and a clever way to do pow().Indeed! The other weird issue is the growing gap between CPU performance and memory performance. In some cases, it's faster to recompute a value than to recall it from DRAM. To the extent that an algorithm can keep everything in registers or L1 cache even if it takes more instructions, it will be faster than code that is constantly pulling data from RAM.

- Cuchulainn
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A concrete example is the double integral that is the bivariate cumulative normal distribution. Most analysis (Drezner, Genz) is delicate and tends to focus on tackling the integral head on (with all the well-known issues.Instead, the integral is the solution of a simplified Goursat PDE (Riemann's method) and we can then unleash all the rigour (positivity, convergence) to create a matrix of values. I suppose it is a bit of lateral thinking. I have not seen this approach done before.This scheme has been tested on Haug's basket examples. And results look good. The core code (a double loop) is about 2-3 lines. If you are interested I can send the code. PM me.// Aziz and Hubbard paper// unable to attach file here.

Last edited by Cuchulainn on May 17th, 2015, 10:00 pm, edited 1 time in total.

- Cuchulainn
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Hypothesis: Nonlinear problems have no closed solution. True/false?

- Traden4Alpha
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QuoteOriginally posted by: CuchulainnHypothesis: Nonlinear problems have no closed solution. True/false?What say you to the good old quadratic formula? Or do you have a restricted definition of "problem"?

- Cuchulainn
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QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnHypothesis: Nonlinear problems have no closed solution. True/false?What say you to the good old quadratic formula? Or do you have a restricted definition of "problem"?You have a special/restricted example of "nonlinear'. To prove something true it must be true for all problems. One swallow makes a summer not.Sometimes you get lucky, e.g. reducing Bernoulli ODE to a linear ODE. This is called serendipity.1) Linear and quadratic polynomial equations are mildly nonlinear. We got lucky. Cardano and Leonardo spent years trying to solve cubics. Besides, you get complex roots but we only want real roots..2) Non-low-order polynomial equations have no closed solution. e.g. solve factorial(x) = x! = y for given y.3) Find closed solution for 1-factor early exercise option problem.

Last edited by Cuchulainn on December 16th, 2015, 11:00 pm, edited 1 time in total.

- Traden4Alpha
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QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnHypothesis: Nonlinear problems have no closed solution. True/false?What say you to the good old quadratic formula? Or do you have a restricted definition of "problem"?You have a special/restricted example of "nonlinear'. To prove something true it must be true for all problems. One swallow makes a summer not.Sometimes you get lucky, e.g. reducing Bernoulli ODE to a linear ODE. This is called serendipity.1) Linear and quadratic polynomial equations are mildly nonlinear. We got lucky. Cardano and Leonardo spent years trying to solve cubics. Besides, you get complex roots but we only want real roots..2) Non-low-order polynomial equations have no closed solution. e.g. solve factorial(x) = x! = y for given y.3) Find closed solution for 1-factor early exercise option problem.Ok, I misinterpreted the quantifier. You are saying "there exist non-linear problems for which no closed form solutions exist." My first reaction is that you are right and it goes beyond examples of problems where we are not sure if a closed form solution might exist if only we were lucky or clever enough to find it.A good example of the non-existence of a closed form solution can be found in certain cellular automata (a nonlinear discrete-valued differential equation on a grid) which can be shown to be capable of universal computation (i.e., one can configure the automata such that it simulates the logic gates of a computer and the system can do any computation based on a program encoded in the initial conditions or coefficients in the system). That implies that at least some non-linear problems map to the halting problem. And I bet there are ways to tie non-linear problems to Godelian undecidability, too. If the system's outcome is undecidable, how can it have a closed form solution?Then there's Stephen Wolfram's contention that there are some systems in which the system itself is the most compact representation of the system and the fastest solution process is to turn the crank on the system rather than look for some magic equation that expresses the end state.But my second reaction is to question what constitutes a valid closed-form solution. It seems to be a compact representation of the solution -- a short string of function symbols that define the values or set of values of the unknowns that solve the original equation. But what are the constraints on the set of function symbols that we might use to create a closed form solution? Is it limited to arithmetic operations? Can we also use square roots, logs, exponentials, and trigonometric functions? What about hyperbolic functions or maybe Bessel functions? I would hypothesize that the history of closed form solutions has been one of creating new symbols and much as it is one of finding ways to write a solution with the set of known function symbols. And yet those symbols encode increasingly complex sets of underlying numerical operations.If one restricted every closed form solutions to the four arithmetic functions(-,+,*,/), then one might be inclined to conclude that almost no nonlinear problems have closed form solutions (e.g., can one write a nice closed form solution for the quadratic formula without using square root or any other function beyond simple arithmetic?). But if one is allowed to construct new symbols (square root, ln, sin, sinh, Bessel, etc.), then perhaps every nonlinear problem has a closed form solution if we could just invent the right function symbol that encodes key elements of that solution. There might even be a generalized halting problem function that lets one write the solution to a universal computation cellular automata in more compact symbol.Thus, if the definition of a closed form solution is not closed, the question of existence such solutions may be open.

- Cuchulainn
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QuoteOk, I misinterpreted the quantifier. You are saying "there exist non-linear problems for which no closed form solutions exist." Yes. But not in the way you have written it. "there exist non-linear problems for which closed form solutions exist. But these are usually the easy/trivial ones." QuoteIf one restricted every closed form solutions to the four arithmetic functions(-,+,*,/), then one might be inclined to conclude that almost no nonlinear problems have closed form solutions (e.g., can one write a nice closed form solution for the quadratic formula without using square root or any other function beyond simple arithmetic?). But if one is allowed to construct new symbols (square root, ln, sin, sinh, Bessel, etc.), then perhaps every nonlinear problem has a closed form solution if we could just invent the right function symbol that encodes key elements of that solution. There might even be a generalized halting problem function that lets one write the solution to a universal computation cellular automata in more compact symbol.Yes, that is a closed solution I would say.But sometime a series solution in which each term solves a PDE is considered closed. I don't agree. Using the word 'closed' can be very comforting. BTW I am not the one who bandies that word around, so those who use it should define haha!

Last edited by Cuchulainn on December 16th, 2015, 11:00 pm, edited 1 time in total.

- Cuchulainn
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QuoteThe answer is usually to start with a model that has a so-called closed-form or exact solution. Need I remind that the first closed-form model that attracted alot of attention was the Black-Scholes model. The many advantages of exact solutions seem obvious to me. Closed and computable! (BUT one snag: we must compute N(x) numerically at the moment until @list1 comes up with a closed formula )

Last edited by Cuchulainn on December 16th, 2015, 11:00 pm, edited 1 time in total.

It also make sense to specify the meaning of the closed form. If a solution is given by an ode or pde whether it can be classified as closed form? In math if the solution is presented by a formula it is called as analytic solution. The closed formed notion I read first in financial mathematics and it still remains as a somewhat fuzzy notion.

- Cuchulainn
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QuoteOriginally posted by: list1It also make sense to specify the meaning of the closed form. If a solution is given by an ode or pde whether it can be classified as closed form? In math if the solution is presented by a formula it is called as analytic solution. The closed formed notion I read first in financial mathematics and it still remains as a somewhat fuzzy notion.Regarding a solution, we can distinguish between its qualitative and quantitative properties. The former includes:Existence, uniqueness, smoothness, growth properties, maximum principlesThe latter includesA formula that allows us to compute a solutionHers is one Classification

Last edited by Cuchulainn on January 15th, 2016, 11:00 pm, edited 1 time in total.

It seems that existence, uniqueness do not properties of a solution and Classification does not define 'closed form' notion.

- Cuchulainn
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QuoteOriginally posted by: list1It seems that existence, uniqueness do not properties of a solution and Classification does not define 'closed form' notion.But you agree with the other definitions in that link?

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: list1It seems that existence, uniqueness do not properties of a solution and Classification does not define 'closed form' notion.But you agree with the other definitions in that link?Yes but classification relates to a given solution. This is strategies for differential equations. Here 'differential' probably can be omitted.

- Cuchulainn
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QuoteOriginally posted by: list1QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: list1It seems that existence, uniqueness do not properties of a solution and Classification does not define 'closed form' notion.But you agree with the other definitions in that link?Yes but classification relates to a given solution. This is strategies for differential equations. Here 'differential' probably can be omitted.There is not much hope if we cannot agree about the meaning of English words. I'm not going to dragged into one of these endless discussions/QuoteClassification: systematic arrangement in groups or categories according to established criteria; specifically

Last edited by Cuchulainn on January 15th, 2016, 11:00 pm, edited 1 time in total.

My original question was the meaning of the closed form solution. Others are not important

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