I would like to calculate an integral, which is determined by two functions: I(T) = ∫0T i( f(t), g(t)) dt where f and g solves ordinary differential equations and i is known.
The obvious approach would be to derive a differential equation for I and the solve it alongside f and g (which can be done, but is numerically expensive in my case). In my case, however, f solves an equation with an initial condition f(0) and g and equation with a final condition g(T).
My best guess at the moment would be to solve f and g on a grid using a standard ODE solver and then use a standard method for numerical integration with equally spaced t-coordinates or some kind of quadrature rule (basically anything described by Numerical Recipes).
Does anyone have a better solution? That is, a method that takes the specific type of ode solver and its accuracy into account.
Many advanced ODE solvers come with a feature called "dense output". The ODE solver gives you not only the values of f and g on a grid (as specified beforehand), but allows you to use its result to find the values at any time. Combining this with an adaptive quadrature rule should give you an answer to whatever precision you need.
Related
First, excuse me for not providing a minimal working example, it is that I just can't think of one, really. I'll just give some pieces of code and ask my question "in principle".
I'm doing thermophysical properties calculation with a real gas model (Peng-Robinson) and here I am having problems when translating a model, where I use pressure p and specific enthalpy h as inputs to calculate all other properties. When it comes to calculating the temperature T, it is linked to the enthalpy h via an equation called departure function, which is itself a function of T. In Modelica it looks like this:
Dh_real = R_m*T*(Z - 1) + (T*dadT - a)/(sqrt(8)*b)*log((Z + (1 + sqrt(2))*B)/(Z + (1 - sqrt(2))*B));
Here a, dadT and Z are also temperature-dependent scalars and partly calculated using matrix operations (dadT) or polynomial-root-calculation (Z) in functions, b and B are parameters.
Calculating the enthalpy from an input temperature (in another model) is straightforward and working fine, the solver can solve the departure function analytically. The other direction has to be solved numerically and this is, I think, why Dymola gives me this error, when translating.
Cannot find differentiation function:
DadT_Unique2([some parameters and T])
with respect to time
Failed to differentiate the equation
dadT = DadT_Unique2([some parameters and T]);
in order to reduce the DAE index.
Failed to reduce the DAE index.
Now DadT is a function within the model, where I use some simple matrix operations to calculate dadT from some parameters and the temperature T. Obviously, Dymola is in need of the derivative of some internal _Unique2-function.
I couldn't find anything in the specification nor in the web about this. Can I provide a derivative of the functions somehow? I tried the smoothOrder-annotation, but without effect. How can I deal with this?
This is not a full answer, but a list of interesting links that you should read:
Michael Tiller on annotation(derivative=dxyz) and other annotations:
http://book.xogeny.com/behavior/functions/func_annos/#derivative
Claytex on numerical Jacobians and flag Hidden.PrintFailureToDifferentiate:
http://www.claytex.com/blog/how-can-i-make-my-models-run-faster/
Two related questions here on StackOverflow:
Dymola solving stationary equation systems for Media-Model
Two-Phase Modelica Media example
Some related Modelica conference papers:
https://modelica.org/events/Conference2005/online_proceedings/Session1/Session1c2.pdf
http://dx.doi.org/10.3384/ecp15118647
http://dx.doi.org/10.3384/ecp15118653
Cubic equation of state, generalized form (table 4.2)
https://books.google.de/books?id=_Op6DQAAQBAJ&pg=PA187
Solving cubic equations of state:
http://dx.doi.org/10.1002/aic.690480421
https://books.google.com/books?id=dd410GGw8wUC&pg=PA48
https://books.google.com/books?id=1rOA5I6kQ7gC&pg=PA620 (Appendix C)
Rewriting partial derivatives:
https://scholar.google.com/scholar?cluster=3379879976574799663
Beloew Hyperlink shows Orthogonal Functions.
I used different commands in maple but i can't apply these Integral expressions in Maple.
How can i integrate such conditional Integrals ??? (For Example the Integral with red box around it)
Orthogonal Functions
(This is more of a math Question than a programming Question, so it probably should've gone to math.stackexchange.com.)
You need to use an assuming clause to tell Maple that m and n are integer, and you need to use option AllSolutions to int to tell it to do a case-by-case analysis of the parameters. For example,
int(sin(n*Pi*x/L)*sin(m*Pi*x/L), x= 0..L, AllSolutions)
assuming n::posint, m::posint, L>0;
I've assumed positivity of all parameters simply to reduce the number of cases presented in Maple's answer.
I am attempting to fit a circle to some data. This requires numerically solving a set of three non-linear simultaneous equations (see the Full Least Squares Method of this document).
To me it seems that the NEWTON function provided by IDL is fit for solving this problem. NEWTON requires the name of a function that will compute the values of the equation system for particular values of the independent variables:
FUNCTION newtfunction,X
RETURN, [Some function of X, Some other function of X]
END
While this works fine, it requires that all parameters of the equation system (in this case the set of data points) is hard coded in the newtfunction. This is fine if there is only one data set to solve for, however I have many thousands of data sets, and defining a new function for each by hand is not an option.
Is there a way around this? Is it possible to define functions programmatically in IDL, or even just pass in the data set in some other manner?
I am not an expert on this matter, but if I were to solve this problem I would do the following. Instead of solving a system of 3 non-linear equations to find the three unknowns (i.e. xc, yc and r), I would use an optimization routine to converge to a solution by starting with an initial guess. For this steepest descent, conjugate gradient, or any other multivariate optimization method can be used.
I just quickly derived the least square equation for your problem as (please check before use):
F = (sum_{i=1}^{N} (xc^2 - 2 xi xc + xi^2 + yc^2 - 2 yi yc + yi^2 - r^2)^2)
Calculating the gradient for this function is fairly easy, since it is just a summation, and therefore writing a steepest descent code would be trivial, to calculate xc, yc and r.
I hope it helps.
It's usual to use a COMMON block in these types of functions to pass in other parameters, cached values, etc. that are not part of the calling signature of the numeric routine.
I have a function f(x) = 1/(x + a+ b*I*sign(x)) and I want to calculate the
integral of
dx dy dz f(x) f(y) f(z) f(x+y+z) f(x-y - z)
over the entire R^3 (b>0 and a,- b are of order unity). This is just a representative example -- in practice I have n<7 variables and 2n-1 instances of f(), n of them involving the n integration variables and n-1 of them involving some linear combintation of the integration variables. At this stage I'm only interested in a rough estimate with relative error of 1e-3 or so.
I have tried the following libraries :
Steven Johnson's cubature code: the hcubature algorithm works but is abysmally slow, taking hundreds of millions of integrand evaluations for even n=2.
HintLib: I tried adaptive integration with a Genz-Malik rule, the cubature routines, VEGAS and MISER with the Mersenne twister RNG. For n=3 only the first seems to be somewhat viable option but it again takes hundreds of millions of integrand evaluations for n=3 and relerr = 1e-2, which is not encouraging.
For the region of integration I have tried both approaches: Integrating over [-200, 200]^n (i.e. a region so large that it essentially captures most of the integral) and the substitution x = sinh(t) which seems to be a standard trick.
I do not have much experience with numerical analysis but presumably the difficulty lies in the discontinuities from the sign() term. For n=2 and f(x)f(y)f(x-y) there are discontinuities along x=0, y=0, x=y. These create a very sharp peak around the origin (with a different sign in the various quadrants) and sort of 'ridges' at x=0,y=0,x=y along which the integrand is large in absolute value and changes sign as you cross them. So at least I know which regions are important. I was thinking that maybe I could do Monte Carlo but somehow "tell" the algorithm in advance where to focus. But I'm not quite sure how to do that.
I would be very grateful if you had any advice on how to evaluate the integral with a reasonable amount of computing power or how to make my Monte Carlo "idea" work. I've been stuck on this for a while so any input would be welcome. Thanks in advance.
One thing you can do is to use a guiding function for your Monte Carlo integration: given an integral (am writing it in 1D for simplicity) of ∫ f(x) dx, write it as ∫ f(x)/g(x) g(x) dx, and use g(x) as a distribution from which you sample x.
Since g(x) is arbitrary, construct it such that (1) it has peaks where you expect them to be in f(x), and (2) such that you can sample x from g(x) (e.g., a gaussian, or 1/(1+x^2)).
Alternatively, you can use a Metropolis-type Markov chain MC. It will find the relevant regions of the integrand (almost) by itself.
Here are a couple of trivial examples.
I want to invert a 4x4 matrix. My numbers are stored in fixed-point format (1.15.16 to be exact).
With floating-point arithmetic I usually just build the adjoint matrix and divide by the determinant (e.g. brute force the solution). That worked for me so far, but when dealing with fixed point numbers I get an unacceptable precision loss due to all of the multiplications used.
Note: In fixed point arithmetic I always throw away some of the least significant bits of immediate results.
So - What's the most numerical stable way to invert a matrix? I don't mind much about the performance, but simply going to floating-point would be to slow on my target architecture.
Meta-answer: Is it really a general 4x4 matrix? If your matrix has a special form, then there are direct formulas for inverting that would be fast and keep your operation count down.
For example, if it's a standard homogenous coordinate transform from graphics, like:
[ux vx wx tx]
[uy vy wy ty]
[uz vz wz tz]
[ 0 0 0 1]
(assuming a composition of rotation, scale, translation matrices)
then there's an easily-derivable direct formula, which is
[ux uy uz -dot(u,t)]
[vx vy vz -dot(v,t)]
[wx wy wz -dot(w,t)]
[ 0 0 0 1 ]
(ASCII matrices stolen from the linked page.)
You probably can't beat that for loss of precision in fixed point.
If your matrix comes from some domain where you know it has more structure, then there's likely to be an easy answer.
I think the answer to this depends on the exact form of the matrix. A standard decomposition method (LU, QR, Cholesky etc.) with pivoting (an essential) is fairly good on fixed point, especially for a small 4x4 matrix. See the book 'Numerical Recipes' by Press et al. for a description of these methods.
This paper gives some useful algorithms, but is behind a paywall unfortunately. They recommend a (pivoted) Cholesky decomposition with some additional features too complicated to list here.
I'd like to second the question Jason S raised: are you certain that you need to invert your matrix? This is almost never necessary. Not only that, it is often a bad idea. If you need to solve Ax = b, it is more numerically stable to solve the system directly than to multiply b by A inverse.
Even if you have to solve Ax = b over and over for many values of b, it's still not a good idea to invert A. You can factor A (say LU factorization or Cholesky factorization) and save the factors so you're not redoing that work every time, but you'd still solve the system each time using the factorization.
You might consider doubling to 1.31 before doing your normal algorithm. It'll double the number of multiplications, but you're doing a matrix invert and anything you do is going to be pretty tied to the multiplier in your processor.
For anyone interested in finding the equations for a 4x4 invert, you can use a symbolic math package to resolve them for you. The TI-89 will do it even, although it'll take several minutes.
If you give us an idea of what the matrix invert does for you, and how it fits in with the rest of your processing we might be able to suggest alternatives.
-Adam
Let me ask a different question: do you definitely need to invert the matrix (call it M), or do you need to use the matrix inverse to solve other equations? (e.g. Mx = b for known M, b) Often there are other ways to do this w/o explicitly needing to calculate the inverse. Or if the matrix M is a function of time & it changes slowly then you could calculate the full inverse once, & there are iterative ways to update it.
If the matrix represents an affine transformation (many times this is the case with 4x4 matrices so long as you don't introduce a scaling component) the inverse is simply the transpose of the upper 3x3 rotation part with the last column negated. Obviously if you require a generalized solution then looking into Gaussian elimination is probably the easiest.