I'm implementing a program for rational b-spline motion, and am looking for some references about sing Bohm's algorithm for knot addition. Essentially, this is used to split the overall curve into multiple segments to avoid having to multiply B-spline functions (motion between points). I am having some difficulty finding references on this, and appreciate any information!
You should be able to find plenty of references by googling "Boehm knot insertion". For example, the section 1.4.3 of this article describes Boehm's knot insertion algorithm.
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I am trying to get the theory behind LabVIEW's Constrained Nonlinear Optimization VI. There description provides how to use it but not which optimization algorithms works behind it.
Here is an overview of the optimization algorithms but it simply states
Solves a general nonlinear optimization problem with nonlinear equality constraint and nonlinear inequality constraint bounds using a sequential quadratic programming method.
I suspect that it is a wrapper for multiple algorithms depending on the inputs... I want to know if it uses a Levenberg-Marquardt or a Downhill-Simplex or other theory. It is not even said if it is trust-region or line search and how the bounds are ensured (e.g. by reflection)... In other languages, the documentation often refers to a paper from which I can take the original theory. This is what I am looking for. Can anyone help (or do I have to contact the NI support)? Thx
(using LabVIEW 2017 and 2018 32bit)
Recently, I have been trying to learn a bit about CPLEX and was hoping someone could help me understand the complexity when solving for integer vs. binary constraints.
For example, say we are trying to allocate a pie around 10 people for maximum utility, where each person has a utility that is linear with the amount of pie they receive. However, we want to introduce the constraint that at least 3 people have to get a bit of pie.
What's the difference between thinking of this as a single integer constraint (number_of_people_with_pie >= 3) vs. 10 binary variables (person_1_has_pie + person_2_has_pie + ... person_10_has_pie >= 3)? I would imagine the former is simplest but wonder if there is any benefits to forming the problem in terms of binary variables?
In addition to this, any recommended reading for better understanding MIP and CPLEX would be greatly appreciated, especially in better understanding where the problem becomes NP or in what situations simplex struggles to find the global minima.
Thanks!
I agree with Alex and Erwin's comment that this really depends on what you want to model. For this particular model I disagree with Alex: to me it makes more sense to use one decision variable per person, otherwise it may become hard to figure out which person gets how much of the pie.
A problem becomes NP hard as soon as you add integrality or SOS constraints. A good reading for MIP in general is Alex Schrijver's "Theory of Integer and Linear Programming". That should cover all the topics you need for an in-depth understanding of things.
It really depends on the case but in yours I would use 1 decision variable rather than 10.
Sometimes, that's not obvious and trying and measuring can prove oneself right or wrong. And that's one of the reason why using high modeling languages can help. (Abstract modeling languages such as OPL)
I recommend a MOOC on cognitive class : https://cognitiveclass.ai/courses/mathematical-optimization-for-business-problems/
and the OPL language manual : https://www.ibm.com/support/knowledgecenter/SSSA5P_12.7.0/ilog.odms.studio.help/pdf/opl_languser.pdf
Does anyone know of a library (any language, though preferably python/R/matlab) for parametric curve fitting, i.e. if you have a set of points in the plane {(x_i,y_i)} you can find parameter estimates for two (polynomial) functions y=f_y(t) and x=f_x(t) for some (arc-length?) parametrization t? This is especially useful if you have some multi-valued function (e.g. a circle) for which regression wouldn't work.
There are a number of papers detailing algorithms (e.g. 'Parametric Curve Fitting', Grossman 1971) but I can't find any corresponding software that would save a lot of time coding up.
For future reference, I ended up using the princurve library in R based on principal curves by Trevor Hastie.
CGAL's manual suggests that there is no such implementation, if you know anything more let me know
N
This sounds like two questions, rather than one:
(1) How to intersect two planar cubic Bezier curves
(2) How to offset a planar cubic Bezier curve
I don't understand the statement that "there is no implementation". Maybe they just mean that there is no implementation in CGAL.
Both problems require numerical methods or approximation -- neither one has a closed-form solution.
But they are both well-known problems with many workable (approximate) solutions available. Searching for "intersect Bezier curves" or "offset Bezier curve" will return dozens of useful references and code samples.
If you want to solve these problems using CGAL ...
(1) Intersection. This is basically a root-finding problem, and CGAL has a bivariate root finder called AlgebraicKernel_d_2::Solve_2.
(2) Offsetting. Could perhaps be done using the Minkowski sum functions. Look at approximated_offset_2, for example. The result will be polygonal, but that might be good enough for your purposes. This is really a curve approximation problem, for which CGAL has no tools, as far as I can see.
For anyone interested:
"There is a CGAL implementation of an exact arrangement of Bezier curves ( http://www.cgal.org/Manual/latest/doc_html/cgal_manual/Arrangement_on_surface_2/Chapter_main.html#Subsection_32.6.7 )" cheers Iddo Hanniel
I'm trying to implement long division for bignums. I can't use a library like GMP unfortunately due to the limitations of embedded programming. Besides, i want the intellectual exercise of learning how to implement it. So far i've got addition and multiplication done using any-length arrays of bytes (so each byte is like a base-256 digit).
I'm just trying to get started on implementing division / modulus and i want to know where to start? I've found lots of highly-optimised (aka unreadable) code on the net, which doesn't help me, and i've found lots of highly-technical mathematical whitepapers from which I can't bridge the gap between theory and implementation.
If someone could recommend a popular algorithm, and point me to a simple to understand explanation of it that leans towards implmenentation, that'd be fantastic.
-edit: I need algorithms which work when the dividend is ~4000bits, and divisor is ~2000bits
-edit: Will this algorithm work with base-256 ? http://courses.cs.vt.edu/~cs1104/BuildingBlocks/divide.030.html
-edit: Is this the algorithm (newton division) i should really be using? http://en.wikipedia.org/wiki/Division_(digital)#Newton.E2.80.93Raphson_division
If you want to learn, then start with the pencil and paper method you used in elementary school. Believe it or not, that is essentially the same O(n^2) algorithm that is used in most bignum libraries for numbers that are in the range you are looking for. The tricky first step is called "quotient estimation", and that will probably be the hardest to understand. Once you understand that, the rest should come easy.
A good reference is Knuth's "Seminumerical Algorithms". He has many discussions about different ways to do quotient estimation both in the text and in the exercises. That book has chapters devoted to bignum implementations.
Are you using the void Four1(long double[],int,int) in your code and then convolving and then doing a inverse transform well I got multiplication to work but when I tried to do division the same way it spat out one result then quit so I cannot help but if you have the tome called "Numeric Recipes in C++" go to near the end and you will find what you are looking for actually it starts on Page 916 to 926.
This question is over 2 years old, but for this size numbers you can look at the OpenSSL source code. It does RSA with this size numbers so has lots of math routines optimized for 1000 to 4000 bit numbers.