# Spline

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1. ## [Solved] 2D linear regression problem

I recommend lapack, or its C interface, lapacke. The function you are looking for is "LAPACK_dgels", http://www.netlib.org/lapack/lapacke.html
2. ## Upper and lower bound for NURBS length

Uh, you need to help me out here... how are these guys going to help me? Save asking them, I mean...? Perhaps I should re-state my intentions. What I currently do in order to determine the length of a nurbs curve is to split it into the non-empty intervals defined by the knot vectors and perform a piece-wise numerical integration (my special thanks go to Gauss & Legendre...). Now I split the intervals in half and perform another iteration. The difference between the larger and the sum of the two corresponding smaller intervals are my error estimates. Until the sum of error estimates is smaller than some ?, I recursively split the interval with the largest error estimate. For all curves I have tested so far this works well and all, however, I lack a (mathematically sound!) argument as to the "real" error bound. Or another approach that has such a guarantee - although my current method seems to give good results, an error bound would be rather large due to my derivate estimation alone. And while De Casteljau has defined a numerically sound algorithm to evaluate Bezier curves, and the NURBS basis can, for some degenerate cases, degenerate to Bernstein polynomials, I fail to see the connection to my problem and would appreciate any more concrete hints. Kind regards Spline
3. ## LALR/LR Parser Generation

So, you read some grammar G1, say, in BNF, defining L(G1), and your program is now accepting L(G1). Then, on the fly, you read G2, and, by some magic trickery as it were, you now accept words from L(G2). Well, nice. But (setting aside the fact that I can think of neither practical application nor theoretical value) how would that do something useful? You need to take, when you encounter a statement, a form of semantic action. Say after playing with your example of addition and substraction, you have the idea to allow assignments, i.e., you have an assignment-expression: additive-expression | postfix-expression ASSIGNMENT_OP assignment_expression ...now you would recognize "a = b = 2+3*sqrt(5);" as an assignment-expression. But what would you *do*? How do you intend to take a semantic action of something you don't have any semantics associated with? There is actually a reason that we have a C-compiler and an Ada-compiler and so on and not one meta-compiler that needs to be fed with a grammar, and then can compile any given language. Setting my reservations regarding the usefulness and feasibility aside, my answer to your initial question still stands: you need, for the grammar you specified, one token look-ahead. Oh, and by the way: yacc does, by and large, not operate on characters, but on tokens. So unicode should not be a problem; as for (f)lex, google found me this: http://stackoverflow.com/questions/9611682/flexlexer-support-for-unicode And I highly recomend it, even if you stick with your original plan, to parse the grammar itself and create the parser for that grammar.
4. ## LALR/LR Parser Generation

Well, this is the kind of situation where you need one token look-ahead. It is, in most cases, sufficient. Whether it is or not depends, as you already noticed, on the language you want to parse (and the way you formalize it). C, for example, can be nicely parsed with an LALR(1)-parser (that is, with one exception: the "dangling else-problem", which is a shift-reduce conflict, but that one can be resolved by preferring a shift). C++ is something different entirely. Parsing it with a LALR(1)-parser has been attempted, but the result is... not very nice. Another type of parser, GLR, would be a better choice. So it really depends on what you want to do. If your language is not more complex than C - and given your question, it probably isn't - LALR(1) might be the right choice for you. What are you using to generate the parser? If you feed your grammar to yacc (or its GNU-implementation, bison), it will work nicely.
5. ## Upper and lower bound for NURBS length

Hi all, although NURBS seem to be used widely, I can find only little information on length approximation of NURBS curves on the web. It can very well be that I simply didn't feed google with the right search terms, in which case I appologize for the noise and kindly ask for a hint as to what these terms might be. To my actual problem... given a NURBS curve C of degree d, defined on the real (well, in terms of implementation, rational) interval [0;1]. For 0<=a,b<=1 and ?>0 find l, u: l <= length of C in the interval [a;b] <= u and u-l <= 2?. Differential geometry says that I have to compute the integral of the speed vector at each point. Well, nice try, but NURBS cannot be differentiated analytically. So a numerical solution is required. However using the "default" algorithms like Gaussian quadrature and difference quotiont (likely to require an exact rational arithmetic to work at all) seems to be a bit wasteful. There must be better algorithms out there, or am I missing something?