| mankind_tweezer | Oh yeah, I know how it goes. The multiplication is not component-wise, I think. |
| _llll_ | or something very much like it |
| mankind_tweezer | It should be (x0,x1,x2,...) . (y0,y1,y2,...) = (x0y0, x1y0+x0y1, x2y0+x1y1+x0y2, ...) We have to check that's well defined under the condition we wrote. But I have a good feeling about it. |
| |Steve| | Okay, now how did you ever come up with that? And in what sense is that natural? I think he means categorically. |
| thermoplyae | It's natural multiplication for the ring of polynomials |
| action | thermoplyae interjects where he can |
| thermoplyae | interjects where he can |
| mankind_tweezer | It's natural! Imagine your ring is the ring of integers, and you are filtering it by powers of 10 |
| |Steve| | That's a good point. |
| mankind_tweezer | Or, indeed, the ring of polynomials and you're filtering it by powers of x |
| |Steve| | Okay, so an element of R/F_1 looks like r+F_1 and an element of F_1/F_2 looks like f+F_2. How does one multiply those? rf+F_1, maybe? (I probably shouldn't have used f, but I just mean an element of the group.) No, that doesn't seem right. It should be rf+F_2. |
| LisaLaptop | I like this math problem- it starts with "Textbook authors and publishers work very hard to minimize the number of errors in a text..." and about 3 problems before that, there are 2 errors in the problem. |
| |Steve| | LisaLaptop: Fitting. So was the condition F_k F_j \subset F_{k+j} correct? Is I^k I^J \subset I^{k+j}? |
| TRWBW | LisaLaptop: you don't know how deep it goes. but if it makes you feel better, the really top-notch math books when you think you've found a mistake it means you don't get something. |
| LisaLaptop | TRWBW, so that either means this is a crappy math book or that the teacher didn't get something |
| cn28h | Anyone know if there is another name for partial pivoting? (LU factorization) I'm having a hard time figuring out what I'm doing wrong and I can't really find much useful info on google looking for partial pivoting |
| mankind_tweezer | Sure, I^k I^j = I^{k+j} even. |
| |Steve| | Was I right about I^k = { i^k | i \in I }? |
| TRWBW | cn28h: pivoting adds a P matrix into the mix. or two, for full pivoting. where P is a permutation |
| mankind_tweezer | I think so, seems to me that that should be the definition of I^k. |
| cn28h | TRWBW, yeah, I've got the idea but I think I am confused somewhere in the process of how to do this. Now I've got the LU factorization itself down just fine (using Doolittle). But for some reason, and I've checked my arithmetic like 3 times, I keep getting a messed up answer |
| |Steve| | mankind_tweezer: It's not obvious to me why that would be an ideal in its own right. |
| mankind_tweezer | I think the condition was probably the right one. The task is to write down the formal expression I wrote above, and check it is well defined, i.e. independent of the choice of representative. |
| cn28h | TRWBW, to do this I should just permute the row with the greteast element in each column before I do the next LU step, right?( that is, of the rows that is rows below the next pivot) |
| mankind_tweezer | For commutative rings it's OK at least, right? |
| |Steve| | I'm not totally sure. |
| TRWBW | cn28h: switching around the columns is writing M'=M*P where P is a permutation matrix |
| |Steve| | Take something like R = Z[x], and I = (x). |
| TRWBW | cn28h: switching around the rows is writing M'=P*M where P is a permutation matrix |
| |Steve| | Is I^2 the same as (x^2)? |
| cn28h | TRWBW, hm, well I'm permuting the rows using PA = LU |
| TRWBW | cn28h: sounds like you got the basic idea, not sure what i can tell you |