Sorry, I didn't realize that course was offered next year. I read through
"Matrices and Linear Algebra" when I was in high school. And used
Friedberg, Insel, Spence's "Linear Algebra" in college.
On Tue, Dec 4, 2012 at 4:37 PM, Alexander Solla <alex.so...@gmail.com>wrote:
> Well, an m x n matrix corresponds to a linear transformation in at most
> min{m,n} dimensions. In particular, this means that a 2x2 matrix
> corresponds to a plane, line, or the origin of 3-space, as a linear
> subspace. Which of those the matrix corresponds to depends on the matrix's
> "rank", which is the number of linearly independent columns (or rows) in
> the matrix.
>
> Do you really need to know /which/ plane or line a matrix corresponds to?
> If so, reduce it using Gaussian elimination and, if appropriate, compute
> its eigenvectors or span. Otherwise, think of it as a generic
> plane/line/0-point.
>
> Outer products represent more of these simple facts about linear algebra.
> The product of an mx1 and 1xn matrices is an mxn matrix with rank at most
> 1. Trouble visualizing this means you are missing the essential facts (for
> the general picture as the product as a line or origin), or requires some
> computational details -- reducing the matrix using Gaussian elimination and
> determining its span.
>
> As I said, I don't mean to be harsh, but playing with a vector algebra
> package without understanding vectors is like playing with a calculator
> without understanding multiplication. You're better off learning what
> multiplication represents first, before using a machine to do it fast. So,
> I can humbly recommend taking a course on the subject. For example,
> https://www.coursera.org/course/matrix
>
>
> On Tue, Dec 4, 2012 at 4:13 PM, Clark Gaebel <cgae...@uwaterloo.ca> wrote:
>
>> No. But that doesn't stop me from being curious with Accelerate. Might
>> you have a better explaination for what's happening here than Trevor's?
>>
>> - Clark
>>
>>
>> On Tue, Dec 4, 2012 at 7:08 PM, Alexander Solla <alex.so...@gmail.com>wrote:
>>
>>> I don't mean to be blunt, but have you guys taken a course in linear
>>> algebra?
>>>
>>>
>>> On Mon, Dec 3, 2012 at 9:21 PM, Trevor L. McDonell <
>>> tmcdon...@cse.unsw.edu.au> wrote:
>>>
>>>> As far as I am aware, the only description is in the Repa paper. I you
>>>> are right, it really should be explained properly somewhere…
>>>>
>>>> At a simpler example, here is the outer product of two vectors [1].
>>>>
>>>> vvProd :: (IsNum e, Elt e) => Acc (Vector e) -> Acc (Vector e) -> Acc
>>>> (Matrix e)
>>>> vvProd xs ys = A.zipWith (*) xsRepl ysRepl
>>>> where
>>>> n = A.size xs
>>>> m = A.size ys
>>>>
>>>> xsRepl = A.replicate (lift (Z :. All :. m )) xs
>>>> ysRepl = A.replicate (lift (Z :. n :. All)) ys
>>>>
>>>> If we then `A.fold (+) 0` the matrix, it would reduce along each row
>>>> producing a vector. So the first element of that vector is going to be
>>>> calculated as (xs[0] * ys[0] + xs[0] * ys[1] + … xs[0] * ys[m-1]). That's
>>>> the idea we want for our matrix multiplication … but I agree, it is
>>>> difficult for me to visualise as well.
>>>>
>>>> I do the same sort of trick with the n-body demo to get all n^2
>>>> particle interactions.
>>>>
>>>> -Trev
>>>>
>>>>
>>>> [1]: http://en.wikipedia.org/wiki/Outer_product#Vector_multiplication
>>>>
>>>>
>>>>
>>>> On 04/12/2012, at 3:41 AM, Clark Gaebel <cgae...@uwaterloo.ca> wrote:
>>>>
>>>> Ah. I see now. Silly Haskell making inefficient algorithms hard to
>>>> write and efficient ones easy. It's actually kind of annoying when
>>>> learning, but probably for the best.
>>>>
>>>> Is there a good write-up of the algorithm you're using somewhere? The
>>>> Repa paper was very brief in its explaination, and I'm having trouble
>>>> visualizing the mapping of the 2D matricies into 3 dimensions.
>>>>
>>>> - Clark
>>>>
>>>>
>>>> On Mon, Dec 3, 2012 at 2:06 AM, Trevor L. McDonell <
>>>> tmcdon...@cse.unsw.edu.au> wrote:
>>>>
>>>>> Hi Clark,
>>>>>
>>>>> The trick is that most accelerate operations work over
>>>>> multidimensional arrays, so you can still get around the fact that we are
>>>>> limited to flat data-parallelism only.
>>>>>
>>>>> Here is matrix multiplication in Accelerate, lifted from the first
>>>>> Repa paper [1].
>>>>>
>>>>>
>>>>> import Data.Array.Accelerate as A
>>>>>
>>>>> type Matrix a = Array DIM2 a
>>>>>
>>>>> matMul :: (IsNum e, Elt e) => Acc (Matrix e) -> Acc (Matrix e) -> Acc
>>>>> (Matrix e)
>>>>> matMul arr brr
>>>>> = A.fold (+) 0
>>>>> $ A.zipWith (*) arrRepl brrRepl
>>>>> where
>>>>> Z :. rowsA :. _ = unlift (shape arr) :: Z :. Exp Int :. Exp
>>>>> Int
>>>>> Z :. _ :. colsB = unlift (shape brr) :: Z :. Exp Int :. Exp
>>>>> Int
>>>>>
>>>>> arrRepl = A.replicate (lift $ Z :. All :. colsB :.
>>>>> All) arr
>>>>> brrRepl = A.replicate (lift $ Z :. rowsA :. All :.
>>>>> All) (A.transpose brr)
>>>>>
>>>>>
>>>>> If you use github sources rather than the hackage package, those
>>>>> intermediate replicates will get fused away.
>>>>>
>>>>>
>>>>> Cheers,
>>>>> -Trev
>>>>>
>>>>> [1] http://www.cse.unsw.edu.au/~chak/papers/KCLPL10.html
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> On 03/12/2012, at 5:07 PM, Clark Gaebel <cgae...@uwaterloo.ca> wrote:
>>>>>
>>>>> Hello cafe,
>>>>>
>>>>> I've recently started learning about cuda and hetrogenous programming,
>>>>> and have been using accelerate [1] to help me out. Right now, I'm running
>>>>> into trouble in that I can't call parallel code from sequential code.
>>>>> Turns
>>>>> out GPUs aren't exactly like Repa =P.
>>>>>
>>>>> Here's what I have so far:
>>>>>
>>>>> import qualified Data.Array.Accelerate as A
>>>>> import Data.Array.Accelerate ( (:.)(..)
>>>>> , Acc
>>>>> , Vector
>>>>> , Scalar
>>>>> , Elt
>>>>> , fold
>>>>> , slice
>>>>> , constant
>>>>> , Array
>>>>> , Z(..), DIM1, DIM2
>>>>> , fromList
>>>>> , All(..)
>>>>> , generate
>>>>> , lift, unlift
>>>>> , shape
>>>>> )
>>>>> import Data.Array.Accelerate.Interpreter ( run )
>>>>>
>>>>> dotP :: (Num a, Elt a) => Acc (Vector a) -> Acc (Vector a) -> Acc
>>>>> (Scalar a)
>>>>> dotP xs ys = fold (+) 0 $ A.zipWith (*) xs ys
>>>>>
>>>>> type Matrix a = Array DIM2 a
>>>>>
>>>>> getRow :: Elt a => Int -> Acc (Matrix a) -> Acc (Vector a)
>>>>> getRow n mat = slice mat . constant $ Z :. n :. All
>>>>>
>>>>> -- Naive matrix multiplication:
>>>>> --
>>>>> -- index (i, j) is equal to the ith row of 'a' `dot` the jth row of 'b'
>>>>> matMul :: A.Acc (Matrix Double) -> A.Acc (Matrix Double) -> A.Acc
>>>>> (Matrix Double)
>>>>> matMul a b' = A.generate (constant $ Z :. nrows :. ncols) $
>>>>> \ix ->
>>>>> let (Z :. i :. j) = unlift ix
>>>>> in getRow i a `dotP` getRow j b
>>>>> where
>>>>> b = A.transpose b' -- I assume row indexing is faster than
>>>>> column indexing...
>>>>> (Z :. nrows :. _ ) = unlift $ shape a
>>>>> (Z :. _ :. ncols) = unlift $ shape b
>>>>>
>>>>>
>>>>> This, of course, gives me errors right now because I'm calling getRow
>>>>> and dotP from within the generation function, which expects Exp[ression]s,
>>>>> not Acc[elerated computation]s.
>>>>>
>>>>> So maybe I need to replace that line with an inner for loop? Is there
>>>>> an easy way to do that with Accelerate?
>>>>>
>>>>> Thanks for your help,
>>>>> - Clark
>>>>>
>>>>> [1] http://hackage.haskell.org/package/accelerate
>>>>> _______________________________________________
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>>>>>
>>>>>
>>>>>
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>>>>>
>>>>
>>>>
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>
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