I do not know what I'm supposed to do here I am a beginner
let interest = 0.02 // Interest rate is 2%
let periods = 12.0 // Number of times the interest is compounded per year
let years = 10.0 // Number of years the money is borrowed for
// Calculate compound interest using the formula from:
// http://en.wikipedia.org/wiki/Compound_interest#Compound_Interest
let compound investment =
***I am supposed to put something here***
let res = compound 1000.0
shouldEqual (round res) 1221.0
printfn "Compound interest (full): %f" res
printfn "Compound interest (two digits): %.2f" res
If i you right understand:
let interest = 0.02 // Interest rate is 2%
let periods = 12.0 // Number of times the interest is compounded per year
let years = 10.0 // Number of years the money is borrowed for
let investment = 1000.0 //principal amount (initial investment)
// Calculate compound interest using the formula from:
// http://en.wikipedia.org/wiki/Compound_interest#Compound_Interest
let compound P j n t =
P*((1.0+j/n)**(n*t))
let res = compound investment interest periods years
//shouldEqual (round res) 1221.0 ? What this?
printfn "Compound interest (full): %f" res
printfn "Compound interest (two digits): %.2f" res
But I'm surprised that I saw this question here
Related
I'm trying to make a cross product between two vectors R^n , is there any way to do this in the most optimized way?
I have looked on the accelerate library but still can not find anything
A cross product can exist in Rn if and only if n=0, 1, 3 or 7
Source: http://www.math.csusb.edu/faculty/pmclough/CP.pdf
So no, you certainly won't find any library that does that. If you meant the element-wise, you can use Accelerate. Here is a short test:
import Accelerate
let n = 10_000_000
let a = (0..<n).map{ _ in Double(arc4random()) / Double(UInt32.max) }
let b = (0..<n).map{ _ in Double(arc4random()) / Double(UInt32.max) }
print("A: [\(a.prefix(10).map{ "\($0)" }.joinWithSeparator(", ")), ...]")
print("B: [\(b.prefix(10).map{ "\($0)" }.joinWithSeparator(", ")), ...]")
var result = [Double](count: n, repeatedValue: 0)
let start = mach_absolute_time()
vDSP_vmulD(a, 1, b, 1, &result, 1, UInt(n))
let stop = mach_absolute_time()
let time = Double(stop - start) / Double(NSEC_PER_SEC)
print("Time: \(time) for \(n) elements")
print("Result: [\(result.prefix(10).map{ "\($0)" }.joinWithSeparator(", ")), ...]")
Output:
A: [0.269752697849123, 0.851672558312228, 0.0668649589798564, 0.0955562389212559, 0.255900985620893, 0.93693982901446, 0.085282990495973, 0.732230591525377, 0.588338787804437, 0.952581417968632, ...]
B: [0.750105029379508, 0.0454008649209051, 0.863010750120275, 0.308104009904923, 0.700024090637459, 0.327355608653127, 0.679469040520366, 0.666848364208557, 0.0567599588671606, 0.623293806245386, ...]
Time: 0.024393279 for 10000000 elements
Result: [0.202342855345318, 0.0386666707767751, 0.0577051784059674, 0.0294412603830718, 0.179136854752495, 0.306712507998386, 0.0579471517250063, 0.488286772182162, 0.0333940853957349, 0.593738097764296, ...]
0.024 seconds for 10 million elements is pretty fast
Parallel execution
If you are looking for the Elementwise product operation
a = (1.0, 2.0)
b = (3.0, 4.0)
a * b = (a1*b1, a2*b2) = (3.0, 8.0)
and you want the fasted possible performance available on iOS you should use the simd framework (Single Instruction Multiple Data).
import simd
let v0 = float2(1.0, 2.0)
let v1 = float2(3.0, 4.0)
let res = v0 * v1
print(res) // float2(3.0, 8.0)
Why is simd so fast?
Without simd, calculating a*b would required the execution of 2 steps
calculate a1 * b1 and put the result into res1
calculate a2 * b2 and put the result into res2
On the other hand, using simd both operations are done in parallel. This is possible because the 2 steps have the same operation with different data. This is exactly what simd does allow you do to.
More
From Wikipedia
Single instruction, multiple data (SIMD), is a class of parallel computers in Flynn's taxonomy. It describes computers with multiple processing elements that perform the same operation on multiple data points simultaneously.
Thus, such machines exploit data level parallelism, but not concurrency: there are simultaneous (parallel) computations, but only a single process (instruction) at a given moment.
SIMD is particularly applicable to common tasks like adjusting the contrast in a digital image or adjusting the volume of digital audio.
Most modern CPU designs include SIMD instructions in order to improve the performance of multimedia use.
I've been toying around with F# lately and wrote this little snippet below, it just creates a number of randomized 3d-vectors, puts them into a list, maps each vector to its length and sums up all those values.
Running the program (as a Release Build .exe, not interactive), the binary consumes in this particular case (10 mio vectors) roughly 550 MB RAM. One Vec3 object should account for 12 bytes (or 16 assuming some alignment takes place). Even if you do the rough math with 32 bytes to account for some book-keeping overhead (bytes per object*10 mio) / 1024 / 1024) you're still 200 MB off the actual consumption. Naively i'd assume to have 10 mio * 4 bytes per single in the end, since the Vec3 objects are 'mapped away'.
My guess so far: either i keep one (or several) copy/copies of my list somewhere and i'm not aware of that, or some intermediate results get never garbage collected? I can't imagine that inheriting from System.Object brings in so much overhead.
Could someone point me into the right direction with this?
TiA
type Vec3(x: single, y: single, z:single) =
let mag = sqrt(x*x + y*y + z*z)
member self.Magnitude = mag
override self.ToString() = sprintf "[%f %f %f]" x y z
let how_much = 10000000
let mutable rng = System.Random()
let sw = new System.Diagnostics.Stopwatch()
sw.Start()
let random_vec_iter len =
let mutable result = []
for x = 1 to len do
let mutable accum = []
for i = 1 to 3 do
accum <- single(rng.NextDouble())::accum
result <- Vec3(accum.[0], accum.[1], accum.[2])::result
result
sum_len_func = List.reduce (fun x y -> x+y)
let map_to_mag_func = List.map (fun (x:Vec3) -> x.Magnitude)
[<EntryPoint>]
let main argv =
printfn "Hello, World"
let res = sum_len_func (map_to_mag_func (random_vec_iter(how_much)))
printfn "doing stuff with %i items took %i, result is %f" how_much (sw.ElapsedMilliseconds) res
System.Console.ReadKey() |> ignore
0 // return an integer exit code
First, your vec is a ref type not a value type (not a struct). So you hold a pointer on top of your 12 bytes (12+16). Then the list is a single-linked list, so another 16 bytes for a .net ref. Then, your List.map will create an intermediate list.
Consider the following code:
let dl = 9.5 / 11.
let min = 21.5 + dl
let max = 40.5 - dl
let a = [ for z in min .. dl .. max -> z ] // should have 21 elements
let b = a.Length
"a" should have 21 elements but has got only 20 elements. The "max - dl" value is missing. I understand that float numbers are not precise, but I hoped that F# could work with that. If not then why F# supports List comprehensions with float iterator? To me, it is a source of bugs.
Online trial: http://tryfs.net/snippets/snippet-3H
Converting to decimals and looking at the numbers, it seems the 21st item would 'overshoot' max:
let dl = 9.5m / 11.m
let min = 21.5m + dl
let max = 40.5m - dl
let a = [ for z in min .. dl .. max -> z ] // should have 21 elements
let b = a.Length
let lastelement = List.nth a 19
let onemore = lastelement + dl
let overshoot = onemore - max
That is probably due to lack of precision in let dl = 9.5m / 11.m?
To get rid of this compounding error, you'll have to use another number system, i.e. Rational. F# Powerpack comes with a BigRational class that can be used like so:
let dl = 95N / 110N
let min = 215N / 10N + dl
let max = 405N / 10N - dl
let a = [ for z in min .. dl .. max -> z ] // Has 21 elements
let b = a.Length
Properly handling float precision issues can be tricky. You should not rely on float equality (that's what list comprehension implicitely does for the last element). List comprehensions on float are useful when you generate an infinite stream. In other cases, you should pay attention to the last comparison.
If you want a fixed number of elements, and include both lower and upper endpoints, I suggest you write this kind of function:
let range from to_ count =
assert (count > 1)
let count = count - 1
[ for i = 0 to count do yield from + float i * (to_ - from) / float count]
range 21.5 40.5 21
When I know the last element should be included, I sometimes do:
let a = [ for z in min .. dl .. max + dl*0.5 -> z ]
I suspect the problem is with the precision of floating point values. F# adds dl to the current value each time and checks if current <= max. Because of precision problems, it might jump over max and then check if max+ε <= max (which will yield false). And so the result will have only 20 items, and not 21.
After running your code, if you do:
> compare a.[19] max;;
val it : int = -1
It means max is greater than a.[19]
If we do calculations the same way the range operator does but grouping in two different ways and then compare them:
> compare (21.5+dl+dl+dl+dl+dl+dl+dl+dl) ((21.5+dl)+(dl+dl+dl+dl+dl+dl+dl));;
val it : int = 0
> compare (21.5+dl+dl+dl+dl+dl+dl+dl+dl+dl) ((21.5+dl)+(dl+dl+dl+dl+dl+dl+dl+dl));;
val it : int = -1
In this sample you can see how adding 7 times the same value in different order results in exactly the same value but if we try it 8 times the result changes depending on the grouping.
You're doing it 20 times.
So if you use the range operator with floats you should be aware of the precision problem.
But the same applies to any other calculation with floats.
Hello everyone
I have converted a project in C# to F# that paints the Mandelbrot set.
Unfortunately does it take around one minute to render a full screen so I am try to find some ways to speed it up.
It is one call that take almost all of the time:
Array.map (fun x -> this.colorArray.[CalcZ x]) xyArray
xyArray (double * double) [] => (array of tuple of double)
colorArray is an array of int32 length = 255
CalcZ is defined as:
let CalcZ (coord:double * double) =
let maxIterations = 255
let rec CalcZHelper (xCoord:double) (yCoord:double) // line break inserted
(x:double) (y:double) iters =
let newx = x * x + xCoord - y * y
let newy = 2.0 * x * y + yCoord
match newx, newy, iters with
| _ when Math.Abs newx > 2.0 -> iters
| _ when Math.Abs newy > 2.0 -> iters
| _ when iters = maxIterations -> iters
| _ -> CalcZHelper xCoord yCoord newx newy (iters + 1)
CalcZHelper (fst coord) (snd coord) (fst coord) (snd coord) 0
As I only use around half of the processor capacity is an idea to use more threads and specifically Array.Parallel.map, translates to system.threading.tasks.parallel
Now my question
A naive solution, would be:
Array.Parallel.map (fun x -> this.colorArray.[CalcZ x]) xyArray
but that took twice the time, how can I rewrite this to take less time, or can I take some other way to utilize the processor better?
Thanks in advance
Gorgen
---edit---
the function that is calling CalcZ looks like this:
let GetMatrix =
let halfX = double bitmap.PixelWidth * scale / 2.0
let halfY = double bitmap.PixelHeight * scale / 2.0
let rect:Mandelbrot.Rectangle =
{xMax = centerX + halfX; xMin = centerX - halfX;
yMax = centerY + halfY; yMin = centerY - halfY;}
let size:Mandelbrot.Size =
{x = bitmap.PixelWidth; y = bitmap.PixelHeight}
let xyList = GenerateXYTuple rect size
let xyArray = Array.ofList xyList
Array.map (fun x -> this.colorArray.[CalcZ x]) xyArray
let region:Int32Rect = new Int32Rect(0,0,bitmap.PixelWidth,bitmap.PixelHeight)
bitmap.WritePixels(region, GetMatrix, bitmap.PixelWidth * 4, region.X, region.Y);
GenerateXYTuple:
let GenerateXYTuple (rect:Rectangle) (pixels:Size) =
let xStep = (rect.xMax - rect.xMin)/double pixels.x
let yStep = (rect.yMax - rect.yMin)/double pixels.y
[for column in 0..pixels.y - 1 do
for row in 0..pixels.x - 1 do
yield (rect.xMin + xStep * double row,
rect.yMax - yStep * double column)]
---edit---
Following a suggestion from kvb (thanks a lot!) in a comment to my question, I built the program in Release mode. Building in the Relase mode generally speeded up things.
Just building in Release took me from 50s to around 30s, moving in all transforms on the array so it all happens in one pass made it around 10 seconds faster. At last using the Array.Parallel.init brought me to just over 11 seconds.
What I learnt from this is.... Use the release mode when timing things and using parallel constructs...
One more time, thanks for the help I have recieved.
--edit--
by using SSE assember from a native dll I have been able to slash the time from around 12 seconds to 1.2 seconds for a full screen of the most computational intensive points. Unfortunately I don't have a graphics processor...
Gorgen
Per the comment on the original post, here is the code I wrote to test the function. The fast version only takes a few seconds on my average workstation. It is fully sequential, and has no parallel code.
It's moderately long, so I posted it on another site: http://pastebin.com/Rjj8EzCA
I'm suspecting that the slowdown you are seeing is in the rendering code.
I don't think that the Array.Parallel.map function (which uses Parallel.For from .NET 4.0 under the cover) should have trouble parallelizing the operation if it runs a simple function ~1 million times. However, I encountered some weird performance behavior in a similar case when F# didn't optimize the call to the lambda function (in some way).
I'd try taking a copy of the Parallel.map function from the F# sources and adding inline. Try adding the following map function to your code and use it instead of the one from F# libraries:
let inline map (f: 'T -> 'U) (array : 'T[]) : 'U[]=
let inputLength = array.Length
let result = Array.zeroCreate inputLength
Parallel.For(0, inputLength, fun i ->
result.[i] <- f array.[i]) |> ignore
result
As an aside, it looks like you're generating an array of coordinates and then mapping it to an array of results. You don't need to create the coordinate array if you use the init function instead of map: Array.Parallel.init 1000 (fun y -> Array.init 1000 (fun x -> this.colorArray.[CalcZ (x, y)]))
EDIT: The following may be inaccurate:
Your problem could be that you call a tiny function a million times, causing the scheduling overhead to overwhelm that actual work you're doing. You should partition the array into much larger chunks so that each individual task takes a millisecond or so. You can use an array of arrays so that you would call Array.Parallel.map on the outer arrays and Array.map on the inner arrays. That way each parallel operation will operate on a whole row of pixels instead of just a single pixel.
I'm trying to experiment with software defined radio concepts. From this article I've tried to implement a GPU-parallelism Discrete Fourier Transform.
I'm pretty sure I could pre-calculate 90 degrees of the sin(i) cos(i) and then just flip and repeat rather than what I'm doing in this code and that that would speed it up. But so far, I don't even think I'm getting correct answers. An all-zeros input gives a 0 result as I'd expect, but all 0.5 as inputs gives 78.9985886f (I'd expect a 0 result in this case too). Basically, I'm just generally confused. I don't have any good input data and I don't know what to do with the result or how to verify it.
This question is related to my other post here
open Microsoft.ParallelArrays
open System
// X64MulticoreTarget is faster on my machine, unexpectedly
let target = new DX9Target() // new X64MulticoreTarget()
ignore(target.ToArray1D(new FloatParallelArray([| 0.0f |]))) // Dummy operation to warm up the GPU
let stopwatch = new System.Diagnostics.Stopwatch() // For benchmarking
let Hz = 50.0f
let fStep = (2.0f * float32(Math.PI)) / Hz
let shift = 0.0f // offset, once we have to adjust for the last batch of samples of a stream
// If I knew that the periodic function is periodic
// at whole-number intervals, I think I could keep
// shift within a smaller range to support streams
// without overflowing shift - but I haven't
// figured that out
//let elements = 8192 // maximum for a 1D array - makes sense as 2^13
//let elements = 7240 // maximum on my machine for a 2D array, but why?
let elements = 7240
// need good data!!
let buffer : float32[,] = Array2D.init<float32> elements elements (fun i j -> 0.5f) //(float32(i * elements) + float32(j)))
let input = new FloatParallelArray(buffer)
let seqN : float32[,] = Array2D.init<float32> elements elements (fun i j -> (float32(i * elements) + float32(j)))
let steps = new FloatParallelArray(seqN)
let shiftedSteps = ParallelArrays.Add(shift, steps)
let increments = ParallelArrays.Multiply(fStep, steps)
let cos_i = ParallelArrays.Cos(increments) // Real component series
let sin_i = ParallelArrays.Sin(increments) // Imaginary component series
stopwatch.Start()
// From the documentation, I think ParallelArrays.Multiply does standard element by
// element multiplication, not matrix multiplication
// Then we sum each element for each complex component (I don't understand the relationship
// of this, or the importance of the generalization to complex numbers)
let real = target.ToArray1D(ParallelArrays.Sum(ParallelArrays.Multiply(input, cos_i))).[0]
let imag = target.ToArray1D(ParallelArrays.Sum(ParallelArrays.Multiply(input, sin_i))).[0]
printf "%A in " ((real * real) + (imag * imag)) // sum the squares for the presence of the frequency
stopwatch.Stop()
printfn "%A" stopwatch.ElapsedMilliseconds
ignore (System.Console.ReadKey())
I share your surprise that your answer is not closer to zero. I'd suggest writing naive code to perform your DFT in F# and seeing if you can track down the source of the discrepancy.
Here's what I think you're trying to do:
let N = 7240
let F = 1.0f/50.0f
let pi = single System.Math.PI
let signal = [| for i in 1 .. N*N -> 0.5f |]
let real =
seq { for i in 0 .. N*N-1 -> signal.[i] * (cos (2.0f * pi * F * (single i))) }
|> Seq.sum
let img =
seq { for i in 0 .. N*N-1 -> signal.[i] * (sin (2.0f * pi * F * (single i))) }
|> Seq.sum
let power = real*real + img*img
Hopefully you can use this naive code to get a better intuition for how the accelerator code ought to behave, which could guide you in your testing of the accelerator code. Keep in mind that part of the reason for the discrepancy may simply be the precision of the calculations - there are ~52 million elements in your arrays, so accumulating a total error of 79 may not actually be too bad. FWIW, I get a power of ~0.05 when running the above single precision code, but a power of ~4e-18 when using equivalent code with double precision numbers.
Two suggestions:
ensure you're not somehow confusing degrees with radians
try doing it sans-parallelism, or just with F#'s asyncs for parallelism
(In F#, if you have an array of floats
let a : float[] = ...
then you can 'add a step to all of them in parallel' to produce a new array with
let aShift = a |> (fun x -> async { return x + shift })
|> Async.Parallel |> Async.RunSynchronously
(though I expect this might be slower that just doing a synchronous loop).)