I'm using the law of cosines in a program and it seems to be a slow point of my code. This is the line of code I have:
Ans = Math.Sqrt(A ^ 2 + B ^ 2 - 2 * A * B * Math.Cos(C - D))
Where A to D are double variables that change every time called. This function seems to take around 2000 ticks to run. I've looked at using the small angle approximation which is where if (C-D) is small enough you can use cos(C-D) = 1 - ((C-D)^2)/2. Unfortunately this turned out to be slower overall than the original code. I've looked at any sort of relationship that can be used to simplify the calculation but A and C are related in a complex way and B and D are related in the same way, there's no relationship between A and B or between C and D.
I've thought about using a lookup function for all the values of (C-D) but my accuracy is currently to at least 6 significant figures and I would prefer to stay at that level as that's the accuracy of my input data, in short this means around a million vaules in the look up and this is only 1 section of the function. I've thought about having a look up for all four values (A, B, C and D) but I'm not sure how to implement that.
I've also already multithreaded this application and attempted use of GPGPU (GPGPU ended up being slower due to the time spent loading in and out of GPU memory).
So, my question is how do I speed up this function.
Thanks in advanced!
The following runs in less than 1/3 the time
ans = Math.Sqrt(a * a + b * b - 2 * a * b * Math.Cos(c - d))
Here's the code that proves it:
Dim sw1 As New Stopwatch
Dim sw2 As New Stopwatch
Dim ans, a, b, c, d As Double
a = 5
b = 10
c = 4
d = 2
sw1.Start()
For x As Integer = 1 To 10000
ans = Math.Sqrt(a ^ 2 + b ^ 2 - 2 * a * b * Math.Cos(c - d))
Next
sw1.Stop()
sw2.Start()
For y As Integer = 1 To 10000
ans = Math.Sqrt(a * a + b * b - 2 * a * b * Math.Cos(c - d))
Next
sw2.Stop()
Console.WriteLine(sw1.ElapsedTicks)
Console.WriteLine(sw2.ElapsedTicks)
Console.WriteLine(sw2.ElapsedTicks * 100 / sw1.ElapsedTicks)
Related
I have two bits of code in VBA for Excel. One calculates the A-squared statistic for the Anderson-Darling test, this bit of code calculates the P value of the A-squared statistic. I am curious if there is a more concise way or more efficient way to calculate this value in VBA:
Function AndDarP(AndDar, Elements)
'Calculates P value for level of significance for the
'Anderson-Darling Test for Normality
'AndDar is the Anderson-Darling Test Statistic
'Elements is the count of elements used in the
'Anderson-Darling test statistic.
'based on calculations at
'http://www.kevinotto.com/RSS/Software/Anderson-Darling%20Normality%20Test%20Calculator.xls
'accessed 21 May 2010
'www.kevinotto.com
'kevin_n_otto#yahoo.com
'Version 6.0
'Permission to freely distribute and modify when properly
'referenced and contact information maintained.
'
'"Keep in mind the test assumes normality, and is looking for sufficient evidence to reject normality.
'That is, a large p-value (often p > alpha = 0.05) would indicate normality.
' * * *
'Test Hypotheses:
'Ho: Data is sampled from a population that is normally distributed
'(no difference between the data and normal data).
'Ha: Data is sampled from a population that is not normally distributed"
Dim M As Double
M = AndDar * (1 + 0.75 / Elements + 2.25 / Elements ^ 2)
Select Case M
Case Is < 0.2
AndDarP = 1 - Exp(-13.436 + 101.14 * M - 223.73 * M ^ 2)
Case Is < 0.34
AndDarP = 1 - Exp(-8.318 + 42.796 * M - 59.938 * M ^ 2)
Case Is < 0.6
AndDarP = Exp(0.9177 - 4.279 * M - 1.38 * M ^ 2)
Case Is < 13
AndDarP = Exp(1.2937 - 5.709 * M + 0.0186 * M ^ 2)
Case Else
AndDarP = 0
End Select
End Function
The outer loop executes n times while the inner loop executes ? So the total time is n*something.
Do i need to learn summation,if yes then any book to refer?
for(int i=1;i<=n;i++)
for(int j=1;j<=n;j+=i)
printf("*");
This question can be approached by inspection:
n = 16
i | j values | # terms
1 | 1, 2, ..., 16 | n
2 | 1, 3, 5, ..., 16 | n / 2
.. | .. | n / 3
16 | 16 | n / n
In the above table, i is the outer loop value, and j values show the iterations of the inner loop. By inspection, we can see that the loops will take n * (1 + 1/2 + 1/3 + ... + 1/n) steps. This is a bounded harmonic series. As this Math Stack Exchange article shows, there is no closed form for the above expression in terms of n. However, as this SO article shows, there is an upper bound of O(n*ln(n)).
So, the running time for your two loops is O(n*ln(n)).
I believe the time complexity of that is O(n*log(n)). Here is why:
Let us pick some arbitrary natural number i and see how many steps the inner loop takes for this given i. Well for this i, you are going from j=1 to j<=n with a jump of i in between. So basically you are doing this summation many steps:
summation = 1 + (1+i) + (1+2i) + ... (1+ki)
where k is the largest integer such that 1+ki <= n. That is, k is the number of steps and this is what we want to solve for. Well we can solve for k in the equality resulting in k <= (n-1)/i and thus k = ⌊(n-1)/i⌋. That is, k is the floor function/integer division of (n-1)/i. Since we are dealing with time complexities, this floor function doesn't matter so we will just say k = n/i for simplicity. This is the number of steps that the inner loop will take for a given i. So we basically need to add all these for i = 1 to i <= n.
So numsteps will be this addition:
numsteps = n/1 + n/2 + n/3 + ... n/n
= n(1 + 1/2 + 1/3 + ... 1+n)
So we need to find the sum of 1 + 1/2 + ... 1/n to finish this. There is actually no good closed form for this sum but it is on the order of ln(n). You can read more about this here. You can also guess this since the integral from 1 to n of 1/x is ln(n). Again, since we are dealing with time complexity, we can just use ln(n) to represent its complexity. Thus we have:
numsteps = n(ln(n))
And so the time complexity is O(n*log(n)).
Edit: My bad, i was calculating the sum :P
I have an equation that can be used to find the gun elevation for artillery, using the range, muzzle velocity and change in altitude in a game called Arma 3. The equation looks like this:
With g being the acceleration due to gravity (9.80665), V being the muzzle velocity, X being the range and Y being the change in altitude (called DAlt in my code).
I'm trying to convert it to a line of code so that I can make a program to calculate the elevation based on given coordinates. However I'm having trouble with it. I currently have this:
If rdoLow.Checked = True Then
Elevation = Math.Atan(((Velocity ^ 2) - (Math.Sqrt((Velocity ^ 4) - (G) * (G * (Range ^ 2) + (2 * DAlt * (Velocity ^ 2)))))) / G * Range)
Else
Elevation = Math.Atan(((Velocity ^ 2) + (Math.Sqrt((Velocity ^ 4) - (G) * (G * (Range ^ 2) + 2 * DAlt * (Velocity ^ 2))))) / G * Range)
End If
Which isn't particularly nice looking but as far as I can tell, it should work. However when I put in the values that the video I got the equation from used, I got a different answer. So there must be something wrong with my equation.
I've tried breaking it in to various parts as separate variables and calculating them, then using those variables in the overall equation, and that still didn't work and gave me an answer that was wrong in another way.
So I'm currently at a loss on how to fix it, starting to wonder if the way that vb handles long equations is different or something.
Any help is much appreciated.
You haven't given any sample data, so I can't verify that this gives the correct answer, but the last part of your equation is missing some parentheses.
Elevation = Math.Atan(((Velocity ^ 2) + Math.Sqrt((Velocity ^ 4) - (G * ((G * (Range ^ 2)) + (2 * DAlt * (Velocity ^ 2)))))) / (G * Range))
Note the parenthesis around the last G * Range.
Multiplication and division have equal precedence, so they are evaluated from left-to-right. See Operator Precedence in Visual Basic.
You were dividing everything by G, then multiplying the result by Range, whereas you needed to multiply G by Range, then divide everything by the result of that.
You can see the difference in this simple example:
Console.WriteLine(3 / 4 * 5) ' Prints 3.75
Console.WriteLine(3 / (4 * 5)) ' Prints 0.15
Out of curiosity I tried the problem. In order to have test data I found this web site, Range Tables For Mortars. I tested with the '82mm Mortar - Far' that has an initial velocity of 200 m/s. One problem I had, and don't know if I fixed it correctly, was that the first part of the equation was returning negative numbers... I also solved for the ±. To test I created a form with a button to perform the calculation, a textbox to enter the distance, and two labels to show the angles. This is what I came up with.
Private Sub Button1_Click(sender As Object, e As EventArgs) Handles Button1.Click
'A - launch angle
'Target
' r - range
' y - altitude
'g - gravity 9.80665 m/s^2
'v - launch speed e.g. 50 m/s
'
'
'Formula
'from - https://en.wikipedia.org/wiki/Trajectory_of_a_projectile#Angle_required_to_hit_coordinate_.28x.2Cy.29
'in parts
'parts - px
' p1 = sqrt( v^4 - g * (g * r^2 + 2 * y * v^2) )
' p2 = v^2 ± p1 note plus / minus
' p3 = p2 / g * r
'
' A = arctan(p3)
Dim Ap, Am, r, y As Double
Dim g As Double = 9.80665
Dim v As Double
Dim p1, p2p, p2m, p3p, p3m As Double
If Not Double.TryParse(TextBox1.Text, r) Then Exit Sub
y = 0
v = 200 '82mm Mortar - Far velocity
p1 = v ^ 4 - g * (g * r ^ 2 + 2 * y * v ^ 2)
If p1 < 0 Then
Debug.WriteLine(p1)
p1 = Math.Abs(p1) 'square root does not like negative numbers
End If
p1 = Math.Sqrt(p1)
'plus / minus
p2p = v ^ 2 + p1
p2m = v ^ 2 - p1
p3p = p2p / (g * r)
p3m = p2m / (g * r)
Const radiansToDegrees As Double = 180 / Math.PI
Ap = Math.Atan(p3p) * radiansToDegrees
Am = Math.Atan(p3m) * radiansToDegrees
Label1.Text = Ap.ToString("n3")
Label2.Text = Am.ToString("n3")
End Sub
Using the web site to verify the calculations the code seem correct.
Writing long formulas in a bunch of nested parentheses serves no purpose, unless you are going for confusion.
I'm having a problem with some vba code.
I have a if statement that doesn't treat the same content equally.
e.g: 0,1 equals 0,1, but a re-run 0,1 does not equal 0,1
(this values are shown by MVBA)
The code is long so before posting it i would like to know if it's possible to see the machine perspective in a if statement (hex, ascii...). This because, although the debug is telling me they are the same (through msgbox, vartype, etc), the if statement is not activated.
pseudo code:
x = 0,0000001 * 1*10^6 (which equals 0,1)
y = 0,0001 * 1*10^3 (which also equals 0,1)
if statement:
x doesn't enter
y does
end if
This is because the floating-point implementation may not be able to represent those number accurately due to the fact that they are encoded in a base 2 representation.
If you want to compare them, I would suggest using Cdec (wich converts to Decimal, a VBA custom base 10 floating-point)
Debug.Print (0.0000001 * 1 * 10 ^ 6) = (0.0001 * 1 * 10 ^ 3) ' False
Debug.Print CDec(0.0000001 * 1 * 10 ^ 6) = CDec(0.0001 * 1 * 10 ^ 3) ' True
While they both display 0.1, in fact 0.0000001 * 1 * 10 ^ 6 flaoting-point value is 0x3FB9999999999999 whereas 0.0001 * 1 * 10 ^ 3 returns 0x3FB999999999999A.
I'd recommend reading What Every Computer Scientist Should Know About Floating-Point Arithmetic
This is the opposite of what most "random order" questions are about.
I want to select data from a database in random order. But I want to be able to repeat certain selects, getting the same order again.
Current (random) select:
SELECT custId, rand() as random from
(
SELECT DISTINCT custId FROM dummy
)
Using this, every key/row gets a random number. Ordering those ascending results in a random order.
But I want to repeat this select, getting the very same order again. My idea is to calculate a random number (r) once per session (e.g. "4") and use this number to shuffle the data in some way.
My first idea:
SELECT custId, custId * 4 as random from
(
SELECT DISTINCT custId FROM dummy
)
(in real life "4" would be something like 4005226664240702)
This results in a different number for each line but the same ones every run. By changing "r" to 5 all numbers will change.
The problem is: multiplication is not sufficient here. It just increases the numbers but keeps the order the same. Therefore I need some other kind of arithmetic function.
More abstract
Starting with my data (A-D). k is the key and r is the random number currently used:
k r
A = 1 4
B = 2 4
C = 3 4
D = 4 4
Doing some calculation using k and r in every line I want to get something like:
k r
A = 1 4 --> 12
B = 2 4 --> 13
C = 3 4 --> 11
D = 4 4 --> 10
The numbers can be whatever they want, but when I order them ascending I want to get a different order than the initial one. In this case D, C, A, B, E.
Setting r to 7 should result in a different order (C, A, B, D):
k r
A = 1 7 --> 56
B = 2 7 --> 78
C = 3 7 --> 23
D = 4 7 --> 80
Every time I use r = 7 should result in the same numbers => same order.
I'm looking for a mathematical function to do the calculation with k and r. Seeding the RAND() function is not suitable because it's not supported by some databases we support
Please note that r is already a randomly generated number
Background
One Table - Two data consumers. One consumer will get random 5% of the table, the other one the other 95%. They don't just get the data but a generated SQL. So there are two SQL's which must not select the same data twice but still random.
You could try and implement the Multiply-With-Carry PseudoRandomNumberGenerator. The C version goes like this (source: Wikipedia):
m_w = <choose-initializer>; /* must not be zero, nor 0x464fffff */
m_z = <choose-initializer>; /* must not be zero, nor 0x9068ffff */
uint get_random()
{
m_z = 36969 * (m_z & 65535) + (m_z >> 16);
m_w = 18000 * (m_w & 65535) + (m_w >> 16);
return (m_z << 16) + m_w; /* 32-bit result */
}
In SQL, you could create a table Random, with two columns to contain w and z, and one ID column to identify each session. Perhaps your vendor supports variables and you need not bother with the table.
Nonetheless, even if we use a table, we immediately run into trouble cause ANSI SQL doesn't support unsigned INTs. In SQL Server I could switch to BIGINT, unsure if your vendor supports that.
CREATE TABLE Random (ID INT, [w] BIGINT, [z] BIGINT)
Initialize a new session, say number 3, by inserting 1 into z and the seed into w:
INSERT INTO Random (ID, w, z) VALUES (3, 8921, 1);
Then each time you wish to generate a new random number, do the computations:
UPDATE Random
SET
z = (36969 * (z % 65536) + z / 65536) % 4294967296,
w = (18000 * (w % 65536) + w / 65536) % 4294967296
WHERE ID = 3
(Note how I have replaced bitwise operands with div and mod operations and how, after computing, you need to mod 4294967296 to stay within the proper 32 bits unsigned int range.)
And select the new value:
SELECT(z * 65536 + w) % 4294967296
FROM Random
WHERE ID = 3
SQLFiddle demo
Not sure if this applies in non-SQL Server, but typically when you use a RAND() function, you can specify a seed. Everytime you specify the same seed, the randomization will be the same.
So, it sounds like you just need to store the seed number and use that each time to get the same set of random numbers.
MSDN Article on RAND
Each vendor has solved this in its own way. Creating your own implementation will be hard, since random number generation is difficult.
Oracle
dbms_random can be initialized with a seed: http://docs.oracle.com/cd/B19306_01/appdev.102/b14258/d_random.htm#i998255
SQL Server
First call to RAND() can provide a seed: http://technet.microsoft.com/en-us/library/ms177610.aspx
MySql
First call to RAND() can provide a seed: http://dev.mysql.com/doc/refman/4.1/en/mathematical-functions.html#function_rand
Postgresql
Use SET SEED or SELECT setseed() : http://www.postgresql.org/docs/8.3/static/sql-set.html