I want, for example, for Mathematica to generate 7 + 5f if I write the expression (2+f) (3+f). I always want f^2 to be computed as 1 (or any other value I assign to it) but for f to be a special undefined symbol. If I define f^2:=1 I get a Tag Power is protected error message.
I am a Mathematica newbie, self taught, so please try to answer this in as elementary fashion as possible.
For the record, I am trying to define Clifford algebra operations in n-dimensional space-time and being able to make an assignment like this would tremendously simplify the task.
Generalized to all symbols e1,e2,e3,...,en
x = (a + a1 e1 + a2 e2 + a3 e3 + a4 e1 e2 - a5 e1 e3 + a6 e2 e3 +
a7 e1 e2 e3);
y = (b + b1 e1 + b2 e2 + b3 e3 + b4 e1 e2 - b5 e1 e3 + b6 e2 e3 +
b7 e1 e2 e3);
ReplaceAll[
Expand[x y],
Power[e_, 2] /; First[Characters[ToString[e]]] === "e" -> 1
]
This way which I have just learned from #Edmund is more elegant:
Expand[(2 + e1)(3 + e2)] /.Power[s_Symbol,2]/; StringStartsQ["e"]#SymbolName[s]->1
6 + 3 e1 + 2 e2 + e1 e2
ReplaceAll[Expand[(2 + f) (3 + f)], Power[f, 2] -> 1]
7 + 5 f
Related
I have problem with clusterization of clients.
I have a dataset with columns such as name, address, email, phone, etc. (in a example A,B,C). Each row has unique identifier (ID). I need to assign CLUSTER_ID (X) to each row. In one cluster all rows have one or more the same attributes as other rows. So clients with ID=1,2,3 have the same A attribute and clients with ID=3,10 have the same B attribute then ID=1,2,3,10 should be in the same cluster.
How can I solve this problem using SQL?
If it's not possible how to write the algorithm (pseudocode)?
The performance is very important, because the dataset contains milions of rows.
Sample Input:
ID A B C
1 A1 B3 C1
2 A1 B2 C5
3 A1 B10 C10
4 A2 B1 C5
5 A2 B8 C1
6 A3 B1 C4
7 A4 B6 C3
8 A4 B3 C5
9 A5 B7 C2
10 A6 B10 C3
11 A8 B5 C4
Sample Output:
ID A B C X
1 A1 B3 C1 1
2 A1 B2 C5 1
3 A1 B10 C10 1
4 A2 B1 C5 1
5 A2 B8 C1 1
6 A3 B1 C4 1
7 A4 B6 C3 1
8 A4 B3 C5 1
9 A5 B7 C2 2
10 A6 B10 C3 1
11 A8 B5 C4 1
Thanks for any help.
A possible way is by repeating updates for the empty X.
Start with cluster_id 1.
F.e. by using a variable.
SET #CurrentClusterID = 1
Take the top 1 record, and update it's X to 1.
Now loop an update for all records with an empty X,
and that can be linked to a record with X = 1 and that has the same A or B or C
Disclaimer:
The statement will vary depending on the RDBMS.
This is just intended as pseudo-code.
WHILE (<<some check to see if there were records updated>>)
BEGIN
UPDATE yourtable t
SET t.X = #CurrentClusterID
WHERE t.X IS NULL
AND EXISTS (
SELECT 1 FROM yourtable d
WHERE d.X = #CurrentClusterID
AND (d.A = t.A OR d.B = t.B OR d.C = t.C)
);
END
Loop that till it updates 0 records.
Now repeat the method for the other clusters, till there are no more empty X in the table.
1) Increase the #CurrentClusterID by 1
2) Update the next top 1 record with an empty X to the new #CurrentClusterID
3) Loop the update till no-more updates were done.
An example test on db<>fiddle here for MS Sql Server.
Task
I want to calculate the permanent P of a NxN matrix for N up to 100. I can make use of the fact that the matrix features only M=4 (or slightly more) different rows and cols. The matrix might look like
A1 ... A1 B1 ... B1 C1 ... C1 D1 ... D1 |
... | r1 identical rows
A1 ... A1 B1 ... B1 C1 ... C1 D1 ... D1 |
A2 ... A2 B2 ... B2 C2 ... C2 D2 ... D2
...
A2 ... A2 B2 ... B2 C2 ... C2 D2 ... D2
A3 ... A3 B3 ... B2 C2 ... C2 D2 ... D2
...
A3 ... A3 B3 ... B3 C3 ... C3 D3 ... D3
A4 ... A4 B4 ... B4 C4 ... C4 D4 ... D4
...
A4 ... A4 B4 ... B4 C4 ... C4 D4 ... D4
---------
c1 identical cols
and c and r are the multiplicities of cols and rows. All values in the matrix are laying between 0 and 1 and are encoded as double precision floating-point numbers.
Algorithm
I tried to use the Ryser formula to calculate the permanent. For the formula, one needs to first calculate the sum of each row and multiply all the row sums. For the matrix above this yields
S0 = (c1 * A1 + c2 * B1 + c3 * C1 + c4 * D1)^r1 * ...
* (c1 * A4 + c2 * B4 + c3 * C4 + c4 * D4)^r4
As a next step the same is done with col 1 deleted
S1 = ((c1-1) * A1 + c2 * B1 + c3 * C1 + c4 * D1)^r1 * ...
* ((c1-1) * A4 + c2 * B4 + c3 * C4 + c4 * D4)^r4
and this number is subtracted from S0.
The algorithm continues with all possible ways to delete single and group of cols and the products of the row sums of the remaining matrix are added (even number of cols deleted) and subtracted (odd number of cols deleted).
The task can be solved relative efficiently if one makes use of the identical cols (for example the result S1 will pop up exactly c1 times).
Problem
Even if the final result is small the values of the intermediate results S0, S1, ... can reach values up to N^N. A double can hold this number but the absolute precision for such big numbers is below or on the order of the expected overall result. The expected result P is on the order of c1!*c2!*c3!*c4! (actually I am interested in P/(c1!*c2!*c3!*c4!) which should lay between 0 and 1).
I tried to arrange the additions and subtractions of the values S in a way that the sums of the intermediate results are around 0. This helps in the sense that I can avoid intermediate results that are exceeding N^N, but this improves things only a little bit. I also thought about using logarithms for the intermediate results to keep the absolute numbers down - but the relative accuracy of the encoded numbers will be still bounded by the encoding as floating point number and I think I will run into the same problem. If possible, I want to avoid the usage of data types that are implementing a variable-precision arithmetic for performance reasons (currently I am using matlab).
I am creating a dashboard using Excel Powerquery(aka. M), in which I need to create a measure which requires rolling up values for last 12 months for two dimension
Example:
Input:
D1 | D2 | MonthYear(D3) | Value
A1 B1 Mar2016 1
A2 B1 Mar2016 2
A3 B1 Mar2016 3
A1 B1 Apr2016 4
A2 B1 Apr2016 5
A3 B1 Apr2016 6
A1 B1 May2016 7
A2 B1 May2016 8
A3 B1 May2016 9
Output:
D1 | D2 | MonthYear(D3) | Value
A1 B1 Mar2016 1
A2 B1 Mar2016 2
A3 B1 Mar2016 3
A1 B1 Apr2016 4+1
A2 B1 Apr2016 5+2
A3 B1 Apr2016 6+3
A1 B1 May2016 7+4+1
A2 B1 May2016 8+5+2
A3 B1 May2016 9+6+3
Also sum should be done only for last 12 months if more data is available. ANy help is appreciated
I covered a very similar scenario to this in my demo file: Power Query demo - Running Total.xlsx
You can download it from my OneDrive and review the steps:
https://1drv.ms/f/s!AGLFDsG7h6JPgw4
Basically you add an Index, Group By the "group columns" (in your scenario D1 and D2) and create an "All Rows" Aggregate column. Then you Copy the "All Rows" column, Expand both "All Rows" columns, Filter and finally Group By and Sum to create the Running Total.
The only bit of code is the Added column to produce a true/false column for the filter, e.g.
[Index] >= [#"All Rows - Copy.Index"]
I am writing a piece of subroutine in NEON for image processing which does color swapping, i.e., I sequentialy load the R,G,B channels from an array, and depending on some configuration, permute some of them.
There are as maximum 6 permutes
(RGB) -> { (RGB),(RBG),(GRB),(GBR),(BRG),(BGR) }
The most efficient way would be to have a separate subroutine for each case and the corresponding VSWP instructions. As the Subroutine will do several other things, I would prefer to keep everything in just one sub, even if it is not so efficient,
Also have read that conditional execution and branching is not advisable. So, if I want to have it in a block with branchless code, the only thing coming to my mind is
New_R = a(0)*R+a(1)*G+a(2)*B
New_G = a(3)*R+a(4)*G+a(5)*B
New_B = a(6)*R+a(7)*G+a(8)*B
where only one a(i) in each row and column will be =1 each time, and the rest will be =0
Question: Any smarter way to do it, having in mind that it has to be coded to NEON?
VTBL.8 is the most powerful tool in NEON to swap bytes.
Loading 3x8 bytes to registers d0,d1,d2 would look like
R G B R G B R G | B R G B R G B R | G B R G B R G B |
0 1 2 3 4 5 6 7 8 9 a b c d e f .... 17
VTBL d3, { d0,d1,d2 }, d6 ;; select bytes to d3 from d0,d1,d2 based on d6
VTBL d4, { d0,d1,d2 }, d7
VTBL d5, { d0,d1,d2 }, d8
where d6,d7,d8 encode the positions to read in the new bytes.
e.g. '0 1 2 3 4 5 6 7' for the original permutation and '0 2 1 3 5 4 6 8', '7 ...' to swap G and B. The constant vectors d6..d8 need to be loaded just once in the beginning of the routine.
Another possibility is to encode the following sequence with interleaved read;
VLD3.8 { d0,d1,d2 }, [r0] ; // Read R, G, B to separate registers
VLD3.8 { d3,d4,d5 }, [r0] ; // Make a second copy (or use some other instruction)
VBIT d3, d1, d6 ; // d3 is now either R or G
VBIT d4, d2, d7 ; // d4 is now either G or B
VBIT d5, d0, d8 ; // d5 is now either B or R
VBIT d0, d4, d9 ; // d0 is now R or (G or B)
VBIT d1, d5, d10 ; // d1 is now G or (B or R)
VBIT d2, d3, d11 ; // d2 is now B or (R or G)
Even though 6 registers for the condition codes are used in the example, 3 independent registers should be enough -- one can also use VBIF if reversed logic needs to be used.
For example, I have a polynomial y＝a_0＋a_1 x + ..... + a_50 x^50. Since I know that the high-order terms are imposing negligible effects on the evaluation of y, I want to cut off them and have something like y＝a_0＋a_1 x + ..... + a_10 x^10, the first eleven terms. How can I realize this?
I thank you all in advance.
In[1]:= y = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4;
y /. x^b_ /; b >= 3 -> 0
Out[2]= a0 + a1 x + a2 x^2
The mathematically proper approach..
Series[ a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4, {x, 0, 2}] // Normal
-> a0 + a1 x + a2 x^2
If your polynomial is actually as simple as shown, with a term for every power of x and none others, you can simply use Take or Part to extract only those terms that you want because of the automatic ordering (in Plus) that Mathematica uses. For example:
exp1 = Expand[(1 + x)^9]
Take[exp1, 5]
1 + 9 x + 36 x^2 + 84 x^3 + 126 x^4 + 126 x^5 + 84 x^6 + 36 x^7 + 9 x^8 + x^9
1 + 9 x + 36 x^2 + 84 x^3 + 126 x^4
If it is not you will need something else. Bill's replacement rule is one concise and efficient method. For more complex manipulations you may wish to decompose the polynomial using CoefficientArrays, CoefficientRules, or CoefficientList.
There is a shortcut to the previous answers which is even more symbolic. You write, say,
y[x_] = a0 + a1 x + a2 x^2 + a3 x^3 + a4 x^4 + a5 x^5;
y[x] + O[x]^3
which gives you,
a0 + a1 x + a2 x^2 + O[x]^3