The kronecker-symbol for two integers n,m is defined as 1 if n=m and 0 otherwise.
Is there an inbuilt kronecker-symbol in maple or an easy way to implement it?
Sorry, if I should have overlooked something trivial, the maple-homepage is down at the moment and google did not return anything helpful.
Edit: I Just found something useful here:
restart:
delta := table(symmetric,identity);
simplify(delta[1,2]);
simplify(delta[1,1]);
delta := TABLE(symmetric, identity, [])
0
1
However it does not seem to simplify even trivial expressions with symbols:
simplify(delta[n,n]);
simplify(delta[n,n+1]);
delta[n, n]
delta[n, n + 1]
where one would expect to obtain 1 and 0, respectively.
Edit 2: I also tried it this way:
restart:
delta := proc(n,m):
if n=m then 1 else 0 fi;
end;
delta(1,2);
delta(1,1);
delta(n,n);
0
1
1
However, this also works poorly for symbols, as
delta(n,m);
returns 0.
Try this
delta:= (m,n)->
`if`(evalb(m < n)::truefalse, `if`(m=n,1,0), 'procname'(m,n))
;
There is a built-in function called KroneckerDelta in the Physics package. Try this
with(Physics):
KroneckerDelta[i,j];
where i and j take integer values. This gives 1 when i=j and 0 when i =/= j.
I prefer something like this: delta:=(mu,nu)->eval(evalb(mu=nu),[true=1,false=0])
Related
I want to implement a constraint depending on the change of values in my binary decision variable, x, over "time".
I am trying to implement a minimum operating time constraint for a unit commitment optimization problem for power systems. x is representing the unit activation where 0 and 1 show that a power unit, n, at a certain time, t, respectively is shut off or turned on.
For this, indicator constraints seem to be a promising solution and with the inspiration of a similar problem the implementation seemed quite straightforward.
So, since boolean operators are introduced (! and ¬), I prematurely wanted to express the change in a boolean way:
#constraint(m, xx1[n=1:N,t=2:T], (!x[n,t-1] && x[n,t]) => {next(t, 1) + next(t, 2) == 2})
Saying: if unit was deactivated before but now is on, then demand the unit to be active for the next 2 times.
Where next(t, i) = x[((t - 1 + i) % T) + 1].
I got the following error:
LoadError: MethodError: no method matching !(::VariableRef)
Closest candidates are:
!(!Matched::Missing) at missing.jl:100
!(!Matched::Bool) at bool.jl:33
!(!Matched::Function) at operators.jl:896
I checked that the indicator constraint is working properly with a single term only.
Question: Is this possible or is there another obvious solution?
Troubleshooting and workarounds: I have tried the following (please correct me if my diagnosis is wrong):
Implement change as an expression: indicator constraints only work with binary integer variables.
Implement change as another variable relating to x. I have found a solution but it is quite sketchy, which is documented in a Julia discourse. The immediate problem, found from the solution, is that indicator constraints do not work as bi-implication but only one way, LHS->RHS. Please see the proper approach given by #Oscar Dowson.
You can get the working code from github.
The trick is to find constraint(s) that have an equivalent truth-table:
# Like
(!x[1] && x[2]) => {z == 1}
# Is equivalent to:
z >= -x[1] + x[2]
# Proof
-x[1] + x[2] = sum <= z
--------------------------
- 0 + 0 = 0 <= 0
- 1 + 0 = -1 <= 0
- 0 + 1 = 1 <= 1
- 1 + 1 = 0 <= 0
I was recommended MOSEK Modeling Cookbook to help working out the correct formulation of constraints.
See eventually the thread here from where I got the answer for further details.
I'm doing analysis on binary data. Suppose I have two uint8 data values:
a = uint8(0xAB);
b = uint8(0xCD);
I want to take the lower two bits from a, and whole content from b, to make a 10 bit value. In C-style, it should be like:
(a[2:1] << 8) | b
I tried bitget:
bitget(a,2:-1:1)
But this just gave me separate [1, 1] logical type values, which is not a scalar, and cannot be used in the bitshift operation later.
My current solution is:
Make a|b (a or b):
temp1 = bitor(bitshift(uint16(a), 8), uint16(b));
Left shift six bits to get rid of the higher six bits from a:
temp2 = bitshift(temp1, 6);
Right shift six bits to get rid of lower zeros from the previous result:
temp3 = bitshift(temp2, -6);
Putting all these on one line:
result = bitshift(bitshift(bitor(bitshift(uint16(a), 8), uint16(b)), 6), -6);
This is doesn't seem efficient, right? I only want to get (a[2:1] << 8) | b, and it takes a long expression to get the value.
Please let me know if there's well-known solution for this problem.
Since you are using Octave, you can make use of bitpack and bitunpack:
octave> a = bitunpack (uint8 (0xAB))
a =
1 1 0 1 0 1 0 1
octave> B = bitunpack (uint8 (0xCD))
B =
1 0 1 1 0 0 1 1
Once you have them in this form, it's dead easy to do what you want:
octave> [B A(1:2)]
ans =
1 0 1 1 0 0 1 1 1 1
Then simply pad with zeros accordingly and pack it back into an integer:
octave> postpad ([B A(1:2)], 16, false)
ans =
1 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0
octave> bitpack (ans, "uint16")
ans = 973
That or is equivalent to an addition when dealing with integers
result = bitshift(bi2de(bitget(a,1:2)),8) + b;
e.g
a = 01010111
b = 10010010
result = 00000011 100010010
= a[2]*2^9 + a[1]*2^8 + b
an alternative method could be
result = mod(a,2^x)*2^y + b;
where the x is the number of bits you want to extract from a and y is the number of bits of a and b, in your case:
result = mod(a,4)*256 + b;
an extra alternative solution close to the C solution:
result = bitor(bitshift(bitand(a,3), 8), b);
I think it is important to explain exactly what "(a[2:1] << 8) | b" is doing.
In assembly, referencing individual bits is a single operation. Assume all operations take the exact same time and "efficient" a[2:1] starts looking extremely inefficient.
The convenience statement actually does (a & 0x03).
If your compiler actually converts a uint8 to a uint16 based on how much it was shifted, this is not a 'free' operation, per se. Effectively, what your compiler will do is first clear the "memory" to the size of uint16 and then copy "a" into the location. This requires an extra step (clearing the "memory" (register)) that wouldn't normally be needed.
This means your statement actually is (uint16(a & 0x03) << 8) | uint16(b)
Now yes, because you're doing a power of two shift, you could just move a into AH, move b into AL, and AH by 0x03 and move it all out but that's a compiler optimization and not what your C code said to do.
The point is that directly translating that statement into matlab yields
bitor(bitshift(uint16(bitand(a,3)),8),uint16(b))
But, it should be noted that while it is not as TERSE as (a[2:1] << 8) | b, the number of "high level operations" is the same.
Note that all scripting languages are going to be very slow upon initiating each instruction, but will complete said instruction rapidly. The terse nature of Python isn't because "terse is better" but to create simple structures that the language can recognize so it can easily go into vectorized operations mode and start executing code very quickly.
The point here is that you have an "overhead" cost for calling bitand; but when operating on an array it will use SSE and that "overhead" is only paid once. The JIT (just in time) compiler, which optimizes script languages by reducing overhead calls and creating temporary machine code for currently executing sections of code MAY be able to recognize that the type checks for a chain of bitwise operations need only occur on the initial inputs, hence further reducing runtime.
Very high level languages are quite different (and frustrating) from high level languages such as C. You are giving up a large amount of control over code execution for ease of code production; whether matlab actually has implemented uint8 or if it is actually using a double and truncating it, you do not know. A bitwise operation on a native uint8 is extremely fast, but to convert from float to uint8, perform bitwise operation, and convert back is slow. (Historically, Matlab used doubles for everything and only rounded according to what 'type' you specified)
Even now, octave 4.0.3 has a compiled bitshift function that, for bitshift(ones('uint32'),-32) results in it wrapping back to 1. BRILLIANT! VHLL place you at the mercy of the language, it isn't about how terse or how verbose you write the code, it's how the blasted language decides to interpret it and execute machine level code. So instead of shifting, uint32(floor(ones / (2^32))) is actually FASTER and more accurate.
What I am trying is getting binary value of a number e.g
de2bi(234)
Which results me in having this answer :
0 1 0 1 0 1 1 1
now what I want is that is its reverse order without changing its values like this :
11101010
i have tried bitrevorder() function but i am not having my desired answer. any help and suggestions will be appreciated.
Example:
>>de2bi(234)
ans = 0 1 0 1 0 1 1 1
>> fliplr(ans)
ans =
1 1 1 0 1 0 1 0
Use the function fliplr. It can be used to reverse the order of array.
Try using the flag 'left-msb' (according to the documentation in http://www.mathworks.com/help/comm/ref/de2bi.html)
The commands below show how to convert a decimal integer to base three without specifying the number of columns in the output matrix. They also show how to place the most significant digit on the left instead of on the right.
t = de2bi(12,[],3) % Convert 12 to base 3.
tleft = de2bi(12,[],3,'left-msb') % Significant digit on left
The output is
t =
0 1 1
tleft =
1 1 0
You just need to use the 'left-msb' option in de2bi:
>>de2bi(234, 'left-msb')
ans =
1 1 1 0 1 0 1 0
You can use a more simple command called dec2bin which produces the desired result:
>> dec2bin(234)
ans =
11101010
Here is the docs: http://www.mathworks.com/help/matlab/ref/dec2bin.html?refresh=true
While this is an old question, I needed to do the same thing for a CRC checksum and feel I should share the results.
In my case I need to reverse 16bit numbers, so, I've tried three methods:
1) Using fliplr() to reverse as per the suggestions:
uint16(bin2dec(fliplr(dec2bin(data,16))))
To test out the speed I decided to try and checksum 12MB of data. Using the above code in my CRC, it took 2000 seconds to complete! Most of this time was performing the bit reversal.
2) I then devised a more optimal solution, though not a one line code it is optimised for speed:
reverse = uint16(0);
for i=1:16
reverse = bitor(bitshift(reverse,1), uint16(bitand(forward,1)));
forward = bitshift(forward,-1);
end
Using the same CRC code, but with this used instead of (1), it took a little over 500 seconds to complete, so already it makes the CRC calculations four times faster!
3) That is still too much time for my liking, so instead I moved everything to a mex function. This allows the use of code from the bit twiddling examples that are floating around for optimum performance. I moved the whole CRC code to the mex function and used the following two other functions to do the bit reversal.
unsigned char crcBitReverse8(unsigned char forward) {
return (unsigned char)(((forward * 0x0802LU & 0x22110LU) | (forward * 0x8020LU & 0x88440LU)) * 0x10101LU >> 16);
}
unsigned short crcBitReverse16(unsigned short forward) {
unsigned char inByte0 = (forward & 0xFF);
unsigned char inByte1 = (forward & 0xFF00) >> 8;
return (unsigned short )((crcBitReverse8(inByte0) << 8) | (crcBitReverse8(inByte1)));
}
Just for comparison, it took just 0.14 seconds to compute the CRC for the same 12MB data chunk (and there is no mistake in the calculation, the CRC checksums for all three methods match what is expected).
So basically, if you have to do this a lot of times (e.g. for CRC) I seriously suggest you write a mex function for doing the reversal. For such a simple operation, native MATLAB code is embarrassing slow.
why not use bitget?
>> bitget( 234, 8:-1:1 )
ans =
1 1 1 0 1 0 1 0
I am taking coursera course and,for an assignment, I have written a code to count the change of an amount given a list of denominations. A doing a lot of research, I found explanations of various algorithms. In the recursive implementation, one of the base cases is if the amount money is 0 then the count is 1. I don't understand why but this is the only way the code works. I feel that is the amount is 0 then there is no way to make change for it and I should throw an exception. The code look like:
function countChange(amount : Int, denoms :List[Int]) : Int = {
if (amount == 0 ) return 1 ....
Any explanation is much appreciated.
To avoid speaking specifically about the Coursera problem, I'll refer to a simpler but similar problem.
How many outcomes are there for 2 coin flips? 4.
(H,H),(H,T),(T,H),(T,T)
How many outcomes are there for 1 coin flip? 2.
(H),(T)
How many outcomes are there for 0 coin flips? 1.
()
Expressing this recursively, how many outcomes are there for N coin flips? Let's call it f(N) where
f(N) = 2 * f(N - 1), for N > 0
f(0) = 1
The N = 0 trivial (base) case is chosen so that the non-trivial cases, defined recursively, work out correctly. Since we're doing multiplication in this example and the identity element for multiplication is 1, it makes sense to choose that as the base case.
Alternatively, you could argue from combinatorics: n choose 0 = 1, 0! = 1, etc.
I'm trying to write a program that cleans data, using Matlab. This program takes in the max and min that the data can be, and throws out data that is less than the min or greater than the max. There looks like a small issue with the cleaning part. This case ONLY happens when the minimum range of the variable being checked is 0. If this is the case, for one reason or another, the program won't throw away data points that are between 0 and -1. I've been trying to fix this for some time now, and noticed that this is the only case where this happens, and if you try to run a SQL query selecting data that is < 0, it will leave out data between 0 and -1, so effectively the same error as what's happening to me. Wondering if anyone might recognize this and know what it could be.
I would write such a function as:
function data = cleanseData(data, limits)
limits = sort(limits);
data = data( limits(1) <= data & data <= limits(2) );
end
an example usage:
a = rand(100,1)*10;
b = cleanseData(a, [-2 5]);
c = cleanseData(a, [0 -1]);
-1 is less than 0, so 0 should be the max value. And if this is the case it will keep points between -1 and 0 by your definition of the cleaning operation:
and throws out data that is less than the min or greater than the max.
If you want to throw away (using the above definition)
data points that are between 0 and -1
then you need to set 0 as the min value and -1 as the max value --- which does not make sense.
Also, I think you mean
and throws out data that is less than the min AND greater than the max.
It may be that the floats are getting casted to ints before the comparison. I don't know matlab, but in python int(-0.5)==0, which could explain the extra data points getting in. You can test this by setting the min to -1, if you then also get values from -1 to -2 then you'll need to make sure casting isn't being done.
If I try to mimic your situation with SQL, and run the following query against a datatable that has 1.00, 0.00, -0.20, -0.80. -1.00, -1.20 and -2.00 in the column SomeVal, it correctly returns -0.20 and -0.80, which is as expected.
SELECT SomeVal
FROM SomeTable
WHERE (SomeVal < 0) AND (SomeVal > - 1)
The same is true for MatLab. Perhaps there's an error in your code. Dheck the above statement with your own SELECT statement to see if something's amiss.
I can imagine such a bug if you do something like
minimum = 0
if minimum and value < minimum