In my code, I need to divide each values of a matrix by the values of another. I could use A./B but some elements in B are 0. I know that if B(i,j) = 0 so A(i,j) = 0 too and I want to have 0/0 = 0. So I wrote a function div and I use bsxfun but I don't have 0, I have NaN :
A = [1,0;1,1];
B = [1,0;1,2];
function n = div(a,b)
if(b==0)
n = 0;
else
n = a./b;
end
end
C = bsxfun(#div,A,B);
Why not just replace the unwanted values after?
C=A./B;
C(A==0 & B==0)=0;
You could do C(isnan(C))=0;, but this will replace all NaN, even the ones not created by 0/0. If zeros always happen together then just C(B==0)=0; will do
If you know your non-zero values in B are never smaller than a very small number eps (for example 1e-300), a simple trick is to add eps to B. All non-zero values are unchanged, while all zero values become eps. When dividing 0/eps you get the wished result.
The reason this is happening is because bsxfun doesn't process the arrays element-wise. Consequently, your function doesn't get two scalars in. It is actually called only once. Your if statement does not work for non-scalar values of b.
Replacing bsxfun with arrayfun will call your function with scalar inputs, and will yield the expected result:
>> C = arrayfun(#div,A,B)
C =
1.0000 0
1.0000 0.5000
Nonetheless, either of the other two answers will be more efficient:
>> C = A./B;
>> C(B==0) = 0 % Ander's answer
C =
1.0000 0
1.0000 0.5000
or
C = A./(B+eps) % user10259794's answer
C =
1.0000 0
1.0000 0.5000
Related
I am using Matlab's backslash operator to solve a system of equations written as two matrices M1 and M2. These two matrices are square and tridiagonal, and so I have defined them as sparse. For example, with the dimensions of each being 5x5, they are defined as follows, with the values in each entry being dependent on some constant a:
N = 5;
a = 1e10;
M1 = spdiags([-a*ones(N,1)... % Sub diagonal
(1 + 2*a)*ones(N,1)... % Main Diagonal
-a*ones(N,1)],... % Super diagonal
-1:1,N,N);
M2 = spdiags([+a*ones(N,1)...
(1 - 2*a)*ones(N,1)...
+a*ones(N,1)],...
-1:1,N,N);
M_out = M1\M2;
So for example, M1 looks like the following in full form:
>> full(M1)
ans =
1.0e+10 *
2.0000 -1.0000 0 0 0
-1.0000 2.0000 -1.0000 0 0
0 -1.0000 2.0000 -1.0000 0
0 0 -1.0000 2.0000 -1.0000
0 0 0 -1.0000 2.0000
Now, if I examine the number of non-zero entries in the result M_out, then I can see they are all non-zero, which is fine:
>> nnz(M_out)
ans =
25
The problem is that I also need to do this for larger values of the constant a. However, if, for example, a=1e16 instead, then the off-diagonal entries of M_out are automatically set to zero, presumably because they have become too small:
>> nnz(M_out)
ans =
5
Is there a better way in Matlab of going about this problem of inverting sparse matrices? Or am I using the backslash operator in the wrong way?
If the size of your matrices doesn't grow too much, I recommend doing a full symbolic computation:
N = 5;
syms a
M1 = diag(-a*ones(N-1,1),-1) + diag((1 + 2*a)*ones(N,1),0) + diag(-a*ones(N-1,1),+1);
M2 = diag(+a*ones(N-1,1),-1) + diag((1 - 2*a)*ones(N,1),0) + diag(+a*ones(N-1,1),+1);
M_out = M1\M2;
M_num_1e10 = subs(M_out,a,1e10);
M_num_1e16 = subs(M_out,a,1e16);
vpa(M_num_1e10)
vpa(M_num_1e16)
In that case, you will need the Symbolic Math Toolbox. If you don't have it, I think you should considerer migrating to Python and work with SymPy.
EDIT:
Considering the way you defined your problem, you need extended precision for your computations. The double precision isn't enough. For example, in double precision (1e16+1) has to be rounded to (1e16), in other words (1e16+1)-(1e16) is equal to zero. So your problem starts in the main diagonal of your matrices. MATLAB only provides extended precision through its symbolic toolbox.
If you want to stick with double precision, you may extend the double precision yourself relying on the so called double-double arithmetic. I say that you will have to do it by yourself because I don't think there is a open source double-double library for MATLAB.
I have a matrix X with 3 columns. For the porous of the question X=randn(5,3).
I want to normalize the columns of X S.T. each column will have a 0 mean and a 1 std. I was using the following code:
X=(X-mean(X))./std(X);
I am getting an std of 1. My mean, however, is a very small value close to 0 but not essential 0. I tried playing a bit with the numbers to find an explanation:
X=1:15;
X=reshape(X,[5 3]);
mean(X-mean(X));
Which gives me 0 value for each column.
X=1:15;
X=reshape(X,[5 3]);
mean((X-mean(X))./std(X));
Which does not. But 0/anything is still 0. what am I missing?
Why am I not getting 0 values?
Are the values I am getting good enough for a pre-clustering algorithm normalization?
Here is a version that does what I think you're trying to do... you need to replicate the matrix because X-mean(X) isn't valid (if you're using the standard implementation)-- you can't subtract a 1x3 from a 5x3.
r = 5; c = 3;
X=randn(r,c);
Xm=repmat(mean(X),r,1);
Xstd = repmat(std(X),r,1);
Xn = (X-Xm)./Xstd;
mean(Xn)
std(Xn)
For me this prints out
ans =
1.0e-16 *
-0.6661 0 0.4441
ans =
1.0000 1.0000 1.0000
Which seems like exactly what you're looking for... note the 1e-16 multiplier on the mean values... this is essentially 0, with some floating point error.
I am trying to write a method that checks if a matrix is orthogonal and return TRUE if it is or FALSE if it isn't My problem is that my isequal() is not working how I want it to. Basically I can do the check in two ways based on the two formulas:
ONE way is check to see if the transpose of matrix R is equal to the inverse of matrix R. If they are equal then it is orthogonal. (R'=inv(R))
ANOTHER way is to check and see if matrix R times the transpose of matrix R equals the Identity matrix of R. (R'R=I) If yes then the matrix is orthogonal. I have most been using isequal() but it keeps yielding false. Can someone look at my code and tell me why this would be so?
I use Z=orth(randn(3,3)) to generate random orthogonal matrix and i call my method isortho(Z)
function R = isortho(r)
%isortho(R), which returns true if R is orthogonal matrix, otherwise returns false.
if ismatrix(r) && size(r,1)==size(r,2) %checks if input is square matrix
'------'
trans=transpose(r)
inverted=inv(r)
isequal(trans,inverted)
trans==inverted
isequal(transpose(r),inv(r)) %METHOD ONE
i=size(r,1);
I=eye(i) %creating Identity matrix based on size of r
r*transpose(r)
r*transpose(r)==I %METHOD TWO
%check if transpose of r is times inverse of r equals Identity matrix of r
if (r*transpose(r)==I)
R= 'True';
else
R= 'False';
end
end
end
this is my output:
>> isortho(Z)
ans =
------
trans =
-0.2579 -0.7291 -0.6339
0.8740 0.1035 -0.4747
0.4117 -0.6765 0.6106
inverted =
-0.2579 -0.7291 -0.6339
0.8740 0.1035 -0.4747
0.4117 -0.6765 0.6106
ans = ////isequal(trans,inverted) which yielded 0 false
0
ans = ////trans==inverted
0 1 0
1 0 0
0 1 1
ans = ////isequal(transpose(r),inv(r))
0
I =
1 0 0
0 1 0
0 0 1
ans =
1.0000 0 0.0000
0 1.0000 0.0000
0.0000 0.0000 1.0000
ans =
1 1 0
1 1 0
0 0 1
ans =
False
>>
could someone help me fix this or tell my why the isequal() is failing when matrix inverted and trans appear to be the same?
As stated in the comments, you are running into computer precision issues. For more detail see Why is 24.0000 not equal to 24.0000 in MATLAB? and http://matlabgeeks.com/tips-tutorials/floating-point-comparisons-in-matlab/. This is not a Matlab specific thing, it's a computer thing, and you just have to deal with it.
In your case, you are trying to see whether two things are equal, but the two things are the result of a lot of floating point operations. So they will virtually never be exactly the same, but should always be very close. So, set a tolerance, say 1e-12, and say that the two things are equal if some measure of their difference is below that tolerance, e.g.:
norm(r.'-inv(r))<tol
Which finds the 2-norm of the difference between the two matrices, and then if it is less that tol, this will evaluate to 1, or true.
If I set tol=1e-12, then everything works well. If I set tol=1e-15, everything works well. But if I set tol=1e-16, then everything stops working! This is because the amount of computer precsion error is larger than 1e-16, so the answer to norm(r.'-inv(r)) cannot be accurate to that tolerance. The smallest amount Matlab can distinguish between on my computer is roughly 2.2x10^(-16), so you have to ensure that you tolerance is set well above this value. Setting tol too large will, of course, mean you say some non-orthogonal matrices are orthogonal, but I would not expect tol=1e-14 to give you any significant issues.
I am developing an image processing application for object detection.
At some point I am using log of the generalized eigenvalues vector of two square covariance matrices.
Assume that I have a 9x9 covariance matrix a.
a = rand(9, 9)%just generating random matrix for testing problem easily
b = eig(a, a)%generalized eigenvalues vector containing nine values equal to 1
%so we have b = [1.000, 1.000, 1.000 ... (9 times)]
c = log(b(:)) %we know b contains values of 1. and log(1) is 0.
Even though we know and can debug to see that b contains elements with value 1 and log(1) is 0, content of c is:
1.0e-014 *
0.0222
0.1110
0.0222
0.0222
-0.0777
0
0.0222
0.0888
0
This in my case.
Anyone knows why doesn't c have values of 0?
Thanks.
As #OliCharlesworth commented the values of b aren't really 1. I did the exact same as you did and got the following for b:
b =
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
But when I opened b up in the variable explorer I got the following:
You'll see that there's only really one value of 1 and not 1.000 meaning that there are some trailing values that aren't shown in MatLab. Hence you would get the following for c:
c =
1.0e-15 *
0.2220
-0.4441
0.2220
-0.2220
0.2220
0
0.2220
-0.1110
-0.1110
Notice the 1.0e-15 that's where trailing values are being found.
Trailing zeros after a number, like
1.0000
indicates, that it is not exactly this value.
Try
format long
to see trailing digits up to 15.
If this isn't exactly enough, try out the print command
sprintf('%.50f',b(1))
to see 50 trailing digits of the value b(1).
Thus, the numbers are not exactly 1, neither log becomes exactly 0.
I want to delete several specific values from a matrix (if they exist). It is highly probable that there are multiple copies of the values in the matrix.
For example, consider an N-by-2 matrix intersections. If the pairs of values [a b] and [c d] exist as rows in that matrix, I want to delete them.
Let's say I want to delete rows like [-2.0 0.5] and [7 7] in the following matrix:
intersections =
-4.0000 0.5000
-2.0000 0.5000
2.0000 3.0000
4.0000 0.5000
-2.0000 0.5000
So that after deletion I get:
intersections =
-4.0000 0.5000
2.0000 3.0000
4.0000 0.5000
What's the most efficient/elegant way to do this?
Try this one-liner (where A is your intersection matrix and B is the value to remove):
A = [-4.0 0.5;
-2.0 0.5;
2.0 3.0;
4.0 0.5;
-2.0 0.5];
B = [-2.0 0.5];
A = A(~all(A == repmat(B,size(A,1),1),2),:);
Then just repeat the last line for each new B you want to remove.
EDIT:
...and here's another option:
A = A((A(:,1) ~= B(1)) | (A(:,2) ~= B(2)),:);
WARNING: The answers here are best used for cases where small floating point errors are not expected (i.e. with integer values). As noted in this follow-up question, using the "==" and "~=" operators can cause unwanted results. In such cases, the above options should be modified to use relational operators instead of equality operators. For example, the second option I added would be changed to:
tolerance = 0.001; % Or whatever limit you want to set
A = A((abs(A(:,1)-B(1)) > tolerance) | (abs(A(:,2)-B(2)) > tolerance),:);
Just a quick head's up! =)
SOME RUDIMENTARY TIMING:
In case anyone was really interested in efficiency, I just did some simple timing for three different ways to get the subindex for the matrix (the two options I've listed above and Fanfan's STRMATCH option):
>> % Timing for option #1 indexing:
>> tic; for i=1:10000, index = ~all(A == repmat(B,size(A,1),1),2); end; toc;
Elapsed time is 0.262648 seconds.
>> % Timing for option #2 indexing:
>> tic; for i=1:10000, index = (A(:,1) ~= B(1)) | (A(:,2) ~= B(2)); end; toc;
Elapsed time is 0.100858 seconds.
>> % Timing for STRMATCH indexing:
>> tic; for i=1:10000, index = strmatch(B,A); end; toc;
Elapsed time is 0.192306 seconds.
As you can see, the STRMATCH option is faster than my first suggestion, but my second suggestion is the fastest of all three. Note however that my options and Fanfan's do slightly different things: my options return logical indices of the rows to keep, and Fanfan's returns linear indices of the rows to remove. That's why the STRMATCH option uses the form:
A(index,:) = [];
while mine use the form:
A = A(index,:);
However, my indices can be negated to use the first form (indexing rows to remove):
A(all(A == repmat(B,size(A,1),1),2),:) = []; % For option #1
A((A(:,1) == B(1)) & (A(:,2) == B(2)),:) = []; % For option #2
The simple solution here is to look to set membership functions, i.e., setdiff, union, and ismember.
A = [-4 0.5;
-2 0.5;
2 3;
4 0.5;
-2 0.5];
B = [-2 .5;7 7];
See what ismember does with the two arrays. Use the 'rows' option.
ismember(A,B,'rows')
ans =
0
1
0
0
1
Since we wish to delete rows of A that are also in B, just do this:
A(ismember(A,B,'rows'),:) = []
A =
-4 0.5
2 3
4 0.5
Beware that set membership functions look for an EXACT match. Integers or multiples of 1/2 such as are in A satisfy that requirement. They are exactly represented in floating point arithmetic in MATLAB.
Had these numbers been real floating point numbers, I'd have been more careful. There I'd have used a tolerance on the difference. In that case, I might have computed the interpoint distance matrix between the two sets of numbers, removing a row of A only if it fell within some given distance of one of the rows of B.
You can also abuse the strmatch function to suit your needs: the following code removes all occurences of a given row b in a matrix A
A(strmatch(b, A),:) = [];
If you need to delete more than one row, such as all rows from matrix B, iterate over them:
for b = B'
A(strmatch(b, A),:) = [];
end
Not sure when this function was introduced (using 2012b) but you can just do:
setdiff(A, B, 'rows')
ans =
-4.0000 0.5000
2.0000 3.0000
4.0000 0.5000
Based on:
A = [-4.0 0.5;
-2.0 0.5;
2.0 3.0;
4.0 0.5;
-2.0 0.5];
B = [-2.0 0.5];