anonymous
  • anonymous
let A and B be two symmetric matrices of the same order. Develop an algorithm to compute C=A+B,taking advantage of symmetry for each matrix.Your algorithm should overwrite B with C. What is the flop count?
Mathematics
katieb
  • katieb
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myininaya
  • myininaya
So we know that this means A[i,j]=A[j,i] , B[i,j]=B[j,i], and C[i,j]=C[j,i]=A[i,j]+B[i,j] A and B are symmetrical. This means A+B is symmetrical.
anonymous
  • anonymous
okay, but how to write the pseudo code for the algorithm?
myininaya
  • myininaya
You need to make a loop I think the loops usually use something like for...do

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anonymous
  • anonymous
okay..it is correct if i do like this : Input: Dim n for \[a _{ij}\], j>=i and \[b _{ij}\],j>=i where i,j=1,...,n Output : B=C Step 1 : for j=1,...,n for i=1,...,j \[b _{ij}^{new}=a _{ij}+b _{ij}\]
anonymous
  • anonymous
but, i don't have any idea to make this as programming.
myininaya
  • myininaya
I'm thinking... Does your algorithm use the fact that A and B are symmetrical? We should be able to do something like Step 1 : for j=1,...,n for i=1,...,j if i does not equal j, then output cij=cji=aij+bij if i equals j, then cij=aij+bij What do you think of this part? Do you think you can use it? Just trying to help.
anonymous
  • anonymous
yes. it's also correct. but do you know how to make it into pseudo code form?such as using mathematica or MATLAB?
myininaya
  • myininaya
err...nope. It has been way to long. I think I could do it with maple after trail and error. But I don't have maple on this computer.
anonymous
  • anonymous
Owh, it's okay. One more question, do you know how to do calculate flop-count? My friend get \[\frac{ n(n+1) }{ 2 }\] is it correct?
myininaya
  • myininaya
actually that is what I got. You want to know how I figured it out... Ok we want to basically know how many operations we did. And I had to look up the definition of flops. So we want to count the number of operations we did. I counted my diagonal elements which were n. Then for the first row we had (n-1) entries left to calculate second row we had (n-2) entries left to calculate third row we had (n-3) entries left to calculate ....... nth row we had (n-n) entries left to calculate So we had n diagonal entries + (n-1)+(n-2)+...+(n-n) other entries to calculate. So we have n+sum(n-i, i=1..n) Do you understand how I got this so far? We can simplify this to what your friend got.
myininaya
  • myininaya
\[n+\sum_{i=1}^{n}(n-i)\] \[n+\sum_{i=1}^{n}n - \sum_{i=1}^{n}i\] \[n+n(n)-\frac{n(n+1)}{2}\] \[n(n+1)-\frac{n(n+1)}{2}=\frac{2n(n+1)-n(n+1)}{2}=\frac{(n+1)(2n-n)}{2}=\frac{1n(n+1)}{2}\]
myininaya
  • myininaya
I counted each entry we had to calculate. I know the diagonal entries will be repeated once but all other entries will be counted twice. So you know if we had a 3 by 3 symmetric matrix. You find the sum of the operations we did by counting the diagonal entries and the entries above the diagonal. So for a 3 b3 symmetric matrix, we have 3 diagonal entries then the number of entries above that diagonal is (3-1)+(3-2)+(3-3) (3-1)=2 that is how many are above the diagonal in the first row (3-2)=1 that is how many are above the diagonal in the second row (3-3)=0 that is how many are above the diagonal in the 3rd row Now I ignored everything below the diagonal because we already performed those operations above the diagonal. We were suppose to use the fact that the matrix was symmetric after all.
myininaya
  • myininaya
So using this we have 3+(3-1)+(3-2)+(3-3) or you can use the simplified formula 3(3+1)/2
myininaya
  • myininaya
Makes sense how I found the flops?
anonymous
  • anonymous
yes. i'm still trying to understand what you explain. =)
myininaya
  • myininaya
[ a11 a12 a13] [ a22 a23] [ a33] Or you could have looked at is in much simpler way lol 3+2+1 =3(3+1)/2
myininaya
  • myininaya
for a 4by 4, we could have said (4+3+2+1)
myininaya
  • myininaya
I made it more complicated than it should have been Do you see what I mean so for an n by n, we do 1+2+...+n
myininaya
  • myininaya
But either way the formula can be simplified to that n(n+1)/2 one
anonymous
  • anonymous
ohhh..now i understand already how to get n(n+1)/2. it's simple but quite tricky. =)
myininaya
  • myininaya
|dw:1381169885319:dw| 1+2+...(n-1)+n \[\sum_{i=1}^{n}i=\frac{n(n+1)}{2}\]
myininaya
  • myininaya
|dw:1381169994491:dw|
anonymous
  • anonymous
woww..you tried so hard to make me understand. thank you so much. now, honestly, i understand already. =) are you a lecturer?
myininaya
  • myininaya
Yes.
anonymous
  • anonymous
oh, i see. until right now, i'm still dont know how to develop an algorithm to compute c=a+b.
myininaya
  • myininaya
Is it because you aren't sure what is acceptable to write exactly?
myininaya
  • myininaya
And also because you want to input in the matlab and see if it works?
anonymous
  • anonymous
yes.
myininaya
  • myininaya
Yeah. I will have to do some review on that part. I'm going to class soon. I can't promise I will look at it tonight but I think @zarkon might know matlab. I could be wrong. No promises. lol.
anonymous
  • anonymous
hehe. It's okay. You helped me a lot today. Actually, i'm going to sleep already because in Malaysia, right now is midnight. Thank you so much for your helped. May God bless you. =)

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