Nume: Vitoga George Patrick
Grupa/Serie: 332CA
Neopt:
I tackled the "unoptimised" version of algorithm using some
additional matrices to store some of the partial results like
A * B, A * A' and A * B * B'.
The multiplication is using the basic "3-for loop way", but I took
advantage that A is a upper triangular matrix, so I selected
only the elements positioned after k in the third for-loop.
When I encountered an opperation with a transposed matrix as one
of the opperands, I saved some computational effort by
replacing the "transpose" opperation with "iterating throungh columns".
The final matrix is made of sum of the partial matrices m_ABBt
and m_AtA.
Opt:
The "optimised" version is an upgrade of "neopt". The update
consists of using pointers to iterate through matrices and also
using "register" for loop related constant variables.
Also, in the last opperation, I removed a for-loop due to the
ability to iterate using pointers, so I iterated up to N * N.
Using all this tricks, I managed to beat the "Unopt" version.
Blas:
This version is the fastest among them all.
The "Basic Linear Algebra Subprograms" is amazing, because is
capable of doing the basic opperations in no time.
I used only the cblas_dtrmm() and cblas_dgemm().
With cblas_dtrmm(), I was able to multiply a triangular matrix
with another matrix and with cblas_dgemm() I was able to multiply
any 2 matrices. Moreover, I even summed 2 matrices using cblas_dgemm()
intern capabilities.
Performances:
An analysis on the input (N = 400) leads us to the following results:
|------------------------------------------------------------------------------|
| NEOPT OPT BLAS |
|------------------------------------------------------------------------------|
|Running time: 1.3291 0.2922 0.0512 |
| |
|Instruction fetches: |
| - I1 1358 1365 1457 |
| - LLi 1325 1327 1382 |
| |
|Data access: |
| - D1 76 mil. 52 mil. 46000 |
| - LLd 100000 93000 40000 |
| |
|Branch prediction: |
| - Miss rate 0.3% 0.3% 0.9% |
|------------------------------------------------------------------------------|
Considering the input size and results, it s clear that Blas and Opt are better
than Neopt.
Results become even clearer when input size is increasing.