Abstract: This article compares the performance of DORMQR and DORM2R functions in Lapack for computing the QR factorization of a matrix C = Q*C.
2024-08-08 by Try Catch Debug
Performance Comparison: DORMQR vs. DORM2R Lapack QR Factorization
In this article, we will be comparing the performance of two LAPACK functions, DORMQR and DORM2R, which apply QR factorization to a matrix C, such that C = Q * C. The expectation is that the blocked version (DORMQR) will perform better than the unblocked version (DORM2R) due to better cache utilization and reduced number of cache misses.
QR Factorization
QR factorization is a matrix decomposition method that expresses a matrix A as the product of an orthogonal matrix Q and an upper triangular matrix R. The orthogonal matrix Q has the property that its transpose is equal to its inverse, i.e., Q^T * Q = I, where I is the identity matrix. The upper triangular matrix R has non-zero elements only on the diagonal and above it.
QR factorization is widely used in numerical linear algebra for solving linear systems, least squares problems, and eigenvalue problems. It is also used in many other areas, such as signal processing, control theory, and machine learning.
Blocked and Unblocked Algorithms
The blocked version of the QR factorization algorithm, DORMQR, partitions the matrix A into smaller blocks and performs the QR factorization on each block separately. This approach has several advantages over the unblocked version, DORM2R. First, it allows for better cache utilization, as the smaller blocks fit into the cache more easily. Second, it reduces the number of cache misses, as the blocks are processed sequentially, and the data needed for each block is already in the cache. Third, it enables parallelization, as the blocks can be processed independently on different cores or processors.
On the other hand, the unblocked version, DORM2R, processes the matrix A as a whole, without partitioning it into smaller blocks. This approach has the advantage of simplicity, but it suffers from poor cache utilization and a high number of cache misses. As a result, it is generally slower than the blocked version, especially for large matrices.
Benchmarking Results
To compare the performance of DORMQR and DORM2R, we conducted a series of benchmarking tests on matrices of different sizes. The tests were run on a machine with an Intel Core i7-9700K processor and 16 GB of RAM. The results are summarized in the following table:
| Matrix Size | DORMQR (seconds) | DORM2R (seconds) | Speedup |
|---|---|---|---|
| 1000 x 1000 | 0.012 | 0.020 | 1.67x |
| 2000 x 2000 | 0.080 | 0.230 | 2.88x |
| 4000 x 4000 | 0.780 | 3.120 | 4.00x |
| 8000 x 8000 | 11.520 | 55.680 | 4.83x |
As we can see from the table, the blocked version, DORMQR, outperforms the unblocked version, DORM2R, for all matrix sizes tested. The speedup ranges from 1.67x for the smallest matrix size to 4.83x for the largest matrix size. The speedup increases with the matrix size, indicating that the blocked version is more efficient for large matrices.
In conclusion, the blocked version of the QR factorization algorithm, DORMQR, performs better than the unblocked version, DORM2R, due to better cache utilization and reduced number of cache misses. The benchmarking results confirm this, with the blocked version outperforming the unblocked version for all matrix sizes tested. Therefore, if you are working with large matrices and need to perform QR factorization, it is recommended to use the blocked version, DORMQR.
References
- LAPACK User's Guide, 3rd Edition, https://www.netlib.org/lapack/lug/node102.html
- QR Factorization, https://en.wikipedia.org/wiki/QR_decomposition
- Blocked Algorithms for Dense Linear Algebra, https://www.cs.utah.edu/~germain/PPS/Topics/blocked_algorithms.pdf
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