The Impact of Compilation Flags and Choosing Single - or -Double-Precision Variables in Linear Systems Solvers

BRUM, R. C.; DE CASTRO, M. C. S.; FARIA, C. O.

doi:10.5540/tcam.2023.024.02.00319

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Articles • Trends Comput. Appl. Math. 24 (2) • Apr-Jun 2023 • https://doi.org/10.5540/tcam.2023.024.02.00319 copy

The Impact of Compilation Flags and Choosing Single - or -Double-Precision Variables in Linear Systems Solvers

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

This paper intends to show the impact of compiler optimization flags and the variable’s precision on direct methods to solve linear systems. The chosen six methods are simple direct methods, so our work could be a study for new researchers in this field. The methods are LU Decomposition, LDU Decomposition, Gaussian Elimination, Gauss-Jordan Elimination, Cholesky Decomposition, and QR Decomposition using the Gram-Schmidt orthogonalization process. Our study showed a huge difference in time between single-and double-precision in all methods, but the error encountered in the single-precision was not so high. Also, the best flags to these methods were the ‘-O3’ and the ‘-Ofast’ ones.

Keywords:
linear systems; compiler flags; optimization; variable precision

Method	# FLOPs	Reference
Gaussian Elimination	O(n³/3)	(Trefethen, 1985) ¹⁶16 L.N. Trefethen. Three mysteries of Gaussian elimination. ACM SIGNUM Newsletter, 20(4) (1985), 2-5. doi:10.1145/1057954.1057955. https://doi.org/10.1145/1057954.1057955...
Gauss-Jordan Elimination	O(n³/2)	(Meyer, 2000) ¹¹11 C.D. Meyer. “Matrix analysis and applied linear algebra”, volume 71. Siam (2000).
LU Decomposition	O(n³/3)	(Trahan et al, 2006) ¹⁵15 J. Trahan, A. Kaw & K. Martin. Computational time for finding the inverse of a matrix: LU decomposition vs. naive gaussian elimination. University of South Florida, (2006).
LDU Decomposition	O(n³/3)	(Meyer, 2000) ¹¹11 C.D. Meyer. “Matrix analysis and applied linear algebra”, volume 71. Siam (2000).
Cholesky Decomposition	O(n³/3)	(Higham, 2009) ⁸8 N.J. Higham. Cholesky factorization. Wiley Interdisciplinary Reviews: Computational Statistics, 1(2) (2009), 251-254. doi:10.1002/wics.18. https://doi.org/10.1002/wics.18...
QR Decomposition (using GS process)	O(2n³/3)	(Trefethen, 1985) ¹⁶16 L.N. Trefethen. Three mysteries of Gaussian elimination. ACM SIGNUM Newsletter, 20(4) (1985), 2-5. doi:10.1145/1057954.1057955. https://doi.org/10.1145/1057954.1057955...

	GCC version 5.4.0		GCC version 7.1.0
Flag	Single-precision	Double-precision	Single-precision	Double-precision
-O0	284.24 s	290.12 s	286.43 s	296.84 s
-O	108.49 s	118.74 s	108.24 s	117.98 s
-O1	108.43 s	118.80 s	107.69 s	118.04 s
-O2	59.12 s	80.34 s	59.10 s	76.65 s
-O3	46.29 s	78.98 s	45.91 s	70.99 s
-Ofast	46.85 s	78.96 s	61.40 s	79.57 s
-Og	106.61 s	118.12 s	107.63 s	117.78 s
-Os	77.27 s	79.53 s	64.14 s	79.33 s

	GCC version 5.4.0		GCC version 7.1.0
Flag	Single-precision	Double-precision	Single-precision	Double-precision
-O0	284.45 s	290.67 s	288.24 s	297.53 s
-O	107.82 s	113.24 s	108.07 s	119.13 s
-O1	107.86 s	121.35 s	108.10 s	119.16 s
-O2	59.31 s	71.79 s	59.47 s	80.94 s
-O3	46.17 s	64.99 s	46.55 s	80.10 s
-Ofast	47.45 s	79.48 s	52.65 s	114.12 s
-Og	109.71 s	117.34 s	107.40 s	118.95 s
-Os	74.91 s	86.43 s	63.69 s	87.22 s

	GCC version 5.4.0		GCC version 7.1.0
Flag	Single-precision	Double-precision	Single-precision	Double-precision
-O0	410.45 s	413.15 s	417.31 s	421.74 s
-O	139.99 s	147.88 s	139.67 s	149.60 s
-O1	139.30 s	147.93 s	139.74 s	150.08 s
-O2	205.27 s	316.08 s	205.34 s	317.09 s
-O3	204.88 s	316.82 s	205.79 s	317.40 s
-Ofast	205.56 s	316.90 s	207.51 s	317.12 s
-Og	139.93 s	146.45 s	138.20 s	147.11 s
-Os	204.24 s	316.86 s	204.45 s	316.87 s

	GCC version 5.4.0		GCC version 7.1.0
Flag	Single-precision	Double-precision	Single-precision	Double-precision
-O0	799.43 s	823.06 s	821.80 s	833.27 s
-O	342.39 s	467.30 s	338.41 s	463.62 s
-O1	342.83 s	466.87 s	340.40 s	463.72 s
-O2	323.91 s	441.57 s	323.96 s	441.29 s
-O3	323.88 s	441.49 s	324.43 s	442.05 s
-Ofast	322.22 s	442.22 s	322.51 s	442.19 s
-Og	342.02 s	463.40 s	340.54 s	464.37 s
-Os	323.45 s	441.58 s	317.42 s	440.68 s

	GCC version 5.4.0		GCC version 7.1.0
Flag	Single-precision	Double-precision	Single-precision	Double-precision
-O0	122.38 s	124.96 s	121.85 s	124.12 s
-O	30.60 s	33.93 s	30.17 s	33.66 s
-O1	30.51 s	33.91 s	30.17 s	33.62 s
-O2	30.47 s	34.04 s	30.14 s	33.81 s
-O3	30.54 s	34.30 s	30.22 s	33.35 s
-Ofast	19.55 s	40.95 s	19.68 s	34.81 s
-Og	47.80 s	49.64 s	47.55 s	49.64 s
-Os	31.34 s	35.05 s	30.98 s	35.70 s

	GCC version 5.4.0		GCC version 7.1.0
Flag	Single-precision	Double-precision	Single-precision	Double-precision
-O0	970.45 s	978.21 s	972.52 s	985.82 s
-O	202.20 s	215.15 s	203.25 s	219.55 s
-O1	202.25 s	215.15 s	203.21 s	219.49 s
-O2	202.23 s	213.81 s	234.20 s	238.09 s
-O3	168.37 s	228.64 s	188.12 s	212.06 s
-Ofast	81.72 s	149.63 s	78.63 s	152.67 s
-Og	224.93 s	238.42 s	224.26 s	245.42 s
-Os	224.47 s	234.26 s	223.36 s	240.93 s

Flag	Single-precision	Double-precision
-O0	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-O	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-O1	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-O2	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-O3	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-Ofast	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-Og	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-Os	1.50 × 10⁻⁴	1.67 × 10⁻¹³

Flag	Single-precision	Double-precision
-O0	-NaN	-NaN
-O	-NaN	-NaN
-O1	-NaN	-NaN
-O2	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-O3	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-Ofast	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-Og	-NaN	-NaN
-Os	1.50 × 10⁻⁴	1.67 × 10⁻¹³

Flag	Single-precision	Double-precision
-O0	-NaN	1.78 × 10⁻²
-O	-NaN	1.78 × 10⁻²
-O1	-NaN	1.78 × 10⁻²
-O2	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-O3	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-Ofast	1.50 × 10⁻⁴	1.67 × 10⁻¹³
-Og	-NaN	1.78 × 10⁻²
-Os	1.50 × 10⁻⁴	1.67 × 10⁻¹³

	Flag ’-O3’		Flag ’-Ofast’
# of unknowns	GCC v. 5.4.0	GCC v. 7.1.0	GCC v. 5.4.0	GCC v. 7.1.0
5000	1.706	1.546	1.685	1.296
10000	1.832	1.864	1.780	1.330
15000	1.930	1.862	1.841	1.378
20000	1.813	1.977	1.609	1.413

	Flag ’-O3’		Flag ’-Ofast’
# of unknowns	GCC v. 5.4.0	GCC v. 7.1.0	GCC v. 5.4.0	GCC v. 7.1.0
5000	1.408	1.721	1.675	2.168
10000	1.835	1.870	1.628	2.271
15000	1.633	1.905	1.517	2.370
20000	1.990	1.947	1.914	2.003

	Flag ’-O3’		Flag ’-Ofast’
# of unknowns	GCC v. 5.4.0	GCC v. 7.1.0	GCC v. 5.4.0	GCC v. 7.1.0
5000	1.546	1.542	1.542	1.528
10000	1.557	1.559	1.554	1.546
15000	1.561	1.561	1.547	1.549
20000	1.558	1.562	1.555	1.563

	Flag ‘-O3’		Flag ‘-Ofast’
# of unknowns	GCC v. 5.4.0	GCC v. 7.1.0	GCC v. 5.4.0	GCC v. 7.1.0
5000	1.363	1.363	1.372	1.371
10000	1.379	1.376	1.369	1.382
15000	1.328	1.313	1.390	1.389
20000	1.398	1.377	1.398	1.376

	Flag ‘-O3’		Flag ‘-Ofast’
# of unknowns	GCC v. 5.4.0	GCC v. 7.1.0	GCC v. 5.4.0	GCC v. 7.1.0
5000	1.123	1.104	2.095	1.769
10000	1.143	1.156	2.177	1.903
15000	1.191	1.192	2.495	2.104
20000	1.215	1.201	1.868	2.368

	Flag ‘-O3’		Flag ‘-Ofast’
# of unknowns	GCC v. 5.4.0	GCC v. 7.1.0	GCC v. 5.4.0	GCC v. 7.1.0
5000	1.358	1.127	1.831	1.942
10000	1.177	1.152	1.748	2.022

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[1] ^*Corresponding author: Cristiane Oliveira Faria - E-mail: cofaria@ime.uerj.br