A Parallel Preconditioned Modified Conjugate Gradient Method for Large Sylvester Matrix Equation

Table 2

Numerical results of Example 8.

	𝑝	4	8	12	16
Parameter iterative method (𝛼=−0.0621, 0.0621, −0.0314, 0.0314)	𝑇 (s)	—	—	—	—
	𝐾	—	—	—	—
	𝐸1	—	—	—	—
	Δ	—	—	—	—
MCG algorithm	𝑇 (s)	612.0169	336.6452	237.2380	179.1261
	𝐾	331	331	331	331
	𝐸1	0.23	0.21	0.20	0.20
	Δ	8.35𝑒−11	8.35𝑒−11	8.35𝑒−11	8.35𝑒−11
The algorithm in the paper (𝛼=−0.0621)	𝑇 (s)	144.6455	79.1208	54.7878	41.6944
	𝐾	80	79	81	81
	𝐸1	1	0.91	0.88	0.87
	Δ	3.64𝑒−11	5.31𝑒−11	9.44𝑒−11	9.44𝑒−11
The algorithm in the paper (𝛼=0.0621)	𝑇 (s)	221.1802	120.1056	84.8724	64.2965
	𝐾	134	133	133	133
	𝐸1	1	0.92	0.87	0.86
	Δ	7.70𝑒−11	9.87𝑒−11	9.87𝑒−11	9.87𝑒−11
The algorithm in the paper (𝛼=−0.0314)	𝑇 (s)	169.8777	91.0232	65.0475	48.8154
	𝐾	104	103	103	103
	𝐸1	1	0.93	0.87	0.87
	Δ	8.07𝑒−11	6.30𝑒−11	6.30𝑒−11	6.30𝑒−11
The algorithm in the paper (𝛼=0.0314)	𝑇 (s)	245.1565	132.7123	93.1277	70.7914
	𝐾	157	154	154	154
	𝐸1	1	0.92	0.88	0.87
	Δ	7.03𝑒−11	5.91𝑒−11	5.91𝑒−11	5.91𝑒−11

Note:
(1) Here 𝑝, 𝑇 (s),  𝐾, 𝐸1, and Δ represent the process number, time, iteration number, absolute parallel efficiency, and error, respectively.
(2) Because the computational time of one processor is too long, we take the minimal time with one of multiprocessors as a reference to calculate the relative speedup and relative parallel efficiency.