Convergent Homotopy Analysis Method for Solving Linear Systems

Table 2

The basic method is convergent (Theorem 1).

Matrix	Method	𝜌(𝐺)	Convergence interval	ℏopt	𝜌(𝐺ℏopt)
pde225	SOR (𝑤=𝑟=1)	0.9776	(−1, −0.0743)	−0.7423	0.7768
pde225	AOR (𝑟=1, 𝑤=0.8)	0.8053	(−1, −0.8096)	−0.9400	0.7739
pde225	SSOR (𝑤=𝑟=0.5)	0.8048	(−1.4441, −1)	−1.2941	0.7474
pde225	SAOR (𝑟=0.5, 𝑤=1)	0.7773	(−1, −0.6038)	−0.8920	0.6673
cage5	SOR (𝑤=𝑟=1)	0.3388	(−1.2814, −1)	−1.1591	0.2314
cage5	AOR (𝑟=1.2, 𝑤=1.9)	0.9240	(−1, −0.0589)	−0.68	0.3355
cage5	SSOR (𝑤=𝑟=0.5)	0.6427	(−1.7235, −1)	−1.5235	0.4557
cage5	SAOR (𝑟=0.1, 𝑤=0.7)	0.5590	(−1.5609, −1)	−1.3756	0.3890
pivtol	SOR (𝑤=𝑟=0.1)	0.9487	(−3.9226, −1)	−2.7526	0.8588
pivtol	AOR (𝑟=0.3, 𝑤=0.6)	0.9958	(−1, −0.0128)	−0.7300	0.7626
pivtol	SSOR (𝑤=𝑟=0.2)	0.8005	(−1.4850, −1)	−1.3450	0.7317
pivtol	SAOR (𝑟=0.1, 𝑤=0.4)	0.9190	(−1, −0.2198)	−0.83	0.6943
bfwb782	SOR (𝑟=𝑤=1)	0.3732	(−1.2270, −1)	−1.1470	0.3024
bfwb782	AOR (𝑟=1.2, 𝑤=1.9)	0.9942	(−1, −0.0045)	−0.65	0.3988
bfwb782	SSOR (𝑤=𝑟=0.5)	0.5984	(−1.6665, −1)	−1.4704	0.4103
bfwb782	SAOR (𝑟=0.1, 𝑤=0.7)	0.5123	(−1.5123, −1)	−1.3423	0.3453
bfwb782	Richardson	0.999998	(−8.5201×10^4, −1)	−8.0701×10^4	0.8952