Convergent Homotopy Analysis Method for Solving Linear Systems

Table 3

The basic method is divergent (Theorem 2).

Matrix	Method	𝜌(𝐺)	Convergence interval	ℏopt	𝜌(𝐺ℏopt)
pde225	SOR (𝑟=𝜔=1.5)	2.4937	(−0.4318, 0)	−0.3	0.7980
pde225	AOR (𝑟=1, 𝜔=1.5)	1.5542	(−0.6816, 0)	−0.5	0.7745
pde225	SSOR (𝑟=2.01, 𝜔=2.01)	1.1112	(0, 17.9804)	12.0540	0.9388
pde225	SAOR (𝑟=2, 𝜔=3)	4.9844	(0, 0.3408)	0.2510	0.7741
cage5	SOR (𝑟=𝜔=2)	1.0150	(−0.9872, 0)	−0.4	0.7425
cage5	AOR (𝑟=1.2, 𝜔=3)	2.0370	(−0.6584, 0)	−0.43	0.3355
cage5	SSOR (𝑟=2.2, 𝜔=2.2)	1.8715	(0, 2.2948)	2.0280	0.7681
cage5	SAOR (𝑟=2, 𝜔=3)	4.0002	(0, 0.6666)	0.458	0.3748
pivtol	SOR (𝑟=𝜔=1)	4.3002	(−0.3773, 0)	−0.33	0.7778
pivtol	AOR (𝑟=0.467, 𝜔=0.981)	1.5280	(−0.7566, 0)	−0.58	0.6766
pivtol	SSOR (𝑟=𝜔=0.5)	2.1132	(−0.6424, 0)	−0.55	0.7306
pivtol	SAOR (𝑟=0.1, 𝜔=0.7)	2.6670	(−0.5454, 0)	−0.47	0.7235
bfwb782	Jacobi	1.0554	(−0.9731, 0)	−0.81305	0.6794
bfwb782	AOR (𝑟=1.2, 𝜔=3)	2.1398	(−0.6352, 0)	−0.4	0.3996
bfwb782	SSOR (𝜔=𝑟=2.2)	2.1411	(0, 1.7528)	1.5550	0.7748
bfwb782	SAOR (𝑟=2, 𝜔=3)	4.0000	(0, 0.6667)	0.45	0.3570