Table 1.

The various steps of the proposed TOA for the first iteration in sphere function solving.

	Step 1		Step 2	Step 3	Step 4		Step 5				Step 6			Final
	$\mathit{X}$		F(X)	S	X		XM		X		X		F(X)
	${\mathit{x}}_1$	${\mathit{x}}_2$			${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_1^{\mathit{M}}$	${\mathit{x}}_2^{\mathit{M}}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_1$	${\mathit{x}}_2$
$X_1$	98.12666	72.68003	14911.23		19.65399	20.55081	14.63504	17.21673	17.16322	8.697356	17.16322	8.697356	370.22	3.97 × 10−55
$X_2$	37.61062	17.84777	1733.102		29.87663	11.62747	15.89913	12.95704	21.92269	5.767422	21.92269	5.767422	513.8675	8.85 × 10−56
$X_3$	19.00723	−69.7136	5221.263		10.5817	−38.3855	17.90698	10.5605	10.17539	−32.2784	10.17539	−32.2784	1145.432	1.84 × 10−57
$X_4$	51.43423	49.23254	5069.323		25.26888	26.46832	15.97408	−0.14922	24.50555	22.1087	24.50555	22.1087	1089.317	1.71 × 10−57
$X_5$	14.63504	17.21673	510.6002	$X_5$	12.52158	17.21673	17.16322	8.697356	9.967937	14.45587	9.967937	14.45587	308.332	4.15 × 10−57
$X_6$	59.62064	51.23127	6179.263		52.33316	48.86162	22.99721	0.240222	−22.3172	4.418826	−22.3172	4.418826	517.582	1.35 × 10−55
$X_7$	84.65901	−87.3821	14802.78		32.22874	56.36313	16.52373	0.837165	7.995059	−0.20421	7.995059	−0.20421	63.96267	6.64 × 10−54
$X_8$	54.2485	−17.3096	3242.523		45.23606	−5.51647	9.916097	3.280799	19.61827	0.864246	19.61827	0.864246	385.6233	1.88 × 10−56
$X_9$	−46.1119	77.9884	8208.503		−35.6225	67.49466	11.12887	2.97873	−5.91138	19.88028	−5.91138	19.88028	430.1698	5.42 × 10−58
$X_{10}$	89.51302	75.22706	13671.69		77.52489	65.93948	9.235507	4.85668	56.40553	47.04885	56.40553	47.04885	5395.179	9.51 × 10−56
Best Solution: $x_1=-$2.0213 × 10−29, $x_2=-$1.1566 × 10−29 and $F\left(X\right)=$ 5.4232 × 10−58