Numerical strategies for recursive least squares solutions to the matrix equation AX = B

Figures & data

Table 1. Execution times in seconds of the recursive solution of the OPP when one single column is added.

	Alg. 5.1	Alg. 5.2	${\displaystyle \frac{\mathrm{A}\mathrm{l}\mathrm{g}.5.1}{\mathrm{A}\mathrm{l}\mathrm{g}.5.2}}$
m	afresh	recursive
(a) Panel A: Orthogonal Procrustes updating with n = 1000 and k = 1.
50	239	6	37
100	273	11	26
250	285	33	9
500	387	178	2
(b) Panel B: Orthogonal Procrustes updating with n = 5000, k = 1.
50	25723	7	3701
100	25510	14	1800
250	25985	71	366
500	26973	307	88

Notes: Results are presented where the matrices in the original Procrustes have m = 50, 100, 250, 500 rows and n = 1000 or 5000 columns and k = 1 column is added.

Table 2. Execution times in seconds of the recursive solution of the OPP when new columns are added.

	Alg. 5.1	Alg. 5.2	${\displaystyle \frac{\mathrm{A}\mathrm{l}\mathrm{g}.5.1}{\mathrm{A}\mathrm{l}\mathrm{g}.5.2}}$
k	afresh	recursive
(a) Panel A: Orthogonal Procrustes updating with m = 100 and n = 1000.
5	2.0	0.1	24
25	2.1	0.1	22
100	2.7	0.2	15
250	3.7	0.5	8
500	6.5	1.6	4
(b) Panel B: Orthogonal Procrustes updating with m = 500 and n = 10, 000.
5	2216.9	11.3	196
25	2117.3	11.3	187
100	2069.8	14.1	147
250	3.7	0.5	8
500	6.5	1.6	4

Notes: Results are presented where the matrices in the original OPP have m = 100 or 500 and n = 1000 and k = 5, 25, 100, 250, 500 new columns are added.

Table 3. Total execution times in seconds of the recursive solution of the OPP when a new single row of data is added 100 times.

	Alg. 5.1	Alg. 5.3	${\displaystyle \frac{\mathrm{A}\mathrm{l}\mathrm{g}.5.1}{\mathrm{A}\mathrm{l}\mathrm{g}.5.3}}$
m	afresh	recursive
(a) Panel A: Orthogonal Procrustes updating with n = 256 and l = 1.
2000	10	6.1	2
5000	20.9	6.1	3
1000	39.3	6.1	6
15,000	57.1	6.1	9
20,000	74.9	6.1	12
(b) Panel B: Orthogonal Procrustes updating with n = 1024 and l = 1.
2000	369.8	371.9	1
5000	546.3	371.9	1
10,000	823.4	371.9	2
15,000	1090.7	371.9	3
20,000	1369.8	371.9	4

Notes: Results are presented when the matrices in the original OPP have dimensions m = 2000, 5000, 10, 000, 15, 000, 20, 000 and n = 256, 1024, and during the updating one extra row is added (l = 1).