Table 4 Residual and CPU time in seconds for solving Example 3.4 by four methods for different sizes.

From: Two fast algorithms for finding the solution of the lower Hessenberg quasi-Toeplitz linear system from Markov chain

n

Back-slash

QR

LU

Our Algorithm 1

Our Algorithm 2

RES

Time (s)

RES

Time (s)

RES

Time (s)

RES

Time (s)

RES

Time(s)

\(2^{10}\)

3.2196e−15

0.0210

3.2196e−15

0.0168

5.6621e−15

0.0591

2.8866e−15

0.0098

4.1078e−15

0.0091

\(2^{11}\)

4.3299e−15

0.1247

4.3299e−15

0.1016

5.6621e−15

0.4107

4.9960e−15

0.0122

4.9960e−15

0.0112

\(2^{12}\)

6.1062e−15

0.7124

6.1062e−15

0.5967

9.6589e−15

2.7468

5.5511e−15

0.0360

5.5511e−15

0.0271

\(2^{13}\)

1.0325e−14

4.7019

1.0325e−14

4.4022

1.3434e−14

21.0452

4.8295e−15

0.0538

4.8850e−15

0.0397

\(2^{14}\)

1.2879e−14

33.5569

1.2879e−14

33.3971

2.9643e−14

166.9035

3.9413e−15

0.1067

3.9302e−15

0.0700