Table 2 Residual and CPU time in seconds for solving Example 3.2 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)

\(2^{10}\)

6.6613e−16

0.0235

6.6613e−16

0.0215

4.3299e−15

0.3554

4.3299e−15

0.0084

4.1078e−15

0.0065

\(2^{11}\)

5.5511e−16

0.1272

5.5511e−16

0.0995

1.2212e−15

1.3883

4.4964e−15

0.0132

4.2744e−15

0.0096

\(2^{12}\)

7.7716e−16

0.6192

7.7716e−16

0.5956

1.3323e−15

5.5822

4.2744e−15

0.0288

4.2188e−15

0.0244

\(2^{13}\)

6.6613e−16

4.7716

6.6613e−16

4.2961

4.1078e−15

27.7111

4.1078e−15

0.0576

4.2188e−15

0.0466

\(2^{14}\)

7.7716e−16

33.2503

7.7716e−16

32.6906

1.6653e−15

185.3402

1.2212e−14

0.1109

1.2323e−14

0.0682