A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm
Abstract
:1. Introduction
2. Comparing the Derivations
2.1. Common Starting Point
2.2. Lubich Derivation
2.3. Mohankumar–Carrington Derivation
3. Test Calculations for Real-Time Propagation
4. Imaginary Time Propagation
4.1. An Error Bound from a Geometric Series
4.2. An Error Bound from the First Term in Equation (27)
4.3. Test Calculations
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
References
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Time-Step | 689.11 | 661.6 | 566.49 | 526.1 | 463.57 | 421.5 | 378.35 | 339.3 |
Iterations | 200 | 208 | 200 | 215 | 200 | 220 | 200 | 220 |
12 | 1.298 × 10 | 1.081 × 10 | 4.925 × 10 | 2.029 × 10 | 2.319 × 10 | 3.604 × 10 | 3.287 × 10 | 7.618 × 10 |
23 | 1.149 × 10 | 4.520 × 10 | 3.644 × 10 | 6.550 × 10 | 3.314 × 10 | 2.538 × 10 | 8.968 × 10 | 1.617 × 10 |
58 | 4.238 × 10 | 3.929 × 10 | 6.506 × 10 | 1.114 × 10 | 2.821 × 10 | 5.802 × 10 | 4.718 × 10 | 1.077 × 10 |
89 | 6.756 × 10 | 1.854 × 10 | 2.126 × 10 | 1.396 × 10 | 7.604 × 10 | 2.442 × 10 | 7.601 × 10 | 1.121 × 10 |
120 | 9.161 × 10 | 3.006 × 10 | 4.231 × 10 | 8.909 × 10 | 1.044 × 10 | 1.852 × 10 | 5.667 × 10 | 1.712 × 10 |
200 | 4.398 × 10 | 2.536 × 10 | 3.462 × 10 | 3.918 × 10 | 9.433 × 10 | 1.276 × 10 | 1.619 × 10 | 1.741 × 10 |
400 | 1.931 × 10 | 5.332 × 10 | 3.017 × 10 | 3.502 × 10 | 1.824 × 10 | 2.030 × 10 | 9.238 × 10 | 1.605 × 10 |
Percentage Increase | |||
---|---|---|---|
1 × 10 | 6.891 × 10 | 6.616 × 10 | 4.15 |
1 × 10 | 5.665 × 10 | 5.261 × 10 | 7.68 |
1 × 10 | 4.632 × 10 | 4.215 × 10 | 9.97 |
1 × 10 | 3.784 × 10 | 3.393 × 10 | 11.5 |
1 × 10 | 3.083 × 10 | 2.739 × 10 | 12.6 |
1 × 10 | 2.509 × 10 | 2.214 × 10 | 13.3 |
n | m | c | Err | Equation (33) | Equation (32) | SL Error |
---|---|---|---|---|---|---|
100 | 12 | 8 | 9.848 × 10 | 3.125 × 10 | 4.243 × 10 | × 10 |
100 | 22 | 8 | 6.149 × 10 | 1.196 × 10 | 1.341 × 10 | 3.545 × 10 |
100 | 22 | 18 | 1.484 × 10 | 9.009 × 10 | 8.576 × 10 | 1.437 × 10 |
200 | 12 | 6 | 7.919 × 10 | 2.357 × 10 | 3.027 × 10 | 1.848 × 10 |
200 | 20 | 8 | 4.160 × 10 | 1.404 × 10 | 1.602 × 10 | 3.545 × 10 |
200 | 20 | 40 | 2.594 × 10 | 9.672 × 10 | --- | 2.476 × 10 |
400 | 12 | 5 | 1.213 × 10 | 4.136 × 10 | 5.155 × 10 | 4.219 × 10 |
400 | 20 | 5 | 2.685 × 10 | 4.638 × 10 | 5.209 × 10 | 4.219 × 10 |
1200 | 100 | 20 | 1.751 × 10 | 7.231 × 10 | 6.541 × 10 | 1.766 × 10 |
1200 | 100 | 40 | 2.842 × 10 | 8.706 × 10 | 4.467 × 10 | 1.191 × 10 |
4000 | 12 | 15 | 1.224 × 10 | 3.731 × 10 | 8.790 × 10 | 2.094 × 10 |
4000 | 32 | 15 | 4.078 × 10 | 1.105 × 10 | 1.064 × 10 | 1.096 × 10 |
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Mohankumar, N.; Carrington, T. A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm. Mathematics 2019, 7, 1109. https://doi.org/10.3390/math7111109
Mohankumar N, Carrington T. A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm. Mathematics. 2019; 7(11):1109. https://doi.org/10.3390/math7111109
Chicago/Turabian StyleMohankumar, N., and Tucker Carrington. 2019. "A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm" Mathematics 7, no. 11: 1109. https://doi.org/10.3390/math7111109
APA StyleMohankumar, N., & Carrington, T. (2019). A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm. Mathematics, 7(11), 1109. https://doi.org/10.3390/math7111109