A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra
Abstract
:1. Introduction
2. The Padé-Laplace Theory
3. Ill-Posedness of the PL Method
4. Regularization of the PL Method
Deconvolution of Noisy Data by RPL
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
References
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Padé approximant [0/1] |
2-norm condition number = 1 |
Infinity-norm condition number = 1 |
Padé approximant [1/2] |
2-norm condition number = 1.31 × 104 |
Infinity-norm condition number = 1.43 × 104 |
Padé approximant [2/3] |
2-norm condition number = 1.33 × 108 |
Infinity-norm condition number = 1.55 × 104 |
Padé approximant [3/4] |
2-norm condition number = 6.85 × 1010 |
Infinity-norm condition number = 8.05 × 1010 |
Padé approximant [0/1] |
2-norm condition number = 1 |
Infinity-norm condition number = 1 |
Padé approximant [1/2] |
2-norm condition number = 3.73 × 104 |
Infinity-norm condition number = 3.91 × 104 |
Padé approximant [2/3] |
2-norm condition number = 1.96 × 109 |
Infinity-norm condition number = 2.18 × 109 |
Padé approximant [3/4] |
2-norm condition number = 5.77 × 1012 |
Infinity-norm condition number = 6.69 × 1012 |
Padé approximant [0/1] |
2-norm condition number = 1 |
Infinity-norm condition number = 1 |
Padé approximant [1/2] |
2-norm condition number = 1.2 × 104 |
Infinity-norm condition number = 1.22 × 104 |
Padé approximant [2/3] |
2-norm condition number = 3.38 × 1010 |
Infinity-norm condition number = 3.59 × 1010 |
Padé approximant [3/4] |
2-norm condition number = 3.42 × 1015 |
Infinity-norm condition number = 3.65 × 1015 |
Padé approximant [0/1] |
2-norm condition number = 1 |
Infinity-norm condition number = 1 |
Padé approximant [1/2] |
2-norm condition number = 4.02 × 103 |
Infinity-norm condition number = 5.06 × 103 |
Padé approximant [2/3] |
2-norm condition number = 1.29 × 106 |
Infinity-norm condition number = 1.75 × 106 |
Padé approximant [3/4] |
2-norm condition number = 4.94 × 107 |
Infinity-norm condition number = 6.31 × 107 |
Padé Approximant | 2-Norm Condition Number | Infinity-Norm Condition Number |
---|---|---|
[0/1] | 1 | 1 |
[1/2] | 146 | 177 |
[2/3] | 2.97 × 103 | 4.53 × 103 |
[3/4] | 1.14 × 105 | 1.87 × 105 |
[4/5] | 6.31 × 105 | 1.09 × 106 |
[5/6] | 4.28 × 107 | 7.73 × 107 |
[6/7] | 8.14 × 109 | 1.53 × 1010 |
[7/8] | 1.15 × 1013 | 2.3 × 1013 |
[8/9] | 3.96 × 1015 | 8.12 × 1015 |
[9/10] | 1.56 × 1017 | 2.13 × 1017 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[0/1] | −7.3144688 × 10−3 | 0.0000000 × 100 | 1.8677469 × 102 | 0.0000000 × 100 |
[1/2] [1/2] | −5.4778310 × 10−2 −4.0862030 × 10−3 | 0.0000000 × 100 0.0000000 × 100 | 4.7728408 × 101 1.5183015 × 102 | 0.0000000 × 100 0.0000000 × 100 |
[2/3] [2/3] [2/3] | −3.1173129 × 100 −5.0003912 × 10−2 −4.0097848 × 10−3 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 2.4292682 × 101 4.7223963 × 101 1.5029938 × 102 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[3/4] [3/4] [3/4] [3/4] | 5.3238003 × 10−1 −5.4445912 × 10−2 −3.3152082 × 10−2 −3.9990211 × 10−3 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −1.7648630 × 100 4.1242607 × 101 7.2435270 × 100 1.4995893 × 102 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[4/5] [4/5] [4/5] [4/5] [4/5] | −1.6210147 × 10−2 −1.6210147 × 10−2 −6.0000000 × 10−2 −4.0000000 × 10−2 −4.0000000 × 10−3 | 7.7526429 × 100 −7.7526429 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 4.9834776 × 102 4.9834776 × 102 3.0000001 × 101 1.8999999 × 101 1.5000000 × 102 | 2.1295890 × 10−2 −2.1295890 × 10−2 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[5/6] [5/6] [5/6] [5/6] [5/6] [5/6] | −1.9041159 × 10−2 −1.9041159 × 10−2 −6.7767944 × 10−1 −5.9999995 × 10−1 −3.9999998 × 10−2 −4.0000000 × 10−3 | 7.7689403 × 100 −7.7689403 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 5.0048358 × 102 5.0048358 × 102 5.9734657 × 10−4 3.0000007 × 101 1.8999993 × 101 1.5000000 × 102 | 2.0717494 × 10−1 −2.0717494 × 10−1 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] | −2.1324889 × 10−2 −2.1324889 × 10−2 1.8612450 × 100 1.0175524 × 10−1 −5.9999997 × 10−2 −3.9999998 × 10−2 4.6051089 × 10−2 4.6051089 × 10−2 −4.0000000 × 10−3 3.2810383 × 10−3 3.2810383 × 10−3 | 7.7191644 × 100 −7.7191644 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 2.7461804 × 10−2 −2.7461804 × 10−2 0.0000000 × 100 1.3004577 × 10−2 −1.3004577 × 10−2 | 4.9388824 × 102 4.9388824 × 102 −1.9332353 × 10−2 −1.5513358 × 10−10 3.0000004 × 101 1.8999995 × 101 −2.8790985 × 10−10 −2.8790985 × 10−10 1.5000000 × 102 3.4151907 × 10−9 3.4151907 × 10−9 | 3.8817019 × 10−1 −3.8817019 × 10−1 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 4.1379327 × 10−10 −4.1379327 × 10−10 0.0000000 × 100 5.8441591 × 10−9 −5.8441591 × 10−9 |
[11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] | −4.7391958 × 10−2 −4.7391958 × 10−2 4.5673491 × 100 1.1502705 × 10−1 −5.9999997 × 10−2 −3.9999998 × 10−2 4.2291942 × 10−2 4.2291942 × 10−2 3.3846007 × 10−3 3.3846007 × 10−3 −4.0000000 × 10−3 4.4403675 × 10−2 | 7.6874054 × 100 −7.6874054 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 2.6291712 × 10−2 −2.6291712 × 10−2 1.2973154 × 10−2 −1.2973154 × 10−2 0.0000000 × 100 0.0000000 × 100 | 4.8822528 × 102 4.8822528 × 102 −1.2573896 × 100 −1.2626689 × 10−10 3.0000004 × 101 1.8999995 × 101 −3.1523708 × 10−10 −3.1523708 × 10−10 3.3215604 × 10−9 3.3215604 × 10−9 1.5000000 × 102 −6.9425412 × 10−10 | 3.0558821 × 100 −3.0558821 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 4.8228821 × 10−10 −4.8228821 × 10−10 5.8739554 × 10−9 −5.8739554 × 10−9 0.0000000 × 100 0.0000000 × 100 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] [10/11] | −2.7077041 × 101 −2.7077041 × 101 −6.1805903 × 100 −6.1805903 × 100 −2.5554003 × 10−1 −2.5554003 × 10−1 −1.5365995 × 10−2 −1.5365995 × 10−2 −6.0005910 × 10−2 −4.0019812 × 10−2 −4.0002474 × 10−3 | 9.1626768 × 101 −9.1626768 × 101 2.1050940 × 101 −2.1050940 × 101 1.2306962 × 101 −1.2306962 × 101 6.2841161 × 100 −6.2841161 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 7.2543691 × 103 7.2543691 × 103 5.1467900 × 102 5.1467900 × 102 1.8078545 × 102 1.8078545 × 102 1.9913166 × 102 1.9913166 × 102 2.9976264 × 101 1.9019637 × 101 1.5000415 × 102 | 7.8190760 × 102 −7.8190760 × 102 3.8373116 × 102 −3.8373116 × 102 4.6513878 × 101 −4.6513878 × 101 −1.9550354 × 10−1 1.9550354 × 10−1 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] [11/12] | 1.9960781 × 102 −1.0407991 × 101 −1.0407991 × 101 −3.5638166 × 10−1 −3.5638166 × 10−1 −1.5318715 × 10−2 −1.5318715 × 10−2 −5.7955252 × 10−1 −5.7955252 × 10−1 −5.9916572 × 10−2 −3.9744843 × 10−2 −3.9973272 × 10−3 | 0.0000000 × 100 2.3945985 × 101 −2.3945985 × 101 1.2371355 × 101 −1.2371355 × 101 6.2841195 × 100 −6.2841195 × 100 1.1399858 × 100 −1.1399858 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −1.0851438 × 104 7.6825966 × 102 7.6825966 × 102 1.9394604 × 102 1.9394604 × 102 1.9912256 × 102 1.9912256 × 102 −4.0087042 × 10−7 −4.0087042 × 10−7 3.0311253 × 101 1.8736139 × 101 1.4994814 × 102 | 0.0000000 × 100 9.8681718 × 102 −9.8681718 × 102 5.7945460 × 101 −5.7945460 × 101 −2.152848 × 10−1 2.1528487 × 10−1 5.6765099 × 10−7 −5.6765099 × 10−7 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[12/13] [12/13] [12/13] [12/13] [12/13] [12/13] [12/13] [12/13] [12/13] [12/13] [12/13] [12/13] [12/13] | 1.7618498 × 102 −9.9833268 × 100 −9.9833268 × 100 −3.4500329 × 10−1 −3.4500329 × 10−1 −1.5318064 × 10−2 −1.5318064 × 10−2 1.1196162 × 100 1.1196162 × 100 −2.6054392 × 10−1 −6.000049 × 10−2 −3.9997497 × 10−2 −3.9999174 × 10−3 | 0.0000000 × 100 2.4442457 × 101 −2.4442457 × 101 1.2393825 × 101 −1.2393825 × 101 6.2840948 × 100 −6.2840948 × 100 2.1693108 × 100 −2.1693108 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −8.9677724 × 103 8.3504040 × 102 8.3504040 × 102 1.9620433 × 102 1.9620433 × 102 1.9911357 × 102 1.9911357 × 102 −1.0252367 × 10−10 −1.0252367 × 10−10 −8.7665167 × 10−6 3.0003785 × 101 1.9006856 × 101 1.5005297 × 102 | 0.0000000 × 100 9.5099219 × 102 −9.5099219 × 102 5.5609160 × 101 −5.5609160 × 101 −2.1027043 × 10−1 2.1027043 × 10−1 5.1378617 × 10−11 −5.1378617 × 10−11 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] [16/17] | −5.5030888 × 101 −5.5030888 × 101 −2.9897761 × 100 −2.9897761 × 100 −3.4405403 × 10−2 −3.4405403 × 10−2 8.8670363 × 100 −1.4329960 × 10−1 −1.4329960 × 10−1 −1.2296947 × 10−2 −1.2296947 × 10−2 −6.0183356 × 10−2 −4.0560106 × 10−2 −4.0063800 × 10−3 3.6260369 × 100 2.9143057 × 100 3.0000759 × 100 | 8.2151753 × 101 −8.2151753 × 101 2.0046116 × 101 −2.0046116 × 101 1.2522466 × 101 −1.2522466 × 101 0.0000000 × 100 6.2347883 × 100 −6.2347883 × 100 6.2827305 × 100 −6.2827305 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 1.0428004 × 104 1.0428004 × 104 3.8418716 × 102 3.8418716 × 102 1.9188035 × 102 1.9188035 × 102 −2.2666940 × 10−8 4.4928375 × 100 4.4928375 × 100 1.9448884 × 102 1.9448884 × 102 2.9308374 × 101 1.9596862 × 101 1.5009758 × 102 −5.5123535 × 10−14 −1.1941466 × 10−13 −1.0784734 × 10−13 | 5.7438302 × 103 −5.7438302 × 103 2.5860369 × 102 −2.5860369 × 102 7.4973331 × 100 −7.4973331 × 100 0.0000000 × 100 −2.1898894 × 100 2.1898894 × 100 2.1886636 × 100 −2.1886636 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] [17/18] | −1.8613610 × 102 −8.1919803 × 101 −3.8053205 × 100 −3.8053205 × 100 4.3311772 × 10−2 4.3311772 × 10−2 1.1180539 × 101 −2.5486699 × 10−1 −2.5486699 × 10−1 −1.3851621 × 10−2 −1.3851621 × 10−2 −6.0255523 × 10−2 −4.0772993 × 10−2 −4.0088550 × 10−3 1.0171588 × 100 2.9535681 × 100 2.9535681 × 100 3.0011943 × 100 | 0.0000000 × 100 0.0000000 × 100 1.8744806 × 101 −1.8744806 × 101 1.2476304 × 101 −1.2476304 × 101 0.0000000 × 100 6.4629459 × 100 −6.4629459 × 100 6.2849665 × 100 −6.2849665 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 1.0840509 × 10−1 −1.0840509 × 10−1 0.0000000 × 100 | 6.2585091 × 104 −2.5169979 × 104 3.0764988 × 102 3.0764988 × 102 1.7795390 × 102 1.7795390 × 102 −1.2838628 × 10−6 2.8160877 × 10−1 2.8160877 × 10−1 1.9866901 × 102 1.9866901 × 102 2.9023392 × 101 1.9842716 × 101 1.5017946 × 102 3.0059349 × 10−13 −1.1383378 × 10−13 −1.1383378 × 10−13 −1.0771318 × 10−13 | 0.0000000 × 100 0.0000000 × 100 3.3948578 × 102 −3.3948578 × 102 2.2826081 × 100 −2.2826081 × 100 0.0000000 × 100 2.1566794 × 100 −2.1566794 × 100 −2.0693506 × 100 2.0693506 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 1.3943289 × 10−14 −1.3943289 × 10−14 0.0000000 × 100 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[0/1] | −4.6310442 × 10−2 | 0.0000000 × 100 | 2.5831230 × 101 | 0.0000000 × 100 |
[1/2] [1/2] | −5.3432749 × 10−2 −1.5461139 × 10−2 | 0.0000000 × 100 0.0000000 × 100 | 2.3078996 × 101 3.0173646 × 100 | 0.0000000 × 100 0.0000000 × 100 |
[2/3] [2/3] [2/3] | 7.5235319 × 10−1 −4.9867328 × 10−2 −1.6476801 × 10−3 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −1.1384877 × 10−1 2.5009292 × 101 9.6566322 × 10−1 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[3/4] [3/4] [3/4] [3/4] | −4.8573755 × 10−2 −4.8573755 × 10−2 −5.0000000 × 10−2 −2.0000000 × 10−3 | 1.5501904 × 101 −1.5501904 × 101 0.0000000 × 100 0.0000000 × 100 | 6.5083699 × 101 6.5083699 × 101 2.5000000 × 101 1.0000000 × 100 | 1.7638061 × 10−3 −1.7638061 × 10−3 0.0000000 × 100 0.0000000 × 100 |
[4/5] [4/5] [4/5] [4/5] [4/5] | −7.3878465 × 10−2 −7.3878465 × 10−2 5.7111411 × 100 −5.0000000 × 10−2 −2.0000000 × 10−3 | 1.5415245 × 101 −1.5415245 × 101 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 6.4287423 × 101 6.4287423 × 101 −1.9725482 × 10−2 2.5000000 × 101 1.0000000 × 100 | 1.3368022 × 10−1 −1.3368022 × 10−1 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[5/6] [5/6] [5/6] [5/6] [5/6] [5/6] | −4.8802383 × 10−2 −4.8802383 × 10−2 1.2369563 × 100 −5.0388491 × 10−2 −4.9999999 × 10−2 −2.0000000 × 10−3 | 1.5484391 × 101 −1.5484391 × 101 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 6.4936263 × 101 6.4936263 × 101 −5.6204172 × 10−6 5.0487277 × 10−5 2.4999948 × 101 1.0000000 × 100 | 2.7538122 × 10−3 −2.7538122 × 10−3 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[6/7] [6/7] [6/7] [6/7] [6/7] [6/7] [6/7] | −4.9203724 × 10−2 −4.9203724 × 10−2 1.4834847 × 100 −1.5036091 × 10−2 −1.5036091 × 10−2 −5.0000000 × 10−2 −2.0000000 × 10−3 | 1.5479164 × 101 −1.5479164 × 101 0.0000000 × 100 6.5817021 × 10−2 −6.5817021 × 10−2 0.0000000 × 100 0.0000000 × 100 | 6.4891961 × 101 6.4891961 × 101 −1.6903900 × 10−5 6.0655888 × 10−11 6.0655888 × 10−11 2.5000000 × 101 1.0000000 × 100 | 4.4870559 × 10−3 −4.4870559 × 10−3 0.0000000 × 100 1.2683360 × 10−10 −1.2683360 × 10−10 0.0000000 × 100 0.0000000 × 100 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[0/1] | −2.8012545 × 10−1 | 0.0000000 × 100 | 2.1185167 × 104 | 0.0000000 × 100 |
[1/2] [1/2] | −7.9820851 × 10−1 −1.6193872 × 10−1 | 0.0000000 × 100 0.0000000 × 100 | 7.7648726 × 103 1.4211874 × 104 | 7.7648726 × 103 1.4211874 × 104 |
[2/3] [2/3] [2/3] | 1.0900893 × 101 −4.7531904 × 10−1 −8.8258329 × 10−2 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −7.5260406 × 102 1.3473832 × 104 7.9883055 × 103 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[3/4] [3/4] [3/4] [3/4] | 9.5438875 × 100 −4.8570632 × 10−1 −9.3996711 × 10−2 3.1160963 × 10−1 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −5.9804862 × 102 1.3123770 × 104 8.3615769 × 103 1.9220357 × 100 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[4/5] [4/5] [4/5] [4/5] [4/5] | 5.9042779 × 100 5.9042779 × 100 −5.5327113 × 10−1 −1.8322055 × 10−1 −2.9933826 × 10−2 | 1.6583941 × 101 −1.6583941 × 101 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 2.1860469 × 103 2.1860469 × 103 1.0385843 × 104 8.5114077 × 103 2.7148235 × 103 | −9.4864888 × 102 9.4864888 × 102 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[5/6] [5/6] [5/6] [5/6] [5/6] [5/6] | 2.2977538 × 100 2.2977538 × 100 −6.1888009 × 10−1 −4.0317904 × 10−1 −1.4563794 × 10−1 −1.6809887 × 10−2 | 6.3404844 × 101 −6.3404844 × 101 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 3.5971774 × 104 3.5971774 × 104 6.3141033 × 103 5.9648905 × 103 7.5396158 × 103 1.8221989 × 103 | −1.5224157 × 103 1.5224157 × 103 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[6/7] [6/7] [6/7] [6/7] [6/7] [6/7] [6/7] | −4.1810290 × 10−1 −4.1810290 × 10−1 −7.0161494 × 10−1 −5.0095620 × 10−1 −2.0110761 × 10−1 −1.0085442 × 10−1 −1.0099624 × 10−2 | 7.7962332 × 101 −7.7962332 × 101 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 5.4818688 × 104 5.4818688 × 104 2.3126253 × 103 8.5888020 × 103 5.5989555 × 103 3.7854835 × 103 1.3572038 × 103 | 5.2449314 × 101 −5.2449314 × 101 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[7/8] [7/8] [7/8] [7/8] [7/8] [7/8] [7/8] [7/8] | −4.5025997 × 10−1 −4.5025997 × 10−1 2.1128167 × 101 −6.9998756 × 10−1 −4.9999418 × 10−1 −1.9999568 × 10−1 −9.9997141 × 10−2 −9.9997170 × 10−3 | 7.6911645 × 101 −7.6911645 × 101 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 5.3301980 × 104 5.3301980 × 104 −6.3893270 × 100 2.3583179 × 103 8.5597566 × 103 5.6501371 × 103 3.7247848 × 103 1.3499779 × 103 | 8.5024460 × 101 −8.5024460 × 101 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[0/1] | −4.6532813 × 10−2 | 0.0000000 × 100 | 2.6046383 × 101 | 0.0000000 × 100 |
[1/2] [1/2] | −4.9493508 × 10−2 4.3303640 × 10−3 | 0.0000000 × 100 0.0000000 × 100 | 2.5618333 × 101 6.1072333 × 10−1 | 0.0000000 × 100 0.0000000 × 100 |
[2/3] [2/3] [2/3] | −2.4527859 × 10−1 −5.2343439 × 10−2 −7.3617836 × 10−3 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −4.1120854 × 10−1 2.4939697 × 101 1.6356657 × 100 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[3/4] [3/4] [3/4] [3/4] | 9.2095511 × 10−2 9.2095511 × 10−2 −5.0339820 × 10−2 −2.5538760 × 10−3 | 1.4129833 × 10−1 −1.4129833 × 10−1 0.0000000 × 100 0.0000000 × 100 | −1.8488909 × 10−3 −1.8488909 × 10−3 2.5216832 × 101 1.0215389 × 100 | −4.2125394 × 10−3 4.2125394 × 10−3 0.0000000 × 100 0.0000000 × 100 |
[4/5] [4/5] [4/5] [4/5] [4/5] | 3.4435350 × 10−1 3.3736775 × 10−2 3.3736775 × 10−2 −4.9906591 × 10−2 −1.5734160 × 10−3 | 0.0000000 × 100 1.6601193 × 10−1 −1.6601193 × 10−1 0.0000000 × 100 0.0000000 × 100 | −3.5226274 × 10−3 9.1900664 × 10−3 9.1900664 × 10−3 2.5247336 × 101 9.1880906 × 10−1 | 0.0000000 × 100 −2.2368378 × 10−2 2.2368378 × 10−2 0.0000000 × 100 0.0000000 × 100 |
[5/6] [5/6] [5/6] [5/6] [5/6] [5/6] | 1.9989881 × 10−1 1.9989881 × 10−1 −4.9507620 × 10−3 −4.9507620 × 10−3 −5.0495725 × 10−2 −2.1235336 × 10−3 | 4.7267992 × 10−1 −4.7267992 × 10−1 1.1479799 × 10−1 −1.1479799 × 10−1 0.0000000 × 100 0.0000000 × 100 | 6.2996699 × 10−2 6.2996699 × 10−2 1.1328981 × 10−2 1.1328981 × 10−2 2.5398527 × 101 9.8862834 × 10−1 | 9.9681731 × 10−2 −9.9681731 × 10−2 3.5395658 × 10−2 −3.5395658 × 10−2 0.0000000 × 100 0.0000000 × 100 |
[6/7] [6/7] [6/7] [6/7] [6/7] [6/7] [6/7] | −4.7121082 × 10−1 −4.7121082 × 10−1 −9.8173414 × 10−2 −1.6574766 × 10−2 −1.6574766 × 10−2 −4.6842552 × 10−2 −1.4529226 × 10−3 | 8.8269035 × 10−1 −8.8269035 × 10−1 0.0000000 × 100 7.8229861 × 10−2 −7.8229861 × 10−2 0.0000000 × 100 0.0000000 × 100 | 8.3216962 × 10−1 8.3216962 × 10−1 4.7499294 × 100 −6.7687562 × 10−2 −6.7687562 × 10−2 2.1736778 × 101 8.7883033 × 10−1 | 2.1437565 × 100 −2.1437565 × 100 0.0000000 × 100 −5.7532139 × 10−2 5.7532139 × 10−2 0.0000000 × 100 0.0000000 × 100 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[0/1] | 9.5517727 × 10−3 | 0.0000000 × 100 | 3.0833546 × 101 | 0.0000000 × 100 |
[1/2] [1/2] | −5.3318240 × 10−2 −3.9710375 × 10−3 | 0.0000000 × 100 0.0000000 × 100 | −5.5129353 × 101 6.9940225 × 101 | 0.0000000 × 100 0.0000000 × 100 |
[2/3] [2/3] [2/3] | −4.8722175 × 10−2 −1.4227833 × 10−2 −2.8106907 × 10−3 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −6.3175619 × 101 2.3348338 × 101 5.4854799 × 101 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[3/4] [3/4] [3/4] [3/4] | 3.4837912 × 10−1 −5.0134633 × 10−2 −8.8230772 × 10−3 −1.9566787 × 10−3 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −3.1650844 × 10−2 −5.9897040 × 101 3.5668241 × 101 3.9174859 × 101 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[4/5] [4/5] [4/5] [4/5] [4/5] | 1.7194367 × 10−1 1.7194367 × 10−1 −5.0000000 × 10−2 −9.0000000 × 10−3 −2.0000000 × 10−3 | 7.7165657 × 100 −7.7165657 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 3.7289858 × 101 3.7289858 × 101 −6.0000000 × 101 3.5000000 × 101 4.0000000 × 101 | 7.0358919 × 10−3 −7.0358919 × 10−3 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[5/6] [5/6] [5/6] [5/6] [5/6] [5/6] | 1.7196744 × 10−1 1.7196744 × 10−1 −5.0000000 × 10−2 −4.1165123 × 10−2 −9.0000000 × 10−3 −2.0000000 × 10−3 | 7.7153057 × 100 −7.7153057 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 3.7277667 × 101 3.7277667 × 101 −5.9999998 × 101 −2.1944340 × 10−6 3.5000000 × 101 4.0000000 × 101 | 6.9193200 × 10−3 −6.9193200 × 10−3 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[6/7] [6/7] [6/7] [6/7] [6/7] [6/7] [6/7] | 1.7193555 × 10−1 1.7193555 × 10−1 7.6658423 × 10−3 7.6658423 × 10−3 −5.0000000 × 10−2 −9.0000000 × 10−3 −2.0000000 × 10−3 | 7.7163519 × 100 −7.7163519 × 100 6.8951426 × 10−2 −6.8951426 × 10−2 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 3.7287794 × 101 3.7287794 × 101 −7.5593396 × 10−11 −7.5593396 × 10−11 −6.0000000 × 101 3.5000000 × 101 4.0000000 × 101 | 7.0750599 × 10−3 −7.0750599 × 10−3 −1.3222430 × 10−11 1.3222430 × 10−11 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[7/8] [7/8] [7/8] [7/8] [7/8] [7/8] [7/8] [7/8] | 1.7194602 × 10−1 1.7194602 × 10−1 9.3410071 × 10−2 −5.0000000 × 10−2 1.9105494 × 10−3 1.9105494 × 10−3 −9.0000000 × 10−3 −2.0000000 × 10−3 | 7.7159127 × 100 −7.7159127 × 100 0.0000000 × 100 0.0000000 × 100 5.3649894 × 10−2 −5.3649894 × 10−2 0.0000000 × 100 0.0000000 × 100 | 3.7283544 × 101 3.7283544 × 101 −4.8135249 × 10−11 −6.0000000 × 101 3.6900374 × 10−10 3.6900374 × 10−10 3.5000000 × 101 4.0000000 × 101 | 7.0238724 × 10−3 −7.0238724 × 10−3 0.0000000 × 100 0.0000000 × 100 −2.1846720 × 10−10 2.1846720 × 10−10 0.0000000 × 100 0.0000000 × 100 |
Level | Re(beta) | Im(beta) | Re(alpha) | Im(alpha) |
---|---|---|---|---|
[3/4] [3/4] [3/4] [3/4] | −1.6916657 × 10−1 −5.2380624 × 10−2 −6.6573511 × 10−3 −1.2001808 × 10−3 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 4.4097682 × 10−1 −5.8372837 × 101 4.6990984 × 101 2.5637257 × 101 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[4/5] [4/5] [4/5] [4/5] [4/5] | −1.4839124 × 10−1 −5.2498616 × 10−2 −6.5669995 × 10−3 −1.1390847 × 10−3 6.3259167 × 10−2 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | 5.5301494 × 10−1 −5.8412725 × 101 4.7777205 × 101 2.4774822 × 101 −4.0520354 × 10−10 | 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[5/6] [5/6] [5/6] [5/6] [5/6] [5/6] | 1.5252952 × 10−1 −5.1367483 × 10−2 1.4373202 × 10−2 1.4373202 × 10−2 −7.8249136 × 10−3 −1.7322604 × 10−3 | 0.0000000 × 100 0.0000000 × 100 4.8906437 × 10−2 −4.8906437 × 10−2 0.0000000 × 100 0.0000000 × 100 | −3.7967646 × 10−4 −5.9007827 × 101 2.8204331 × 10−3 2.8204331 × 10−3 3.9143448 × 101 3.4508292 × 101 | 0.0000000 × 100 0.0000000 × 100 −1.6883927 × 10−3 1.6883927 × 10−3 0.0000000 × 100 0.0000000 × 100 |
[6/7] [6/7] [6/7] [6/7] [6/7] [6/7] [6/7] | 1.0670606 × 10−1 1.0670606 × 10−1 −2.6417235 × 10−2 −2.6417235 × 10−2 −4.8651982 × 10−2 −9.2794132 × 10−3 −2.0087154 × 10−3 | 1.6000709 × 10−1 −1.6000709 × 10−1 5.3662647 × 10−2 −5.3662647 × 10−2 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 | −8.9423565 × 10−3 −8.9423565 × 10−3 8.8029576 × 10−2 8.8029576 × 10−2 −6.1368344 × 101 3.5660235 × 101 4.0347352 × 101 | 1.2589839 × 10−2 −1.2589839 × 10−2 −4.2733350 × 10−1 4.2733350 × 10−1 0.0000000 × 100 0.0000000 × 100 0.0000000 × 100 |
[7/8] [7/8] [7/8] [7/8] [7/8] [7/8] [7/8] [7/8] | 3.2939684 × 10−1 4.4906532 × 10−2 4.4906532 × 10−2 −5.0840652 × 10−2 −1.2343101 × 10−2 −1.2343101 × 10−2 −8.8362213 × 10−3 −1.9621520 × 10−3 | 0.0000000 × 100 1.1239422 × 10−1 −1.1239422 × 10−1 0.0000000 × 100 4.0179838 × 10−2 −4.0179838 × 10−2 0.0000000 × 100 0.0000000 × 100 | −1.9983042 × 10−2 4.4950812 × 10−3 4.4950812 × 10−3 −6.0463730 × 101 1.0214574 × 10−1 1.0214574 × 10−1 3.5620994 × 101 3.9226211 × 101 | 0.0000000 × 100 −4.1878055 × 10−3 4.1878055 × 10−3 0.0000000 × 100 8.3652044 × 10−2 −8.3652044 × 10−2 0.0000000 × 100 0.0000000 × 100 |
[8/9] [8/9] [8/9] [8/9] [8/9] [8/9] [8/9] [8/9] [8/9] | 8.7551404 × 10−1 4.8439223 × 10−3 4.8439223 × 10−3 −6.2372719 × 10−2 −3.7786151 × 10−2 1.1713447 × 10−2 1.1713447 × 10−2 −9.5422085 × 10−3 −2.0198601 × 10−3 | 0.0000000 × 100 1.2261909 × 10−1 −1.2261909 × 10−1 0.0000000 × 100 0.0000000 × 100 3.6263972 × 10−2 −3.6263972 × 10−2 0.0000000 × 100 0.0000000 × 100 | −4.1265290 × 10−1 6.7327908 × 10−2 6.7327908 × 10−2 −3.3812097 × 101 −2.9331610 × 101 8.1928119 × 10−5 8.1928119 × 10−5 3.6568214 × 101 4.0687095 × 101 | 0.0000000 × 100 1.1109084 × 10−2 −1.1109084 × 10−2 0.0000000 × 100 0.0000000 × 100 3.9570379 × 10−4 −3.9570379 × 10−4 0.0000000 × 100 0.0000000 × 100 |
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Shams Es-haghi, S.; Gardner, D.J. A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra. Molecules 2021, 26, 4838. https://doi.org/10.3390/molecules26164838
Shams Es-haghi S, Gardner DJ. A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra. Molecules. 2021; 26(16):4838. https://doi.org/10.3390/molecules26164838
Chicago/Turabian StyleShams Es-haghi, Siamak, and Douglas J. Gardner. 2021. "A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra" Molecules 26, no. 16: 4838. https://doi.org/10.3390/molecules26164838