To paraphrase and possibly expand on my candidate statements, there are two primary reasons I would like to serve the CAS as a board member. We live in a very exciting era, especially as it relates to technological innovation. However, it is also a critical time for the field. Computing power and data collection have grown in ways that would have been unthinkable outside of science fiction even twenty-five years ago. This makes it a wonderful time to be an actuary. I’ve often claimed that actuaries were “doing data science” before that term was invented. While some of the methods and tools may have changed in the recent past, the concept of understanding all elements of risk—quantitative and qualitative—remains one of our core strengths. We have been dealing with most of the problems of data collection, scrubbing, imputation, and analysis for over a century. We are not just “data scientists for insurance”—we are experts in applying the critical combination of statistical techniques, business specific domain knowledge, and awareness of the limitations of any model or methodology.

This leads to one of the issues about which I feel strongly. The proliferation of data-analysis-type jobs is both a boon and a burden. We can, and have, taken advantage of advances in statistical theory, practice, algorithms, and computing power to name just a few. Yet we may also be in danger of being pigeonholed, if not rendered obsolete, if we do not evolve. Data capture, amalgamation, cleaning, and analysis have undergone a veritable sea change in the last decade, and actuaries need to be current, or at least conversant, with these techniques. As a member of the board, I would help the society “ride the wave” of the digital revolution and not only keep but enhance the value that actuaries have and bring to our principals. I believe that in order to stay relevant and valuable in the 21st century, especially in light of other North American actuarial organizations entering the general insurance arena, we need to expand the actuarial toolkit beyond its traditional boundaries. I was a member of the Education Structure Task Force which reviewed the then current syllabus with an eye towards the baseline of what the actuary of the next 25 years should know. I continued as a member of both the Statistics and CERA implementation task forces. The current exam S is one result of these task forces, and serves to better prepare actuaries of the 21st century for the problems they will face. I would like to see similar improvements to the education of actuaries in more advanced statistical concepts, especially the introduction of Bayesian hierarchical models and having actuaries aware of basic unsupervised learning techniques. Not all education needs to be officially enshrined in the syllabus, though. I do not want another exam for the sake of an exam; I do not think we need another exam. Rather, I want the CAS to encourage actuaries to write not only theoretical papers, but papers and monographs which present practical solutions and methods in a clear fashion. These can be used by all actuaries to expand their capabilities and thus enhance their value.

A second related issue which I feel is important is that I believe that increasing the visibility of the actuarial role in non-traditional risk management venues is in the best interest of the CAS and its members. New corporate and risk paradigms have emerged in the past decade, and existing non-insurance entities have seen value in actuarial services. New corporate concepts such as the proliferation of peer-to-peer companies have need of risk management services, and some already use actuaries. Traditional companies are now exposed to both new kinds of risk, such as cyber, as well as increased exposure to risks previously considered *de minimis*, such as domestic terrorism against property and infrastructure. Even non-insurance, non-classical risk bearing entities have made use of actuarial services, as the NFL did as part of its response to the brain trauma lawsuit. I believe that increasing awareness about the specific value actuaries bring to risk management roles in non-traditional venues will both solidify and expand opportunities for current and future actuaries.

Overall, I consider myself extremely fortunate to have this career at the crossroads of statistics and problem solving at this time. The CAS has been nurturing and advancing the science, art, education, and standards of casualty actuarial practice for over a century, providing the framework in which we actuaries perform with integrity and competence. For that alone we all owe the CAS a debt of gratitude. As a predominantly volunteer-driven and run organization, we can demonstrate that gratitude by giving of our time and expertise to the CAS and all of its members. More recently, I was privileged to be a member of the nominating committee for the past two years, and saw first-hand the care and concern that the CAS leadership has regarding the growth and health of the organization. The strategic thinking and attention to the short, medium, and long-term needs of the society and its members inspired me. Serving on the board would allow me to both give back to the CAS and “pay forward” the debt of gratitude I owe the society and all its members.

I would certainly appreciate your support come August 1, but more importantly, I would appreciate it if you took the time to review the candidates—and they are all interested in bettering the CAS and its members—and make an informed decision for whom to vote come August. This is your society, and as recent political events have shown, elections matter! Please use your vote, and use it as best you see fit. Thank you.

]]>set.seed(4987) library(microbenchmark) library(Matrix) A <- matrix(rnorm(1e6, 1e3, 1e2), ncol = 1e3) B <- matrix(rnorm(1e6, 1e3, 1e2), ncol = 1e3) A <- crossprod(A, A) A <- A * 1000 / mean(A) colnames(A) <- colnames(B) <- NULL options(scipen=4, digits = 3) BLAS <- microbenchmark( sort(c(as.vector(A), as.vector(B))), det(A), A %*% B, t(A) %*% B, crossprod(A, B), solve(A), solve(A, t(B)), solve(B), chol(A), chol(B, pivot = TRUE), qr(A, LAPACK = TRUE), svd(A), eigen(A, symmetric = TRUE), eigen(A, symmetric = FALSE), eigen(B, symmetric = FALSE), lu(A), fft(A), Hilbert(3000), toeplitz(A[1:500, 1]), princomp(A), times=25L, unit='ms', control = list(order = 'block') ) NotBLAS <- microbenchmark( A + 2, A - 2, A * 2, A / 2, A * 0.5, A ^ 2, sqrt(A[1:10000]), sin(A[1:10000]), A + B, A - B, A * B, A / B, A[1:1e5] %% B[1:1e5], A[1:1e5] %/% B[1:1e5], times = 5000L, unit='ms', control = list(order = 'block') )

My machine is a i7-2600K overclocked to 4.65Ghz with 16GB RAM, running Win7 64. I’ve also used RStudio Version 0.99.893 for each of these tests.

I ran the tests on four versions of R:

- Plain vanilla R version 3.2.4 Revised (2016-03-16 r70336) installed from the binary on CRAN
- The same version of R as #1, but with Rblas.dll replaced by one based on OpenBLAS 2.17 compiled specifically for the SandyBridge CPU (as part of the #3)
- R compiled from source using the gcc 4.9.3 toolchain, passing
`-march=native`

to EOPTS and linking to a multi-threaded (4 max) SandyBridge-specific OpenBLAS 2.17 - Same as #3, but activating link-time optimization for core R and the BLAS

This time I was a bit wiser, and saved the results to RDS objects so that I could combine them into one R session and compare them. I’ll post the raw output at the bottom of the page, but I think it’s more intuitive to look at graphical comparisons. The takeaway is if you can **install a fast BLAS!** The other optimization options had minor effects, sometimes better sometimes worse, but using a fast BLAS made a significant difference. Adding a `Platform`

variable to the four sets of `BLAS`

and `NotBLAS`

outputs and then combining them in one session allowed for the generation of comparison plots. The call for these graphs, for those following along at home, is:

ggplot(BLAS) + geom_boxplot(aes(y = time / 1e6, color = Platform, x = expr)) + scale_y_continuous(labels = comma) + ylab('milliseconds') + coord_flip() ggplot(BLAS) + geom_jitter(aes(y = time / 1e6, color = Platform, x = expr), alpha = 0.3) + scale_y_continuous(labels = comma) + ylab('milliseconds') + coord_flip() ggplot(NotBLAS) + geom_boxplot(aes(y = time / 1e6, color = Platform, x = expr)) + scale_y_continuous(labels = comma) + ylab('milliseconds') + coord_flip()

You may also have to right-click and sellect “View Image” to get a better picture. First is a boxplot comparison for the BLAS related functions:

As there are only 25 samples for each function, a jitter plot may be a bit simpler:

It is clear that just adding the BLAS makes a noticeable difference for matrix-related functionality. The other optimizations tend to provide some benefit, but nothing significant.

The non-BLAS related functions, understandably, don’t respond the same way to the BLAS. As I ran 5000 samples per expression, a jitter plot would not be that clear, even with a low alpha.

What is seen here, in my opinion, that using the local architecture sometimes improves performance, such as for division, and other retards it slightly, such as for `sin`

and `sqrt`

. I think the latter two depend on the optimization as well. Default `R`

calls for `-O3`

for C code. The magnitude of the improvements seems to be greater than the change in those times where there is an impediment, at least when eyeballing the plot. If I recall from previous tests, strangely, it may be slightly more efficient to pass `-mtune = native`

, at least for my test suite. I’m not sure why, as arch implies tune. I’d have to recompile `R`

yet again to test it, though. Also, with 5000 iterations per expression, garbage collection has to be run on occasion, which accounts for those outlier points.

In summation, if you deal with heavy calculation in R, specifically matrix-related, you will notice significant speed improvement by replacing the vanilla `Rblas.dll`

with a faster one. Compiling or tuning for a specific architecture may show some small increases, but does not return the same results.

While it’s actually easier now to compile OpenBLAS for R and in R on Windows, my instructions are a bit dated, and so I’ll have to update those eventually. I have considered hosting some pre-compiled Rblas files, as Dr. Ei-ji Nakama does, but I’ve held back for two reasons 1) I have to ensure I apply the proper licenses and 2) I’m a bit concerned about someone suing or the like if for whatever reason, a blas didn’t work properly. I have disclaimers and all, but you never know 8-).

In any event, I’m always interested in hearing comments about your experience with compiling R or a fast BLAS for Windows, and especially if you found any of my previous posts helpful. Thanks!

R version 3.2.4 Revised (2016-03-16 r70336) (pure vanilla) Unit: milliseconds BLAS expr Min LQ Median UQ Max Mean SD CV n (fctr) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (int) 1 sort(c(as.vector(A), as.vector(B))) 264.56 266.56 267.13 270.50 327.81 270.39 12.15 0.04492 25 2 det(A) 134.83 135.77 136.47 137.74 206.93 139.83 14.12 0.10095 25 3 A %*% B 465.64 473.39 492.22 504.79 546.33 493.55 22.62 0.04583 25 4 t(A) %*% B 468.96 501.26 506.00 535.00 564.67 514.08 24.71 0.04807 25 5 crossprod(A, B) 737.52 746.47 759.38 768.71 813.29 762.19 18.66 0.02449 25 6 solve(A) 607.80 615.18 621.48 639.45 679.81 628.14 18.80 0.02992 25 7 solve(A, t(B)) 847.49 853.39 855.29 859.28 881.07 858.57 8.89 0.01036 25 8 solve(B) 617.06 621.25 622.45 624.65 683.30 625.33 12.56 0.02008 25 9 chol(A) 116.61 117.19 117.40 118.45 123.19 118.46 1.99 0.01676 25 10 chol(B, pivot = TRUE) 2.38 2.45 2.52 2.59 6.91 3.15 1.51 0.48091 25 11 qr(A, LAPACK = TRUE) 423.32 424.46 425.25 426.35 432.92 425.62 2.07 0.00487 25 12 svd(A) 2219.86 2242.51 2265.10 2294.88 2372.62 2277.14 45.47 0.01997 25 13 eigen(A, symmetric = TRUE) 939.63 946.36 952.16 963.85 1020.42 959.00 19.78 0.02063 25 14 eigen(A, symmetric = FALSE) 3623.95 3650.20 3657.46 3675.48 3740.09 3662.55 28.50 0.00778 25 15 eigen(B, symmetric = FALSE) 4072.00 4127.98 4175.12 4228.25 4344.13 4183.00 77.04 0.01842 25 16 lu(A) 137.73 142.06 144.51 147.26 210.54 156.60 26.47 0.16901 25 17 fft(A) 112.27 116.98 119.89 123.29 135.01 120.63 5.20 0.04308 25 18 Hilbert(3000) 134.83 196.37 197.50 199.45 386.30 201.86 46.23 0.22905 25 19 toeplitz(A[1:500, 1]) 4.92 5.06 5.09 5.45 10.49 5.80 1.72 0.29759 25 20 princomp(A) 1834.20 1855.31 1903.78 1947.02 2127.45 1923.42 81.44 0.04234 25 NotBLAS expr Min LQ Median UQ Max Mean SD CV n (fctr) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (int) 1 A + 2 1.719 1.891 1.922 2.012 74.96 2.542 2.5880 1.018 5000 2 A - 2 1.723 1.910 1.965 2.066 72.86 2.628 2.6389 1.004 5000 3 A * 2 1.725 1.900 1.948 2.049 80.89 2.604 2.6526 1.019 5000 4 A/2 3.014 3.162 3.179 3.205 74.44 3.776 2.5372 0.672 5000 5 A * 0.5 1.718 1.884 1.908 1.971 74.80 2.538 2.5780 1.016 5000 6 A^2 1.718 1.896 1.939 2.036 73.41 2.587 2.6111 1.009 5000 7 sqrt(A[1:10000]) 0.110 0.111 0.111 0.112 5.30 0.116 0.0833 0.717 5000 8 sin(A[1:10000]) 0.364 0.365 0.365 0.366 1.35 0.368 0.0429 0.117 5000 9 A + B 1.306 1.384 1.448 1.542 68.83 1.582 1.8367 1.161 5000 10 A - B 1.308 1.355 1.432 1.539 67.87 1.571 1.8412 1.172 5000 11 A * B 1.310 1.410 1.471 1.571 69.73 1.610 1.8548 1.152 5000 12 A/B 4.767 4.779 4.789 4.853 66.93 4.946 1.7605 0.356 5000 13 A[1:100000]%%B[1:100000] 2.528 2.544 2.558 2.583 65.80 2.639 1.2633 0.479 5000 14 A[1:100000]%/%B[1:100000] 2.271 2.296 2.318 2.370 65.33 2.423 1.2646 0.522 5000 ---------- R version 3.2.4 Revised (2016-03-16 r70336) (pure vanilla) With SandyBridge-specific OpenBLAS (2.17) Unit: milliseconds BLAS expr Min LQ Median UQ Max Mean SD CV n (fctr) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (int) 1 sort(c(as.vector(A), as.vector(B))) 264.10 265.06 265.77 266.93 323.4 268.39 11.52 0.0429 25 2 det(A) 14.31 14.57 15.26 16.67 20.0 15.82 1.43 0.0903 25 3 A %*% B 19.25 21.34 26.83 28.49 84.1 27.78 12.68 0.4563 25 4 t(A) %*% B 26.46 30.04 32.20 38.76 43.6 34.25 5.37 0.1567 25 5 crossprod(A, B) 17.63 21.14 22.82 27.48 35.5 24.68 5.10 0.2066 25 6 solve(A) 40.67 45.92 49.58 51.48 107.1 51.29 12.37 0.2412 25 7 solve(A, t(B)) 46.32 50.88 51.63 56.40 63.2 53.00 4.24 0.0799 25 8 solve(B) 47.09 51.22 51.72 56.44 118.6 56.32 13.71 0.2435 25 9 chol(A) 6.59 6.65 6.77 7.01 10.6 7.38 1.39 0.1890 25 10 chol(B, pivot = TRUE) 2.36 2.47 2.50 2.58 6.8 3.25 1.58 0.4851 25 11 qr(A, LAPACK = TRUE) 70.19 71.59 72.47 74.69 79.1 72.96 2.22 0.0304 25 12 svd(A) 375.74 378.56 389.32 415.33 444.7 399.66 26.19 0.0655 25 13 eigen(A, symmetric = TRUE) 171.72 175.04 176.94 177.97 239.8 179.80 13.20 0.0734 25 14 eigen(A, symmetric = FALSE) 849.09 859.77 872.76 891.57 915.7 876.86 20.89 0.0238 25 15 eigen(B, symmetric = FALSE) 1011.80 1065.85 1070.08 1075.52 1085.1 1065.76 18.14 0.0170 25 16 lu(A) 17.91 18.91 20.78 22.62 82.8 32.68 25.43 0.7782 25 17 fft(A) 110.52 111.17 113.16 113.34 113.8 112.33 1.18 0.0105 25 18 Hilbert(3000) 128.99 194.32 194.70 195.01 377.4 196.80 45.44 0.2309 25 19 toeplitz(A[1:500, 1]) 4.77 4.96 5.03 5.04 10.0 5.58 1.64 0.2937 25 20 princomp(A) 285.96 302.37 316.33 329.08 400.5 322.00 29.13 0.0905 25 NotBLAS expr Min LQ Median UQ Max Mean SD CV n (fctr) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (int) 1 A + 2 1.712 1.887 1.904 1.938 72.64 2.514 2.6240 1.044 5000 2 A - 2 1.723 1.893 1.913 1.962 68.52 2.540 2.5268 0.995 5000 3 A * 2 1.721 1.894 1.918 1.974 68.97 2.543 2.5326 0.996 5000 4 A/2 3.016 3.173 3.192 3.229 73.36 3.804 2.5315 0.665 5000 5 A * 0.5 1.724 1.900 1.939 2.049 73.70 2.590 2.6048 1.006 5000 6 A^2 1.722 1.890 1.910 1.974 71.55 2.543 2.5479 1.002 5000 7 sqrt(A[1:10000]) 0.110 0.111 0.111 0.112 5.59 0.116 0.0876 0.753 5000 8 sin(A[1:10000]) 0.364 0.365 0.366 0.366 1.32 0.370 0.0408 0.110 5000 9 A + B 1.310 1.336 1.378 1.447 62.35 1.512 1.7318 1.146 5000 10 A - B 1.310 1.345 1.411 1.491 63.70 1.542 1.7465 1.133 5000 11 A * B 1.312 1.349 1.421 1.501 65.96 1.550 1.7753 1.145 5000 12 A/B 4.766 4.781 4.804 4.959 73.53 4.992 1.8146 0.364 5000 13 A[1:100000]%%B[1:100000] 2.526 2.541 2.550 2.591 63.56 2.638 1.2305 0.467 5000 14 A[1:100000]%/%B[1:100000] 2.272 2.294 2.310 2.361 66.49 2.399 1.2681 0.529 5000 ---------- R version 3.2.4 Patched (2016-03-28 r70390) -- "Very Secure Dishes" Compiled with -march=native With SandyBridge-specific OpenBLAS (2.17) BLAS expr Min LQ Median UQ Max Mean SD CV n (fctr) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (int) 1 sort(c(as.vector(A), as.vector(B))) 266.79 267.24 268.46 271.46 329.37 272.04 12.36 0.0454 25 2 det(A) 14.24 14.43 15.12 16.94 18.63 15.69 1.43 0.0911 25 3 A %*% B 20.95 24.25 28.79 31.89 34.96 28.26 4.21 0.1490 25 4 t(A) %*% B 26.70 29.53 33.74 41.83 103.68 39.71 18.48 0.4652 25 5 crossprod(A, B) 17.65 19.46 28.94 31.99 36.97 26.53 6.56 0.2474 25 6 solve(A) 41.47 45.97 49.44 52.75 56.85 49.38 4.19 0.0849 25 7 solve(A, t(B)) 46.34 51.00 53.22 55.86 113.34 55.53 12.45 0.2242 25 8 solve(B) 47.90 52.74 54.88 57.05 126.30 58.01 15.05 0.2595 25 9 chol(A) 6.57 6.72 6.93 7.43 12.90 7.74 1.84 0.2377 25 10 chol(B, pivot = TRUE) 2.37 2.47 2.49 2.53 6.26 3.22 1.53 0.4755 25 11 qr(A, LAPACK = TRUE) 70.84 71.54 73.10 74.91 78.28 73.59 2.19 0.0298 25 12 svd(A) 375.93 384.60 399.36 454.94 563.73 421.43 49.29 0.1170 25 13 eigen(A, symmetric = TRUE) 170.29 174.23 177.88 185.33 244.08 182.23 14.50 0.0796 25 14 eigen(A, symmetric = FALSE) 839.89 851.24 861.85 874.42 961.61 870.93 31.10 0.0357 25 15 eigen(B, symmetric = FALSE) 985.55 1045.25 1097.99 1142.28 1200.26 1093.09 56.17 0.0514 25 16 lu(A) 18.25 20.03 21.52 23.14 91.58 34.15 27.10 0.7936 25 17 fft(A) 106.88 110.80 112.50 119.96 132.55 115.51 6.66 0.0576 25 18 Hilbert(3000) 133.10 197.69 201.83 204.57 411.96 205.26 50.69 0.2470 25 19 toeplitz(A[1:500, 1]) 5.04 5.20 5.35 5.49 10.66 5.94 1.65 0.2773 25 20 princomp(A) 326.08 336.14 341.65 349.99 425.50 354.56 31.37 0.0885 25 NotBLAS expr Min LQ Median UQ Max Mean SD CV n (fctr) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (int) 1 A + 2 1.731 1.890 1.911 1.956 73.12 2.538 2.6108 1.0286 5000 2 A - 2 1.726 1.884 1.903 1.943 70.36 2.527 2.5691 1.0168 5000 3 A * 2 1.727 1.891 1.910 1.945 70.49 2.531 2.5708 1.0156 5000 4 A/2 2.040 2.193 2.210 2.236 71.35 2.816 2.5552 0.9075 5000 5 A * 0.5 1.726 1.902 1.939 2.016 74.95 2.591 2.6501 1.0227 5000 6 A^2 1.727 1.886 1.907 1.966 71.32 2.541 2.5889 1.0188 5000 7 sqrt(A[1:10000]) 0.509 0.516 0.516 0.517 4.36 0.521 0.0670 0.1285 5000 8 sin(A[1:10000]) 0.715 0.720 0.721 0.722 1.72 0.725 0.0416 0.0573 5000 9 A + B 1.331 1.356 1.378 1.459 65.85 1.522 1.8010 1.1831 5000 10 A - B 1.331 1.357 1.383 1.461 64.68 1.527 1.7970 1.1765 5000 11 A * B 1.332 1.354 1.372 1.431 65.46 1.513 1.8009 1.1905 5000 12 A/B 2.412 2.423 2.432 2.460 68.39 2.580 1.8492 0.7168 5000 13 A[1:100000]%%B[1:100000] 2.917 2.999 3.005 3.027 67.03 3.084 1.2797 0.4150 5000 14 A[1:100000]%/%B[1:100000] 2.683 2.759 2.769 2.816 67.74 2.861 1.2999 0.4543 5000 ---------- R version 3.2.4 Patched (2016-03-28 r70390) -- "Very Secure Dishes" Compiled with -march=native and partial LTO (R core, blas, and lapack only) With SandyBridge-specific OpenBLAS (2.17) BLAS expr Min LQ Median UQ Max Mean SD CV n (fctr) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (int) 1 sort(c(as.vector(A), as.vector(B))) 263.84 265.83 266.39 268.22 323.52 269.49 11.48 0.0426 25 2 det(A) 16.55 17.15 17.72 19.73 23.69 18.55 1.82 0.0982 25 3 A %*% B 19.55 21.52 25.02 33.33 96.59 29.20 15.17 0.5195 25 4 t(A) %*% B 27.71 33.26 35.59 38.03 45.20 35.47 4.33 0.1220 25 5 crossprod(A, B) 17.81 21.49 25.98 28.99 34.36 25.94 4.49 0.1730 25 6 solve(A) 48.38 51.94 54.40 57.04 112.31 56.72 12.03 0.2122 25 7 solve(A, t(B)) 54.93 56.65 60.59 63.24 72.91 61.05 5.25 0.0859 25 8 solve(B) 55.50 58.85 60.88 62.99 121.31 63.46 12.50 0.1969 25 9 chol(A) 8.57 8.69 8.80 8.98 12.21 9.40 1.28 0.1361 25 10 chol(B, pivot = TRUE) 2.35 2.44 2.51 2.62 6.87 3.18 1.60 0.5035 25 11 qr(A, LAPACK = TRUE) 71.16 72.38 73.57 74.85 83.43 74.41 2.93 0.0393 25 12 svd(A) 376.02 380.71 392.98 437.93 490.97 407.28 34.08 0.0837 25 13 eigen(A, symmetric = TRUE) 173.96 176.57 179.35 181.38 259.73 183.29 16.79 0.0916 25 14 eigen(A, symmetric = FALSE) 852.68 856.26 860.10 864.31 926.66 865.16 16.63 0.0192 25 15 eigen(B, symmetric = FALSE) 987.24 993.14 1001.10 1020.62 1079.71 1010.79 24.32 0.0241 25 16 lu(A) 19.18 20.01 21.76 24.21 26.51 22.11 2.44 0.1103 25 17 fft(A) 106.65 107.09 108.58 110.35 116.92 108.98 2.36 0.0216 25 18 Hilbert(3000) 125.60 191.73 192.18 194.44 314.63 195.15 33.81 0.1732 25 19 toeplitz(A[1:500, 1]) 4.99 5.18 5.21 5.24 10.22 5.79 1.62 0.2801 25 20 princomp(A) 285.97 296.12 300.58 317.75 370.74 315.50 30.97 0.0982 25 NotBLAS expr Min LQ Median UQ Max Mean SD CV n (fctr) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (dbl) (int) 1 A + 2 1.828 1.995 2.022 2.091 74.52 2.615 2.4756 0.9468 5000 2 A - 2 1.807 1.973 1.995 2.064 70.60 2.636 2.5624 0.9721 5000 3 A * 2 1.827 1.991 2.009 2.054 71.57 2.637 2.5618 0.9714 5000 4 A/2 2.147 2.293 2.309 2.335 68.18 2.912 2.4859 0.8537 5000 5 A * 0.5 1.817 1.990 2.007 2.049 68.35 2.628 2.5134 0.9564 5000 6 A^2 1.795 1.964 1.980 2.011 67.96 2.596 2.5082 0.9660 5000 7 sqrt(A[1:10000]) 0.531 0.535 0.536 0.536 2.51 0.540 0.0474 0.0878 5000 8 sin(A[1:10000]) 0.719 0.726 0.726 0.727 1.70 0.728 0.0413 0.0566 5000 9 A + B 1.326 1.355 1.391 1.545 66.25 1.690 1.8104 1.0713 5000 10 A - B 1.321 1.365 1.437 1.602 68.45 1.726 1.8618 1.0788 5000 11 A * B 1.327 1.366 1.439 1.586 63.71 1.719 1.8044 1.0499 5000 12 A/B 2.435 2.463 2.530 2.560 65.87 2.804 1.8057 0.6439 5000 13 A[1:100000]%%B[1:100000] 2.987 3.002 3.007 3.025 63.89 3.092 1.2231 0.3956 5000 14 A[1:100000]%/%B[1:100000] 2.718 2.732 2.736 2.745 63.70 2.809 1.2202 0.4344 5000

]]>

`{align}`

environment. If you notice any malformed math or missing text, please drop me a note. ∞
]]>I recently had the opportunity to use Padé approximants. There is a lot of good information available on line on the theory and applications of using Padé approximants, but I had trouble finding a good example explaining just how to calculate the co-efficients.

Hearken back to undergraduate calculus for a moment. For a given function, its Taylor series is the “best” polynomial representations of that function. If the function is being evaluated at 0, the Taylor series representation is also called the Maclaurin series. The error is proportional to the first “left-off” term. Also, the series is only a good estimate in a small radius around the point for which it is calculated (e.g. 0 for a Maclaurin series).

Padé approximants estimate functions as the quotient of two polynomials. Specifically, given a Taylor series expansion of a function of order , there are two polynomials, of order and of order , such that , called the Padé approximant of order , “agrees” with the original function in order . You may ask, “but the Taylor series from whence it is derived is also of order ?” And you would be correct. However, the Padé approximant seems to consistently have a wider radius of convergence than its parent Taylor series, and, being a quotient, is composed of lower-degree polynomials. With the normalization that the first term of is always 1, there is a set of linear equations which will generate the unique Padé approximant coefficients. Letting be the coefficients for the Taylor series, one can solve:

(1)

remembering that all and are 0.

There is a lot of research on the theory of Padé approximants and Padé tables, how they work, their relationship to continued fractions, and why they work so well. For example, the interested reader is directed to Baker (1975), Van Assche (2006), Wikipedia, and MathWorld for more.

The function will be used as the example. This function has a special representation in almost all computer languages, often called `log1p(x)`

, as naïve implementation as `log(1 + x)`

will suffer catastrophic floating point errors for near 0.

The Maclaurin series expansion for is:

The code below will compare a Padé[3,3] approximant with the 6-term Maclaurin series, which would actually be the Padé[6/0] approximant. First to calculate the coefficients. We know the Maclaurin coefficients, they are . Therefore, the system of linear equations looks like this:

(2)

Rewriting in terms of the known coefficients, we get:

(3)

We can solve this in R using `solve`

:

A <- matrix(c(0, 0, 1, -.5 ,1 / 3 , -.25, .2, 0, 0, 0, 1, -.5, 1 / 3, -.25, 0, 0, 0, 0, 1, -.5, 1 / 3, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0), ncol = 7) B <- c(0, -1, .5, -1/3, .25, -.2, 1/6) P_Coeff <- solve(A, B) print(P_Coeff)

## [1] 1.5000000 0.6000000 0.0500000 0.0000000 1.0000000 1.0000000 0.1833333

Now we can create the estimating functions:

ML <- function(x){x - .5 * x ^ 2 + x ^ 3 / 3 - .25 * x ^ 4 + .2 * x ^ 5 - x ^ 6 / 6} PD33 <- function(x){ NUMER <- x + x ^ 2 + 0.1833333333333333 * x ^ 3 DENOM <- 1 + 1.5 * x + .6 * x ^ 2 + 0.05 * x ^ 3 return(NUMER / DENOM) }

Let’s compare the behavior of these functions around 0 with the naïve and sophisticated implementations of in R.

library(dplyr) library(ggplot2) library(tidyr) D <- seq(-1e-2, 1e-2, 1e-6) RelErr <- tbl_df(data.frame(X = D, Naive = (log(1 + D) - log1p(D)) / log1p(D), MacL = (ML(D) - log1p(D)) / log1p(D), Pade = (PD33(D) - log1p(D)) / log1p(D))) RelErr2 <- gather(RelErr, Type, Error, -X) RelErr2 %>% group_by(Type) %>% summarize(MeanError = mean(Error, na.rm = TRUE)) %>% knitr::kable(digits = 18)

Type | MeanError |
---|---|

Naïve | -4.3280e-15 |

MacL | -2.0417e-14 |

Pade | -5.2000e-17 |

Graphing the relative error in a small area around 0 shows the differing behaviors. First, against the naïve implementation, both estimates do much better.

ggplot(RelErr2, aes(x = X)) + geom_point(aes(y = Error, colour = Type), alpha = 0.5)

But when compared one against the other, the Pade approximant (blue) shows better behavior than the Maclaurin (red) and it’s relative error stays below EPS for a wider swath.

ggplot(RelErr, aes(x = X)) + geom_point(aes(y = MacL), colour = 'red', alpha = 0.5) + geom_point(aes(y = Pade), colour = 'blue', alpha = 0.5)

Just for fun, restricting the y axis to that above, overlaying the naïve formulation (green) looks like this:

ggplot(RelErr, aes(x = X)) + geom_point(aes(y = Naive), colour = 'green', alpha = 0.5) + scale_y_continuous(limits = c(-1.5e-13, 0)) + geom_point(aes(y = MacL), colour = 'red', alpha = 0.5) + geom_point(aes(y = Pade), colour = 'blue', alpha = 0.5)

There are certainly more efficient and elegant ways to calculate Padé approximants, but I found this exercise helpful, and I hope you do as well!

- Baker, G. A. Essentials of Padé Approximants Academic Press, 1975
- Van Assche, W. Pade and Hermite-Pade approximation and orthogonality ArXiv Mathematics e-prints, 2006