Impl special functions

.vscode/settings.json

@@ -1,5 +1,7 @@
 {
   "cSpell.words": [
+    "Abramowitz",
+    "acosh",
     "aiᵢⱼ",
     "alphai",
     "alphar",
@@ -10,6 +12,7 @@
     "blasint",
     "brusselator",
     "bweuler",
+    "cmath",
     "condest",
     "Demmel",
     "devel",
@@ -37,6 +40,7 @@
     "Fehlberg",
     "fftw",
     "Flannery",
+    "frexp",
     "FSAL",
     "fweuler",
     "gfortran",
@@ -51,6 +55,7 @@
     "jobvr",
     "jobz",
     "lcov",
+    "ldexp",
     "ldvl",
     "ldvr",
     "ldvt",
@@ -65,6 +70,7 @@
     "memcheck",
     "Merson",
     "msgpass",
+    "msun",
     "nelt",
     "Nørsett",
     "nstage",
@@ -75,6 +81,9 @@
     "powi",
     "PREORDER",
     "pthread",
+    "pzero",
+    "qone",
+    "qzero",
     "Radau",
     "RINFOG",
     "rowcol",
@@ -83,6 +92,7 @@
     "rustup",
     "rwork",
     "Schur",
+    "Stegun",
     "substeps",
     "Teukolsky",
     "tgamma",
@@ -113,5 +123,9 @@
     "zscal",
     "zsyrk",
     "zticks"
-  ]
+  ],
+  "rust-analyzer.linkedProjects": [
+    "./russell_lab/Cargo.toml"
+  ],
+  "rust-analyzer.showUnlinkedFileNotification": false
 }

russell_lab/c_code/math_functions.c

@@ -3,3 +3,6 @@
 double c_erf(double x) { return erf(x); }
 double c_erfc(double x) { return erfc(x); }
 double c_gamma(double x) { return tgamma(x); }
+double c_ln_gamma(double x) { return lgamma(x); }
+double c_frexp(double x, int *exp) { return frexp(x, exp); }
+double c_ldexp(double frac, int exp) { return ldexp(frac, exp); }

russell_lab/data/reference/README.md

@@ -0,0 +1,14 @@
+# Reference data
+
+This directory contains auxiliary data files for testing.
+
+The files starting with `as-` correspond to results from Reference # 1 (Abramowitz and Stegun, 1972)
+
+The `.txt` files are tables for visual inspection of results whereas .cmp files are for automatic "unit testing".
+
+The `.py` files can be used to re-generate all tables (using SciPy).
+
+## References
+
+[1] Abramowitz M, Stegun IA (1972) Handbook of Mathematical Functions with Formulas, Graphs,
+    and Mathematical Tables. U.S. Department of Commerce, NIST

russell_lab/data/reference/as-17-elliptic-integrals-table17.5-small.cmp

@@ -0,0 +1,26 @@
+                    phi                      k                      F
+  0.000000000000000e+00  8.715574274765817e-02  0.000000000000000e+00
+  5.235987755982988e-01  8.715574274765817e-02  5.237709509615746e-01
+  7.853981633974483e-01  8.715574274765817e-02  7.859411089683984e-01
+  1.047197551196598e+00  8.715574274765817e-02  1.048367145544390e+00
+  1.570796326794897e+00  8.715574274765817e-02  1.573792130924768e+00
+  0.000000000000000e+00  3.420201433256687e-01  0.000000000000000e+00
+  5.235987755982988e-01  3.420201433256687e-01  5.262839905622028e-01
+  7.853981633974483e-01  3.420201433256687e-01  7.939814299617325e-01
+  1.047197551196598e+00  3.420201433256687e-01  1.065968913708522e+00
+  1.570796326794897e+00  3.420201433256687e-01  1.620025899124204e+00
+  0.000000000000000e+00  7.660444431189780e-01  0.000000000000000e+00
+  5.235987755982988e-01  7.660444431189780e-01  5.378676499559212e-01
+  7.853981633974483e-01  7.660444431189780e-01  8.343124726282660e-01
+  1.047197551196598e+00  7.660444431189780e-01  1.164316364623915e+00
+  1.570796326794897e+00  7.660444431189780e-01  1.935581096004722e+00
+  0.000000000000000e+00  8.660254037844386e-01  0.000000000000000e+00
+  5.235987755982988e-01  8.660254037844386e-01  5.422291098035529e-01
+  7.853981633974483e-01  8.660254037844386e-01  8.512237490711855e-01
+  1.047197551196598e+00  8.660254037844386e-01  1.212596615254979e+00
+  1.570796326794897e+00  8.660254037844386e-01  2.156515647499643e+00
+  0.000000000000000e+00  1.000000000000000e+00  0.000000000000000e+00
+  5.235987755982988e-01  1.000000000000000e+00  5.493061443340548e-01
+  7.853981633974483e-01  1.000000000000000e+00  8.813735870195429e-01
+  1.047197551196598e+00  1.000000000000000e+00  1.316957896924816e+00
+  1.570796326794897e+00  1.000000000000000e+00                    inf

russell_lab/data/reference/as-17-elliptic-integrals-table17.5a.txt

@@ -0,0 +1,68 @@
+alp|phi           0           5          10          15          20          25          30
+      0  0.00000000  0.08726646  0.17453293  0.26179939  0.34906585  0.43633231  0.52359878
+      2  0.00000000  0.08726660  0.17453400  0.26180298  0.34907428  0.43634855  0.52362636
+      4  0.00000000  0.08726700  0.17453721  0.26181374  0.34909952  0.43639719  0.52370903
+      6  0.00000000  0.08726767  0.17454255  0.26183163  0.34914148  0.43647806  0.52384653
+      8  0.00000000  0.08726860  0.17454999  0.26185656  0.34919998  0.43659086  0.52403839
+
+     10  0.00000000  0.08726980  0.17455949  0.26188842  0.34927479  0.43673518  0.52428402
+     12  0.00000000  0.08727124  0.17457102  0.26192707  0.34936558  0.43691046  0.52458259
+     14  0.00000000  0.08727294  0.17458451  0.26197234  0.34947200  0.43711606  0.52493314
+     16  0.00000000  0.08727487  0.17459991  0.26202402  0.34959358  0.43735119  0.52533449
+     18  0.00000000  0.08727703  0.17461714  0.26208189  0.34972983  0.43761496  0.52578529
+
+     20  0.00000000  0.08727940  0.17463611  0.26214568  0.34988016  0.43790635  0.52628399
+     22  0.00000000  0.08728199  0.17465675  0.26221511  0.35004395  0.43822422  0.52682887
+     24  0.00000000  0.08728477  0.17467895  0.26228985  0.35022048  0.43856733  0.52741799
+     26  0.00000000  0.08728773  0.17470261  0.26236958  0.35040901  0.43893430  0.52804924
+     28  0.00000000  0.08729086  0.17472762  0.26245392  0.35060870  0.43932364  0.52872029
+
+     30  0.00000000  0.08729413  0.17475386  0.26254249  0.35081868  0.43973377  0.52942863
+     32  0.00000000  0.08729755  0.17478119  0.26263487  0.35103803  0.44016296  0.53017153
+     34  0.00000000  0.08730108  0.17480950  0.26273064  0.35126576  0.44060939  0.53094608
+     36  0.00000000  0.08730472  0.17483864  0.26282934  0.35150083  0.44107115  0.53174916
+     38  0.00000000  0.08730844  0.17486848  0.26293052  0.35174218  0.44154622  0.53257745
+
+     40  0.00000000  0.08731222  0.17489887  0.26303369  0.35198869  0.44203247  0.53342745
+     42  0.00000000  0.08731606  0.17492967  0.26313836  0.35223920  0.44252769  0.53429546
+     44  0.00000000  0.08731992  0.17496073  0.26324403  0.35249254  0.44302960  0.53517761
+     46  0.00000000  0.08732379  0.17499189  0.26335020  0.35274748  0.44353584  0.53606986
+     48  0.00000000  0.08732766  0.17502300  0.26345633  0.35300280  0.44404396  0.53696798
+
+     50  0.00000000  0.08733149  0.17505392  0.26356191  0.35325724  0.44455151  0.53786765
+     52  0.00000000  0.08733528  0.17508448  0.26366643  0.35350955  0.44505593  0.53876438
+     54  0.00000000  0.08733901  0.17511455  0.26376936  0.35375845  0.44555469  0.53965358
+     56  0.00000000  0.08734265  0.17514397  0.26387020  0.35400269  0.44604519  0.54053059
+     58  0.00000000  0.08734620  0.17517259  0.26396842  0.35424101  0.44652487  0.54139069
+
+     60  0.00000000  0.08734962  0.17520029  0.26406355  0.35447217  0.44699117  0.54222911
+     62  0.00000000  0.08735291  0.17522691  0.26415509  0.35469497  0.44744153  0.54304111
+     64  0.00000000  0.08735605  0.17525232  0.26424258  0.35490823  0.44787348  0.54382197
+     66  0.00000000  0.08735902  0.17527640  0.26432556  0.35511081  0.44828459  0.54456704
+     68  0.00000000  0.08736182  0.17529903  0.26440362  0.35530160  0.44867252  0.54527182
+
+     70  0.00000000  0.08736442  0.17532010  0.26447634  0.35547958  0.44903502  0.54593192
+     72  0.00000000  0.08736681  0.17533949  0.26454334  0.35564377  0.44936997  0.54654316
+     74  0.00000000  0.08736898  0.17535712  0.26460428  0.35579326  0.44967539  0.54710162
+     76  0.00000000  0.08737092  0.17537289  0.26465883  0.35592721  0.44994944  0.54760364
+     78  0.00000000  0.08737262  0.17538672  0.26470671  0.35604488  0.45019046  0.54804587
+
+     80  0.00000000  0.08737408  0.17539854  0.26474766  0.35614560  0.45039699  0.54842534
+     82  0.00000000  0.08737528  0.17540830  0.26478147  0.35622880  0.45056775  0.54873947
+     84  0.00000000  0.08737622  0.17541594  0.26480795  0.35629402  0.45070168  0.54898608
+     86  0.00000000  0.08737689  0.17542142  0.26482697  0.35634086  0.45079795  0.54916348
+     88  0.00000000  0.08737730  0.17542473  0.26483842  0.35636908  0.45085595  0.54927042
+
+     90  0.00000000  0.08737744  0.17542583  0.26484225  0.35637850  0.45087533  0.54930614
+
+alp|phi           0           5          10          15          20          25          30
+      5  0.00000000  0.08726730  0.17453962  0.26182180  0.34911842  0.43643361  0.52377095
+     15  0.00000000  0.08727387  0.17459198  0.26199739  0.34953092  0.43722998  0.52512754
+     25  0.00000000  0.08728623  0.17469061  0.26232912  0.35031330  0.43874792  0.52772849
+     35  0.00000000  0.08730289  0.17482397  0.26277965  0.35138244  0.44083848  0.53134425
+     45  0.00000000  0.08732185  0.17497630  0.26329709  0.35261989  0.44328233  0.53562273
+
+     55  0.00000000  0.08734084  0.17512935  0.26382007  0.35388123  0.44580113  0.54009390
+     65  0.00000000  0.08735756  0.17526454  0.26428466  0.35501092  0.44808179  0.54419926
+     75  0.00000000  0.08736998  0.17536524  0.26463237  0.35586223  0.44981644  0.54735990
+     85  0.00000000  0.08737659  0.17541895  0.26481840  0.35631976  0.45075457  0.54908352

russell_lab/data/reference/as-17-elliptic-integrals-table17.5b.txt

@@ -0,0 +1,68 @@
+alp|phi          35          40          45          50          55          60
+      0  0.61086524  0.69813170  0.78539816  0.87266463  0.95993109  1.04719755
+      2  0.61090819  0.69819436  0.78548509  0.87278045  0.96008037  1.04738465
+      4  0.61103691  0.69838220  0.78574574  0.87312784  0.96052821  1.04794603
+      6  0.61125108  0.69869484  0.78617974  0.87370649  0.96127450  1.04888194
+      8  0.61155010  0.69913161  0.78678644  0.87451593  0.96231911  1.05019278
+
+     10  0.61193318  0.69969159  0.78756494  0.87555545  0.96366180  1.05187911
+     12  0.61239927  0.70037358  0.78851403  0.87682412  0.96530224  1.05394160
+     14  0.61294707  0.70117608  0.78963221  0.87832076  0.96723998  1.05638099
+     16  0.61357504  0.70209730  0.79091768  0.88004389  0.96947438  1.05919813
+     18  0.61428140  0.70313511  0.79236827  0.88199174  0.97200462  1.06239384
+
+     20  0.61506406  0.70428706  0.79398143  0.88416214  0.97482960  1.06596891
+     22  0.61592071  0.70555037  0.79575422  0.88655254  0.97794790  1.06992405
+     24  0.61684871  0.70692183  0.79768324  0.88915992  0.98135773  1.07425976
+     26  0.61784515  0.70839788  0.79976461  0.89198071  0.98505681  1.07897628
+     28  0.61890682  0.70997451  0.80199389  0.89501076  0.98904227  1.08407347
+
+     30  0.62003018  0.71164728  0.80436610  0.89824524  0.99331059  1.08955067
+     32  0.62121138  0.71341124  0.80687558  0.90167852  0.99785743  1.09540656
+     34  0.62244622  0.71526098  0.80951599  0.90530415  1.00267749  1.10163899
+     36  0.62373019  0.71719052  0.81228024  0.90911465  1.00776438  1.10824474
+     38  0.62505840  0.71919335  0.81516039  0.91310148  1.01311039  1.11521933
+
+     40  0.62642563  0.72126235  0.81814765  0.91725487  1.01870633  1.12255667
+     42  0.62782630  0.72338982  0.82123227  0.92156370  1.02454127  1.13024880
+     44  0.62925446  0.72556741  0.82440346  0.92601535  1.03060230  1.13828546
+     46  0.63070385  0.72778615  0.82764941  0.93059558  1.03687427  1.14665369
+     48  0.63216783  0.73003640  0.83095712  0.93528835  1.04333948  1.15533731
+
+     50  0.63363946  0.73230789  0.83431247  0.94007568  1.04997735  1.16431636
+     52  0.63511149  0.73458970  0.83770010  0.94493756  1.05676412  1.17356652
+     54  0.63657639  0.73687028  0.84110344  0.94985177  1.06367248  1.18305833
+     56  0.63802636  0.73913751  0.84450468  0.95479381  1.07067128  1.19275649
+     58  0.63945343  0.74137870  0.84788483  0.95973682  1.07772516  1.20261907
+
+     60  0.64084944  0.74358071  0.85122375  0.96465156  1.08479434  1.21259662
+     62  0.64220613  0.74572998  0.85450024  0.96950647  1.09183436  1.22263139
+     64  0.64351520  0.74781266  0.85769220  0.97426773  1.09879601  1.23265660
+     66  0.64476839  0.74981471  0.86077677  0.97889946  1.10562535  1.24259576
+     68  0.64595751  0.75172208  0.86373057  0.98336406  1.11226392  1.25236238
+
+     70  0.64707458  0.75352078  0.86652996  0.98762253  1.11864920  1.26185988
+     72  0.64811189  0.75519716  0.86915135  0.99163506  1.12471530  1.27098218
+     74  0.64906209  0.75673800  0.87157159  0.99536166  1.13039401  1.27961482
+     76  0.64991829  0.75813076  0.87376830  0.99876287  1.13561610  1.28763696
+     78  0.65067414  0.75936376  0.87572037  1.00180067  1.14031304  1.29492436
+
+     80  0.65132394  0.76042640  0.87740833  1.00443942  1.14441892  1.30135321
+     82  0.65186270  0.76130931  0.87881481  1.00664678  1.14787262  1.30680495
+     84  0.65228621  0.76200457  0.87992495  1.00839470  1.15062010  1.31117165
+     86  0.65259116  0.76250582  0.88072675  1.00966028  1.15261651  1.31436170
+     88  0.65277510  0.76280846  0.88121143  1.01042658  1.15382828  1.31630510
+
+     90  0.65283658  0.76290965  0.88137359  1.01068319  1.15423455  1.31695790
+
+alp|phi          35          40          45          50          55          60
+      5  0.61113335  0.69852295  0.78594111  0.87338828  0.96086405  1.04836715
+     15  0.61325114  0.70162198  0.79025416  0.87915412  0.96832014  1.05774229
+     25  0.61733857  0.70764702  0.79870514  0.89054388  0.98317128  1.07657042
+     35  0.62308236  0.71621617  0.81088311  0.90718679  1.00518803  1.10489545
+     45  0.62997691  0.72667222  0.82601788  0.92829036  1.03371296  1.14242906
+
+     55  0.63730374  0.73800634  0.84280548  0.95232093  1.06716268  1.18788407
+     65  0.64414930  0.74882464  0.85924936  0.97660210  1.10223077  1.23764210
+     75  0.64950235  0.75745364  0.87269924  0.99710535  1.13306645  1.28370993
+     85  0.65245368  0.76227978  0.88036502  1.00908898  1.15171457  1.31291870

russell_lab/data/reference/as-17-elliptic-integrals-table17.5c.txt

@@ -0,0 +1,68 @@
+alp|phi          65          70          75          80          85          90
+      0  1.13446401  1.22173048  1.30899694  1.39626340  1.48352986  1.57079633
+      2  1.13469294  1.22200477  1.30931959  1.39663672  1.48395543  1.57127495
+      4  1.13537994  1.22282810  1.31028822  1.39775763  1.48523342  1.57271243
+      6  1.13652576  1.22420180  1.31190491  1.39962909  1.48736769  1.57511361
+      8  1.13813158  1.22612810  1.31417314  1.40225598  1.49036470  1.57848658
+
+     10  1.14019906  1.22861010  1.31709778  1.40564522  1.49423361  1.58284280
+     12  1.14273032  1.23165180  1.32068514  1.40980577  1.49898627  1.58819721
+     14  1.14572789  1.23525808  1.32494296  1.41474871  1.50463742  1.59456834
+     16  1.14919471  1.23943470  1.32988047  1.42048728  1.51120474  1.60197853
+     18  1.15313409  1.24418827  1.33550840  1.42703700  1.51870904  1.61045415
+
+     20  1.15754967  1.24952627  1.34183901  1.43441578  1.52717445  1.62002590
+     22  1.16244535  1.25545700  1.34888616  1.44264399  1.53662865  1.63072910
+     24  1.16782525  1.26198957  1.35666531  1.45174466  1.54710309  1.64260414
+     26  1.17369362  1.26913385  1.36519359  1.46174360  1.55863334  1.65569693
+     28  1.18005472  1.27690045  1.37448981  1.47266958  1.57125942  1.67005943
+
+     30  1.18691274  1.28530059  1.38457455  1.48455455  1.58502624  1.68575035
+     32  1.19427162  1.29434605  1.39547013  1.49743384  1.59998406  1.70283594
+     34  1.20213489  1.30404906  1.40720064  1.51134644  1.61618906  1.72139083
+     36  1.21050542  1.31442210  1.41979198  1.52633523  1.63370398  1.74149923
+     38  1.21938520  1.32547772  1.43327179  1.54244734  1.65259894  1.76325618
+
+     40  1.22877499  1.33722824  1.44766938  1.55973441  1.67295226  1.78676913
+     42  1.23867392  1.34968545  1.46301565  1.57825301  1.69485156  1.81215985
+     44  1.24907904  1.36286013  1.47934287  1.59806493  1.71839498  1.83956672
+     46  1.25998475  1.37676148  1.49668438  1.61923762  1.74369264  1.86914755
+     48  1.27138210  1.39139640  1.51507416  1.64184453  1.77086836  1.90108303
+
+     50  1.28325798  1.40676855  1.53454619  1.66596542  1.80006176  1.93558110
+     52  1.29559414  1.42287717  1.55513354  1.69168665  1.83143068  1.97288227
+     54  1.30836604  1.43971560  1.57686709  1.71910125  1.86515414  2.01326657
+     56  1.32154149  1.45726934  1.59977378  1.74830880  1.90143590  2.05706232
+     58  1.33507910  1.47551372  1.62387409  1.77941482  1.94050873  2.10465766
+
+     60  1.34892643  1.49441087  1.64917867  1.81252953  1.98263957  2.15651565
+     62  1.36301803  1.51390609  1.67568359  1.84776547  2.02813570  2.21319469
+     64  1.37727323  1.53392332  1.70336398  1.88523335  2.07735219  2.27537643
+     66  1.39159384  1.55435972  1.73216516  1.92503509  2.13070051  2.34390472
+     68  1.40586195  1.57507940  1.76199085  1.96725237  2.18865839  2.41984165
+
+     70  1.41993796  1.59590624  1.79268736  2.01192798  2.25177995  2.50455008
+     72  1.43365925  1.61661644  1.82402292  2.05903582  2.32070416  2.59981973
+     74  1.44684001  1.63693134  1.85566175  2.10843282  2.39615610  2.70806761
+     76  1.45927266  1.65651218  1.88713308  2.15978295  2.47892739  2.83267258
+     78  1.47073163  1.67495873  1.91779814  2.21243977  2.56980281  2.97856895
+
+     80  1.48098006  1.69181489  1.94682231  2.26527326  2.66935045  3.15338525
+     82  1.48977975  1.70658456  1.97316665  2.31643896  2.77736748  3.36986803
+     84  1.49690410  1.71876033  1.99562118  2.36313736  2.89146664  3.65185597
+     86  1.50215336  1.72786543  2.01290452  2.40153358  3.00370926  4.05275817
+     88  1.50537033  1.73350464  2.02384126  2.42718003  3.09448898  4.74271727
+
+     90  1.50645424  1.73541516  2.02758942  2.43624605  3.13130133         inf
+
+alp|phi          65          70          75          80          85          90
+      5  1.13589544  1.22344604  1.31101537  1.39859928  1.48619317  1.57379213
+     15  1.14740244  1.23727471  1.32732612  1.41751762  1.50780533  1.59814200
+     25  1.17069811  1.26548460  1.36083467  1.45663012  1.55273384  1.64899522
+     35  1.20625660  1.30915104  1.41338702  1.51870347  1.62477858  1.73124518
+     45  1.25446980  1.36971948  1.48788472  1.60847673  1.73081713  1.85407468
+
+     55  1.31490567  1.44840433  1.58817233  1.73347444  1.88296142  2.03471531
+     65  1.38443224  1.54409676  1.71762935  1.90483674  2.10348169  2.30878680
+     75  1.45316359  1.64683711  1.87145396  2.13389514  2.43657614  2.76806315
+     85  1.49977411  1.72372395  2.00498776  2.38364709  2.94868875  3.83174200