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Expand test cases for negative x.

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Add comment on the calculation of derivatives of cot(x).
This commit is contained in:
jzmaddock
2014-10-25 17:28:09 +01:00
parent bda3331ed3
commit dffdb444ee
2 changed files with 25 additions and 3 deletions
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18
include/boost/math/special_functions/detail/polygamma.hpp
View File

@@ -207,7 +207,23 @@
template <class T, class Policy>
T poly_cot_pi(int n, T x, const Policy& pol)
{
// Return n'th derivative of cot(pi*x) at x:
// Return n'th derivative of cot(pi*x) at x, these are simply
// tabulated for up to n = 9, beyond that it is possible to
// calculate coefficients as follows:
//
// The general form of each derivative is:
//
// pi^n * SUM{k=0, n} C[k,n] * cos(k * pi * x) * csc^(n+1)(pi * x)
//
// With constant C[0,1] = -1 and all other C[k,n] = 0;
// Then for each k < n+1:
// C[|1 - k|, n+1] += -(k + n + 2) * C[k, n] / 2;
// C[k + 1, n+1] += -(n + 2 - k) * C[k, n] / 2;
//
// It's worth noting however, that as well as requiring quite a bit
// of storage space, this method has no better accuracy than recursion
// on x to x > 0 when computing polygamma :-(
//
T s = boost::math::sin_pi(x);
switch(n)
{

10
test/test_polygamma.hpp
View File

@@ -138,7 +138,7 @@ void test_polygamma(T, const char* name)
} };
do_test_polygamma<T>(big_data, name, "Mathematica Data - large arguments");
boost::array<boost::array<value_type, 3>, 214> neg_data =
boost::array<boost::array<value_type, 3>, 310> neg_data =
{ {
{ 1, SC_(-12.750), SC_(19.663772856722737612034697464751605) }, { 1, SC_(-12.250), SC_(19.660817549236368273654684043826967) }, { 1, SC_(-11.750), SC_(19.657621376522814505537196503582823) }, { 1, SC_(-11.250), SC_(19.654153659190554029589711115880278) }, { 1, SC_(-10.750), SC_(19.650378280099093364749509767503149) }, { 1, SC_(-10.250), SC_(19.646252424622652795021809881312377) }, { 1, SC_(-9.7500), SC_(19.641724953976865133273035997897957) }, { 1, SC_(-9.2500), SC_(19.636734280660725370869519577921538) }, { 1, SC_(-8.7500), SC_(19.631205558842085383108670448917024) }, { 1, SC_(-8.2500), SC_(19.625046917622010980803778160828770) }, { 1, SC_(-7.7500), SC_(19.618144334352289464741323510141514) }, { 1, SC_(-7.2500), SC_(19.610354539293269015698176691590937) }, { 1, SC_(-6.7500), SC_(19.601495010731061577124257953429755) }, { 1, SC_(-6.2500), SC_(19.591329569019785068016844943671793) }, { 1, SC_(-5.7500), SC_(19.579547136931335925546754524074474) }, { 1, SC_(-5.2500), SC_(19.565729569019785068016844943671793) }, { 1, SC_(-4.7500), SC_(19.549301390239464469970194977760674) }, { 1, SC_(-4.2500), SC_(19.529448389881463072551992335962043) }, { 1, SC_(-3.7500), SC_(19.504980060599575273294294700752364) }, { 1, SC_(-3.2500), SC_(19.474085068082155114074483685443011) }, { 1, SC_(-2.7500), SC_(19.433868949488464162183183589641253) }, { 1, SC_(-2.2500), SC_(19.379410511869137362595193744614609) }, { 1, SC_(-1.7500), SC_(19.301637544529786476232770366500757) }, { 1, SC_(-1.2500), SC_(19.181879647671606498397662880417078) }, { 1, SC_(-0.75000), SC_(18.975106932284888517049096897113002) }, { 1, SC_(-0.25000), SC_(18.541879647671606498397662880417078) },
{ 2, SC_(-12.750), SC_(-124.03079461415823384604153251543681) }, { 2, SC_(-12.250), SC_(124.01896466745858356132308878716344) }, { 2, SC_(-11.750), SC_(-124.03175955222881001960976796032603) }, { 2, SC_(-11.250), SC_(124.01787668541028735821044014586602) }, { 2, SC_(-10.750), SC_(-124.03299241970518808612682102178640) }, { 2, SC_(-10.250), SC_(124.01647202148710491650947992638728) }, { 2, SC_(-9.7500), SC_(-124.03460234084420729198290916496876) }, { 2, SC_(-9.2500), SC_(124.01461482266526541911391108670126) }, { 2, SC_(-8.7500), SC_(-124.03676016548723903560636876475972) }, { 2, SC_(-8.2500), SC_(124.01208782525148933477537240192445) }, { 2, SC_(-7.7500), SC_(-124.03974558822776381694747663647984) }, { 2, SC_(-7.2500), SC_(124.00852603656573370687098416695770) }, { 2, SC_(-6.7500), SC_(-124.04404218787195165891317097369578) }, { 2, SC_(-6.2500), SC_(124.00327776890408296268303058132483) }, { 2, SC_(-5.7500), SC_(-124.05054526159038888901020902683808) }, { 2, SC_(-5.2500), SC_(123.99508576890408296268303058132483) }, { 2, SC_(-4.7500), SC_(-124.06106552130930069964553408642549) }, { 2, SC_(-4.2500), SC_(123.98126436732757934536308673076874) }, { 2, SC_(-3.7500), SC_(-124.07972713378925404561433420306057) }, { 2, SC_(-3.2500), SC_(123.95521103942202265902072971875978) }, { 2, SC_(-2.7500), SC_(-124.11765305971517997154026012898649) }, { 2, SC_(-2.2500), SC_(123.89694977406016558118732052440384) }, { 2, SC_(-1.7500), SC_(-124.21382135423058192495874247308867) }, { 2, SC_(-1.2500), SC_(123.72136678366236036856729308956159) }, { 2, SC_(-0.75000), SC_(-124.58699919679617959259722643810325) }, { 2, SC_(-0.25000), SC_(122.69736678366236036856729308956159) },
@@ -148,7 +148,13 @@ void test_polygamma(T, const char* name)
{ 6, SC_(-12.750), SC_(-1.1791224761262553923199740571475659e7) }, { 6, SC_(-12.250), SC_(1.1791224761212959953338096554813394e7) }, { 6, SC_(-11.750), SC_(-1.1791224761275698941741584362376857e7) }, { 6, SC_(-11.250), SC_(1.1791224761195566737931185183742355e7) }, { 6, SC_(-10.750), SC_(-1.1791224761298983469279962918313528e7) }, { 6, SC_(-10.250), SC_(1.1791224761163997446137740568065356e7) }, { 6, SC_(-9.7500), SC_(-1.1791224761342381822144610498473088e7) }, { 6, SC_(-9.2500), SC_(1.1791224761103426349211734412564015e7) }, { 6, SC_(-8.7500), SC_(-1.1791224761428342570269073362379294e7) }, { 6, SC_(-8.2500), SC_(1.1791224760979163768967238911569317e7) }, { 6, SC_(-7.7500), SC_(-1.1791224761611690548456004706583543e7) }, { 6, SC_(-7.2500), SC_(1.1791224760702370925869563208323974e7) }, { 6, SC_(-6.7500), SC_(-1.1791224762040456784641593897905007e7) }, { 6, SC_(-6.2500), SC_(1.1791224760018512109221395670440442e7) }, { 6, SC_(-5.7500), SC_(-1.1791224763168189258973034501364332e7) }, { 6, SC_(-5.2500), SC_(1.1791224758085776826021395670440442e7) }, { 6, SC_(-4.7500), SC_(-1.1791224766632825018904254258839318e7) }, { 6, SC_(-4.2500), SC_(1.1791224751536137687542195274483895e7) }, { 6, SC_(-3.7500), SC_(-1.1791224779829886169083430020877813e7) }, { 6, SC_(-3.2500), SC_(1.1791224722787982136529824573562902e7) }, { 6, SC_(-2.7500), SC_(-1.1791224848871927321346804506474521e7) }, { 6, SC_(-2.2500), SC_(1.1791224534791825988329541210690757e7) }, { 6, SC_(-1.7500), SC_(-1.1791225454217875175963567504038405e7) }, { 6, SC_(-1.2500), SC_(1.1791222068440904625468941445147639e7) }, { 6, SC_(-0.75000), SC_(-1.1791239778278671029974833461007258e7) }, { 6, SC_(-0.25000), SC_(1.1791071073496904625468941445147639e7) },
{ 7, SC_(-12.750), SC_(3.3035269585550319411369451657990653e8) }, { 7, SC_(-12.250), SC_(3.3035269585550014530679554645726841e8) }, { 7, SC_(-11.750), SC_(3.3035269585549597724076958273392155e8) }, { 7, SC_(-11.250), SC_(3.3035269585549020632656302567379925e8) }, { 7, SC_(-10.750), SC_(3.3035269585548210560734246359421460e8) }, { 7, SC_(-10.250), SC_(3.3035269585547056321166932680182245e8) }, { 7, SC_(-9.7500), SC_(3.3035269585545384621477943726294791e8) }, { 7, SC_(-9.2500), SC_(3.3035269585542919758450034698830934e8) }, { 7, SC_(-8.7500), SC_(3.3035269585539213080586956648886140e8) }, { 7, SC_(-8.2500), SC_(3.3035269585533516103728829633890794e8) }, { 7, SC_(-7.7500), SC_(3.3035269585524545242332002141349801e8) }, { 7, SC_(-7.2500), SC_(3.3035269585510030650375087453009371e8) }, { 7, SC_(-6.7500), SC_(3.3035269585485817969386206988585281e8) }, { 7, SC_(-6.2500), SC_(3.3035269585444002902560781621765444e8) }, { 7, SC_(-5.7500), SC_(3.3035269585368867935011094629708018e8) }, { 7, SC_(-5.2500), SC_(3.3035269585227536550842381621765444e8) }, { 7, SC_(-4.7500), SC_(3.3035269584947086190323815702711063e8) }, { 7, SC_(-4.2500), SC_(3.3035269584354251332378488235637904e8) }, { 7, SC_(-3.7500), SC_(3.3035269583002256126086884537779074e8) }, { 7, SC_(-3.2500), SC_(3.3035269579619261006329391884897976e8) }, { 7, SC_(-2.7500), SC_(3.3035269570114408444331054633801022e8) }, { 7, SC_(-2.2500), SC_(3.3035269539127781220563177006740899e8) }, { 7, SC_(-1.7500), SC_(3.3035269416026348990428605870784761e8) }, { 7, SC_(-1.2500), SC_(3.3035268771818605685450990413016373e8) }, { 7, SC_(-0.75000), SC_(3.3035263686402030648824099487997220e8) }, { 7, SC_(-0.25000), SC_(3.3035184214649965685450990413016373e8) },
{ 8, SC_(-7.7500), SC_(-1.0569114259666913771892699180522388e10) }, { 8, SC_(-7.2500), SC_(1.0569114259666319911021283350705169e10) }, { 8, SC_(-6.7500), SC_(-1.0569114259667313537290849324034796e10) }, { 8, SC_(-6.2500), SC_(1.0569114259665591328976435838084546e10) }, { 8, SC_(-5.7500), SC_(-1.0569114259668699611772332137177045e10) }, { 8, SC_(-5.2500), SC_(1.0569114259662820559674440318084546e10) }, { 8, SC_(-4.7500), SC_(-1.0569114259674567879524502974422220e10) }, { 8, SC_(-4.2500), SC_(1.0569114259649513356345466704581650e10) }, { 8, SC_(-3.7500), SC_(-1.0569114259707322912185335499305285e10) }, { 8, SC_(-3.2500), SC_(1.0569114259560384126678660185038311e10) }, { 8, SC_(-2.7500), SC_(-1.0569114259982263662729459870590150e10) }, { 8, SC_(-2.2500), SC_(1.0569114258563670778105953357268290e10) }, { 8, SC_(-1.7500), SC_(-1.0569114264464825392297531107332441e10) }, { 8, SC_(-1.2500), SC_(1.0569114231281566759079742278380307e10) }, { 8, SC_(-0.75000), SC_(-1.0569114526390508516485165684831300e10) }, { 8, SC_(-0.25000), SC_(1.0569108819622773799079742278380307e10) },
{ 9, SC_(-7.7500), SC_(3.8051374324233954962308727459400485e11) }, { 9, SC_(-7.2500), SC_(3.8051374324233938918730928010849146e11) }, { 9, SC_(-6.7500), SC_(3.8051374324233908537939910023379690e11) }, { 9, SC_(-6.2500), SC_(3.8051374324233848474063291767903138e11) }, { 9, SC_(-5.7500), SC_(3.8051374324233723728009045648294056e11) }, { 9, SC_(-5.2500), SC_(3.8051374324233449483283804413023138e11) }, { 9, SC_(-4.7500), SC_(3.8051374324232805216534792821594812e11) }, { 9, SC_(-4.2500), SC_(3.8051374324231168248427408936422641e11) }, { 9, SC_(-3.7500), SC_(3.8051374324226598999820108764248547e11) }, { 9, SC_(-3.2500), SC_(3.8051374324212293823321496967578170e11) }, { 9, SC_(-2.7500), SC_(3.8051374324160613219689518915140179e11) }, { 9, SC_(-2.2500), SC_(3.8051374323936280896024439692195702e11) }, { 9, SC_(-1.7500), SC_(3.8051374322693593017285422874024520e11) }, { 9, SC_(-1.2500), SC_(3.8051374313023439288413955260640509e11) }, { 9, SC_(-0.75000), SC_(3.8051374187988955981988925091310821e11) }, { 9, SC_(-0.25000), SC_(3.8051370416629108357213955260640509e11) }
{ 9, SC_(-7.7500), SC_(3.8051374324233954962308727459400485e11) }, { 9, SC_(-7.2500), SC_(3.8051374324233938918730928010849146e11) }, { 9, SC_(-6.7500), SC_(3.8051374324233908537939910023379690e11) }, { 9, SC_(-6.2500), SC_(3.8051374324233848474063291767903138e11) }, { 9, SC_(-5.7500), SC_(3.8051374324233723728009045648294056e11) }, { 9, SC_(-5.2500), SC_(3.8051374324233449483283804413023138e11) }, { 9, SC_(-4.7500), SC_(3.8051374324232805216534792821594812e11) }, { 9, SC_(-4.2500), SC_(3.8051374324231168248427408936422641e11) }, { 9, SC_(-3.7500), SC_(3.8051374324226598999820108764248547e11) }, { 9, SC_(-3.2500), SC_(3.8051374324212293823321496967578170e11) }, { 9, SC_(-2.7500), SC_(3.8051374324160613219689518915140179e11) }, { 9, SC_(-2.2500), SC_(3.8051374323936280896024439692195702e11) }, { 9, SC_(-1.7500), SC_(3.8051374322693593017285422874024520e11) }, { 9, SC_(-1.2500), SC_(3.8051374313023439288413955260640509e11) }, { 9, SC_(-0.75000), SC_(3.8051374187988955981988925091310821e11) }, { 9, SC_(-0.25000), SC_(3.8051370416629108357213955260640509e11) },
{SC_(10), SC_(-7.7500), SC_(-1.5220204740668341333676731352020598e13)}, { SC_(10), SC_(-7.2500), SC_(1.5220204740668340669650877835677384e13) }, { SC_(10), SC_(-6.7500), SC_(-1.5220204740668341932700845125388608e13) }, { SC_(10), SC_(-6.2500), SC_(1.5220204740668339422138220784050543e13) }, { SC_(10), SC_(-5.7500), SC_(-1.5220204740668344670625746819834321e13) }, { SC_(10), SC_(-5.2500), SC_(1.5220204740668333038285748986372463e13) }, { SC_(10), SC_(-4.7500), SC_(-1.5220204740668360644738342521168221e13) }, { SC_(10), SC_(-4.2500), SC_(1.5220204740668289586193246215389596e13) }, { SC_(10), SC_(-3.7500), SC_(-1.5220204740668491301932335869743932e13) }, { SC_(10), SC_(-3.2500), SC_(1.5220204740667845482073107110240315e13) }, { SC_(10), SC_(-2.7500), SC_(-1.5220204740670250922735818265720155e13) }, { SC_(10), SC_(-2.2500), SC_(1.5220204740659352776617813040228546e13) }, { SC_(10), SC_(-1.7500), SC_(-1.5220204740723597111914149030851634e13) }, { SC_(10), SC_(-1.2500), SC_(1.5220204740174337594057347065492760e13) }, { SC_(10), SC_(-0.75000), SC_(-1.5220204748421004942502520332720988e13) }, { SC_(10), SC_(-0.25000), SC_(1.5220204428462791119561347065492760e13) },
{SC_(11), SC_(-7.7500), SC_(6.6969403856797204470999075620782555e14)}, { SC_(11), SC_(-7.2500), SC_(6.6969403856797204443997065278152669e14) }, { SC_(11), SC_(-6.7500), SC_(6.6969403856797204385976298181981934e14) }, { SC_(11), SC_(-6.2500), SC_(6.6969403856797204254719282828940321e14) }, { SC_(11), SC_(-5.7500), SC_(6.6969403856797203939795943831775966e14) }, { SC_(11), SC_(-5.2500), SC_(6.6969403856797203131161247792548979e14) }, { SC_(11), SC_(-4.7500), SC_(6.6969403856797200883878751610651220e14) }, { SC_(11), SC_(-4.2500), SC_(6.6969403856797194026913294831009712e14) }, { SC_(11), SC_(-3.7500), SC_(6.6969403856797170626423300519402108e14) }, { SC_(11), SC_(-3.2500), SC_(6.6969403856797079082317494121441662e14) }, { SC_(11), SC_(-2.7500), SC_(6.6969403856796654470987612349915749e14) }, { SC_(11), SC_(-2.2500), SC_(6.6969403856794204628163394590053064e14) }, { SC_(11), SC_(-1.7500), SC_(6.6969403856775315995316280043863158e14) }, { SC_(11), SC_(-1.2500), SC_(6.6969403856557086094467166780182235e14) }, { SC_(11), SC_(-0.75000), SC_(6.6969403851936945358946446654116707e14) }, { SC_(11), SC_(-0.25000), SC_(6.6969403582250925196910686780182235e14) },
{SC_(12), SC_(-7.7500), SC_(-3.2145233093874118337222380549030432e16)}, { SC_(12), SC_(-7.2500), SC_(3.2145233093874118336089459241510371e16) }, { SC_(12), SC_(-6.7500), SC_(-3.2145233093874118338538862264211861e16) }, { SC_(12), SC_(-6.2500), SC_(3.2145233093874118332956585600971684e16) }, { SC_(12), SC_(-5.7500), SC_(-3.2145233093874118346470957452659967e16) }, { SC_(12), SC_(-5.2500), SC_(3.2145233093874118311384271328272970e16) }, { SC_(12), SC_(-4.7500), SC_(-3.2145233093874118410246620594666049e16) }, { SC_(12), SC_(-4.2500), SC_(3.2145233093874118103287175260580644e16) }, { SC_(12), SC_(-3.7500), SC_(-3.2145233093874119174645495148550237e16) }, { SC_(12), SC_(-3.2500), SC_(3.2145233093874114857792705593486958e16) }, { SC_(12), SC_(-2.7500), SC_(-3.2145233093874135691619437169973801e16) }, { SC_(12), SC_(-2.2500), SC_(3.2145233093874008724100861918481841e16) }, { SC_(12), SC_(-1.7500), SC_(-3.2145233093875066825103277125147005e16) }, { SC_(12), SC_(-1.2500), SC_(3.2145233093861362402303729768622063e16) }, { SC_(12), SC_(-0.75000), SC_(-3.2145233094206840811597208557586761e16) }, { SC_(12), SC_(-0.25000), SC_(3.2145233067527970956138307688622063e16) },
{SC_(13), SC_(-7.7500), SC_(1.6715535177261375555623235851859237e18)}, { SC_(13), SC_(-7.2500), SC_(1.6715535177261375555616811505615971e18) }, { SC_(13), SC_(-6.7500), SC_(1.6715535177261375555601152932765871e18) }, { SC_(13), SC_(-6.2500), SC_(1.6715535177261375555560635840337346e18) }, { SC_(13), SC_(-5.7500), SC_(1.6715535177261375555448386655062426e18) }, { SC_(13), SC_(-5.2500), SC_(1.6715535177261375555111931703465213e18) }, { SC_(13), SC_(-4.7500), SC_(1.6715535177261375554006502097069245e18) }, { SC_(13), SC_(-4.2500), SC_(1.6715535177261375549959051229408070e18) }, { SC_(13), SC_(-3.7500), SC_(1.6715535177261375533086111846120836e18) }, { SC_(13), SC_(-3.2500), SC_(1.6715535177261375450685102745473439e18) }, { SC_(13), SC_(-2.7500), SC_(1.6715535177261374960497681856044819e18) }, { SC_(13), SC_(-2.2500), SC_(1.6715535177261371205337428998473234e18) }, { SC_(13), SC_(-1.7500), SC_(1.6715535177261330943278445785436631e18) }, { SC_(13), SC_(-1.2500), SC_(1.6715535177260640528966928029814669e18) }, { SC_(13), SC_(-0.75000), SC_(1.6715535177236684875710325164741107e18) }, { SC_(13), SC_(-0.25000), SC_(1.6715535174521967818565724133494669e18) },
{SC_(20), SC_(-7.7500), SC_(-1.0700016187896297695358366297227405e31)}, { SC_(20), SC_(-7.2500), SC_(1.0700016187896297695358366297227207e31) }, { SC_(20), SC_(-6.7500), SC_(-1.0700016187896297695358366297227919e31) }, { SC_(20), SC_(-6.2500), SC_(1.0700016187896297695358366297225123e31) }, { SC_(20), SC_(-5.7500), SC_(-1.0700016187896297695358366297237267e31) }, { SC_(20), SC_(-5.2500), SC_(1.0700016187896297695358366297178064e31) }, { SC_(20), SC_(-4.7500), SC_(-1.0700016187896297695358366297508349e31) }, { SC_(20), SC_(-4.2500), SC_(1.0700016187896297695358366295346680e31) }, { SC_(20), SC_(-3.7500), SC_(-1.0700016187896297695358366312489970e31) }, { SC_(20), SC_(-3.2500), SC_(1.0700016187896297695358366140480322e31) }, { SC_(20), SC_(-2.7500), SC_(-1.0700016187896297695358368457690989e31) }, { SC_(20), SC_(-2.2500), SC_(1.0700016187896297695358322831891149e31) }, { SC_(20), SC_(-1.7500), SC_(-1.0700016187896297695359814356987569e31) }, { SC_(20), SC_(-1.2500), SC_(1.0700016187896297695260533442006924e31) }, { SC_(20), SC_(-0.75000), SC_(-1.0700016187896297714516730336449055e31) }, { SC_(20), SC_(-0.25000), SC_(1.0700016187896275255700182817756420e31) },
{SC_(23), SC_(-7.7500), SC_(7.2766958095269026379022334905108869e36)}, { SC_(23), SC_(-7.2500), SC_(7.2766958095269026379022334905108869e36) }, { SC_(23), SC_(-6.7500), SC_(7.2766958095269026379022334905108869e36) }, { SC_(23), SC_(-6.2500), SC_(7.2766958095269026379022334905108868e36) }, { SC_(23), SC_(-5.7500), SC_(7.2766958095269026379022334905108866e36) }, { SC_(23), SC_(-5.2500), SC_(7.2766958095269026379022334905108848e36) }, { SC_(23), SC_(-4.7500), SC_(7.2766958095269026379022334905108714e36) }, { SC_(23), SC_(-4.2500), SC_(7.2766958095269026379022334905107503e36) }, { SC_(23), SC_(-3.7500), SC_(7.2766958095269026379022334905093860e36) }, { SC_(23), SC_(-3.2500), SC_(7.2766958095269026379022334904893135e36) }, { SC_(23), SC_(-2.7500), SC_(7.2766958095269026379022334900771270e36) }, { SC_(23), SC_(-2.2500), SC_(7.2766958095269026379022334770834817e36) }, { SC_(23), SC_(-1.7500), SC_(7.2766958095269026379022327513062336e36) }, { SC_(23), SC_(-1.2500), SC_(7.2766958095269026379021422520579094e36) }, { SC_(23), SC_(-0.75000), SC_(7.2766958095269026378642504512809960e36) }, { SC_(23), SC_(-0.25000), SC_(7.2766958095269025158194448897624671e36) }
} };
do_test_polygamma<T>(neg_data, name, "Mathematica Data - negative arguments");