D3 test mpmath matched Yang

src/nmie-impl.hpp

@@ -393,24 +393,7 @@ namespace nmie {
                                std::vector<std::complex<FloatType> >& D1,
                                std::vector<std::complex<FloatType> >& D3) {
     evalDownwardD1(z, D1);
-//    int lnmx = evalKapteynNumberOfLostSignificantDigits(nmax_, z);
-//    std::vector<std::complex<FloatType> > r;
-//    if (lnmx < 4) {
-//      r.resize(nmax_+1);
-//      evalForwardR(z, r);
-//    } else {
-//      int valid_digits = 6;
-//      int nstar = getNStar(nmax_, z, valid_digits);
-//      r.resize(nstar);
-//      evalBackwardR(z,r);
-//    }
-//    convertRtoD1(z, r, D1);
-
-    // TODO: Do we need this check?
-    // if (cabs(D1[0]) > 1.0e15) {
-    //   throw std::invalid_argument("Unstable D1! Please, try to change input parameters!\n");
-    // //printf("Warning: Potentially unstable D1! Please, try to change input parameters!\n");
-    // }
+
 
     // Upward recurrence for PsiZeta and D3 - equations (18a) - (18d)
     PsiZeta_[0] = static_cast<FloatType>(0.5)*(static_cast<FloatType>(1.0) - std::complex<FloatType>(nmm::cos(2.0*z.real()), nmm::sin(2.0*z.real()))

tests/mpmath_riccati_bessel.py

@@ -23,6 +23,9 @@ def ksi(n,z):
 def psi_div_ksi(n,z):
     return psi(n,z)/ksi(n,z)
 
+def psi_mul_ksi(n,z):
+    return psi(n,z)*ksi(n,z)
+
 def psi_div_xi(n,z):
     return psi(n,z)/xi(n,z)
 

tests/mpmath_special_functions_test_generator.py

@@ -90,6 +90,7 @@ class UpdateSpecialFunctionsEvaluations:
         return sorted(n_set)
 
     def get_test_data_nlist(self, z_record, output_dps, n, func):
+        isNeedMoreDPS = False
         x = str(z_record[0])
         mr = str(z_record[1][0])
         mi = str(z_record[1][1])
@@ -106,9 +107,12 @@ class UpdateSpecialFunctionsEvaluations:
                      mp.nstr(mp.fabs(D1nz.real * 10 ** -output_dps), 2) + ',' +
                      mp.nstr(mp.fabs(D1nz.imag * 10 ** -output_dps), 2) +
                      '},')
+            if mp.nstr(D1nz.real, output_dps) == '0.0' \
+                or mp.nstr(D1nz.imag, output_dps) == '0.0':
+                isNeedMoreDPS = True
         except:
-            pass
-        return z_str
+            isNeedMoreDPS = True
+        return z_str, isNeedMoreDPS
 
     def get_test_data(self, Du_test, output_dps, max_num_elements_of_n_list, func, funcname):
         output_list = ['// complex(z), n, complex(f(n,z)), abs_err_real, abs_err_imag',
@@ -126,16 +130,17 @@ class UpdateSpecialFunctionsEvaluations:
             failed_evaluations = 0
             for n in n_list:
                 mp.mp.dps = 20
-                old_z_string = self.get_test_data_nlist(z_record, output_dps, n, func)
+                old_z_string, isNeedMoreDPS = self.get_test_data_nlist(z_record, output_dps, n, func,)
                 mp.mp.dps = 37
-                new_z_string = self.get_test_data_nlist(z_record, output_dps, n, func)
-                while old_z_string != new_z_string:
+                new_z_string, isNeedMoreDPS = self.get_test_data_nlist(z_record, output_dps, n, func)
+                while old_z_string != new_z_string \
+                        or isNeedMoreDPS:
                     new_dps = int(mp.mp.dps * 1.41)
                     if new_dps > 300: break
                     mp.mp.dps = new_dps
                     print("New dps = ", mp.mp.dps, 'n =', n, ' (max ',n_list[-1],') for z =', z, '     ', end='')
                     old_z_string = new_z_string
-                    new_z_string = self.get_test_data_nlist(z_record, output_dps, n, func)
+                    new_z_string, isNeedMoreDPS = self.get_test_data_nlist(z_record, output_dps, n, func)
 
                 if new_z_string != '':
                     output_list.append(new_z_string)
@@ -161,9 +166,10 @@ def main():
                                                  output_dps=16, max_num_elements_of_nlist=51)
                                                  # output_dps=5, max_num_elements_of_nlist=3)
     # sf_evals.run_test(mrb.D1, 'D1')
-    sf_evals.run_test(mrb.D2, 'D2')
-    # sf_evals.run_test(mrb.D3, 'D3')
+    # sf_evals.run_test(mrb.D2, 'D2')
+    sf_evals.run_test(mrb.D3, 'D3')
     # sf_evals.run_test(mrb.psi_div_ksi, 'psi_div_ksi')
+    # sf_evals.run_test(mrb.psi_mul_ksi, 'psi_mul_ksi')
     # sf_evals.run_test(mrb.psi_div_xi, 'psi_div_xi')
     with open(sf_evals.filename, 'w') as out_file:
         out_file.write(sf_evals.get_file_content())

tests/test_spec_functions_data.hpp

@@ -1094,3 +1094,574 @@ D2_test_16digits
 {{3.141592653589793,0.0},18,{-5.639036854808339,0.0},5.6e-16,0.0},
 {{3.141592653589793,0.0},20,{-6.285084985730819,0.0},6.3e-16,0.0},
 };
+
+// complex(z), n, complex(f(n,z)), abs_err_real, abs_err_imag
+std::vector< std::tuple< std::complex<double>, int, std::complex<double>, double, double > >
+D3_test_16digits
+= {
+{{13300.0,-0.1},0,{0.0,1.0},0.0,1.0e-16},
+{{13300.0,-0.1},1,{-3.400439576064988e-13,0.9999999943467692},3.4e-29,1.0e-16},
+{{13300.0,-0.1},2,{-1.02013189877118e-12,0.9999999830403075},1.0e-28,1.0e-16},
+{{13300.0,-0.1},3,{-2.040263875397417e-12,0.9999999660806147},2.0e-28,1.0e-16},
+{{13300.0,-0.1},4,{-3.400439965340277e-12,0.9999999434676906},3.4e-28,1.0e-16},
+{{13300.0,-0.1},5,{-5.100660272406543e-12,0.9999999152015347},5.1e-28,1.0e-16},
+{{13300.0,-0.1},6,{-7.140924926354723e-12,0.9999998812821465},7.1e-28,1.0e-16},
+{{13300.0,-0.1},8,{-1.224158792368965e-11,0.9999997964836711},1.2e-27,1.0e-16},
+{{13300.0,-0.1},10,{-1.870243051444869e-11,0.9999996890722586},1.9e-27,1.0e-16},
+{{13300.0,-0.1},12,{-2.652345467096505e-11,0.9999995590479018},2.7e-27,1.0e-16},
+{{13300.0,-0.1},15,{-4.080533667787133e-11,0.999999321612076},4.1e-27,1.0e-16},
+{{13300.0,-0.1},18,{-5.814764248686488e-11,0.999999033297069},5.8e-27,1.0e-16},
+{{13300.0,-0.1},22,{-8.603139703783761e-11,0.9999985697315893},8.6e-27,1.0e-16},
+{{13300.0,-0.1},27,{-1.28537232356385e-10,0.9999978630764811},1.3e-26,1.0e-16},
+{{13300.0,-0.1},32,{-1.795444131588628e-10,0.9999970150896922},1.8e-26,1.0e-16},
+{{13300.0,-0.1},40,{-2.788389500368772e-10,0.999995364340014},2.8e-26,1.0e-16},
+{{13300.0,-0.1},48,{-3.99897670899495e-10,0.9999933517785008},4.0e-26,1.0e-16},
+{{13300.0,-0.1},59,{-6.018913482575541e-10,0.999989993731453},6.0e-26,1.0e-16},
+{{13300.0,-0.1},71,{-8.691806030714543e-10,0.9999855502377231},8.7e-26,1.0e-16},
+{{13300.0,-0.1},87,{-1.301751636840212e-9,0.999978359198404},1.3e-25,1.0e-16},
+{{13300.0,-0.1},105,{-1.892478559255885e-9,0.999968539275809},1.9e-25,1.0e-16},
+{{13300.0,-0.1},128,{-2.807697724335093e-9,0.9999533258374243},2.8e-25,1.0e-16},
+{{13300.0,-0.1},155,{-4.111763702027196e-9,0.9999316501039903},4.1e-25,1.0e-16},
+{{13300.0,-0.1},189,{-6.106883864825003e-9,0.9998984910892781},6.1e-25,1.0e-16},
+{{13300.0,-0.1},229,{-8.95805814929697e-9,0.9998511110833991},9.0e-25,1.0e-16},
+{{13300.0,-0.1},279,{-1.328871882120813e-8,0.9997791604205924},1.3e-24,1.0e-16},
+{{13300.0,-0.1},338,{-1.949566427437159e-8,0.9996760682893887},1.9e-24,1.0e-16},
+{{13300.0,-0.1},411,{-2.882119692710565e-8,0.99952124896133},2.9e-24,1.0e-16},
+{{13300.0,-0.1},499,{-4.248789333886431e-8,0.9992945106023009},4.2e-24,1.0e-16},
+{{13300.0,-0.1},606,{-6.268784827324142e-8,0.9989597110407947},6.3e-24,1.0e-16},
+{{13300.0,-0.1},736,{-9.254452618551611e-8,0.9984655761319616},9.3e-24,1.0e-16},
+{{13300.0,-0.1},894,{-1.367351871332925e-7,0.9977357768741157},1.4e-23,1.0e-16},
+{{13300.0,-0.1},1086,{-2.022251492259403e-7,0.9966576457442607},2.0e-23,1.0e-16},
+{{13300.0,-0.1},1319,{-2.993332266274127e-7,0.9950664665997103},3.0e-23,1.0e-16},
+{{13300.0,-0.1},1601,{-4.432934379930979e-7,0.9927238197940671},4.4e-23,9.9e-17},
+{{13300.0,-0.1},1945,{-6.593672942603503e-7,0.989243494339538},6.6e-23,9.9e-17},
+{{13300.0,-0.1},2362,{-9.83775545482961e-7,0.9840970601559616},9.8e-23,9.8e-17},
+{{13300.0,-0.1},13560,{-0.1978882692837703,-3.967948672613146e-5},2.0e-17,4.0e-21},
+{{15000.0,-10000.0},0,{0.0,1.0},0.0,1.0e-16},
+{{15000.0,-10000.0},1,{2.84026945162933e-9,0.9999999988164005},2.8e-25,1.0e-16},
+{{15000.0,-10000.0},2,{8.520808375061527e-9,0.9999999964492016},8.5e-25,1.0e-16},
+{{15000.0,-10000.0},3,{1.704161681064366e-8,0.9999999928984033},1.7e-24,1.0e-16},
+{{15000.0,-10000.0},4,{2.840269481889634e-8,0.9999999881640057},2.8e-24,1.0e-16},
+{{15000.0,-10000.0},6,{5.96456598963634e-8,0.9999999751444127},6.0e-24,1.0e-16},
+{{15000.0,-10000.0},7,{7.952754718748665e-8,0.9999999668592176},8.0e-24,1.0e-16},
+{{15000.0,-10000.0},9,{1.278121319805862e-7,0.9999999467380309},1.3e-23,1.0e-16},
+{{15000.0,-10000.0},11,{1.874577982316124e-7,0.9999999218824499},1.9e-23,1.0e-16},
+{{15000.0,-10000.0},13,{2.584645476351414e-7,0.9999998922924764},2.6e-23,1.0e-16},
+{{15000.0,-10000.0},16,{3.862767071525949e-7,0.9999998390305346},3.9e-23,1.0e-16},
+{{15000.0,-10000.0},20,{5.964567324115234e-7,0.9999997514442596},6.0e-23,1.0e-16},
+{{15000.0,-10000.0},24,{8.520811370830121e-7,0.9999996449204608},8.5e-23,1.0e-16},
+{{15000.0,-10000.0},30,{1.320726020447437e-6,0.99999944962697},1.3e-22,1.0e-16},
+{{15000.0,-10000.0},36,{1.891620943893063e-6,0.9999992117242355},1.9e-22,1.0e-16},
+{{15000.0,-10000.0},44,{2.811870049126164e-6,0.9999988282397975},2.8e-22,1.0e-16},
+{{15000.0,-10000.0},54,{4.217807545189578e-6,0.9999982423621463},4.2e-22,1.0e-16},
+{{15000.0,-10000.0},67,{6.470151250788692e-6,0.9999973037777175},6.5e-22,1.0e-16},
+{{15000.0,-10000.0},82,{9.665475868536043e-6,0.9999959722496228},9.7e-22,1.0e-16},
+{{15000.0,-10000.0},100,{1.434344645897526e-5,0.9999940229077095},1.4e-21,1.0e-16},
+{{15000.0,-10000.0},122,{2.131073094626111e-5,0.9999911196408604},2.1e-21,1.0e-16},
+{{15000.0,-10000.0},149,{3.174043095735599e-5,0.9999867736922388},3.2e-21,1.0e-16},
+{{15000.0,-10000.0},182,{4.729993952558016e-5,0.999980290442509},4.7e-21,1.0e-16},
+{{15000.0,-10000.0},223,{7.094066703164427e-5,0.9999704404991336},7.1e-21,1.0e-16},
+{{15000.0,-10000.0},272,{0.0001054581586252006,0.9999560599143091},1.1e-20,1.0e-16},
+{{15000.0,-10000.0},333,{0.0001579606214003589,0.999934189160139},1.6e-20,1.0e-16},
+{{15000.0,-10000.0},407,{0.0002358450668200512,0.9999017510880184},2.4e-20,1.0e-16},
+{{15000.0,-10000.0},497,{0.0003515433469161231,0.9998535770854752},3.5e-20,1.0e-16},
+{{15000.0,-10000.0},608,{0.0005259513321601364,0.9997809874552286},5.3e-20,1.0e-16},
+{{15000.0,-10000.0},742,{0.0007831841585954275,0.9996739907967603},7.8e-20,1.0e-16},
+{{15000.0,-10000.0},907,{0.0011701301961291,0.9995131858706238},1.2e-19,1.0e-16},
+{{15000.0,-10000.0},1108,{0.001746289656337749,0.9992740743002811},1.7e-19,1.0e-16},
+{{15000.0,-10000.0},1354,{0.0026083036501201,0.9989170579410502},2.6e-19,1.0e-16},
+{{15000.0,-10000.0},1655,{0.003898434272384769,0.9983843598666412},3.9e-19,1.0e-16},
+{{15000.0,-10000.0},2022,{0.005823086211357998,0.9975932972803287},5.8e-19,1.0e-16},
+{{15000.0,-10000.0},2471,{0.008705827478245269,0.9964165769276073},8.7e-19,1.0e-16},
+{{15000.0,-10000.0},3019,{0.01301723709079792,0.9946748807501365},1.3e-18,9.9e-17},
+{{15000.0,-10000.0},3688,{0.01947471605308964,0.9921070242409564},1.9e-18,9.9e-17},
+{{15000.0,-10000.0},4506,{0.02918119661762942,0.9883391854624398},2.9e-18,9.9e-17},
+{{15000.0,-10000.0},5506,{0.0438113434957399,0.9828686787236778},4.4e-18,9.8e-17},
+{{15000.0,-10000.0},6727,{0.06591734046001419,0.9750775397377561},6.6e-18,9.8e-17},
+{{15000.0,-10000.0},8219,{0.09949401019510352,0.9643312731804727},9.9e-18,9.6e-17},
+{{15000.0,-10000.0},10042,{0.1506925762625648,0.9504421084014469},1.5e-17,9.5e-17},
+{{15000.0,-10000.0},12270,{0.2287073285360497,0.9349327103661624},2.3e-17,9.3e-17},
+{{15000.0,-10000.0},14991,{0.345508860352488,0.9237817578495397},3.5e-17,9.2e-17},
+{{15000.0,-10000.0},18316,{0.5122082784326123,0.9302050482741219},5.1e-17,9.3e-17},
+{{100000.0,-100000.0},0,{0.0,1.0},0.0,1.0e-16},
+{{100000.0,-100000.0},1,{5.000024999999999e-11,0.9999999999999997},5.0e-27,1.0e-16},
+{{100000.0,-100000.0},2,{1.500007500000001e-10,0.9999999999999992},1.5e-26,1.0e-16},
+{{100000.0,-100000.0},3,{3.000015000000007e-10,0.9999999999999985},3.0e-26,1.0e-16},
+{{100000.0,-100000.0},4,{5.000025000000021e-10,0.9999999999999975},5.0e-26,1.0e-16},
+{{100000.0,-100000.0},5,{7.500037500000051e-10,0.9999999999999963},7.5e-26,1.0e-16},
+{{100000.0,-100000.0},6,{1.05000525000001e-9,0.9999999999999948},1.1e-25,1.0e-16},
+{{100000.0,-100000.0},8,{1.800009000000031e-9,0.999999999999991},1.8e-25,1.0e-16},
+{{100000.0,-100000.0},11,{3.300016500000106e-9,0.9999999999999835},3.3e-25,1.0e-16},
+{{100000.0,-100000.0},14,{5.250026250000272e-9,0.9999999999999738},5.3e-25,1.0e-16},
+{{100000.0,-100000.0},18,{8.550042750000724e-9,0.9999999999999573},8.6e-25,1.0e-16},
+{{100000.0,-100000.0},23,{1.380006900000189e-8,0.9999999999999311},1.4e-24,1.0e-16},
+{{100000.0,-100000.0},29,{2.175010875000471e-8,0.9999999999998915},2.2e-24,1.0e-16},
+{{100000.0,-100000.0},37,{3.515017575001231e-8,0.9999999999998249},3.5e-24,1.0e-16},
+{{100000.0,-100000.0},48,{5.880029400003443e-8,0.9999999999997077},5.9e-24,1.0e-16},
+{{100000.0,-100000.0},61,{9.455047275008891e-8,0.9999999999995317},9.5e-24,1.0e-16},
+{{100000.0,-100000.0},78,{1.540507702502354e-7,0.9999999999992416},1.5e-23,1.0e-16},
+{{100000.0,-100000.0},99,{2.475012375006048e-7,0.9999999999987931},2.5e-23,1.0e-16},
+{{100000.0,-100000.0},126,{4.000520002515681e-7,0.9999999999980797},4.0e-23,1.0e-16},
+{{100000.0,-100000.0},161,{6.520532602541127e-7,0.9999999999969523},6.5e-23,1.0e-16},
+{{100000.0,-100000.0},205,{1.055755278760557e-6,0.9999999999952785},1.1e-22,1.0e-16},
+{{100000.0,-100000.0},262,{1.722658613277119e-6,0.9999999999928704},1.7e-22,1.0e-16},
+{{100000.0,-100000.0},333,{2.780563902816566e-6,0.9999999999899628},2.8e-22,1.0e-16},
+{{100000.0,-100000.0},425,{4.526272631408507e-6,0.9999999999876121},4.5e-22,1.0e-16},
+{{100000.0,-100000.0},541,{7.330586653090414e-6,0.9999999999902157},7.3e-22,1.0e-16},
+{{100000.0,-100000.0},690,{1.191980959932403e-5,1.000000000011442},1.2e-21,1.0e-16},
+{{100000.0,-100000.0},879,{1.933809669012375e-5,1.000000000090291},1.9e-21,1.0e-16},
+{{100000.0,-100000.0},1120,{3.138815693438994e-5,1.000000000335671},3.1e-21,1.0e-16},
+{{100000.0,-100000.0},1427,{5.094415467934417e-5,1.000000001042943},5.1e-21,1.0e-16},
+{{100000.0,-100000.0},1818,{8.267396315355681e-5,1.000000003004152},8.3e-21,1.0e-16},
+{{100000.0,-100000.0},2316,{0.000134154969744219,1.000000008328086},1.3e-20,1.0e-16},
+{{100000.0,-100000.0},2950,{0.000217637333500487,1.000000022595044},2.2e-20,1.0e-16},
+{{100000.0,-100000.0},3758,{0.0003531597950135793,1.000000060595726},3.5e-20,1.0e-16},
+{{100000.0,-100000.0},4788,{0.0005732460753128427,1.000000161440903},5.7e-20,1.0e-16},
+{{100000.0,-100000.0},6100,{0.0009304067579528531,1.000000428180528},9.3e-20,1.0e-16},
+{{84.0,80.0},0,{0.0,1.0},0.0,1.0e-16},
+{{84.0,80.0},1,{-7.380873229998488e-5,0.9999968564160156},7.4e-21,1.0e-16},
+{{84.0,80.0},30,{-0.03434386053057851,0.9991195766939804},3.4e-18,1.0e-16},
+{{84.0,80.0},50,{-0.09402245799827037,1.000353490231355},9.4e-18,1.0e-16},
+{{84.0,80.0},60,{-0.1345473903609001,1.003170792738456},1.3e-17,1.0e-16},
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+{{6.2831853071795862319959269370884,-1.0},6,{0.01459255193802049,0.2667761946934352},1.5e-18,2.7e-17},
+{{6.2831853071795862319959269370884,-1.0},7,{-0.1607940300198065,-0.09188619932509611},1.6e-17,9.2e-18},
+{{6.2831853071795862319959269370884,-1.0},8,{-0.5579504118636914,-0.3387942804714765},5.6e-17,3.4e-17},
+{{6.2831853071795862319959269370884,-1.0},9,{-0.9244229427611135,-0.3579875585692874},9.2e-17,3.6e-17},
+{{6.2831853071795862319959269370884,-1.0},10,{-1.171200204612873,-0.3401289272437849},1.2e-16,3.4e-17},
+{{6.2831853071795862319959269370884,-1.0},11,{-1.37508608462511,-0.3423969579768689},1.4e-16,3.4e-17},
+{{6.2831853071795862319959269370884,-1.0},12,{-1.565395784656673,-0.3551116134192908},1.6e-16,3.6e-17},
+{{6.2831853071795862319959269370884,-1.0},13,{-1.748425846404238,-0.3720163239527605},1.7e-16,3.7e-17},
+{{6.2831853071795862319959269370884,-1.0},14,{-1.926390361766107,-0.391030316077631},1.9e-16,3.9e-17},
+{{6.2831853071795862319959269370884,-1.0},15,{-2.100611149363269,-0.4113324775120068},2.1e-16,4.1e-17},
+{{6.2831853071795862319959269370884,-1.0},16,{-2.271978855394655,-0.4325025916882605},2.3e-16,4.3e-17},
+{{6.2831853071795862319959269370884,-1.0},18,{-2.608469851298484,-0.4765432487673419},2.6e-16,4.8e-17},
+{{6.2831853071795862319959269370884,-1.0},19,{-2.774374125145401,-0.4991481353598902},2.8e-16,5.0e-17},
+{{6.2831853071795862319959269370884,-1.0},20,{-2.939080962634986,-0.5220318339355317},2.9e-16,5.2e-17},
+{{6.2831853071795862319959269370884,-1.0},22,{-3.265643862194712,-0.56843085660439},3.3e-16,5.7e-17},
+{{6.2831853071795862319959269370884,-1.0},23,{-3.427777831706615,-0.5918741791259039},3.4e-16,5.9e-17},
+{{6.2831853071795862319959269370884,-1.0},25,{-3.750257209199445,-0.6391251649246937},3.8e-16,6.4e-17},
+{{6.2831853071795862319959269370884,-1.0},26,{-3.910752977720189,-0.6628981349489083},3.9e-16,6.6e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},0,{0.0,1.0},0.0,1.0e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},1,{-0.01637478695014086,0.9905292850317945},1.6e-18,9.9e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},2,{-0.04980087627017362,0.9722307558129726},5.0e-18,9.7e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},3,{-0.1015090337393254,0.9465663881070201},1.0e-17,9.5e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},4,{-0.172949229136423,0.9161101257264401},1.7e-17,9.2e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},5,{-0.2651206177152717,0.8847668405120703},2.7e-17,8.8e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},6,{-0.3776003562568451,0.8575457214749106},3.8e-17,8.6e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},7,{-0.5077272342885332,0.8395456242436236},5.1e-17,8.4e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},8,{-0.6507173089214951,0.8343944242890056},6.5e-17,8.3e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},9,{-0.8009751014809687,0.8431592161745789},8.0e-17,8.4e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},10,{-0.9537418889305799,0.864554677181108},9.5e-17,8.6e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},11,{-1.105961091834413,0.8960951648054553},1.1e-16,9.0e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},12,{-1.256157424357276,0.9352073393504297},1.3e-16,9.4e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},13,{-1.403875529392405,0.9797877843479927},1.4e-16,9.8e-17},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},14,{-1.549168129100615,1.02830055390916},1.5e-16,1.0e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},15,{-1.692284051392539,1.079676237510791},1.7e-16,1.1e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},16,{-1.833516950412929,1.133178106606117},1.8e-16,1.1e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},17,{-1.973144021666479,1.188294020548408},2.0e-16,1.2e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},19,{-2.248507874013405,1.302018460543744},2.2e-16,1.3e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},20,{-2.384616726790979,1.360171448495856},2.4e-16,1.4e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},22,{-2.654392752188639,1.478316329319233},2.7e-16,1.5e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},23,{-2.788272523600718,1.53810912969502},2.8e-16,1.5e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},25,{-3.05442272767237,1.658785490026429},3.1e-16,1.7e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},27,{-3.318830366257712,1.78059624870022},3.3e-16,1.8e-16},
+{{6.2831853071795862319959269370884,3.1415926535897932384626433832795},29,{-3.581862130100558,1.903267755210881},3.6e-16,1.9e-16},
+{{6.2831853071795862319959269370884,1.0},0,{0.0,1.0},0.0,1.0e-16},
+{{6.2831853071795862319959269370884,1.0},1,{-0.0107103550235295,0.97870468885667},1.1e-18,9.8e-17},
+{{6.2831853071795862319959269370884,1.0},2,{-0.03362513398684257,0.9355922315987426},3.4e-18,9.4e-17},
+{{6.2831853071795862319959269370884,1.0},3,{-0.07215247140260382,0.8699089855443275},7.2e-18,8.7e-17},
+{{6.2831853071795862319959269370884,1.0},4,{-0.1324003193922573,0.7816245319680321},1.3e-17,7.8e-17},
+{{6.2831853071795862319959269370884,1.0},5,{-0.2237229202421749,0.6738669791565766},2.2e-17,6.7e-17},
+{{6.2831853071795862319959269370884,1.0},6,{-0.3566827737268949,0.5572280064731737},3.6e-17,5.6e-17},
+{{6.2831853071795862319959269370884,1.0},7,{-0.534881214573448,0.4520353301037817},5.3e-17,4.5e-17},
+{{6.2831853071795862319959269370884,1.0},8,{-0.7453172985703958,0.3791789032364014},7.5e-17,3.8e-17},
+{{6.2831853071795862319959269370884,1.0},9,{-0.9643357843499773,0.3438258795491612},9.6e-17,3.4e-17},
+{{6.2831853071795862319959269370884,1.0},10,{-1.175625522079248,0.3354693797128221},1.2e-16,3.4e-17},
+{{6.2831853071795862319959269370884,1.0},11,{-1.375335050387865,0.3416989205502342},1.4e-16,3.4e-17},
+{{6.2831853071795862319959269370884,1.0},12,{-1.565392601516738,0.3550436439844526},1.6e-16,3.6e-17},
+{{6.2831853071795862319959269370884,1.0},13,{-1.748423760549718,0.3720116512461041},1.7e-16,3.7e-17},
+{{6.2831853071795862319959269370884,1.0},14,{-1.926390134893502,0.3910300876389843},1.9e-16,3.9e-17},
+{{6.2831853071795862319959269370884,1.0},15,{-2.100611133732458,0.4113324702563825},2.1e-16,4.1e-17},
+{{6.2831853071795862319959269370884,1.0},16,{-2.271978854602206,0.4325025916197443},2.3e-16,4.3e-17},
+{{6.2831853071795862319959269370884,1.0},18,{-2.608469851297546,0.4765432487679804},2.6e-16,4.8e-17},
+{{6.2831853071795862319959269370884,1.0},19,{-2.774374125145379,0.499148135359919},2.8e-16,5.0e-17},
+{{6.2831853071795862319959269370884,1.0},20,{-2.939080962634986,0.5220318339355327},2.9e-16,5.2e-17},
+{{6.2831853071795862319959269370884,1.0},22,{-3.265643862194712,0.56843085660439},3.3e-16,5.7e-17},
+{{6.2831853071795862319959269370884,1.0},23,{-3.427777831706615,0.5918741791259039},3.4e-16,5.9e-17},
+{{6.2831853071795862319959269370884,1.0},25,{-3.750257209199445,0.6391251649246937},3.8e-16,6.4e-17},
+{{6.2831853071795862319959269370884,1.0},26,{-3.910752977720189,0.6628981349489083},3.9e-16,6.6e-17},
+{{3.1415926535897932384626433832795,0.0},0,{0.0,1.0},0.0,1.0e-16},
+{{3.1415926535897932384626433832795,0.0},1,{-0.02928440396155443,0.9080003316496248},2.9e-18,9.1e-17},
+{{3.1415926535897932384626433832795,0.0},2,{-0.111414420013723,0.7161490362627657},1.1e-17,7.2e-17},
+{{3.1415926535897932384626433832795,0.0},3,{-0.3086445141021306,0.4340404684115407},3.1e-17,4.3e-17},
+{{3.1415926535897932384626433832795,0.0},4,{-0.6853420975491589,0.1613084612168837},6.9e-17,1.6e-17},
+{{3.1415926535897932384626433832795,0.0},5,{-1.154547531604494,0.03095979893899376},1.2e-16,3.1e-18},
+{{3.1415926535897932384626433832795,0.0},6,{-1.583565196074497,0.003296559784896674},1.6e-16,3.3e-19},
+{{3.1415926535897932384626433832795,0.0},7,{-1.965821634628246,0.0002268900069871888},2.0e-16,2.3e-20},
+{{3.1415926535897932384626433832795,0.0},8,{-2.324862655057826,1.114344043688996e-5},2.3e-16,1.1e-21},
+{{3.1415926535897932384626433832795,0.0},9,{-2.672097814251784,4.137546472473845e-7},2.7e-16,4.1e-23},
+{{3.1415926535897932384626433832795,0.0},10,{-3.012310405818385,1.206868381771849e-8},3.0e-16,1.2e-24},
+{{3.1415926535897932384626433832795,0.0},11,{-3.347886623965971,2.84447322718975e-10},3.3e-16,2.8e-26},
+{{3.1415926535897932384626433832795,0.0},12,{-3.680202022452916,5.536734412459245e-12},3.7e-16,5.5e-28},
+{{3.1415926535897932384626433832795,0.0},13,{-4.010122340785055,9.05809239443902e-14},4.0e-16,9.1e-30},
+{{3.1415926535897932384626433832795,0.0},14,{-4.338225297702631,1.263667883273076e-15},4.3e-16,1.3e-31},
+{{3.1415926535897932384626433832795,0.0},15,{-4.66491342262249,1.521676227453879e-17},4.7e-16,1.5e-33},
+{{3.1415926535897932384626433832795,0.0},16,{-4.990476814065978,1.598129828453938e-19},5.0e-16,1.6e-35},
+{{3.1415926535897932384626433832795,0.0},17,{-5.315130348653373,1.4770651858539e-21},5.3e-16,1.5e-37},
+{{3.1415926535897932384626433832795,0.0},18,{-5.639036854808338,1.21085227329631e-23},5.6e-16,1.2e-39},
+{{3.1415926535897932384626433832795,0.0},20,{-6.285084985730819,5.832688628117013e-28},6.3e-16,5.8e-44},
+{{3.141592653589793,0.0},0,{0.0,1.0},0.0,1.0e-16},
+{{3.141592653589793,0.0},1,{-0.02928440396155444,0.9080003316496248},2.9e-18,9.1e-17},
+{{3.141592653589793,0.0},2,{-0.111414420013723,0.7161490362627657},1.1e-17,7.2e-17},
+{{3.141592653589793,0.0},3,{-0.3086445141021306,0.4340404684115406},3.1e-17,4.3e-17},
+{{3.141592653589793,0.0},4,{-0.685342097549159,0.1613084612168837},6.9e-17,1.6e-17},
+{{3.141592653589793,0.0},5,{-1.154547531604494,0.03095979893899374},1.2e-16,3.1e-18},
+{{3.141592653589793,0.0},6,{-1.583565196074497,0.003296559784896672},1.6e-16,3.3e-19},
+{{3.141592653589793,0.0},7,{-1.965821634628247,0.0002268900069871886},2.0e-16,2.3e-20},
+{{3.141592653589793,0.0},8,{-2.324862655057826,1.114344043688994e-5},2.3e-16,1.1e-21},
+{{3.141592653589793,0.0},9,{-2.672097814251784,4.13754647247384e-7},2.7e-16,4.1e-23},
+{{3.141592653589793,0.0},10,{-3.012310405818385,1.206868381771847e-8},3.0e-16,1.2e-24},
+{{3.141592653589793,0.0},11,{-3.347886623965971,2.844473227189745e-10},3.3e-16,2.8e-26},
+{{3.141592653589793,0.0},12,{-3.680202022452917,5.536734412459235e-12},3.7e-16,5.5e-28},
+{{3.141592653589793,0.0},13,{-4.010122340785055,9.058092394439002e-14},4.0e-16,9.1e-30},
+{{3.141592653589793,0.0},14,{-4.338225297702632,1.263667883273073e-15},4.3e-16,1.3e-31},
+{{3.141592653589793,0.0},15,{-4.664913422622491,1.521676227453876e-17},4.7e-16,1.5e-33},
+{{3.141592653589793,0.0},16,{-4.990476814065978,1.598129828453934e-19},5.0e-16,1.6e-35},
+{{3.141592653589793,0.0},17,{-5.315130348653373,1.477065185853896e-21},5.3e-16,1.5e-37},
+{{3.141592653589793,0.0},18,{-5.639036854808339,1.210852273296306e-23},5.6e-16,1.2e-39},
+{{3.141592653589793,0.0},20,{-6.285084985730819,5.832688628116996e-28},6.3e-16,5.8e-44},
+};
+
+// complex(z), n, complex(f(n,z)), abs_err_real, abs_err_imag
+std::vector< std::tuple< std::complex<double>, int, std::complex<double>, double, double > >
+psi_mul_ksi_test_16digits
+= {
+{{84.0,80.0},0,{0.5,1.624140276402612e-70},5.0e-17,1.6e-86},
+{{84.0,80.0},1,{0.500001811514801,-3.71139617775679e-5},5.0e-17,3.7e-21},
+{{84.0,80.0},30,{0.4999484578754718,-0.01729402460398425},5.0e-17,1.7e-18},
+{{84.0,80.0},50,{0.4956325284553509,-0.04692224582862037},5.0e-17,4.7e-18},
+{{84.0,80.0},60,{0.4897893479849209,-0.0661984546136782},4.9e-17,6.6e-18},
+{{84.0,80.0},70,{0.4803262801451547,-0.08725408858529796},4.8e-17,8.7e-18},
+{{84.0,80.0},75,{0.4740593514549392,-0.09803187176761216},4.7e-17,9.8e-18},
+{{84.0,80.0},80,{0.466738701850348,-0.1087220143709562},4.7e-17,1.1e-17},
+{{84.0,80.0},85,{0.4583908913749478,-0.1191299019766889},4.6e-17,1.2e-17},
+{{84.0,80.0},90,{0.4490824803820364,-0.129071077565662},4.5e-17,1.3e-17},
+{{84.0,80.0},99,{0.4302529666196181,-0.1452856299266772},4.3e-17,1.5e-17},
+{{84.0,80.0},116,{0.3900835621894661,-0.1682281347837009},3.9e-17,1.7e-17},
+{{84.0,80.0},130,{0.355978786440762,-0.1791010472323009},3.6e-17,1.8e-17},
+};