Derivative of the incomplete Gamma function and the sample of a Gamma random variable.

In addition to igamma(a, x), this code implements:
* igamma_der_a(a, x) = d igamma(a, x) / da -- derivative of igamma with respect to the parameter
* gamma_sample_der_alpha(alpha, sample) -- reparameterization derivative of a Gamma(alpha, 1) random variable sample with respect to the alpha parameter

The derivatives are computed by forward mode differentiation of the igamma(a, x) code. Although gamma_sample_der_alpha can be implemented via igamma_der_a, a separate function is more accurate and efficient due to analytical cancellation of some terms. All three functions are implemented by a method parameterized with "mode" that always computes the derivatives, but does not return them unless required by the mode. The compiler is expected to (and, based on benchmarks, does) skip the unnecessary computations depending on the mode.

1 files changed, 92 insertions, 0 deletions

diff --git a/unsupported/test/special_functions.cpp b/unsupported/test/special_functions.cpp
index 48d0db95e..29ba6203a 100644
--- a/unsupported/test/special_functions.cpp
+++ b/unsupported/test/special_functions.cpp
@@ -375,6 +375,98 @@ template<typename ArrayType> void array_special_functions()
     CALL_SUBTEST(res = i1e(x);
                  verify_component_wise(res, expected););
   }
+
+  /* Code to generate the data for the following two test cases.
+    N = 5
+    np.random.seed(3)
+
+    a = np.logspace(-2, 3, 6)
+    a = np.ravel(np.tile(np.reshape(a, [-1, 1]), [1, N]))
+    x = np.random.gamma(a, 1.0)
+    x = np.maximum(x, np.finfo(np.float32).tiny)
+
+    def igamma(a, x):
+      return mpmath.gammainc(a, 0, x, regularized=True)
+
+    def igamma_der_a(a, x):
+      res = mpmath.diff(lambda a_prime: igamma(a_prime, x), a)
+      return np.float64(res)
+
+    def gamma_sample_der_alpha(a, x):
+      igamma_x = igamma(a, x)
+      def igammainv_of_igamma(a_prime):
+        return mpmath.findroot(lambda x_prime: igamma(a_prime, x_prime) -
+            igamma_x, x, solver='newton')
+      return np.float64(mpmath.diff(igammainv_of_igamma, a))
+
+    v_igamma_der_a = np.vectorize(igamma_der_a)(a, x)
+    v_gamma_sample_der_alpha = np.vectorize(gamma_sample_der_alpha)(a, x)
+  */
+
+  // Test igamma_der_a
+  {
+    ArrayType a(30);
+    ArrayType x(30);
+    ArrayType res(30);
+    ArrayType v(30);
+
+    a << 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0, 1.0,
+        1.0, 1.0, 10.0, 10.0, 10.0, 10.0, 10.0, 100.0, 100.0, 100.0, 100.0,
+        100.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0;
+
+    x << 1.25668890405e-26, 1.17549435082e-38, 1.20938905072e-05,
+        1.17549435082e-38, 1.17549435082e-38, 5.66572070696e-16,
+        0.0132865061065, 0.0200034203853, 6.29263709118e-17, 1.37160367764e-06,
+        0.333412038288, 1.18135687766, 0.580629033777, 0.170631439426,
+        0.786686768458, 7.63873279537, 13.1944344379, 11.896042354,
+        10.5830172417, 10.5020942233, 92.8918587747, 95.003720371,
+        86.3715926467, 96.0330217672, 82.6389930677, 968.702906754,
+        969.463546828, 1001.79726022, 955.047416547, 1044.27458568;
+
+    v << -32.7256441441, -36.4394150514, -9.66467612263, -36.4394150514,
+        -36.4394150514, -1.0891900302, -2.66351229645, -2.48666868596,
+        -0.929700494428, -3.56327722764, -0.455320135314, -0.391437214323,
+        -0.491352055991, -0.350454834292, -0.471773162921, -0.104084440522,
+        -0.0723646747909, -0.0992828975532, -0.121638215446, -0.122619605294,
+        -0.0317670267286, -0.0359974812869, -0.0154359225363, -0.0375775365921,
+        -0.00794899153653, -0.00777303219211, -0.00796085782042,
+        -0.0125850719397, -0.00455500206958, -0.00476436993148;
+
+    CALL_SUBTEST(res = igamma_der_a(a, x); verify_component_wise(res, v););
+  }
+
+  // Test gamma_sample_der_alpha
+  {
+    ArrayType alpha(30);
+    ArrayType sample(30);
+    ArrayType res(30);
+    ArrayType v(30);
+
+    alpha << 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0,
+        1.0, 1.0, 1.0, 10.0, 10.0, 10.0, 10.0, 10.0, 100.0, 100.0, 100.0, 100.0,
+        100.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0;
+
+    sample << 1.25668890405e-26, 1.17549435082e-38, 1.20938905072e-05,
+        1.17549435082e-38, 1.17549435082e-38, 5.66572070696e-16,
+        0.0132865061065, 0.0200034203853, 6.29263709118e-17, 1.37160367764e-06,
+        0.333412038288, 1.18135687766, 0.580629033777, 0.170631439426,
+        0.786686768458, 7.63873279537, 13.1944344379, 11.896042354,
+        10.5830172417, 10.5020942233, 92.8918587747, 95.003720371,
+        86.3715926467, 96.0330217672, 82.6389930677, 968.702906754,
+        969.463546828, 1001.79726022, 955.047416547, 1044.27458568;
+
+    v << 7.42424742367e-23, 1.02004297287e-34, 0.0130155240738,
+        1.02004297287e-34, 1.02004297287e-34, 1.96505168277e-13, 0.525575786243,
+        0.713903991771, 2.32077561808e-14, 0.000179348049886, 0.635500453302,
+        1.27561284917, 0.878125852156, 0.41565819538, 1.03606488534,
+        0.885964824887, 1.16424049334, 1.10764479598, 1.04590810812,
+        1.04193666963, 0.965193152414, 0.976217589464, 0.93008035061,
+        0.98153216096, 0.909196397698, 0.98434963993, 0.984738050206,
+        1.00106492525, 0.97734200649, 1.02198794179;
+
+    CALL_SUBTEST(res = gamma_sample_der_alpha(alpha, sample);
+                 verify_component_wise(res, v););
+  }
 #endif
 }