cavis/libnd4j/include/ops/declarable/helpers/gammaMathFunc.h

/*******************************************************************************
 * Copyright (c) 2015-2018 Skymind, Inc.
 * Copyright (c) 2019 Konduit K.K.
 *
 * This program and the accompanying materials are made available under the
 * terms of the Apache License, Version 2.0 which is available at
 * https://www.apache.org/licenses/LICENSE-2.0.
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
 * WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
 * License for the specific language governing permissions and limitations
 * under the License.
 *
 * SPDX-License-Identifier: Apache-2.0
 ******************************************************************************/

//
// @author Yurii Shyrma (iuriish@yahoo.com)
//

#ifndef LIBND4J_GAMMAMATHFUNC_H
#define LIBND4J_GAMMAMATHFUNC_H

#include <ops/declarable/helpers/helpers.h>
#include "array/NDArray.h"

namespace sd {
namespace ops {
namespace helpers {

    // calculate the digamma function for each element for array
    void diGamma(sd::LaunchContext* context, const NDArray& x, NDArray& z);

	// calculate the polygamma function
    void polyGamma(sd::LaunchContext* context, const NDArray& n, const NDArray& x, NDArray& z);

    // calculate the digamma function for one element
	// implementation is based on serial representation written in terms of the Hurwitz zeta function as polygamma = (-1)^{n+1} * n! * zeta(n+1, x)
	template <typename T>
	_CUDA_HD T diGammaScalar(T x) {

		const int xInt = static_cast<int>(x);

		// negative and zero
		if(x <= 0) {
			if(x == xInt)	// integer
				return DataTypeUtils::infOrMax<T>();
			else
				return diGammaScalar<T>(1 - x) - M_PI / sd::math::nd4j_tan<T,T>(M_PI * x); // use reflection formula psi(1-x) = psi(x) + pi*cot(pi*x)
		}

		// positive integer
		if(x == xInt && xInt <= 20) {		// psi(n) = -Euler_Mascheroni_const + sum_from_k=1_to_n-1( 1/k ), for n = 1,2,3,...inf, we use this formula only for n <= 20 to avoid time consuming sum calculation for bigger n
			T result = -0.577215664901532;
			for (uint i = 1; i <= xInt - 1; ++i) {
				result += static_cast<T>(1) / i;
			}
			return result;
		}

		// positive half-integer
		if(x - xInt == 0.5 && xInt <= 20) {		// psi(n+0.5) = -Euler_Mascheroni_const - 2*ln(2) + sum_from_k=1_to_n( 2/(2*k-1) )	, for n = 1,2,3,...inf, we use this formula only for n <= 20 to avoid time consuming sum calculation for bigger n
			T result = -0.577215664901532 - 2 * sd::math::nd4j_log<T,T>(2);
			for (uint i = 1; i <= xInt; ++i) {
				result += static_cast<T>(2) / (2*i - 1);
			}
			return result;
		}

		// positive, smaller then 5; we should use number > 5 in order to have satisfactory accuracy in asymptotic expansion
		if(x < 5)
			return diGammaScalar<T>(1 + x) - static_cast<T>(1) / x;		 // recurrence formula  psi(x) = psi(x+1) - 1/x.

		// *** other positive **** //

		// truncated expansion formula (from wiki)
		// psi(x) = log(x) - 1/(2*x) - 1/(12*x^2) + 1/(120*x^4) - 1/(252*x^6) + 1/(240*x^8) - 5/(660*x^10) + 691/(32760*x^12) - 1/(12*x^14) + ...

		if(x >= (sizeof(T) > 4 ? 1.e16 : 1.e8))		// if x is too big take into account only log(x)
			return sd::math::nd4j_log<T,T>(x);

		// coefficients used in truncated asymptotic expansion formula
		const T coeffs[7] = {-(T)1/12, (T)1/120, -(T)1/252, (T)1/240, -(T)5/660, (T)691/32760, -(T)1/12};
		// const T coeffs[7] = {-0.0833333333333333, 0.00833333333333333, -0.00396825396825397, 0.00416666666666667, -0.00757575757575758, 0.0210927960927961, -0.0833333333333333};

		const T x2Inv = static_cast<T>(1) / (x * x);
		T result = 0;

		for (int i = 6; i >= 0; --i)
			result = (result + coeffs[i]) * x2Inv;
		return result + sd::math::nd4j_log<T,T>(x) - static_cast<T>(0.5) / x;
	}

}
}
}


#endif //LIBND4J_GAMMAMATHFUNC_H