legendre.h

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/*
 * Copyright 2021 MusicScience37 (Kenta Kabashima)
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://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.
 */
#pragma once
 
#include <limits>
#include <utility>
 
#include "num_collect/base/concepts/integral.h"
#include "num_collect/base/concepts/real_scalar.h"
#include "num_collect/constants/half.h"  // IWYU pragma: keep
#include "num_collect/constants/one.h"   // IWYU pragma: keep
#include "num_collect/constants/two.h"   // IWYU pragma: keep
#include "num_collect/constants/zero.h"  // IWYU pragma: keep
 
namespace num_collect::functions {
 
template <base::concepts::real_scalar F, base::concepts::integral I>
constexpr auto legendre(F x, I n) -> F {
    if constexpr (std::is_signed_v<I>) {
        if (n < constants::zero<I>) {
            return std::numeric_limits<F>::quiet_NaN();
        }
    }
    if (n == constants::zero<I>) {
        return constants::one<F>;
    }
    if (n == constants::one<I>) {
        return x;
    }
    F y_p = constants::zero<F>;
    F y = x;
    F y_m = constants::one<F>;
    for (I i = constants::one<I>; i < n; ++i) {
        y_p =
            (static_cast<F>(constants::two<I> * i + constants::one<I>) * x * y -
                static_cast<F>(i) * y_m) /
            static_cast<F>(i + constants::one<I>);
        y_m = y;
        y = y_p;
    }
    return y;
}
 
template <base::concepts::real_scalar F, base::concepts::integral I>
constexpr auto legendre_with_diff(F x, I n) -> std::pair<F, F> {
    if constexpr (std::is_signed_v<I>) {
        if (n < constants::zero<I>) {
            return {std::numeric_limits<F>::quiet_NaN(),
                std::numeric_limits<F>::quiet_NaN()};
        }
    }
    if (n == constants::zero<I>) {
        return {constants::one<F>, constants::zero<F>};
    }
    if (n == constants::one<I>) {
        return {x, constants::one<F>};
    }
 
    if (x == constants::one<F>) {
        return {constants::one<F>,
            constants::half<F> * static_cast<F>(n) *
                static_cast<F>(n + constants::one<I>)};
    }
    if (x == -constants::one<F>) {
        if (n % constants::two<I> == constants::zero<I>) {
            return {constants::one<F>,
                -constants::half<F> * static_cast<F>(n) *
                    static_cast<F>(n + constants::one<I>)};
        }
        return {-constants::one<F>,
            constants::half<F> * static_cast<F>(n) *
                static_cast<F>(n + constants::one<I>)};
    }
 
    F y_p = constants::zero<F>;
    F y = x;
    F y_m = constants::one<F>;
    for (I i = constants::one<I>; i < n; ++i) {
        y_p =
            (static_cast<F>(constants::two<I> * i + constants::one<I>) * x * y -
                static_cast<F>(i) * y_m) /
            static_cast<F>(i + constants::one<I>);
        y_m = y;
        y = y_p;
    }
 
    F dif = static_cast<F>(n) * (y_m - x * y) / (constants::one<F> - x * x);
 
    return {y, dif};
}
 
}  // namespace num_collect::functions