本征：矢量或矩阵逐项幂？

问题描述

给出一个实数 c 的向量和一个整数 rw 的向量，我想创建一个向量 z 和元素 z_i = c_i ^ rw_i 。我尝试使用组件级函数 pow 来执行此操作，但出现编译器错误。

Given a vector of reals c and a vector of integers rw, I want to create a vector z with elements z_i=c_i^rw_i. I tried to do this using the component-wise function pow, but I get a compiler error.

#include <Eigen/Core>

typedef Eigen::VectorXd RealVector;
typedef Eigen::VectorXi IntVector; // dynamically-sized vector of integers
RealVector c; c << 2, 3, 4, 5;
IntVector rw; rw << 6, 7, 8, 9;
RealVector z = c.pow(rw);    **compile error**

编译器错误为

error C2664: 'const Eigen::MatrixComplexPowerReturnValue<Derived> Eigen::MatrixBase<Derived>::pow(const std::complex<double> &) const': cannot convert argument 1 from 'IntVector' to 'const double &'
      with
      [
          Derived=Eigen::Matrix<double,-1,1,0,-1,1>
      ]
c:\auc\sedanal\LammSolve.h(117): note: Reason: cannot convert from 'IntVector' to 'const double'
c:\auc\sedanal\LammSolve.h(117): note: No user-defined-conversion operator available that can perform this conversion, or the operator cannot be called

此代码有什么问题？并且，假设它可以固定，当c为实矩阵而不是向量时，我将如何执行相同的操作，以计算c中所有元素的 c_ij ^ b_i ？

What is wrong with this code? And, assuming it can be fixed, how would I do the same operation when c is a real matrix instead of a vector, to compute c_ij^b_i for all elements of c?

编译器是Visual Studio 2015，在64位Windows 7下运行。

Compiler is Visual Studio 2015, running under 64-bit Windows 7.

推荐答案

首先， MatrixBase :: pow 是一个计算方矩阵的矩阵幂的函数（如果矩阵具有特征值分解，则该函数是相同的

First of all, MatrixBase::pow is a function that computes the matrix power of a square matrix (if the matrix has an eigenvalue decomposition, it is the same matrix, but with the eigenvalues raised to the given power).

您想要的是元素智能，因为没有 cwisePow MatrixBase 中的函数需要切换到 Array 域。此外，没有幂的整数专业化（这可能是有效的，但只能达到一定的阈值，并且检查每个元素的阈值会浪费计算时间），因此您需要将指数转换为类型

What you want is an element-wise power, which since there is no cwisePow function in MatrixBase, requires switching to the Array-domain. Furthermore, there is no integer-specialization for the powers (this could be efficient, but only up to a certain threshold -- and checking for that threshold for every element would waste computation time), so you need to cast the exponents to the type of your matrix.

还要回答您的奖金问题：

To also answer your bonus-question:

#include <iostream>
#include <Eigen/Core>

int main(int argc, char **argv) {
    Eigen::MatrixXd A; A.setRandom(3,4);
    Eigen::VectorXi b = (Eigen::VectorXd::Random(3)*16).cast<int>();

    Eigen::MatrixXd C = A.array() // go to array domain
        .pow(                 // element-wise power
             b.cast<double>() // cast exponents to double
            .replicate(1, A.cols()).array() // repeat exponents to match size of A
        );

    std::cout << A << '\n' << b << '\n' << C << '\n';
}

本质上，这将称为 C（i，j ）= std :: pow（A（i，j），b（i））每个 i ， j 。如果所有指数都很小，实际上可能比使用
简单嵌套循环（调用专门的 pow（double，int）实现（例如gcc的））要快。 __ builtin_powi ），但是您应该使用实际数据进行基准测试。

Essentially, this will call C(i,j) = std::pow(A(i,j), b(i)) for each i, j. If all your exponents are small, you might actually be faster than that with asimple nested loop that calls a specialized pow(double, int) implementation (like gcc's __builtin_powi), but you should benchmark that with actual data.

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