Question

我需要Ubuntu下的C eigenproblem求解器。为此我从lapack 3.5.0中给了LAPACKE一个镜头并且实际上设法写了下面的示例程序例子我用正交系统和对角矩阵

构造

EV = [
  .6, -.8, 0
  .8,  .6, 0
   0,   0, 1
];
D = [
  2,  0, 0
  0, -3, 0
  0,  0, 0
];

通过产生A：= EV D EV'。

虽然较低的程序运行正常但结果却奇怪不准确。这里是输出结束：

...
Lambda: -3.07386, 0, 1.87386
EV = [
  -0.788205        0       -0.615412
   0.615412        0       -0.788205
  -0               1        0
];
Info: 0

作为文档，我使用了www.physics.orst.edu/~rubin/nacphy/lapack/routines/dsyev.html

我的完整资料来源：

/**
 * eigen.cpp
 * 
 * Given that you put version 3.5.0 into /opt/lapack/ compile this with: 
 * g++ eigen.cpp -I"/opt/lapack/lapack-3.5.0/lapacke/include" \
     -L"/opt/lapack/lapack-3.5.0" -llapacke -llapack -lblas -lcblas
 * The order of included libraries is important!
 */

#include <iostream>
#include <string>
#include <sstream>
// cstdlib needs including before clapack!
#include <cstdlib>
#include <cblas.h>
#include <lapacke.h>

using namespace std;

/** Column major style! */
string matrix2string(int m, int n, const double* A)
{
  ostringstream oss;
  for (int j=0;j<m;j++)
  {
    for (int k=0;k<n;k++)
    {
      oss << A[j+k*m] << "\t";
    }
    oss << endl;
  }
  return oss.str();
}

int main(int argc, char** argv)
{
  //> Preliminaries. -------------------------------------------------
  // Column Major Matrices for setting up the problem.
  double D_orig[9] = {
    2,  0, 0,
    0, -3, 0,
    0,  0, 0
  };
  double EV_orig[9] = {
    3./5., 4./5., 0,
    -4./5., 3./5., 0,
    0, 0, 1
  };
  double A[9] = { 0,0,0,0,0,0,0,0,0 };
  double dummy[9] = { 0,0,0,0,0,0,0,0,0 };
  // A = EV D EV'
  // dummy := D EV'
  // A := EV dummy
  cblas_dgemm(CblasColMajor,CblasNoTrans,CblasTrans,3,3,3,1,&D_orig[0],3,&EV_orig[0],3,0,&dummy[0],3);
  cblas_dgemm(CblasColMajor,CblasNoTrans,CblasNoTrans,3,3,3,1,&EV_orig[0],3,&dummy[0],3,0,&A[0],3);
  cout << "Set up the problem building A = EV D EV'" << endl <<
    "EV = [" << endl << matrix2string(3,3,&EV_orig[0]).c_str() << "];" << endl <<
     "D = [" << endl << matrix2string(3,3,&D_orig[0]).c_str() << "];" << endl <<
     "A = [" << endl << matrix2string(3,3,&A[0]).c_str() << "];" << endl;
  //< ----------------------------------------------------------------
  //> Actual eigen value problem. ------------------------------------
  char jobz = 'V'; // We want both vectors and values.
  char uplo = 'L'; // 'L' means lower triangular input A as opposed to 'U'.
  int N = 3; // Matrix dimension, or as they call it, 'order'.
  // As stated by example ATriL is unnecessary. Just replace all of its
  // occurences with plain A and all is well.
  double ATriL[9] = { A[0], A[1], A[2], A[4], A[5], A[8], 0, 0, 0 }; // Lower Triangle of symmetric A.
    // Note that it is larger than necessary. It will contain the eigenvectors at the end.
  int lda = 3;
  double w[3] = { 0, 0, 0 }; // Container for the eigenvalues.
  int lwork = 15; // Size of the worker array. Set to (NB+2)*N where NB here is the largest blocksize.
    // Note, however, that the definition of NB is more complex.
    // Compare http://ftp.mcs.anl.gov/pub/MINPACK-2/lapack/ilaenv.f
  double work[lwork];
  int info = 0;

  // "double symmetric eigenvalue problem" I presume.
  // lapack_int LAPACKE_dsyev( int matrix_order, char jobz, char uplo, lapack_int n,
  //                           double* a, lapack_int lda, double* w );
  info = LAPACKE_dsyev(LAPACK_COL_MAJOR, jobz, uplo, N, &ATriL[0], lda, &work[0]);
  // Note that the function takes no parameters lwork and w and that the
  // eigenvalues appear to be written into work.
  cout << "Ran dsyev(..) -- presumably 'double symmetric eigenvalue'." << endl <<
    "Lambda: " << work[0] << ", " << work[1] << ", " << work[2] << endl <<
    "EV = [" << endl << matrix2string(3,3,&ATriL[0]) << "];" << endl <<
    "Info: " << info << endl;
  //< ----------------------------------------------------------------
  return EXIT_SUCCESS;
}

最后的实际问题：为什么结果如此不准确，我该怎么做才能改善它们？

Answer 1

结果是准确的 - 对于你给lapack的数据。不幸的是，你没有解决你想要的问题。

具有较低与较高三角形部分的部分仅适用于其他一些（内部？）算法。在您的简单情况下，您不必担心这一点。如果您传递矩阵A而不是ATril，那么您应该没问题。

更详细：您正在构建double ATriL[9]，使其显示为

A[0], A[4], 0, 
A[1], A[5], 0, 
A[2], A[8], 0

来lapack。当你现在告诉它使用这个矩阵的下三角部分作为对称输入（char uplo = 'L';）时，lapack将有效地看到矩阵

A[0], A[1], A[2],        -1.2 2.4 0
A[1], A[5], A[8],   ==    2.4 0   0
A[2], A[8], 0             0   0   0

其特征向量实际上就是你得到的那些。

LAPACKE特征解决方案不准确。怎么改进呢？

1 个答案: