MKL库求解矩阵特征值、特征向量(LAPACKE_dgeev、dsyev)

LAPACK（Linear Algebra PACKage）库，是用Fortran语言编写的线性代数计算库，包含线性方程组求解（\(AX=B\)）、矩阵分解、矩阵求逆、求矩阵特征值、奇异值等。该库用BLAS库做底层运算。

本示例将重点介绍使用LAPACK库求解非对称矩阵与对称矩阵的特征值、特征向量过程。

1 一般矩阵

在LAPACK库中，使用?geev函数计算一般矩阵的特征值及其左/右特征向量，其中右特征向量\(v_j\)满足

\[A*v_j = \lambda_j*v_j
\]

左特征向量\(u_j\)满足

\[u_j^H*A = \lambda_j*u_j^H
\]

其中\(u_j^H\)表示\(u_j\)的共轭转置。

1.1 参数详解

示例将使用MKL库中，LAPACK中的dgeev函数演示。

lapack_int LAPACKE_dgeev( matrix_order,     //(input) 行优先(LAPACK_ROW_MAJOR)或列优先(LAPACK_COL_MAJOR)
                          jobvl, 	    //(input) 指定是否计算左特征向量，"V"：计算；"N"：不计算
                          jobvr,	    //(input) 指定是否计算右特征向量，同上
                          n,		    //(input) 矩阵的阶数
                          a,		    //(input/output) 待求解A矩阵
                          lda,		    //(input) A矩阵的第一维
                          wr,		    //(output) 特征向量实部
                          wi,		    //(output) 特征向量虚部
                          vl,		    //(output) 左特征向量
                          ldvl,		    //(input) 左特征向量第一维
                          vr,		    //(output) 右特征向量
                          ldvr		    //(input) 右特征向量第一维
             			)

1.2 定义待求解矩阵

#include <stdlib.h>
#include <stdio.h>
#include "mkl_lapacke.h"

// 参数
#define N 5
#define LDA N
#define LDVL N
#define LDVR N

MKL_INT n = N, lda = LDA, ldvl = LDVL, ldvr = LDVR, info;

double wr[N], wi[N], vl[LDVL*N], vr[LDVR*N];
double a[LDA*N] = {
    -1.01,  0.86, -4.60,  3.31, -4.81,
    3.98,  0.53, -7.04,  5.29,  3.55,
    3.30,  8.26, -3.89,  8.20, -1.51,
    4.43,  4.96, -7.66, -7.33,  6.18,
    7.31, -6.43, -6.16,  2.47,  5.58
};

1.3 求解特征值、特征向量

LAPACKE_dgeev( LAPACK_ROW_MAJOR, 'V', 'V', n, a, lda, wr, wi, vl, ldvl, vr, ldvr );

与Matlab中使用eig方法求取特征值与特征向量，所得结果相同。

A = [  -1.01,  0.86, -4.60,  3.31, -4.81;
        3.98,  0.53, -7.04,  5.29,  3.55;
        3.30,  8.26, -3.89,  8.20, -1.51;
        4.43,  4.96, -7.66, -7.33,  6.18;
        7.31, -6.43, -6.16,  2.47,  5.58];
[V,D]=eig(A);
Vector = V;
Lambda = diag(D);
display(Lambda);
display(Vector);

完整代码

#include <stdlib.h>
#include <stdio.h>
#include "mkl_lapacke.h"

//print特征值和特征向量
extern void print_eigenvalues( char* desc, MKL_INT n, double* wr, double* wi );
extern void print_eigenvectors( char* desc, MKL_INT n, double* wi, double* v,
      MKL_INT ldv );

#define N 5
#define LDA N
#define LDVL N
#define LDVR N

int main() {

        MKL_INT n = N, lda = LDA, ldvl = LDVL, ldvr = LDVR, info;

        double wr[N], wi[N], vl[LDVL*N], vr[LDVR*N];
        double a[LDA*N] = {
           -1.01,  0.86, -4.60,  3.31, -4.81,
            3.98,  0.53, -7.04,  5.29,  3.55,
            3.30,  8.26, -3.89,  8.20, -1.51,
            4.43,  4.96, -7.66, -7.33,  6.18,
            7.31, -6.43, -6.16,  2.47,  5.58
        };

        printf( "LAPACKE_dgeev (row-major, high-level) Example Program Results\n" );
        // 求解
        info = LAPACKE_dgeev( LAPACK_ROW_MAJOR, 'V', 'V', n, a, lda, wr, wi,
                        vl, ldvl, vr, ldvr );

        if( info > 0 ) {
                printf( "The algorithm failed to compute eigenvalues.\n" );
                exit( 1 );
        }

        print_eigenvalues( "Eigenvalues", n, wr, wi );

        print_eigenvectors( "Left eigenvectors", n, wi, vl, ldvl );

        print_eigenvectors( "Right eigenvectors", n, wi, vr, ldvr );
        exit( 0 );
}

void print_eigenvalues( char* desc, MKL_INT n, double* wr, double* wi ) {
        MKL_INT j;
        printf( "\n %s\n", desc );
   for( j = 0; j < n; j++ ) {
      if( wi[j] == (double)0.0 ) {
         printf( " %6.2f", wr[j] );
      } else {
         printf( " (%6.2f,%6.2f)", wr[j], wi[j] );
      }
   }
   printf( "\n" );
}

void print_eigenvectors( char* desc, MKL_INT n, double* wi, double* v, MKL_INT ldv ) {
        MKL_INT i, j;
        printf( "\n %s\n", desc );
   for( i = 0; i < n; i++ ) {
      j = 0;
      while( j < n ) {
         if( wi[j] == (double)0.0 ) {
            printf( " %6.2f", v[i*ldv+j] );
            j++;
         } else {
            printf( " (%6.2f,%6.2f)", v[i*ldv+j], v[i*ldv+(j+1)] );
            printf( " (%6.2f,%6.2f)", v[i*ldv+j], -v[i*ldv+(j+1)] );
            j += 2;
         }
      }
      printf( "\n" );
   }
}

2 求解对称阵

2.1 参数详解

采用LAPACKE_dsyev方法实现，参数比LAPACKE_dgeev少，含义相似。

lapack_int LAPACKE_dsyev(	matrix_layout,
                             	        jobz, 			//是否计算特征值和特征向量， "V"/"N"
                             	        uplo,			//表示使用A矩阵的上三角或下三角矩阵 "U"/"L"
                             	        n,
                             	        a,
                             	        lda,
                             	        w 			//包含降序的特征值
                         )

完整代码

#include <stdlib.h>
#include <stdio.h>
#include "mkl_lapacke.h"

extern void print_matrix(const char* desc, MKL_INT m, MKL_INT n, double* a, MKL_INT lda);

#define N 5
#define LDA N

int main() {

    MKL_INT n = N, lda = LDA, info;

    double w[N];
    double a[LDA * N] = {
     1.96, - 6.49, - 0.47, - 7.20, - 0.65,
    -6.49,   3.80, - 6.39,   1.50, - 6.34,
    -0.47, - 6.39,   4.17, - 1.51,   2.67,
    -7.20,   1.50, - 1.51,   5.70,   1.80,
    -0.65, - 6.34,   2.67,   1.80, - 7.10,
    };

    printf("LAPACKE_dsyev (row-major, high-level) Example Program Results\n");
    //求解特征值、特征向量
    info = LAPACKE_dsyev(LAPACK_ROW_MAJOR, 'V', 'U', n, a, lda, w);		//'U'表示下三角

    if (info > 0) {
        printf("The algorithm failed to compute eigenvalues.\n");
        exit(1);
    }

    print_matrix("Eigenvalues", 1, n, w, 1);

    print_matrix("Eigenvectors (stored columnwise)", n, n, a, lda);
    exit(0);
} 

void print_matrix(const char* desc, MKL_INT m, MKL_INT n, double* a, MKL_INT lda) {
    MKL_INT i, j;
    printf("\n %s\n", desc);
    for (i = 0; i < m; i++) {
        for (j = 0; j < n; j++) printf(" %6.2f", a[i * lda + j]);
        printf("\n");
    }
}

输出为：