?sy(he)evd

Compute all the eigenvalues and (optional) eigenvectors of a symmetric (Hermitian) matrix. The eigenvectors are calculated by using a divide and conquer algorithm.

That is, matrix A is factorized into , where is a diagonal matrix, the diagonal element is an eigenvalue, Z is an orthogonal matrix, and each column vector of Z is a corresponding eigenvector. That is, , where .

Interface Definition

C interface:

void dsyevd_(const char *jobz, const char *uplo, const int *n, double *a, const int *lda, double *w, double *work, const int *lwork, int *iwork, const int *liwork, int *info);

void ssyevd_(const char *jobz, const char *uplo, const int *n, float *a, const int *lda, float *w, float *work, const int *lwork, int *iwork, const int *liwork, int *info);

void cheevd_(const char *jobz, const char *uplo, const int *n, float _Complex *a, const int *lda, float *w, float _Complex *work, const int *lwork, float *rwork, const int *lrwork, int *iwork, const int *liwork, int *info);

void zheevd_(const char *jobz, const char *uplo, const int *n, double _Complex *a, const int *lda, double *w, double _Complex *work, const int *lwork, double *rwork, const int *lrwork, int *iwork, const int *liwork, int *info);

Fortran interface:

DSYEVD(JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO);

SSYEVD(JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO);

CHEEVD(JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO);

ZHEEVD(JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO);

Parameters

Parameter	Type	Description	Input/Output
jobz	Character	'N': computes only eigenvalues. 'V': computes eigenvalues and eigenvectors at the same time.	Input
uplo	Character	'U': saves the upper triangular matrix of A. 'L': saves the lower triangular matrix of A.	Input
n	Integer	Number of rows or columns in the symmetric matrix A.	Input
a	A single-precision floating-point array for ssyevd A double-precision floating-point array for dsyevd A single-precision complex array for cheevd A double-precision complex array for zheevd	Saves the symmetric matrix to be factorized before this function is called. Saves the eigenvectors after this function is called.	Input/Output
lda	Integer	Leading dimension of matrix A. lda ≥ max(1, n).	Input
w	A single-precision floating-point array for ssyevd/cheevd A double-precision floating-point array for dsyevd/zheevd	Eigenvalues in ascending order. The length is n.	Output
work	A single-precision floating-point array for ssyevd A double-precision floating-point array for dsyevd A single-precision complex array for cheevd A double-precision complex array for zheevd	Workspace array. After this function is called, work[0] returns the optimal lwork value.	Output
lwork	Integer	Length of the work array. If lwork = -1, the optimal work size is queried and the result is saved in work[0]. If lwork ≠ -1: When n ≤ 1, lwork ≥ 1. When jobz = 'N' and n > 1, lwork ≥ 2n + 1. When jobz = 'V' and n > 1, lwork ≥ 1 + 6n + 2nn.	Input
rwork (only for complex types)	A single-precision real type for cheevd A double-precision real type for zheevd	Workspace array. After this function is called, work[0] returns the optimal lwork value.	Output
lrwork (only for the complex type)	A single-precision complex array for cheevd A double-precision complex array for zheevd	Length of the rwork array. If lrwork = -1, the optimal work size is queried and the result is saved in rwork[0]. If lrwork ≠ -1: When n ≤ 1, lwork ≥ 1. When jobz = 'N' and n > 1, lwork ≥ 2n + 1. When jobz = 'V' and n > 1, lwork ≥ 1 + 6n + 2nn.	Input
iwork	Integer array	Temporary storage space. After this interface is called with lwork = -1, iwork[0] is the optimal liwork value.	Output
liwork	Integer	Length of the iwork array. If liwork=-1, the optimal iwork size is queried and the result is saved in iwork[0]. Otherwise: When jobz = 'N' or n ≤ 1, liwork ≥ 1. When jobz = 'V' and n > 1, liwork ≥ 3+5*n.	Input
info	Integer	Execution result: 0: The execution is successful. Smaller than 0: The value of the -info-th parameter is invalid. Greater than 0, the jobz='N' calculation cannot be converged and the info/(n+1)-th eigenvalue cannot be calculated.	Output

Dependencies

#include "klapack.h"

Examples

C interface:

      
       
         
         
               char jobz = 'V'; 
    char uplo = 'L'; 
    int n = 5; 
    int lda = 5; 
    int info = 0; 
    double w[5]; 
    double *work = NULL; 
    double qwork; 
    int lwork = -1; 
    int *iwork = NULL; 
    int qiwork; 
    int liwork = -1; 
    /* 
     * Symmetric A (stored in column-major): 
     *   7.027  8.710  1.015  6.929  7.584 
     *   8.710  0.839  2.469  3.850  0.559 
     *   1.015  2.469  1.930  6.761  7.207 
     *   6.929  3.850  6.761  4.344  4.804 
     *   7.584  0.559  7.207  4.804  6.177 
     */ 
    double a[] = {7.027, 8.710, 1.015, 6.929, 7.584, 
                    8.710, 0.839, 2.469, 3.850, 0.559, 
                    1.015, 2.469, 1.930, 6.761, 7.207, 
                    6.929, 3.850, 6.761, 4.344, 4.804, 
                    7.584, 0.559, 7.207, 4.804, 6.177}; 
    /* Query optimal work size */ 
    dsyevd_(&jobz, &uplo, &n, a, &lda, w, &qwork, &lwork, &qiwork, &liwork, &info); 
    if (info != 0) { 
        return ERROR; 
    } 
    lwork = (int)qwork; 
    work = (double *)malloc(sizeof(double) * lwork); 
    liwork = (int)qiwork; 
    iwork = (int *)malloc(sizeof(int) * liwork); 
    /* Calculate eigenvalues and eigenvectors */ 
    dsyevd_(&jobz, &uplo, &n, a, &lda, w, work, &lwork, iwork, &liwork, &info); 
    free(work); 
    free(iwork); 
    /* 
     * Output: 
     * Eigenvalues (in w) 
     *  -8.8422 -3.3411 1.1888 6.2050 25.1065 
     * Eigenvectors (in a, stored in column-major) 
     *   0.5405  -0.1616  -0.3059  -0.5235  -0.5604 
     *  -0.4913   0.5975   0.2432  -0.4882  -0.3228 
     *   0.4886   0.4989   0.3454   0.5213  -0.3482 
     *  -0.2410  -0.6062   0.5831   0.1038  -0.4729 
     *  -0.4120   0.0219  -0.6228   0.4528  -0.4867 
     */

          

        

      
     

Fortran interface:

        CHARACTER :: jobz = "V" 
        CHARACTER :: uplo = "L" 
        PARAMETER (n = 5) 
        PARAMETER (lda = 5) 
        INTEGER :: info = 0 
        REAL(8) :: w(5); 
        REAL(8) :: qwork(1) 
        REAL(8), ALLOCATABLE :: work(:) 
        INTEGER :: lwork = -1 
        INTEGER :: qiwork(1) 
        REAL(8), ALLOCATABLE :: iwork(:) 
        INTEGER :: liwork = -1 
 
*       Symmetric A (stored in column-major): 
*         7.027  8.710  1.015  6.929  7.584 
*         8.710  0.839  2.469  3.850  0.559 
*         1.015  2.469  1.930  6.761  7.207 
*         6.929  3.850  6.761  4.344  4.804 
*         7.584  0.559  7.207  4.804  6.177 
        REAL(8) :: a(n, n) 
        DATA a / 7.027, 8.710, 1.015, 6.929, 7.584, 
     $           8.710, 0.839, 2.469, 3.850, 0.559, 
     $           1.015, 2.469, 1.930, 6.761, 7.207, 
     $           6.929, 3.850, 6.761, 4.344, 4.804, 
     $           7.584, 0.559, 7.207, 4.804, 6.177 / 
*       Query optimal work size 
        EXTERNAL DSYEVD 
        CALL DSYEVD(jobz, uplo, n, a, lda, w, qwork, lwork, qiwork, 
     $              liwork, info) 
        IF (info.NE.0) THEN 
            CALL EXIT(1) 
        END IF 
        lwork = INT(qwork(1)) 
        liwork = INT(qiwork(1)) 
        ALLOCATE(work(lwork), iwork(liwork)) 
*       Calculate eigenvalues and eigenvectors 
        CALL DSYEVD(jobz, uplo, n, a, lda, w, work, lwork, iwork, 
     $              liwork, info) 
        DEALLOCATE(work, iwork); 
 
*       Output: 
*       Eigenvalues (in w) 
*        -8.8422 -3.3411 1.1888 6.2050 25.1065 
*       Eigenvectors (in a, stored in column-major) 
*         0.5405  -0.1616  -0.3059  -0.5235  -0.5604 
*        -0.4913   0.5975   0.2432  -0.4882  -0.3228 
*         0.4886   0.4989   0.3454   0.5213  -0.3482 
*        -0.2410  -0.6062   0.5831   0.1038  -0.4729 
*        -0.4120   0.0219  -0.6228   0.4528  -0.4867

Parent topic: Functions for Solving Eigenvalue Problems