Developer Reference for Intel® oneAPI Math Kernel Library for C
?spevd
Uses divide and conquer algorithm to compute all eigenvalues and (optionally) all eigenvectors of a real symmetric matrix held in packed storage.
Syntax
lapack_intLAPACKE_sspevd ( intmatrix_layout , charjobz , charuplo , lapack_intn , float*ap , float*w , float*z , lapack_intldz );
lapack_intLAPACKE_dspevd ( intmatrix_layout , charjobz , charuplo , lapack_intn , double*ap , double*w , double*z , lapack_intldz );
Include Files
mkl.h
Description
sspevd dspevd spevd
The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix A (held in packed storage). In other words, it can compute the spectral factorization of A as:
A = Z*Λ*Z^{T} .
Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λ i , and Z is the orthogonal matrix whose columns are the eigenvectors zi . Thus,
A*z_{i} = λ_{i}*z_{i} for i = 1, 2, ..., n .
If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the QL or QR algorithm.
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
jobz
Must be ‘N’ or ‘V’ .
If jobz = 'N' , then only eigenvalues are computed. If jobz = 'V' , then eigenvalues and eigenvectors are computed.
uplo
Must be ‘U’ or ‘L’ .
If uplo = 'U' , ap stores the packed upper triangular part of A . If uplo = 'L' , ap stores the packed lower triangular part of A .
n
The order of the matrix A ( n≥ 0 ).
- ap , work
-
REAL for sspevd DOUBLE PRECISION for dspevd Arrays: ap(*) contains the packed upper or lower triangle of symmetric matrix A , as specified by uplo . The dimension of ap must be max(1, n *( n +1)/2) work is a workspace array, its dimension max(1, lwork) .
ldz
The leading dimension of the output array z .
Constraints: if jobz = 'N' , then ldz≥ 1 ; if jobz = 'V' , then ldz≥ max(1, n) .
lwork
The dimension of the array work . Constraints: if n≤ 1 , then lwork≥ 1 ; if jobz = 'N' and n > 1 , then lwork≥ 2*n ; if jobz = 'V' and n > 1 , then lwork≥n^{2}+ 6*n + 1 . If lwork = -1 , then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla . See Application Notes for details.
iwork
Workspace array, its dimension max(1, liwork) .
liwork
The dimension of the array iwork . Constraints: if n ≤ 1, then liwork≥ 1 ; if jobz = 'N' and n > 1 , then liwork≥ 1 ; if jobz = 'V' and n > 1, then liwork≥ 5*n+3 . If liwork = -1 , then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla . See Application Notes for details.
Output Parameters
- w , z
-
REAL for sspevd DOUBLE PRECISION for dspevd Arrays: w(*) , size at least max(1, n ). If info = 0 , contains the eigenvalues of the matrix A in ascending order. See also info .
z (size max(1, ldz * n )) .
The second dimension of z must be: at least 1 if jobz = 'N' ;at least max(1, n ) if jobz = 'V' . If jobz = 'V' , then this array is overwritten by the orthogonal matrix Z which contains the eigenvectors of A . If jobz = 'N' , then z is not referenced.
- ap
-
On exit, this array is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A.
- work(1)
-
On exit, if info = 0 , then work(1) returns the required lwork .
- iwork(1)
-
On exit, if info = 0 , then iwork(1) returns the required liwork .
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = i , then the algorithm failed to converge; i indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.
If info = -i , the i -th parameter had an illegal value.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
Application Notes
The computed eigenvalues and eigenvectors are exact for a matrix \(A+E\) such that \(||E||_{2} = O(\varepsilon) ||A||_{2}\) , where \(\varepsilon\) is the machine precision.
The complex analogue of this routine is hpevd .
See also syevd for matrices held in full storage, and sbevd for banded matrices.