Developer Reference for Intel® oneAPI Math Kernel Library for C
?sbevd
Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric band matrix using divide and conquer algorithm.
Syntax
lapack_intLAPACKE_ssbevd ( intmatrix_layout , charjobz , charuplo , lapack_intn , lapack_intkd , float*ab , lapack_intldab , float*w , float*z , lapack_intldz );
lapack_intLAPACKE_dsbevd ( intmatrix_layout , charjobz , charuplo , lapack_intn , lapack_intkd , double*ab , lapack_intldab , double*w , double*z , lapack_intldz );
Include Files
mkl.h
Description
ssbevd dsbevd sbevd
The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric band matrix A . 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' , ab stores the upper triangular part of A . If uplo = 'L' , ab stores the lower triangular part of A .
n
The order of the matrix A ( n≥ 0 ).
kd
The number of super- or sub-diagonals in A
( kd≥ 0 ).
- ab , work
-
REAL for ssbevd DOUBLE PRECISION for dsbevd . Arrays:
ab (size at least max(1, ldab * n ) for column major layout and at least max(1, ldab *( kd + 1)) for row major layout) is an array containing either upper or lower triangular part of the symmetric matrix A (as specified by uplo ) in band storage format.
The second dimension of ab must be at least max(1, n ). work is a workspace array, its dimension max(1, lwork) .
ldab
The leading dimension of ab ; must be at least kd +1 for column major layout and n for row major layout .
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≥ 2n ; if jobz = 'V' and n > 1 , then lwork≥ 2*n^{2} + 5*n + 1 . If lwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size 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 job = 'N' and n > 1 , then liwork < 1 ; if job = 'V' and n > 1 , then liwork < 5*n+3 . If liwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size 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 ssbevd DOUBLE PRECISION for dsbevd 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 if job = 'V' and at least 1 if job = 'N' ) .
The second dimension of z must be: at least 1 if job = 'N' ; at least max(1, n ) if job = 'V' . If job = 'V' , then this array is overwritten by the orthogonal matrix Z which contains the eigenvectors of A . The i- th column of Z contains the eigenvector which corresponds to the eigenvalue w(i)w[i - 1] . If job = 'N' , then z is not referenced.
- ab
-
On exit, this array is overwritten by the values generated during the reduction to tridiagonal form.
- work(1)
-
On exit, if lwork > 0 , then work(1) returns the required minimal size of lwork .
- iwork(1)
-
On exit, if liwork > 0 , then iwork(1) returns the required minimal size of 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 hbevd .
See also syevd for matrices held in full storage, and spevd for matrices held in packed storage.