Developer Reference for Intel® oneAPI Math Kernel Library for C
?sygvx
Computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem.
Syntax
lapack_intLAPACKE_ssygvx ( intmatrix_layout , lapack_intitype , charjobz , charrange , charuplo , lapack_intn , float*a , lapack_intlda , float*b , lapack_intldb , floatvl , floatvu , lapack_intil , lapack_intiu , floatabstol , lapack_int*m , float*w , float*z , lapack_intldz , lapack_int*ifail );
lapack_intLAPACKE_dsygvx ( intmatrix_layout , lapack_intitype , charjobz , charrange , charuplo , lapack_intn , double*a , lapack_intlda , double*b , lapack_intldb , doublevl , doublevu , lapack_intil , lapack_intiu , doubleabstol , lapack_int*m , double*w , double*z , lapack_intldz , lapack_int*ifail );
Include Files
mkl.h
Description
ssygvx dsygvx sygvx
The routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
A*x = λ*B*x , A*B*x = λ*x , or B*A*x = λ*x .
Here A and B are assumed to be symmetric and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
itype
Must be 1 or 2 or 3. Specifies the problem type to be solved:
if itype = 1 , the problem type is A*x = λ*B*x ; if itype = 2 , the problem type is A * B * x = λ * x ; if itype = 3 , the problem type is B*A*x = λ*x .
jobz
Must be ‘N’ or ‘V’ .
If jobz = 'N' , then compute eigenvalues only. If jobz = 'V' , then compute eigenvalues and eigenvectors.
range
Must be ‘A’ or ‘V’ or ‘I’ .
If range = 'A' , the routine computes all eigenvalues. If range = 'V' , the routine computes eigenvalues lambda(i)w[i] in the half-open interval: vl<lambda(i)w[i]≤vu . If range = 'I' , the routine computes eigenvalues with indices il to iu .
uplo
Must be ‘U’ or ‘L’ .
If uplo = 'U' , arrays a and b store the upper triangles of A and B ; If uplo = 'L' , arrays a and b store the lower triangles of A and B .
n
The order of the matrices A and B ( n≥ 0 ).
- a , b , work
-
REAL for ssygvx DOUBLE PRECISION for dsygvx . Arrays:
a (size at least max(1, lda * n )) contains the upper or lower triangle of the symmetric matrix A , as specified by uplo .
The second dimension of a must be at least max(1, n ).
b (size at least max(1, ldb * n )) contains the upper or lower triangle of the symmetric positive definite matrix B , as specified by uplo .
The second dimension of b must be at least max(1, n ). work is a workspace array, its dimension max(1, lwork) .
lda
The leading dimension of a ; at least max(1, n ).
ldb
The leading dimension of b ; at least max(1, n ).
- vl , vu
-
REAL for ssygvx DOUBLE PRECISION for dsygvx . If range = 'V' , the lower and upper bounds of the interval to be searched for eigenvalues. Constraint: vl< vu . If range = 'A' or ‘I’ , vl and vu are not referenced.
il , iu
If range = 'I' , the indices in ascending order of the smallest and largest eigenvalues to be returned. Constraint: 1 ≤il≤iu≤n , if n > 0 ; il=1 and iu=0 if n = 0 . If range = 'A' or ‘V’ , il and iu are not referenced.
- abstol
-
REAL for ssygvx DOUBLE PRECISION for dsygvx . The absolute error tolerance for the eigenvalues. See Application Notes for more information.
ldz
The leading dimension of the output array z . Constraints:
ldz≥ 1 ; if jobz = 'V' , ldz≥ max(1, n) for column major layout and ldz ≥ max(1, m ) for row major layout .
lwork
The dimension of the array work ; lwork < max(1, 8n) . If lwork = -1 , then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla .
iwork
Workspace array, size at least max(1, 5 n ).
Output Parameters
- a
-
On exit, the upper triangle (if uplo = 'U' ) or the lower triangle (if uplo = 'L' ) of A , including the diagonal, is overwritten.
- b
-
On exit, if info≤n , the part of b containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U^{T}*U or B = L*L^{T} .
m
The total number of eigenvalues found,
0 ≤m≤n . If range = 'A' , m = n , and if range = 'I' , m = iu-il+1 .
- w , z
-
REAL for ssygvx DOUBLE PRECISION for dsygvx . Arrays:
w , size at least max(1, n ).
The first m elements of w contain the selected eigenvalues in ascending order.
z (size at least max(1, ldz * m ) for column major layout and max(1, ldz * n ) for row major layout) .
The second dimension of z must be at least max(1, m ).
If jobz = 'V' , then if info = 0 , the first m columns of z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i -th column of z holding the eigenvector associated with w [ i - 1] . The eigenvectors are normalized as follows:
if itype = 1 or 2 , Z^{T}*B*Z = I ; if itype = 3 , Z^{T}*inv(B)*Z = I ; If jobz = 'N' , then z is not referenced. If an eigenvector fails to converge, then that column of z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in ifail . Note: you must ensure that at least max(1, m ) columns are supplied in the array z ; if range = 'V' , the exact value of m is not known in advance and an upper bound must be used.
- work(1)
-
On exit, if info = 0 , then work(1) returns the required minimal size of lwork .
ifail
Array, size at least max(1, n ). If jobz = 'V' , then if info = 0 , the first m elements of ifail are zero; if info > 0 , the ifail contains the indices of the eigenvectors that failed to converge. If jobz = 'N' , then ifail is not referenced.
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
If info > 0 , spotrf / dpotrf and ssyevx / dsyevx returned an error code:
If info = i≤n , ssyevx / dsyevx failed to converge, and i eigenvectors failed to converge. Their indices are stored in the array ifail ;
If info = n + i , for 1 ≤i≤n , then the leading minor of order i of B is not positive-definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.
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
An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to \(\text{abstol}+\varepsilon\max(|a|,|b|)\) , where \(\varepsilon\) is the machine precision.
If abstol is less than or equal to zero, then \(\varepsilon ||T||_{1}\) is used as tolerance, where \(T\) is the tridiagonal matrix obtained by reducing \(C\) to tridiagonal form, where \(C\) is the symmetric matrix of the standard symmetric problem to which the generalized problem is transformed. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2* ?lamch (‘S’), not zero.
If this routine returns with info > 0 , indicating that some eigenvectors did not converge, set abstol to 2* ?lamch (‘S’).