Developer Reference for Intel® oneAPI Math Kernel Library for C
?spgvx
Computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem with matrices in packed storage.
Syntax
lapack_intLAPACKE_sspgvx ( intmatrix_layout , lapack_intitype , charjobz , charrange , charuplo , lapack_intn , float*ap , float*bp , floatvl , floatvu , lapack_intil , lapack_intiu , floatabstol , lapack_int*m , float*w , float*z , lapack_intldz , lapack_int*ifail );
lapack_intLAPACKE_dspgvx ( intmatrix_layout , lapack_intitype , charjobz , charrange , charuplo , lapack_intn , double*ap , double*bp , doublevl , doublevu , lapack_intil , lapack_intiu , doubleabstol , lapack_int*m , double*w , double*z , lapack_intldz , lapack_int*ifail );
Include Files
mkl.h
Description
sspgvx dspgvx spgvx
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, stored in packed format, 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 = lambda*B*x ; if itype = 2 , the problem type is A * B * x = lambda * x ; if itype = 3 , the problem type is B*A*x = lambda*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 ap and bp store the upper triangles of A and B ; If uplo = 'L' , arrays ap and bp store the lower triangles of A and B .
n
The order of the matrices A and B ( n≥ 0 ).
- ap , bp , work
-
REAL for sspgvx DOUBLE PRECISION for dspgvx . Arrays: ap(*) contains the packed upper or lower triangle of the symmetric matrix A , as specified by uplo . The size of ap must be at least max(1, n *( n +1)/2). bp(*) contains the packed upper or lower triangle of the symmetric matrix B , as specified by uplo . The size of bp must be at least max(1, n *( n +1)/2). work(*) is a workspace array, size at least max(1, 8 n ).
- vl , vu
-
REAL for sspgvx DOUBLE PRECISION for dspgvx . 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 sspgvx DOUBLE PRECISION for dspgvx . 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 .
iwork
Workspace array, size at least max(1, 5 n ).
Output Parameters
- ap
-
On exit, the contents of ap are overwritten.
- bp
-
On exit, contains the triangular factor U or L from the Cholesky factorization B = U^{T}*U or B = L*L^{T} , in the same storage format as B .
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 sspgvx DOUBLE PRECISION for dspgvx . Arrays: w(*) , size at least max(1, n ). If info = 0 , contains the 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, n ). 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). 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.
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 , spptrf / dpptrf and sspevx / dspevx returned an error code:
If info = i≤n , sspevx / dspevx 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 instead, where \(T\) is the tridiagonal matrix obtained by reducing \(A\) to tridiagonal form. Eigenvalues are 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’).