Visible to Intel only — GUID: GUID-1A7CEAA4-8249-46DB-A1FE-1BEEC1D17DE7
Visible to Intel only — GUID: GUID-1A7CEAA4-8249-46DB-A1FE-1BEEC1D17DE7
?spgvd
Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem with matrices in packed storage using a divide and conquer method.
Syntax
call sspgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info)
call dspgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info)
call spgvd(ap, bp, w [,itype] [,uplo] [,z] [,info])
Include Files
- mkl.fi, lapack.f90
Description
The routine computes all the 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.
If eigenvectors are desired, it uses a divide and conquer algorithm.
Input Parameters
- itype
-
INTEGER. 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
-
CHARACTER*1. Must be 'N' or 'V'.
If jobz = 'N', then compute eigenvalues only.
If jobz = 'V', then compute eigenvalues and eigenvectors.
- uplo
-
CHARACTER*1. 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
-
INTEGER. The order of the matrices A and B (n≥ 0).
- ap, bp, work
-
REAL for sspgvd
DOUBLE PRECISION for dspgvd.
Arrays:
ap(*) contains the packed upper or lower triangle of the symmetric matrix A, as specified by uplo.
The dimension 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 dimension of bp must be at least max(1, n*(n+1)/2).
work is a workspace array, its dimension max(1, lwork).
- ldz
-
INTEGER. The leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥ max(1, n).
- lwork
-
INTEGER.
The dimension of the array work.
Constraints:
If n≤ 1, lwork≥ 1;
If jobz = 'N' and n>1, lwork≥ 2n;
If jobz = 'V' and n>1, lwork≥ 2n2+6n+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
-
INTEGER.
Workspace array, dimension max(1, lwork).
- liwork
-
INTEGER.
The dimension of the array iwork.
Constraints:
If n≤ 1, liwork≥ 1;
If jobz = 'N' and n>1, liwork≥ 1;
If jobz = 'V' and n>1, liwork≥ 5n+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
- ap
-
On exit, the contents of ap are overwritten.
- bp
-
On exit, contains the triangular factor U or L from the Cholesky factorization B = UT*U or B = L*LT, in the same storage format as B.
- w, z
-
REAL for sspgv
DOUBLE PRECISION for dspgv.
Arrays:
w(*), size at least max(1, n).
If info = 0, contains the eigenvalues in ascending order.
z(ldz,*).
The second dimension of z must be at least max(1, n).
If jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:
if itype = 1 or 2, ZT*B*Z = I;
if itype = 3, ZT*inv(B)*Z = I;
If jobz = 'N', then z is not referenced.
- work(1)
-
On exit, if info = 0, then work(1) returns the required minimal size of lwork.
- iwork(1)
-
On exit, if info = 0, then iwork(1) returns the required minimal size of liwork.
- info
-
INTEGER.
If info = 0, the execution is successful.
If info = -i, the i-th argument had an illegal value.
If info > 0, spptrf/dpptrf and sspevd/dspevd returned an error code:
If info = i≤n, sspevd/dspevd failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not converge to zero;
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
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.
Specific details for the routine spgvd interface are the following:
- ap
-
Holds the array A of size (n*(n+1)/2).
- bp
-
Holds the array B of size (n*(n+1)/2).
- w
-
Holds the vector with the number of elements n.
- z
-
Holds the matrix Z of size (n, n).
- itype
-
Must be 1, 2, or 3. The default value is 1.
- uplo
-
Must be 'U' or 'L'. The default value is 'U'.
- jobz
-
Restored based on the presence of the argument z as follows:
jobz = 'V', if z is present,
jobz = 'N', if z is omitted.
Application Notes
If it is not clear how much workspace to supply, use a generous value of lwork (or liwork) for the first run, or set lwork = -1 (liwork = -1).
If lwork (or liwork) has any of admissible sizes, which is no less than the minimal value described, then the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array (work, iwork) on exit. Use this value (work(1), iwork(1)) for subsequent runs.
If lwork = -1 (liwork = -1), then the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work, iwork). This operation is called a workspace query.
Note that if lwork (liwork) is less than the minimal required value and is not equal to -1, then the routine returns immediately with an error exit and does not provide any information on the recommended workspace.