SSYGVX(3)

🎀 NAME 🎀

Okay bestie, welcome to SSYGVX(3) — your fave page from the Library Functions Manual. 💅 The source file is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ssygvx.f, and yes, this is the LAPACK generalized symmetric-definite eigenproblem main character. No cap. ✨

✨ SYNOPSIS ✨

Functions/Subroutines 🦋

subroutine SSYGVX (itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu,
    il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)
    SSYGVX

Function/Subroutine Documentation 👑

subroutine SSYGVX (integer itype, character jobz, character range, character
    uplo, integer n, real, dimension( lda, * ) a, integer lda, real,
    dimension( ldb, * ) b, integer ldb, real vl, real vu, integer il,
    integer iu, real abstol, integer m, real, dimension( * ) w, real,
    dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer
    lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail,
    integer info)
    SSYGVX

Purpose 🫶

SSYGVX computes selected eigenvalues, and optionally eigenvectors, of a real generalized symmetric-definite eigenproblem, in one of these forms: A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric, and B is also positive definite. You can pick eigenvalues and eigenvectors by specifying either a range of values or a range of indices for the desired eigenvalues. 💖

Parameters 🌸

• ITYPE — INTEGER. Specifies the problem type to be solved:

= 1:  A*x = (lambda)*B*x
= 2:  A*B*x = (lambda)*x
= 3:  B*A*x = (lambda)*x

• JOBZ — CHARACTER*1. Tells the routine what to compute:

= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.

• RANGE — CHARACTER*1. Picks the eigenvalue range:

= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval (VL,VU]
       will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.

• UPLO — CHARACTER*1. Says which triangle is stored:

= 'U':  Upper triangle of A and B are stored;
= 'L':  Lower triangle of A and B are stored.

• N — INTEGER. The order of the matrix pencil (A,B). N >= 0. 🎀
• A — REAL array, dimension (LDA, N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. Bestie, she’s gone. 💀

• LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• B — REAL array, dimension (LDB, N). On entry, the symmetric matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix 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. So yeah, B gets a whole new outfit. 🦋

• LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• VL — REAL. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
• VU — REAL. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
• IL — INTEGER. If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
• IU — INTEGER. If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
• ABSTOL — REAL. The absolute error tolerance for the eigenvalues. 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

ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of 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*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). 🔥

• M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
• W — REAL array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order. Slay, sorted. ✨
• Z — REAL array, dimension (LDZ, max(1,M)). If JOBZ = 'N', then Z is not referenced. 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 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: the user 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. Plan ahead, bestie. 🫶

• LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
• WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🎀
• LWORK — INTEGER. The length of the array WORK. LWORK >= max(1,8*N). For optimal efficiency, LWORK >= (NB+3)*N, where NB is the blocksize for SSYTRD returned by ILAENV.

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. Lowkey iconic behavior. 💅

• IWORK — INTEGER array, dimension (5*N).
• IFAIL — INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.
• INFO — INTEGER. The exit tea:

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  SPOTRF or SSYEVX returned an error code:
   <= N:  if INFO = i, SSYEVX failed to converge;
          i eigenvectors failed to converge.  Their indices
          are stored in array IFAIL.
   > N:   if INFO = N + i, for 1 <= i <= N, then the leading
          principal minor of order i of B is not positive.
          The factorization of B could not be completed and
          no eigenvalues or eigenvectors were computed.

Author 💖

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

Contributors 🍓

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 294 of file ssygvx.f 🌈

Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

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Okay bestie, that’s the whole page. Footer sign-off: LAPACK Version 3.12.0 and footer path /home/abui...RC/ssygvx.f(3). No separate date on the page — just that source/version and path combo. Byeee, stay positive definite and slay those eigenvalues 💜👑