SSPGVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 NAME 💅

okay so this is sspgvx.f(3) from the Library Functions Manual — the header path is /home/abui...RC/sspgvx.f(3), and the source file is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sspgvx.f. It's the manual page for SSPGVX, bestie. 💖

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

       subroutine SSPGVX (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu,
           abstol, m, w, z, ldz, work, iwork, ifail, info)
           SSPGVX

🌈 Function/Subroutine Documentation 🌈

   subroutine SSPGVX (integer itype, character jobz, character range, character
       uplo, integer n, real, dimension( * ) ap, real, dimension( * ) bp, 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, dimension( * ) iwork, integer, dimension(
       * ) ifail, integer info)
       SSPGVX

🦋 Purpose 🦋

okay so SSPGVX computes selected eigenvalues, and optionally eigenvectors, of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric, stored in packed storage, 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. slay. ✨

🎀 Parameters 🎀

💖 ITYPE 💖

ITYPE is INTEGER and it 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 🦋

JOBZ is CHARACTER*1:

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

🌸 RANGE 🌸

RANGE is CHARACTER*1:

  • = '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 🎀

UPLO is CHARACTER*1:

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

⭐ N ⭐

N is INTEGER — the order of the matrix pencil (A,B). N >= 0.

💜 AP 💜

AP is REAL array, dimension (N*(N+1)/2).

On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed. literally gone, bestie 💀

🫶 BP 🫶

BP is REAL array, dimension (N*(N+1)/2).

On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:

if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, 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.

🌈 VL 🌈

VL is 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 🦋

VU is 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 💫

IL is 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 💫

IU is 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 🍓

ABSTOL is 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 A to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('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 👑

M is INTEGER — the total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

🌟 W 🌟

W is REAL array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order.

🧚 Z 🧚

Z is 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. 🫶

🎀 LDZ 🎀

LDZ is INTEGER — the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

💜 WORK 💜

WORK is REAL array, dimension (8*N).

💜 IWORK 💜

IWORK is INTEGER array, dimension (5*N).

🌸 IFAIL 🌸

IFAIL is 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 🔥

INFO is INTEGER:

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: SPPTRF or SSPEVX returned an error code:
    • <= N: if INFO = i, SSPEVX 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 269 of file sspgvx.f.

🧚 Author 🧚

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okay bestie, that's the whole page. With all the sparkle, this is LAPACK Version 3.12.0, from /home/abui...RC/sspgvx.f(3). Mwah 💋

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