SRC/sspgvx.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💖 NAME 💖

SRC/sspgvx.f — okay bestie, welcome to the SRC/sspgvx.f page from the Library Functions Manual 🫶 this is the SSPGVX lore, and it’s lowkey iconic.

✨ 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 🌸

SSPGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite eigenproblem, bestie. The forms are 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. 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. It’s the order of the matrix pencil (A,B). N >= 0.

AP

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

On entry, it’s 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 💀

BP

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

On entry, it’s 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, it holds 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', this is 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', this is 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', it’s 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', it’s 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. It’s 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. It’s 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. It’s 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 📌

Definition at line 269 of file sspgvx.f.

🧾 Author 🧾

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