SRC/dsbevx.f(3)

📦
version 3.12.1 · 4 versions
, in arch rolling

💖 NAME

okay so you found SRC/dsbevx.f(3) in the Library Functions Manual, bestie. This is the page for dsbevx, a LAPACK subroutine that computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. You can pick those eigenvalues and eigenvectors either by a range of values or by a range of indices. Periodt.

✨ SYNOPSIS

🎀 Functions/Subroutines

subroutine dsbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu,
    il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices.

🦋 Function/Subroutine Documentation

🌸 subroutine dsbevx (...)

subroutine dsbevx (character jobz, character range, character uplo, integer
    n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab,
    double precision, dimension( ldq, * ) q, integer ldq, double precision
    vl, double precision vu, integer il, integer iu, double precision
    abstol, integer m, double precision, dimension( * ) w, double precision,
    dimension( ldz, * ) z, integer ldz, double precision, dimension( * )
    work, integer, dimension( * ) iwork, integer, dimension( * ) ifail,
    integer info)

DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices.

🌟 Purpose

DSBEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

💅 Parameters

  • JOBZ (CHARACTER*1) — tells the routine what to compute, no cap.
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.
  • RANGE (CHARACTER*1) — how you want to pick the eigenvalues, bestie.
    • = '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) — which triangle of A is stored. iconic little switch.
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N (INTEGER) — The order of the matrix A. N >= 0.
  • KD (INTEGER) — The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
  • AB (DOUBLE PRECISION array, dimension (LDAB, N)) — On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
    • if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    • if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.

  • LDAB (INTEGER) — The leading dimension of the array AB. LDAB >= KD + 1.
  • Q (DOUBLE PRECISION array, dimension (LDQ, N)) — If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.
  • LDQ (INTEGER) — The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
  • VL (DOUBLE PRECISION) — If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU (DOUBLE PRECISION) — 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 (DOUBLE PRECISION) — 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 AB to tridiagonal form.

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*DLAMCH('S').

The source also mentions Demmel and Kahan, LAPACK Working Note #3.

  • 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 (DOUBLE PRECISION array, dimension (N)) — The first M elements contain the selected eigenvalues in ascending order.
  • Z (DOUBLE PRECISION array, dimension (LDZ, max(1,M))) — 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). 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. If JOBZ = 'N', then Z is not referenced. 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 (INTEGER) — The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK (DOUBLE PRECISION array, dimension (7*N)).
  • 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 vibe check at the end.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.

👑 Author

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

🎀 Definition

Definition at line 262 of file dsbevx.f.

🦋 Author

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That’s the whole tea ☕ — this footer is LAPACK Version 3.12.0 and source SRC/dsbevx.f(3), so you know exactly where this page came from. Bye bestie, go slay those eigenvalues 💅✨🫶

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