hbevx(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🎀 hbevx — {hb,sb}evx: eig, bisection 🎀

okay bestie, this is hbevx(3) from the Library Functions Manual — the LAPACK corner of the library-calls universe 💖 and it's literally four Fortran subroutines that all do the same iconic thing for four different data types. eigenvalues, eigenvectors, band matrices, bisection, the whole vibe. let's get into it ✨

✨ NAME ✨

hbevx — {hb,sb}evx: eig, bisection

🦋 SYNOPSIS 🦋

📚 Functions

  • subroutine chbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) — CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • 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
  • subroutine ssbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) — SSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
  • subroutine zhbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) — ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
👀 notice how chbevx and zhbevx (the complex ones) carry an extra rwork argument, and the real ones dsbevx / ssbevx don't. that's not a typo, that's just how they roll 💅

💫 Detailed Description 💫

ngl, this section is basically an empty room with a sign pointing downstairs 🫶 all the actual deets live in the Function Documentation right below — no cap, that's the whole section.

🔥 Function Documentation 🔥

🌸 subroutine chbevx 🌸

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

CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 😤✨

🎯 Purpose

okay so CHBEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A 💜. and the fun part? you get to pick which ones — either by giving it a range of values, or a range of indices. choose your own adventure, but make it linear algebra 🔥

🧾 Parameters

  • 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 is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
  • AB — AB is COMPLEX array, dimension (LDAB, N). On entry, the upper or lower triangle of the Hermitian 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 💀 (bestie, your input array is not surviving this).
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
  • Q — Q is COMPLEX array, dimension (LDQ, N)
    • If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form.
    • If JOBZ = 'N', the array Q is not referenced 🙈
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
  • 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 AB 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').

Also: see by Demmel and Kahan, LAPACK Working Note #3 📖 iconic reading tbh.

  • 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). The first M elements contain the selected eigenvalues in ascending order ⬆️✨
  • Z — Z is COMPLEX 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. plan accordingly bestie 🙏
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is COMPLEX array, dimension (N) 🛠️
  • RWORK — RWORK is REAL array, dimension (7*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: 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 at line 264 of file chbevx.f. 🫶

🌸 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 — same choose-your-own-adventure energy, just with DOUBLE PRECISION this time 🔥

🧾 Parameters

  • 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 is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • KD — (and this is literally where part 1 of the page runs out bestie 😭 the KD deets — and everything after — are coming in the next part, hold that thought 🫶)
  • KD — KD is INTEGER. It's the number of superdiagonals if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. Constraint: KD >= 0. No negatives, bestie. 💅
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, N). On entry, this holds the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows. The j-th column of A lives in the j-th column of AB like this:
    • 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 gets 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. If UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. So yeah, it's a glow-up, not a storage unit. ✨

  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. Constraint: LDAB >= KD + 1. Respect the leading dimension, periodt.
  • Q — Q is DOUBLE PRECISION array, dimension (LDQ, N). If JOBZ = 'V', this is the N-by-N orthogonal matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced. She's only on stage when invited. 🎀
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
  • VL — VL is DOUBLE PRECISION. If RANGE='V', this is the lower bound of the interval to be searched for eigenvalues. Constraint: VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • VU — VU is DOUBLE PRECISION. If RANGE='V', this is the upper bound of the interval to be searched for eigenvalues. Constraint: VL < VU. Not referenced if RANGE = 'A' or 'I'.
  • IL — IL is INTEGER. If RANGE='I', this is the index of the smallest eigenvalue to be returned. Constraint: 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', this is the index of the largest eigenvalue to be returned. Constraint: 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 DOUBLE PRECISION. The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it's 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'). See [by Demmel and Kahan, LAPACK Working Note #3] — yes, the note is right there in the original, bestie. 💀

  • M — M is INTEGER. The total number of eigenvalues found. Constraint: 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. She's counting, not guessing. 🧮
  • W — W is DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order. Sorted, slayed, ascending. 👑
  • Z — Z is 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. Don't play yourself. 🙈
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. Constraint: LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • WORK — WORK is DOUBLE PRECISION array, dimension (7*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. You passed something cursed, bestie. 😭
    • > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL. Not every icon makes it, but the info is there. 💅

🌸 Author

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

Definition at line 262 of file dsbevx.f.

🦋 subroutine ssbevx

subroutine ssbevx (character jobz, character range, character uplo, integer
    n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real,
    dimension( ldq, * ) q, integer ldq, 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)

SSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. Yes, "OTHER" in all caps, she's mysterious like that. 🕶️

✨ Purpose

!> !> SSBEVX 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 — JOBZ is CHARACTER*1:
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors.

Pick your fighter. 💅

  • 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.

It's giving "choose your own adventure" but for eigenvalues. 🌈

  • UPLO — UPLO is CHARACTER*1:
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER. The order of the matrix A. Constraint: N >= 0.
  • KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. Constraint: KD >= 0.
  • AB — AB is REAL 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. Same storage glow-up, just REAL this time. 🍓

  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. Constraint: LDAB >= KD + 1.
  • Q — Q is REAL 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 — LDQ is INTEGER. The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
  • VL — VL is REAL. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Constraint: 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. Constraint: 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. Constraint: 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. Constraint: 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. (And yes, the rest of the ssbevx parameter tea is still steeping — this is where part 2 dips out. More coming, bestie. 🫶)
  • ABSTOL — ABSTOL is REAL</p> and it's your absolute error tolerance for the eigenvalues, bestie. An approximate eigenvalue gets accepted as converged when we can tell it lives in an interval [a,b] whose width is less than or equal to:
ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. And if ABSTOL is less than or equal to zero? Then EPS*|T| gets used in its place, where |T| is the 1-norm of the tridiagonal matrix you get by reducing AB to tridiagonal form. iykyk

Heads up: eigenvalues come out most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. And if this routine returns with INFO>0 — meaning some eigenvectors didn't converge — try setting ABSTOL to 2*SLAMCH('S'). 💅

See by Demmel and Kahan, LAPACK Working Note #3. (no cap, the original page literally has a gap where the title should be — the reference itself is Demmel and Kahan, LAPACK Working Note #3 🫶)

  • M — M is INTEGER</p>, the total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', then M = N, and if RANGE = 'I', then M = IU-IL+1. Math girlies stay winning ✨
  • W — W is REAL array, dimension (N)</p>. The first M elements contain the selected eigenvalues in ascending order. sorted, as a treat 🎀
  • Z — Z is REAL array, dimension (LDZ, max(1,M))</p>. Okay so:
    • 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: you, 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 isn't known in advance, so an upper bound must be used. do NOT skip this 🙈
  • LDZ — LDZ is INTEGER</p>, the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). 🦋
  • WORK — WORK is REAL array, dimension (7*N)</p>. That's seven times N, she's a big girl 💜
  • IWORK — IWORK is INTEGER array, dimension (5*N)</p>. Five times N, equally iconic 👑
  • IFAIL — IFAIL is INTEGER array, dimension (N)</p>. 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. she only shows up when she's needed 🌸
  • INFO — INFO is INTEGER</p>, your exit-status bestie:
    • = 0: successful exit. we love to see it ✅
    • < 0: if INFO = -i, the i-th argument had an illegal value. oops 💀
    • > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL. it happens to the best of us

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 262 of file ssbevx.f.

🦋 subroutine zhbevx (character jobz, character range, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldq, * ) q, integer ldq, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. the complex double-precision sibling has entered the chat 💅✨

✨ Purpose:

!>
!> ZHBEVX computes selected eigenvalues and, optionally, eigenvectors
!> of a complex Hermitian 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.
!>

so basically: complex Hermitian band matrix A, you pick which eigenvalues you want — either by a range of values or a range of indices — and she'll hand over the eigenvalues and, if you asked nicely, the eigenvectors too. 🎀

💖 Parameters

  • JOBZ — JOBZ is CHARACTER*1</p>
    • = 'N': Compute eigenvalues only;
    • = 'V': Compute eigenvalues and eigenvectors. go big or go home
  • RANGE — RANGE is CHARACTER*1</p>
    • = '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. note the half-open interval, she is precise ✨
  • UPLO — UPLO is CHARACTER*1</p>
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored. pick a side bestie
  • N — N is INTEGER</p>, the order of the matrix A. N >= 0. 🦋
  • KD — KD is INTEGER</p>, the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. the band width, the vibe width
  • AB — AB is COMPLEX*16 array, dimension (LDAB, N)</p>. On entry, the upper or lower triangle of the Hermitian 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. she eats your input and leaves something else behind, that's the deal 💀

  • LDAB — LDAB is INTEGER</p>, the leading dimension of the array AB. LDAB >= KD + 1. 🔥
  • Q — Q is COMPLEX*16 array, dimension (LDQ, N)</p>
    • If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form.
    • If JOBZ = 'N', the array Q is not referenced. ghosted 👻
  • LDQ — LDQ is INTEGER</p>, the leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N). 🎀
  • VL — VL is DOUBLE PRECISION</p>. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. strict inequality, she means it
  • VU — VU is DOUBLE PRECISION</p>. 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</p>. 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'. the N = 0 case is a whole mood 🌸
  • IU — IU is INTEGER</p>. 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 DOUBLE PRECISION</p>, 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.

Again, for the people in the back: 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'). 🫶

See by Demmel and Kahan, LAPACK Working Note #3. (same missing-title situation as before, love that for us)

  • M — M is INTEGER</p>, 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 DOUBLE PRECISION array, dimension (N)</p>. The first M elements contain the selected eigenvalues in ascending order. ascending order supremacy ⭐
  • Z — Z is COMPLEX*16 array, dimension (LDZ, max(1,M))</p>
    • 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. this is the part where you allocate generously 😭
  • LDZ — LDZ is INTEGER</p>, the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). 🦋
  • WORK — WORK is COMPLEX*16 array, dimension (N)</p>. 💅
  • RWORK — RWORK is DOUBLE PRECISION array, dimension (7*N)</p>. Seven times N, no notes ✨
  • IWORK — IWORK is INTEGER array, dimension (5*N)</p>. Five times N, also no notes 🎀
  • IFAIL
  • IFAIL — okay so this one is an INTEGER array, dimension (N) 🫶. If JOBZ = 'V', then when INFO = 0, the first M elements of IFAIL are zero — clean, no notes. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge, so it's literally holding the problem children 😭. And if JOBZ = 'N', then IFAIL is not referenced at all — she's not in the group chat 💅.
  • INFO — an INTEGER, and it's the final vibe check:
    • = 0: successful exit, we love a happy ending ✨
    • < 0: if INFO = -i, then the i-th argument had an illegal value — red flag, bestie 🚩
    • > 0: if INFO = i, then i eigenvectors failed to converge, and their indices are stored in array IFAIL — the receipts are right there 🔥
📍 Definition at line 264 of file zhbevx.f.

👑 Author

The author squad, no cap:

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

👑 Author (Doxygen edition)

Generated automatically by Doxygen for LAPACK from the source code.

Okay that's a wrap, bestie 💖 this glow-up comes from LAPACK, Version 3.12.0 — hbevx(3) — and that's the whole page, periodt 🎀✨

📦 where this page comes from