SRC/dspevx.f(3)

💅 SRC/dspevx.f — from the Library Functions Manual, bestie 🎀

okay so this is a page from the Library Functions Manual and it's all about dspevx, a Fortran subroutine that is genuinely doing the most. let's get into it ✨

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🌸 SYNOPSIS

✨ Functions/Subroutines

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

DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. lowkey iconic of it.

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🦋 Function/Subroutine Documentation

subroutine dspevx (character jobz, character range, character uplo, integer
    n, double precision, dimension( * ) ap, 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)

DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices. (yes it says it twice, the header is just like that, we love consistency 💖)

💜 Purpose:

!>
!> DSPEVX computes selected eigenvalues and, optionally, eigenvectors
!> of a real symmetric matrix A in packed storage.  Eigenvalues/vectors
!> can be selected by specifying either a range of values or a range of
!> indices for the desired eigenvalues.
!>

so basically: real symmetric matrix A, packed storage, and you get to choose which eigenvalues you want — either by value range or by index range. main character energy, honestly 👑

🔥 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.
!>

• AP — AP is DOUBLE PRECISION 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, AP is overwritten by values generated during the
!>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
!>          and first superdiagonal of the tridiagonal matrix T overwrite
!>          the corresponding elements of A, and if UPLO = 'L', the
!>          diagonal and first subdiagonal of T overwrite the
!>          corresponding elements of A.
!>

🫶 heads up bestie: AP gets overwritten on exit. your original matrix data is gone-gone. no cap.

• VL — VL is 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 — VU is 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 — 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 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 AP 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.
!>

🍓 real talk: the accuracy sweet spot is ABSTOL = 2*DLAMCH('S'), not zero. and if INFO>0 comes back (some eigenvectors were like "nope, didn't converge"), the move is to set ABSTOL to 2*DLAMCH('S') and try again. also yes, the "See" line really does have a blank where the citation title should be — that's how it is in the source, we're not fixing it, that's canon 💀

• 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 DOUBLE PRECISION array, dimension (N)

!>          If INFO = 0, the selected eigenvalues in ascending order.
!>

• 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.
!>

🌈 important note, bestie: you (the user) have to make sure Z has at least max(1,M) columns. and if RANGE = 'V', M isn't known in advance so you gotta bring an upper bound. plan ahead, queen 👑

• LDZ — LDZ is INTEGER

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

• WORK — WORK is DOUBLE PRECISION 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:  if INFO = i, then i eigenvectors failed to converge.
!>                Their indices are stored in array IFAIL.
!>

⭐ vibe check on INFO: 0 = we ate 💅, negative = you passed a bad argument (the i-th one), positive = that many eigenvectors ghosted you and their indices are chilling in IFAIL.

👑 Author

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

Definition at line 231 of file dspevx.f.

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🧚 Author

Generated automatically by Doxygen for LAPACK from the source code. (not a human, just a little documentation fairy 🧚‍♀️)

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and that's the whole page, bestie — you're officially fluent in dspevx 🫶💖

— LAPACK, Version 3.12.0, SRC/dspevx.f(3) ✨ slay and goodbye ✨