CHEEVR(3)

💅 NAME 💅

okay so bestie, this is the cheevr.f(3) page from the Library Functions Manual ✨. The header shows the truncated path /home/abui...RC/cheevr.f(3) (iykyk), and the full source file is:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cheevr.f

---

🦋 SYNOPSIS 🦋

🎀 Functions/Subroutines 🎀

       subroutine CHEEVR (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
           m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork,
           info)

CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices 💖

---

🧚 Function/Subroutine Documentation 🧚

   subroutine CHEEVR (character jobz, character range, character uplo, integer
       n, complex, dimension( lda, * ) a, integer lda, real vl, real vu,
       integer il, integer iu, real abstol, integer m, real, dimension( * ) w,
       complex, dimension( ldz, * ) z, integer ldz, integer, dimension( * )
       isuppz, complex, dimension( * ) work, integer lwork, real, dimension( *
       ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork,
       integer info)

CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

✨ Purpose ✨

okay so CHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. It's giving “choose your own eigenvalue adventure” 💅

CHEEVR first reduces the matrix A to tridiagonal form T with a call to CHETRD. Then, whenever possible, CHEEVR calls CSTEMR to compute the eigenspectrum using Relatively Robust Representations. CSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows.

For each unreduced block (submatrix) of T:

1. (a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general.
2. (b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d).
3. (c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.
4. (d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see CSTEMR's documentation and:

• Inderjit S. Dhillon and Beresford N. Parlett: Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
• Inderjit Dhillon and Beresford Parlett: SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
• Inderjit Dhillon: , Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1 : 💀 CHEEVR calls CSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and when partial spectrum requests are made.

⚠️ Normal execution of CSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

😭 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.

For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and CSTEIN are called.

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.

A

A is COMPLEX array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 💀

LDA

LDA is INTEGER The leading dimension of the array A. LDA >= 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 A to tridiagonal form.

See by Demmel and Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, , LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.

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 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

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

ISUPPZ

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of CSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the unitary transformations applied by CUNMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1.

WORK

WORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

• LWORK — INTEGER, bestie, and it's the length of the array WORK. Rules: LWORK >= max(1,2*N). And like, if you want it to actually slay performance-wise, LWORK >= (NB+1)*N, where NB is the max of the blocksize for CHETRD and for CUNMTR as returned by ILAENV. ✨

And okay, the special case: if LWORK = -1, then a workspace query is assumed — the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. So like, it's just peeking, not doing the whole thing. 💅

• RWORK — RWORK is REAL array, dimension (MAX(1,LRWORK)). On exit, if INFO = 0, RWORK(1) returns the optimal (and minimal) LRWORK. It literally tells you what it needed. Considerate queen. 👑
• LRWORK — INTEGER. This one's the length of the array RWORK, and LRWORK >= max(1,24*N) — yeah, 24*N, not a typo, don't come for me. 🫶

If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. Same deal as LWORK, just a different diva. 💀

• IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK. See the pattern? They all do this. It's giving "we rehearsed this". 🎀
• LIWORK — INTEGER, the dimension of the array IWORK. LIWORK >= max(1,10*N). 🌸

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. Third time's the charm, bestie. 😭

• INFO — INTEGER, the vibe report. Here's the whole scale:
  • = 0: successful exit 🎉 (we ate, no crumbs)
  • < 0: if INFO = -i, the i-th argument had an illegal value 🙈 (something you passed in was not it)
  • > 0: Internal error 💀 (not your fault, but also not great)

---

👑 Author

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

🌟 Contributors:

• Inderjit Dhillon, IBM Almaden, USA
• Osni Marques, LBNL/NERSC, USA
• Ken Stanley, Computer Science Division, University of California at Berkeley, USA
• Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Literally a whole group chat of legends built this. Respect the roster. 🫡💜

📍 Definition at line 354 of file cheevr.f.

🖨️ Author

Generated automatically by Doxygen for LAPACK from the source code.

---

And that's a wrap on CHEEVR(3), bestie!! 🦋 This whole glow-up is LAPACK Version 3.12.0, straight out of /home/abui...RC/cheevr.f(3) — go forth, pass LWORK = -1, query those workspace sizes like the main character you are, and slay those eigenvalues. Periodt. 💖✨🎀