CGEESX(3)

💅 CGEESX(3) — Library Functions Manual 💅

okay bestie, welcome to the cgeesx.f manual page. it’s giving LAPACK main character energy, and we’re gonna go through every single flag like it’s a group chat. no cap. ✨

🌸 NAME 🌸

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

that’s the file, that’s the name, and yes it’s literally a whole path. slay. 💅

💖 SYNOPSIS 💖

✨ Functions/Subroutines ✨

   subroutine CGEESX (jobvs, sort, select, sense, n, a, lda, sdim, w, vs,
           ldvs, rconde, rcondv, work, lwork, rwork, bwork, info)

CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 🦋

🦋 Function/Subroutine Documentation 🦋

   subroutine CGEESX (character jobvs, character sort, external select,
       character sense, integer n, complex, dimension( lda, * ) a, integer lda,
       integer sdim, complex, dimension( * ) w, complex, dimension( ldvs, * )
       vs, integer ldvs, real rconde, real rcondv, complex, dimension( * )
       work, integer lwork, real, dimension( * ) rwork, logical, dimension( * )
       bwork, integer info)

CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 🌈

🎀 Purpose 🎀

okay so CGEESX computes, for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. this gives the Schur factorization A = Z*T*(Z**H). iconic. 👑

optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (RCONDV). the leading columns of Z form an orthonormal basis for this invariant subspace. the way that all fits together is lowkey beautiful. 🫶

for further explanation of the reciprocal condition numbers RCONDE and RCONDV, see Section 4.10 of the LAPACK Users’ Guide (where these quantities are called s and sep respectively).

a complex matrix is in Schur form if it is upper triangular. periodt. 💅

🔥 Parameters 🔥

🔹 JOBVS

• JOBVS is CHARACTER*1
• = 'N' — Schur vectors are not computed;
• = 'V' — Schur vectors are computed.

🔹 SORT

• SORT is CHARACTER*1
• Specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
• = 'N' — Eigenvalues are not ordered;
• = 'S' — Eigenvalues are ordered (see SELECT).

🔹 SELECT

• SELECT is a LOGICAL FUNCTION of one COMPLEX argument
• SELECT must be declared EXTERNAL in the calling subroutine.
• If SORT = 'S', SELECT is used to select eigenvalues to order to the top left of the Schur form.
• If SORT = 'N', SELECT is not referenced.
• An eigenvalue W(j) is selected if SELECT(W(j)) is true.

🔹 SENSE

• SENSE is CHARACTER*1
• Determines which reciprocal condition numbers are computed.
• = 'N' — None are computed;
• = 'E' — Computed for average of selected eigenvalues only;
• = 'V' — Computed for selected right invariant subspace only;
• = 'B' — Computed for both.
• If SENSE = 'E', 'V' or 'B', SORT must equal 'S'. bestie, that’s not optional. 💀

🔹 N

• N is INTEGER
• The order of the matrix A. N >= 0.

🔹 A

• A is COMPLEX array, dimension (LDA, N)
• On entry, the N-by-N matrix A.
• On exit, A is overwritten by its Schur form T.

🔹 LDA

• LDA is INTEGER
• The leading dimension of the array A. LDA >= max(1,N).

🔹 SDIM

• SDIM is INTEGER
• If SORT = 'N', SDIM = 0.
• If SORT = 'S', SDIM = number of eigenvalues for which SELECT is true.

🔹 W

• W is COMPLEX array, dimension (N)
• W contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.

🔹 VS

• VS is COMPLEX array, dimension (LDVS,N)
• If JOBVS = 'V', VS contains the unitary matrix Z of Schur vectors.
• If JOBVS = 'N', VS is not referenced.

🔹 LDVS

• LDVS is INTEGER
• The leading dimension of the array VS. LDVS >= 1, and if JOBVS = 'V', LDVS >= N.

🔹 RCONDE

• RCONDE is REAL
• If SENSE = 'E' or 'B', RCONDE contains the reciprocal condition number for the average of the selected eigenvalues.
• Not referenced if SENSE = 'N' or 'V'.

🔹 RCONDV

• RCONDV is REAL
• If SENSE = 'V' or 'B', RCONDV contains the reciprocal condition number for the selected right invariant subspace.
• Not referenced if SENSE = 'N' or 'E'.

🔹 WORK

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

🔹 LWORK

• LWORK is INTEGER
• The dimension of the array WORK. LWORK >= max(1,2*N).
• Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM), where SDIM is the number of selected eigenvalues computed by this routine.
• Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < max(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may not be large enough.
• For good performance, LWORK must generally be larger.
• If LWORK = -1, then a workspace query is assumed; the routine only calculates upper bound on the optimal size of the array WORK, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

bestie, LWORK = -1 is the workspace-query vibe, not a regular size. don’t panic, just read the first entry of WORK. 🫶

🔹 RWORK

• RWORK is REAL array, dimension (N)

🔹 BWORK

• BWORK is LOGICAL array, dimension (N)
• Not referenced if SORT = 'N'.

🔹 INFO

• INFO is INTEGER
• = 0 — successful exit
• < 0 — if INFO = -i, the i-th argument had an illegal value.
• > 0 — if INFO = i, and i is
  • <= N — the QR algorithm failed to compute all the eigenvalues; elements 1:ILO-1 and i+1:N of W contain those eigenvalues which have converged; if JOBVS = 'V', VS contains the transformation which reduces A to its partially converged Schur form.
  • = N+1 — the eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned);
  • = N+2 — after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy SELECT=.TRUE. This could also be caused by underflow due to scaling.

👑 Author 👑

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

Definition at line 236 of file cgeesx.f.

🌈 Author 🌈

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