complex16GEeigen(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💖 complex16GEeigen — LAPACK 💖

okay bestie, sit down, hydrate, because this is the complex16GEeigen(3) page and it's a whole group of LAPACK driver routines. Think of it as the group chat for complex16 eigenvalue driver functions for GE matrices. Ten routines, all in one family, all doing the eigenvalues/Schur/vector thing. Let's go 🦋

✨ NAME ✨

complex16GEeigen — complex16 🌸

🎀 SYNOPSIS 🎀

Functions

`zgees`

subroutine zgees (JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS, LDVS,
    WORK, LWORK, RWORK, BWORK, INFO)

ZGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 💅

`zgeesx`

subroutine zgeesx (JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS,
    LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, BWORK, INFO)

ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices ✨

`zgeev`

subroutine zgeev (JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK,
    LWORK, RWORK, INFO)

ZGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 🌸

`zgeevx`

subroutine zgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL,
    VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK,
    RWORK, INFO)

ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 🦋

`zgges`

subroutine zgges (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM,
    ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK,
    INFO)

ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 💜

`zgges3`

subroutine zgges3 (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
    SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
    BWORK, INFO)

ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm) 🔥

`zggesx`

subroutine zggesx (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B,
    LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV,
    WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)

ZGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 👑

`zggev`

subroutine zggev (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL,
    LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)

ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 🍓

`zggev3`

subroutine zggev3 (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL,
    LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)

ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm) ⭐

`zggevx`

subroutine zggevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
    ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM,
    BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO)

ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 💖

🌈 Detailed Description 🌈

This is the group of complex16 eigenvalue driver functions for GE matrices 💅

👑 Function Documentation 👑

subroutine `zgees`

subroutine zgees (character JOBVS, character SORT, external SELECT, integer
    N, complex*16, dimension( lda, * ) A, integer LDA, integer SDIM,
    complex*16, dimension( * ) W, complex*16, dimension( ldvs, * ) VS,
    integer LDVS, complex*16, dimension( * ) WORK, integer LWORK, double
    precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer
    INFO)

ZGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 🎀

Purpose

ZGEES 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). ✨

Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left. The leading columns of Z then form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.

A complex matrix is in Schur form if it is upper triangular. no cap 💅

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*16 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.
- The eigenvalue W(j) is selected if SELECT(W(j)) is true.
N
- N is INTEGER
- The order of the matrix A. N >= 0.
A
- A is COMPLEX*16 array, dimension (LDA,N)
- On entry, the N-by-N matrix A.
- On exit, A has been overwritten by its Schur form T. 💀 (yes bestie, it gets rewritten, that's the vibe)
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*16 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*16 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; if JOBVS = 'V', LDVS >= N.
WORK
- WORK is COMPLEX*16 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).
- For good performance, LWORK must generally be larger.
- If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
RWORK
- RWORK is DOUBLE PRECISION 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 matrix 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.

subroutine `zgeesx`

subroutine zgeesx (character JOBVS, character SORT, external SELECT,
    character SENSE, integer N, complex*16, dimension( lda, * ) A, integer
    LDA, integer SDIM, complex*16, dimension( * ) W, complex*16, dimension(
    ldvs, * ) VS, integer LDVS, double precision RCONDE, double precision
    RCONDV, complex*16, dimension( * ) WORK, integer LWORK, double
    precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer
    INFO)

ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 💜

Purpose

ZGEESX 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). ✨

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. 🦋

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). iykyk 📖

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

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*16 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'.
N
N — INTEGER. It's the order of the matrix A, bestie. And N >= 0, no negativity allowed ✨
A — COMPLEX*16 array, dimension (LDA, N). On entry, it's your N-by-N matrix A. On exit, A gets overwritten by its Schur form T 💅 so like… say goodbye, she's not coming back the way you left her.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N), periodt.
SDIM — INTEGER. If SORT = 'N', then SDIM = 0. If SORT = 'S', then SDIM = the number of eigenvalues for which SELECT is true 🎀
W — COMPLEX*16 array, dimension (N). W holds the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.
VS — COMPLEX*16 array, dimension (LDVS,N). If JOBVS = 'V', VS contains the unitary matrix Z of Schur vectors 🦋. If JOBVS = 'N', VS is not referenced.
LDVS — INTEGER. The leading dimension of the array VS. LDVS >= 1, and if JOBVS = 'V', then LDVS >= N.
RCONDE — DOUBLE PRECISION. 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 — DOUBLE PRECISION. 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 — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK 👑
LWORK — INTEGER. The dimension of the array WORK. LWORK >= max(1,2*N). Also, if SENSE = 'E' or 'V' or 'B', then 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 an 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 🙈
RWORK — DOUBLE PRECISION array, dimension (N).
BWORK — LOGICAL array, dimension (N). Not referenced if SORT = 'N'.
INFO — INTEGER. The vibe check report, and it goes like this:
- = 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.

🌈 `subroutine zgeev` 🌈

subroutine zgeev (character JOBVL, character JOBVR, integer N, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) W,
    complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16,
    dimension( ldvr, * ) VR, integer LDVR, complex*16, dimension( * ) WORK,
    integer LWORK, double precision, dimension( * ) RWORK, integer INFO)

ZGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. She's the main character fr 💅

Purpose

ZGEEV computes, for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. Like, both? Only one? Your call, bestie 🫶

The right eigenvector v(j) of A satisfies

A * v(j) = lambda(j) * v(j)

where lambda(j) is its eigenvalue.

The left eigenvector u(j) of A satisfies

u(j)**H * A = lambda(j) * u(j)**H

where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. That's the glow-up, no cap ✨

Parameters

JOBVL — CHARACTER*1.
- = 'N': left eigenvectors of A are not computed;
- = 'V': left eigenvectors of are computed.
JOBVR — CHARACTER*1.
- = 'N': right eigenvectors of A are not computed;
- = 'V': right eigenvectors of A are computed.
N — INTEGER. The order of the matrix A. N >= 0.
A — COMPLEX*16 array, dimension (LDA,N). On entry, the N-by-N matrix A. On exit, A has been overwritten 🙈
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
W — COMPLEX*16 array, dimension (N). W contains the computed eigenvalues 👑
VL — COMPLEX*16 array, dimension (LDVL,N). If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If JOBVL = 'N', VL is not referenced. And u(j) = VL(:,j), the j-th column of VL.
LDVL — INTEGER. The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', then LDVL >= N.
VR — COMPLEX*16 array, dimension (LDVR,N). If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If JOBVR = 'N', VR is not referenced. And v(j) = VR(:,j), the j-th column of VR.
LDVR — INTEGER. The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', then LDVR >= N.
WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK 💖
LWORK — INTEGER. The dimension of the array WORK. LWORK >= max(1,2*N). For good performance, LWORK must generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
RWORK — DOUBLE PRECISION array, dimension (2*N).
INFO — INTEGER.
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements i+1:N of W contain eigenvalues which have converged.

Author

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

🦋 `subroutine zgeevx` 🦋

subroutine zgeevx (character BALANC, character JOBVL, character JOBVR,
    character SENSE, integer N, complex*16, dimension( lda, * ) A, integer
    LDA, complex*16, dimension( * ) W, complex*16, dimension( ldvl, * ) VL,
    integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, integer
    ILO, integer IHI, double precision, dimension( * ) SCALE, double
    precision ABNRM, double precision, dimension( * ) RCONDE, double
    precision, dimension( * ) RCONDV, complex*16, dimension( * ) WORK,
    integer LWORK, double precision, dimension( * ) RWORK, integer INFO)

ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. It's giving zgeev's overachieving older sister 💅

Purpose

ZGEEVX computes, for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). Obsessed 😍

The right eigenvector v(j) of A satisfies

A * v(j) = lambda(j) * v(j)

where lambda(j) is its eigenvalue.

The left eigenvector u(j) of A satisfies

u(j)**H * A = lambda(j) * u(j)**H

where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. Same energy, every time ✨

Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation D * A * D**(-1), where D is a diagonal matrix, to make its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will 🎀 For further explanation of balancing, see section 4.10.2 of the LAPACK Users' Guide.

Parameters

BALANC — okay bestie, this one's deets are up next…

`BALANC`

okay so BALANC is CHARACTER*1 and it's basically the vibe check for your matrix 💅 — it tells the routine how to diagonally scale and/or permute the input so its eigenvalues behave better conditioned.

'N' — do not diagonally scale or permute. leave it alone, she's fine as she is.
'P' — perform permutations to make the matrix more nearly upper triangular. do not diagonally scale.
'S' — diagonally scale the matrix, i.e. replace A by D*A*D**(-1), where D is a diagonal matrix chosen to make the rows and columns of A more equal in norm. do not permute.
'B' — both diagonally scale and permute A. the whole glow-up ✨

heads up bestie: the computed reciprocal condition numbers are for the matrix after balancing and/or permuting. permuting does not change condition numbers (in exact arithmetic), but balancing does. so that matters, fr fr.

`JOBVL`

JOBVL is CHARACTER*1.

'N' — left eigenvectors of A are not computed;
'V' — left eigenvectors of A are computed.

if SENSE = 'E' or 'B', then JOBVL must = 'V'. non-negotiable, she said what she said.

`JOBVR`

JOBVR is CHARACTER*1.

'N' — right eigenvectors of A are not computed;
'V' — right eigenvectors of A are computed.

if SENSE = 'E' or 'B', then JOBVR must = 'V'. same energy, periodt.

`SENSE`

SENSE is CHARACTER*1 and it decides which reciprocal condition numbers get computed.

'N' — none are computed;
'E' — computed for eigenvalues only;
'V' — computed for right eigenvectors only;
'B' — computed for eigenvalues and right eigenvectors.

if SENSE = 'E' or 'B', then both left and right eigenvectors must also be computed (JOBVL = 'V' and JOBVR = 'V'). yeah, package deal 🎀

`N`

N is INTEGER. it's the order of the matrix A, and N >= 0.

`A`

A is COMPLEX*16 array, dimension (LDA,N).

on entry, it's the N-by-N matrix A.

on exit, A has been overwritten. if JOBVL = 'V' or JOBVR = 'V', then A contains the Schur form of the balanced version of the matrix A. so like… she's not who she was before, and that's okay 💖

`LDA`

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

`W`

W is COMPLEX*16 array, dimension (N) and it holds the computed eigenvalues. main character data fr 🌟

`VL`

VL is COMPLEX*16 array, dimension (LDVL,N).

if JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues.

if JOBVL = 'N', VL is not referenced at all — ghosted 👻

and u(j) = VL(:,j), the j-th column of VL.

`LDVL`

LDVL is INTEGER, the leading dimension of the array VL. LDVL >= 1; and if JOBVL = 'V', then LDVL >= N.

`VR`

VR is COMPLEX*16 array, dimension (LDVR,N).

if JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues.

if JOBVR = 'N', VR is not referenced. gone, reduced to atoms 💀

and v(j) = VR(:,j), the j-th column of VR.

`LDVR`

LDVR is INTEGER, the leading dimension of the array VR. LDVR >= 1; and if JOBVR = 'V', then LDVR >= N.

`ILO`

ILO is INTEGER.

`IHI`

IHI is INTEGER.

ILO and IHI are integer values determined when A was balanced. the balanced A(i,j) = 0 if I > J and J = 1,...,ILO-1 or I = IHI+1,...,N. those are the structural zeros, she's keeping receipts 📝

`SCALE`

SCALE is DOUBLE PRECISION array, dimension (N) and it's got all the tea on the permutations and scaling factors applied when balancing A.

if P(j) is the index of the row and column interchanged with row and column j, and D(j) is the scaling factor applied to row and column j, then:

SCALE(J) = P(J),    for J = 1,...,ILO-1
         = D(J),    for J = ILO,...,IHI
         = P(J)     for J = IHI+1,...,N.

and the order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. order matters, bestie, don't skip steps 🦋

`ABNRM`

ABNRM is DOUBLE PRECISION — the one-norm of the balanced matrix, which is the maximum of the sum of absolute values of elements of any column. basically the biggest column flex 💪

`RCONDE`

RCONDE is DOUBLE PRECISION array, dimension (N), and RCONDE(j) is the reciprocal condition number of the j-th eigenvalue.

`RCONDV`

RCONDV is DOUBLE PRECISION array, dimension (N), and RCONDV(j) is the reciprocal condition number of the j-th right eigenvector.

`WORK`

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)).

on exit, if INFO = 0, then WORK(1) returns the optimal LWORK. she's telling you what she needs, listen 😭

`LWORK`

LWORK is INTEGER — the dimension of the array WORK.

if SENSE = 'N' or 'E', then LWORK >= max(1,2*N). and if SENSE = 'V' or 'B', then LWORK >= N*N+2*N.

for good performance, LWORK must generally be larger. no skimping, she deserves the space 👑

if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. cute little loophole, iykyk.

`RWORK`

RWORK is DOUBLE PRECISION array, dimension (2*N).

`INFO`

INFO is INTEGER.

= 0: successful exit ✨
< 0: if INFO = -i, the i-th argument had an illegal value. one of your args fumbled 🙈
> 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements 1:ILO-1 and i+1:N of W contain eigenvalues which have converged.

Author

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

`zgges`

subroutine zgges (character JOBVSL, character JOBVSR, character SORT,
    external SELCTG, integer N, complex*16, dimension( lda, * ) A, integer
    LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer SDIM,
    complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA,
    complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16,
    dimension( ldvsr, * ) VSR, integer LDVSR, complex*16, dimension( * )
    WORK, integer LWORK, double precision, dimension( * ) RWORK, logical,
    dimension( * ) BWORK, integer INFO)

ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices. yeah. she does a lot. multi-talented 🎀

Purpose

ZGGES computes, for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized complex Schur form (S, T), and optionally left and/or right Schur vectors (VSL and VSR). this gives the generalized Schur factorization:

(A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR. yes bestie, conjugate-transpose, we love her 💜

optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T. the leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces). iconique behaviour ✨

if you only need the generalized eigenvalues, use the driver ZGGEV instead — it's faster. know when to call the right girlie, fr fr.

a generalized eigenvalue for a pair of matrices (A,B) is a scalar w, or a ratio alpha/beta = w, such that A - w*B is singular. it's usually represented as the pair (alpha,beta), because there's a reasonable interpretation for beta=0, and even for both being zero. no division by zero drama here, we planned ahead 💅

a pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers. that's the whole aesthetic.

Parameters

`JOBVSL`

JOBVSL is CHARACTER*1.

'N' — do not compute the left Schur vectors;
'V' — compute the left Schur vectors.

`JOBVSR`

JOBVSR is CHARACTER*1.

'N' — do not compute the right Schur vectors;
'V' — compute the right Schur vectors.

`SORT`

SORT is CHARACTER*1 and specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.

'N' — eigenvalues are not ordered;
'S' — eigenvalues are ordered (see SELCTG).

`SELCTG`

SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments. SELCTG must be declared EXTERNAL in the calling subroutine.

if SORT = 'N', SELCTG is not referenced.

if SORT = 'S', SELCTG is used to select eigenvalues to sort to the top left of the Schur form.

an eigenvalue ALPHA(j)/BETA(j) is selected if SELCTG(ALPHA(j),BETA(j)) is true.

okay so listen: a selected complex eigenvalue may no longer satisfy SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned). in this case INFO is set to N+2 (see INFO below). she's chaotic like that, not her fault 😭

`N`

N is INTEGER — the order of the matrices A, B, VSL, and VSR. N >= 0.

`A`

A is COMPLEX*16 array, dimension (LDA, N).

on entry, it's the first of the pair of matrices.

on exit, A has been overwritten by its generalized Schur form S.

`LDA`

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

`B`

B is COMPLEX*16 array, dimension (LDB, N).

on entry, it's the second of the pair of matrices.

on exit, B has been overwritten by its generalized Schur form T.

`LDB`

LDB is INTEGER — the leading dimension of B. LDB >= max(1,N).

`SDIM`

SDIM is INTEGER.

if SORT = 'N', then SDIM = 0.

if SORT = 'S', then SDIM = number of eigenvalues (after sorting) for which SELCTG is true.

`ALPHA`

ALPHA — is COMPLEX*16 array, dimension (N) — just a lil array of eigenvalues, no cap 💅✨
BETA — is COMPLEX*16 array, dimension (N). On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by ZGGES. The BETA(j) will be non-negative real. 🍓

Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). 💅 So don't be out here dividing by zero, bestie, it's giving risky vibes 😭

VSL — is COMPLEX*16 array, dimension (LDVSL,N). If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'. 🦋 left Schur vectors are lowkey iconic
LDVSL — is INTEGER. The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL = 'V', LDVSL >= N. 🌸
VSR — is COMPLEX*16 array, dimension (LDVSR,N). If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'. 💜 right Schur vectors, periodt
LDVSR — is INTEGER. The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. ✨
WORK — is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🎀
LWORK — is INTEGER. The dimension of the array WORK. LWORK >= max(1,2*N). For good performance, LWORK must generally be larger. 🫶 like, bigger is better, no cap If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 🔥
RWORK — is DOUBLE PRECISION array, dimension (8*N). 🌈
BWORK — is LOGICAL array, dimension (N). Not referenced if SORT = 'N'. 💀
INFO — is INTEGER
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value. 🚩
- =1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. 😭
- > N:
  - =N+1: other than QZ iteration failed in ZHGEQZ
  - =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling.
  - =N+3: reordering failed in ZTGSEN. 💀

👑 Author 👑

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

🦋 `subroutine zgges3` 🦋

   subroutine zgges3 (character JOBVSL, character JOBVSR, character SORT,
       external SELCTG, integer N, complex*16, dimension( lda, * ) A, integer
       LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer SDIM,
       complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA,
       complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16,
       dimension( ldvsr, * ) VSR, integer LDVSR, complex*16, dimension( * )
       WORK, integer LWORK, double precision, dimension( * ) RWORK, logical,
       dimension( * ) BWORK, integer INFO)

ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm) 🔥

🎯 Purpose: 🎯

ZGGES3 computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized complex Schur form (S, T), and optionally left and/or right Schur vectors (VSL and VSR). This gives the generalized Schur factorization

        (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR. ✨

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T. The leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces). 💅

(If only the generalized eigenvalues are needed, use the driver ZGGEV instead, which is faster.) 🏎️

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero. 🍓

A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers. 👑

📝 Parameters 📝

JOBVSL — is CHARACTER*1
- = 'N': do not compute the left Schur vectors;
- = 'V': compute the left Schur vectors. 🦋
JOBVSR — is CHARACTER*1
- = 'N': do not compute the right Schur vectors;
- = 'V': compute the right Schur vectors. 💜
SORT — is CHARACTER*1. Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
- = 'N': Eigenvalues are not ordered;
- = 'S': Eigenvalues are ordered (see SELCTG). 🌸
SELCTG — is a LOGICAL FUNCTION of two COMPLEX*16 arguments. SELCTG must be declared EXTERNAL in the calling subroutine. If SORT = 'N', SELCTG is not referenced. If SORT = 'S', SELCTG is used to select eigenvalues to sort to the top left of the Schur form. An eigenvalue ALPHA(j)/BETA(j) is selected if SELCTG(ALPHA(j),BETA(j)) is true. 🎀

Note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case INFO is set to N+2 (See INFO below). 😭

N — is INTEGER. The order of the matrices A, B, VSL, and VSR. N >= 0. ✨
A — is COMPLEX*16 array, dimension (LDA, N). On entry, the first of the pair of matrices. On exit, A has been overwritten by its generalized Schur form S. 🔥
LDA — is INTEGER. The leading dimension of A. LDA >= max(1,N). 💅
B — is COMPLEX*16 array, dimension (LDB, N). On entry, the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T. 🍓
LDB — is INTEGER. The leading dimension of B. LDB >= max(1,N). 🌈
SDIM — is INTEGER. If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which SELCTG is true. 👑
ALPHA — is COMPLEX*16 array, dimension (N). 🦋
BETA — is COMPLEX*16 array, dimension (N). On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by ZGGES3. The BETA(j) will be non-negative real. 💜

Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). 💅

VSL — is COMPLEX*16 array, dimension (LDVSL,N). If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'. 🌸
LDVSL — is INTEGER. The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL = 'V', LDVSL >= N. 🎀
VSR — is COMPLEX*16 array, dimension (LDVSR,N). If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'. ✨
LDVSR — is INTEGER. The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. 🔥
WORK — is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🫶
LWORK — is INTEGER. The dimension of the array WORK. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 💀
RWORK — is DOUBLE PRECISION array, dimension (8*N). 🌈
BWORK — is LOGICAL array, dimension (N). Not referenced if SORT = 'N'. 💜
INFO —

Ok so INFO is an INTEGER and honestly it's the ✨receipt✨ for everything that just went down — read it like your phone's notification stack:

= 0: successful exit 🎉 everything's cute, we ate, no notes
< 0: if INFO = -i, the i-th argument had an illegal value. yeah that one's on the caller bestie, argument number i showed up wrong 🙈
=1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. so it's giving "not a full slay, but the tail end of the eigenvalues still eats" 💅
> N:
- =N+1: other than QZ iteration failed in ZLAQZ0 💀
- =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling. roundoff literally said "not today" 😭 and scaling can be the culprit too, so watch out.
- =N+3: reordering failed in ZTGSEN 💔

👑 Author 👑

Giving credit where it's due, because we always do — this one was cooked up by:

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

🦋 subroutine zggesx 🦋

subroutine zggesx (character JOBVSL, character JOBVSR, character SORT,
    external SELCTG, character SENSE, integer N, complex*16, dimension( lda,
    * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB,
    integer SDIM, complex*16, dimension( * ) ALPHA, complex*16, dimension( *
    ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL,
    complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, double precision,
    dimension( 2 ) RCONDE, double precision, dimension( 2 ) RCONDV,
    complex*16, dimension( * ) WORK, integer LWORK, double precision,
    dimension( * ) RWORK, integer, dimension( * ) IWORK, integer LIWORK,
    logical, dimension( * ) BWORK, integer INFO)

so basically ZGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices ✨ iconic behaviour honestly.

✨ Purpose ✨

okay so ZGGESX computes, for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the complex Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization

     (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR. conjugate-transpose? obsessed, she's doing the most and we love her 🫶

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). lowkey the glow-up pipeline of matrix decompositions 🌈

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or for both being zero.

A pair of matrices (S,T) is in generalized complex Schur form if T is upper triangular with non-negative diagonal and S is upper triangular. periodt 💅

🎀 Parameters 🎀

`JOBVSL`

JOBVSL is CHARACTER*1

= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.

`JOBVSR`

JOBVSR is CHARACTER*1

= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.

`SORT`

SORT is CHARACTER*1

Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.

= 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELCTG).

`SELCTG`

SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments

SELCTG must be declared EXTERNAL in the calling subroutine.

If SORT = 'N', SELCTG is not referenced.

If SORT = 'S', SELCTG is used to select eigenvalues to sort to the top left of the Schur form.

Note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case INFO is set to N+3 see INFO below). yeah bestie, sorting can ✨change✨ the vibes of an ill-conditioned eigenvalue, that's not a bug that's just the drama 😭

`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 deflating subspaces only;
= 'B': Computed for both.

If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

`N`

N is INTEGER

The order of the matrices A, B, VSL, and VSR. N >= 0.

`A`

A is COMPLEX*16 array, dimension (LDA, N)

On entry, the first of the pair of matrices.

On exit, A has been overwritten by its generalized Schur form S. she gets a whole new personality, love that for her 💖

`LDA`

LDA is INTEGER

The leading dimension of A. LDA >= max(1,N).

`B`

B is COMPLEX*16 array, dimension (LDB, N)

On entry, the second of the pair of matrices.

On exit, B has been overwritten by its generalized Schur form T.

`LDB`

LDB is INTEGER

The leading dimension of B. LDB >= max(1,N).

`SDIM`

SDIM is INTEGER

If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which SELCTG is true.

`ALPHA`

ALPHA is COMPLEX*16 array, dimension (N)

`BETA`

BETA is COMPLEX*16 array, dimension (N)

On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T). BETA(j) will be non-negative real.

Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). so like... do NOT just divide them and act surprised, bestie 💀

`VSL`

VSL is COMPLEX*16 array, dimension (LDVSL,N)

If JOBVSL = 'V', VSL will contain the left Schur vectors.

Not referenced if JOBVSL = 'N'.

`LDVSL`

LDVSL is INTEGER

The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N.

`VSR`

VSR is COMPLEX*16 array, dimension (LDVSR,N)

If JOBVSR = 'V', VSR will contain the right Schur vectors.

Not referenced if JOBVSR = 'N'.

`LDVSR`

LDVSR is INTEGER

The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.

`RCONDE`

RCONDE is DOUBLE PRECISION array, dimension ( 2 )

If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues.

Not referenced if SENSE = 'N' or 'V'.

`RCONDV`

RCONDV is DOUBLE PRECISION array, dimension ( 2 )

If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition number for the selected deflating subspaces.

Not referenced if SENSE = 'N' or 'E'.

`WORK`

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. the receipts are literally in the first slot, iconic 📬

`LWORK`

LWORK is INTEGER

The dimension of the array WORK.

If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else LWORK >= MAX(1,2*N). 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.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. yes queen, -1 is the "just tell me the size, no commitment" setting 🙈

`RWORK`

RWORK is DOUBLE PRECISION array, dimension ( 8*N )

Real workspace.

`IWORK`

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

`LIWORK`

LIWORK is INTEGER

The dimension of the array IWORK.

If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise LIWORK >= N+2.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. same energy, different array 🎀

`BWORK`

BWORK is LOGICAL array, dimension (N)

Not referenced if SORT = 'N'.

`INFO`

INFO — INFO is INTEGER
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value.
- = 1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
- > N:
  - =N+1: other than QZ iteration failed in ZHGEQZ
  - =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling.
  - =N+3: reordering failed in ZTGSEN.

okay so bestie, if your INFO comes back > N, the QZ saga did NOT go how we wanted 💀 but don't panic — the numbers you got are still the numbers you got, fr fr.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌸 subroutine `zggev` 🌸

subroutine zggev (character JOBVL, character JOBVR, integer N, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B,
    integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( *
    ) BETA, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16,
    dimension( ldvr, * ) VR, integer LDVR, complex*16, dimension( * ) WORK,
    integer LWORK, double precision, dimension( * ) RWORK, integer INFO)

ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 💅

✨ Purpose ✨

ZGGEV computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero.

The right generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies

             A * v(j) = lambda(j) * B * v(j).

The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies

             u(j)**H * A = lambda(j) * u(j)**H * B

where u(j)**H is the conjugate-transpose of u(j).

🎀 Parameters 🎀

JOBVL — JOBVL is CHARACTER*1
- = 'N': do not compute the left generalized eigenvectors;
- = 'V': compute the left generalized eigenvectors.
JOBVR — JOBVR is CHARACTER*1
- = 'N': do not compute the right generalized eigenvectors;
- = 'V': compute the right generalized eigenvectors.
N — N is INTEGER. The order of the matrices A, B, VL, and VR. N >= 0.
A — A is COMPLEX*16 array, dimension (LDA, N). On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. 🫶 (yes bestie, it's gone, it gets clobbered, that's the vibe.)
LDA — LDA is INTEGER. The leading dimension of A. LDA >= max(1,N).
B — B is COMPLEX*16 array, dimension (LDB, N). On entry, the matrix B in the pair (A,B). On exit, B has been overwritten.
LDB — LDB is INTEGER. The leading dimension of B. LDB >= max(1,N).
ALPHA — ALPHA is COMPLEX*16 array, dimension (N)
BETA — BETA is COMPLEX*16 array, dimension (N). On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues.

Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).

VL — VL is COMPLEX*16 array, dimension (LDVL,N). If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'.
LDVL — LDVL is INTEGER. The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
VR — VR is COMPLEX*16 array, dimension (LDVR,N). If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'.
LDVR — LDVR is INTEGER. The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
WORK — WORK is COMPLEX*16 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). For good performance, LWORK must generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

RWORK — RWORK is DOUBLE PRECISION array, dimension (8*N)
INFO — INFO is INTEGER
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value.
- =1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
- > N:
  - =N+1: other then QZ iteration failed in ZHGEQZ,
  - =N+2: error return from ZTGEVC.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine `zggev3` 🦋

subroutine zggev3 (character JOBVL, character JOBVR, integer N, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B,
    integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( *
    ) BETA, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16,
    dimension( ldvr, * ) VR, integer LDVR, complex*16, dimension( * ) WORK,
    integer LWORK, double precision, dimension( * ) RWORK, integer INFO)

ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm) 🔥

✨ Purpose ✨

ZGGEV3 computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

The right generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies

             A * v(j) = lambda(j) * B * v(j).

The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies

             u(j)**H * A = lambda(j) * u(j)**H * B

where u(j)**H is the conjugate-transpose of u(j).

🎀 Parameters 🎀

JOBVL — JOBVL is CHARACTER*1
- = 'N': do not compute the left generalized eigenvectors;
- = 'V': compute the left generalized eigenvectors.
JOBVR — JOBVR is CHARACTER*1
- = 'N': do not compute the right generalized eigenvectors;
- = 'V': compute the right generalized eigenvectors.
N — N is INTEGER. The order of the matrices A, B, VL, and VR. N >= 0.
A — A is COMPLEX*16 array, dimension (LDA, N). On entry, the matrix A in the pair (A,B). On exit, A has been overwritten.
LDA — LDA is INTEGER. The leading dimension of A. LDA >= max(1,N).
B — B is COMPLEX*16 array, dimension (LDB, N). On entry, the matrix B in the pair (A,B). On exit, B has been overwritten.
LDB — LDB is INTEGER. The leading dimension of B. LDB >= max(1,N).
ALPHA — ALPHA is COMPLEX*16 array, dimension (N)
BETA — BETA is COMPLEX*16 array, dimension (N). On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues.

VL — VL is COMPLEX*16 array, dimension (LDVL,N). If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'.
LDVL — LDVL is INTEGER. The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
VR
VR — a COMPLEX*16 array, dimension (LDVR,N). If JOBVR = 'V', the right generalized eigenvectors v(j) get stacked one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector is scaled so that its biggest component has abs(real part) + abs(imag. part) = 1. And if JOBVR = 'N'? Not referenced, bestie, we literally do not look at it 💅
LDVR — INTEGER, the leading dimension of the matrix VR. The rules are LDVR >= 1, and if JOBVR = 'V' then LDVR >= N.
WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, then WORK(1) serves up the optimal LWORK like it's nothing ✨
LWORK — INTEGER, the dimension of the array WORK.

If LWORK = -1, a workspace query is assumed: the routine only calculates the optimal size of the WORK array, returns that value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. Sneaky lil thing but we respect it 🫶

RWORK — DOUBLE PRECISION array, dimension (8*N).
INFO — INTEGER. The receipts, in order:
- = 0: successful exit, we ate 💅
- < 0: if INFO = -i, the i-th argument had an illegal value
- = 1,...,N: the QZ iteration failed 😭. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
- > N: =N+1: other than QZ iteration failed in ZHGEQZ; =N+2: error return from ZTGEVC.

✍️ Author

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

🦋 subroutine `zggevx` 🦋

subroutine zggevx (character BALANC, character JOBVL, character JOBVR,
    character SENSE, integer N, complex*16, dimension( lda, * ) A, integer
    LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16,
    dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16,
    dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * )
    VR, integer LDVR, integer ILO, integer IHI, double precision, dimension(
    * ) LSCALE, double precision, dimension( * ) RSCALE, double precision
    ABNRM, double precision BBNRM, double precision, dimension( * ) RCONDE,
    double precision, dimension( * ) RCONDV, complex*16, dimension( * )
    WORK, integer LWORK, double precision, dimension( * ) RWORK, integer,
    dimension( * ) IWORK, logical, dimension( * ) BWORK, integer INFO)

ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 🌈

🎯 Purpose

ZGGEVX computes, for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues — and, if you want them, the left and/or right generalized eigenvectors. Iconic behavior honestly.

Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). So many little guys, and they all matter 🧚

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda, or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It's usually represented as the pair (alpha,beta), because there's a reasonable interpretation for beta=0, and even for both of them being zero. Yes, both zero. We don't judge 💀

The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j) .

The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

u(j)**H * A  = lambda(j) * u(j)**H * B.

where u(j)**H is the conjugate-transpose of u(j). Conjugate-transpose, say it with your chest 💜

🛠️ Parameters

BALANC — CHARACTER*1. Tells the routine which balance option to perform:
- = 'N': do not diagonally scale or permute;
- = 'P': permute only;
- = 'S': scale only;
- = 'B': both permute and scale.

Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. That's the whole plot twist 🔥

JOBVL — CHARACTER*1:
- = 'N': do not compute the left generalized eigenvectors;
- = 'V': compute the left generalized eigenvectors.
JOBVR — CHARACTER*1:
- = 'N': do not compute the right generalized eigenvectors;
- = 'V': compute the right generalized eigenvectors.
SENSE — CHARACTER*1. Determines which reciprocal condition numbers get computed:
- = 'N': none are computed;
- = 'E': computed for eigenvalues only;
- = 'V': computed for eigenvectors only;
- = 'B': computed for eigenvalues and eigenvectors.
N — INTEGER. The order of the matrices A, B, VL, and VR. N >= 0.
A — COMPLEX*16 array, dimension (LDA, N). On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains the first part of the complex Schur form of the 'balanced' versions of the input A and B. She got a whole makeover 💅
LDA — INTEGER. The leading dimension of A. LDA >= max(1,N).
B — COMPLEX*16 array, dimension (LDB, N). On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains the second part of the complex Schur form of the 'balanced' versions of the input A and B. Matching set, obviously 🎀
LDB — INTEGER. The leading dimension of B. LDB >= max(1,N).
ALPHA — COMPLEX*16 array, dimension (N).
BETA — COMPLEX*16 array, dimension (N). On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues.

Note: the quotient ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio ALPHA/BETA. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). Do NOT just divide them, bestie, I'm begging 😭

VL — COMPLEX*16 array, dimension (LDVL,N). If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'.
LDVL — INTEGER. The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
VR — COMPLEX*16 array, dimension (LDVR,N). If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'.
LDVR — INTEGER. The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
ILO — INTEGER.
IHI — INTEGER. ILO and IHI are integer values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
LSCALE — DOUBLE PRECISION array, dimension (N). Details of the permutations and scaling factors applied to the left side of A and B. If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then

LSCALE(j) = PL(j)  for j = 1,...,ILO-1
          = DL(j)  for j = ILO,...,IHI
          = PL(j)  for j = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

RSCALE — DOUBLE PRECISION array, dimension (N). Details of the permutations and scaling factors applied to the right side of A and B. If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then

RSCALE(j) = PR(j)  for j = 1,...,ILO-1
          = DR(j)  for j = ILO,...,IHI
          = PR(j)  for j = IHI+1,...,N

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

ABNRM — DOUBLE PRECISION. The one-norm of the balanced matrix A.
BBNRM — DOUBLE PRECISION. The one-norm of the balanced matrix B.
RCONDE — DOUBLE PRECISION array, dimension (N). If SENSE = 'E' or 'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. If SENSE = 'N' or 'V', RCONDE is not referenced.
RCONDV — DOUBLE PRECISION array, dimension (N). If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway. If SENSE = 'N' or 'E', RCONDV is not referenced.
WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🫶
LWORK —

`LWORK`

okay bestie, LWORK is an INTEGER, and it's literally the dimension of the WORK array. the rule is LWORK >= max(1,2*N). no cap.

if SENSE = 'E', then you need LWORK >= max(1,4*N).

if SENSE = 'V' or SENSE = 'B', then it's giving LWORK >= max(1,2*N*N+2*N). yes, that's the whole vibe.

if LWORK = -1, then a workspace query is assumed. the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. iconic behavior, honestly.

`RWORK`

RWORK is a DOUBLE PRECISION array, dimension (lrwork). lrwork must be at least max(1,6*N) if BALANC = 'S' or BALANC = 'B', and at least max(1,2*N) otherwise. it's the real workspace, periodt.

`IWORK`

IWORK is an INTEGER array, dimension (N+2). if SENSE = 'E', IWORK is not referenced. so if that's your setting, she's just there for moral support. 💅

`BWORK`

BWORK is a LOGICAL array, dimension (N). if SENSE = 'N', BWORK is not referenced. okay? okay.

`INFO`

INFO is an INTEGER, and it's the final vibe check:

= 0: successful exit. we love a happy ending. ✨
< 0: if INFO = -i, the i-th argument had an illegal value. not the move, bestie.
= 1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
> N:
- =N+1: other than QZ iteration failed in ZHGEQZ.
- =N+2: error return from ZTGEVC.

✨ Author ✨

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

💖 Further Details 💖

Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, second, applying diagonal similarity transformation to the rows and columns to make the rows and columns as close in norm as possible. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide.

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is

chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

An approximate error bound for the angle between the i-th computed eigenvector VL(i) or VR(i) is given by

EPS * norm(ABNRM, BBNRM) / DIF(i).

For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole page, bestie. this was LAPACK version 3.11.0, dated Sun Nov 27 2022, and it's the complex16GEeigen(3) manual page. go off, you understood the assignment. 💖✨

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

complex16GEeigen(3)

💖 complex16GEeigen — LAPACK 💖

✨ NAME ✨

🎀 SYNOPSIS 🎀

Functions

zgees

zgeesx

zgeev

zgeevx

zgges

zgges3

zggesx

zggev

zggev3

zggevx

🌈 Detailed Description 🌈

👑 Function Documentation 👑

subroutine zgees

Purpose

Parameters

Author

subroutine zgeesx

Purpose

Parameters

Author

🌈 subroutine zgeev 🌈

Purpose

Parameters

Author

🦋 subroutine zgeevx 🦋

Purpose

Parameters

BALANC

JOBVL

JOBVR

SENSE

N

A

LDA

W

VL

LDVL

VR

LDVR

ILO

IHI

SCALE

ABNRM

RCONDE

RCONDV

WORK

LWORK

RWORK

INFO

Author

zgges

Purpose

Parameters

JOBVSL

JOBVSR

SORT

SELCTG

N

A

LDA

B

LDB

SDIM

ALPHA

👑 Author 👑

🦋 subroutine zgges3 🦋

🎯 Purpose: 🎯

📝 Parameters 📝

👑 Author 👑

🦋 subroutine zggesx 🦋

✨ Purpose ✨

🎀 Parameters 🎀

JOBVSL

JOBVSR

SORT

`zgees`

`zgeesx`

`zgeev`

`zgeevx`

`zgges`

`zgges3`

`zggesx`

`zggev`

`zggev3`

`zggevx`

subroutine `zgees`

subroutine `zgeesx`

🌈 `subroutine zgeev` 🌈

🦋 `subroutine zgeevx` 🦋

`BALANC`

`JOBVL`

`JOBVR`

`SENSE`

`N`

`A`

`LDA`

`W`

`VL`

`LDVL`

`VR`

`LDVR`

`ILO`

`IHI`

`SCALE`

`ABNRM`

`RCONDE`

`RCONDV`

`WORK`

`LWORK`

`RWORK`

`INFO`

`zgges`

`JOBVSL`

`JOBVSR`

`SORT`

`SELCTG`

`N`

`A`

`LDA`

`B`

`LDB`

`SDIM`

`ALPHA`

🦋 `subroutine zgges3` 🦋

`JOBVSL`

`JOBVSR`

`SORT`

`SELCTG`

`SENSE`

`N`

`A`

`LDA`

`B`

`LDB`

`SDIM`

`ALPHA`

`BETA`

`VSL`

`LDVSL`

`VSR`

`LDVSR`

`RCONDE`

`RCONDV`

`WORK`

`LWORK`

`RWORK`

`IWORK`

`LIWORK`

`BWORK`

`INFO`

🌸 subroutine `zggev` 🌸

🦋 subroutine `zggev3` 🦋

🦋 subroutine `zggevx` 🦋

`LWORK`

`RWORK`