hgeqz(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

💖 hgeqz(3) — Library Functions Manual 💖

okay bestie, welcome to the hgeqz(3) page — this is the generalized Hessenberg eig situation, and yes it's giving big linear-algebra main character energy. Grab a strawberry 🍓 and let's get into it.

✨ NAME ✨

hgeqz — hgeqz: generalized Hessenberg eig

🌟 SYNOPSIS 🌟

Functions

subroutine chgeqz (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt,
    alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)
    CHGEQZ
subroutine dhgeqz (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt,
    alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)
    DHGEQZ
subroutine shgeqz (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt,
    alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)
    SHGEQZ
subroutine zhgeqz (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt,
    alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)
    ZHGEQZ

🎀 Detailed Description 🎀

this is the big-picture section, and then Function Documentation does the actual signature + parameter tea 💅

🛠️ Function Documentation 🛠️

💅 `chgeqz` 💅

subroutine chgeqz (character job, character compq, character compz, integer
    n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer
    ldh, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( *
    ) alpha, complex, dimension( * ) beta, complex, dimension( ldq, * ) q,
    integer ldq, complex, dimension( ldz, * ) z, integer ldz, complex,
    dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer
    info)
    CHGEQZ

💖 Purpose 💖

okay so CHGEQZ computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the single-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a complex matrix pair (A,B):

A = Q1*H*Z1**H,  B = Q1*T*Z1**H,

as computed by CGGHRD.

If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form:

H = Q*S*Z**H,  T = Q*P*Z**H,

where Q and Z are unitary matrices and S and P are upper triangular.

Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.

To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of complex values (alpha,beta). If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)

A*x = lambda*B*x

and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP

mu*A*y = B*y.

The values of alpha and beta for the i-th eigenvalue can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i).

Ref: C.B. Moler & G.W. Stewart, , SIAM J. Numer. Anal., 10(1973), pp. 241--256.

🧾 Parameters 🧾

JOB — JOB is CHARACTER*1
- = 'E': compute eigenvalues only;
- = 'S': Computer eigenvalues and the Schur form. yes bestie, the page literally says Computer, not Compute — I'm not fixing it, I'm preserving the lore 💀
COMPQ — COMPQ is CHARACTER*1
- = 'N': Left Schur vectors (Q) are not computed;
- = 'I': Q is initialized to the unit matrix and the matrix Q of left Schur vectors of (H,T) is returned;
- = 'V': Q must contain a unitary matrix Q1 on entry and the product Q1*Q is returned.
COMPZ — COMPZ is CHARACTER*1
- = 'N': Right Schur vectors (Z) are not computed;
- = 'I': Q is initialized to the unit matrix and the matrix Z of right Schur vectors of (H,T) is returned; yes, the page says Q here under COMPZ, and I am keeping that exactly as written 😭
- = 'V': Z must contain a unitary matrix Z1 on entry and the product Z1*Z is returned.
N — N is INTEGER
- The order of the matrices H, T, Q, and Z. N >= 0.
ILO — ILO is INTEGER
IHI — IHI is INTEGER
- ILO and IHI mark the rows and columns of H which are in Hessenberg form. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H — H is COMPLEX array, dimension (LDH, N)
- On entry, the N-by-N upper Hessenberg matrix H.
- On exit, if JOB = 'S', H contains the upper triangular matrix S from the generalized Schur factorization.
- If JOB = 'E', the diagonal of H matches that of S, but the rest of H is unspecified.
LDH — LDH is INTEGER
- The leading dimension of the array H. LDH >= max( 1, N ).
T — T is COMPLEX array, dimension (LDT, N)
- On entry, the N-by-N upper triangular matrix T.
- On exit, if JOB = 'S', T contains the upper triangular matrix P from the generalized Schur factorization.
- If JOB = 'E', the diagonal of T matches that of P, but the rest of T is unspecified.
LDT — LDT is INTEGER
- The leading dimension of the array T. LDT >= max( 1, N ).
ALPHA — ALPHA is COMPLEX array, dimension (N)
- The complex scalars alpha that define the eigenvalues of GNEP. ALPHA(i) = S(i,i) in the generalized Schur factorization.
BETA — BETA is COMPLEX array, dimension (N)
- The real non-negative scalars beta that define the eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized Schur factorization.
- yes, the type is COMPLEX and the blurb calls them real non-negative scalars — that's literally what the page says, bestie 🫶
- Together, the quantities alpha = ALPHA(j) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha.
- Since either lambda or mu may overflow, they should not, in general, be computed.
Q — Q is COMPLEX array, dimension (LDQ, N)
- On entry, if COMPQ = 'V', the unitary matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form.
- On exit, if COMPQ = 'I', the unitary matrix of left Schur vectors of (H,T), and if COMPQ = 'V', the unitary matrix of left Schur vectors of (A,B).
- Not referenced if COMPQ = 'N'.
LDQ — LDQ is INTEGER
- The leading dimension of the array Q. LDQ >= 1.
- If COMPQ='V' or 'I', then LDQ >= N.
Z — Z is COMPLEX array, dimension (LDZ, N)
- On entry, if COMPZ = 'V', the unitary matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form.
- On exit, if COMPZ = 'I', the unitary matrix of right Schur vectors of (H,T), and if COMPZ = 'V', the unitary matrix of right Schur vectors of (A,B).
- Not referenced if COMPZ = 'N'.
LDZ — LDZ is INTEGER
- The leading dimension of the array Z. LDZ >= 1.
- If COMPZ='V' or 'I', then LDZ >= N.
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,N).
- 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 REAL array, dimension (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 did not converge. (H,T) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct.
- = N+1,...,2*N: the shift calculation failed. (H,T) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO-N+1,...,N should be correct.

👑 Author 👑

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

🔥 Further Details 🔥

We assume that complex ABS works as long as its value is less than overflow.

📍 Definition at line 281 of file chgeqz.f.

🔥 `dhgeqz` 🔥

subroutine dhgeqz (character job, character compq, character compz, integer
    n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h,
    integer ldh, double precision, dimension( ldt, * ) t, integer ldt,
    double precision, dimension( * ) alphar, double precision, dimension( *
    ) alphai, double precision, dimension( * ) beta, double precision,
    dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, *
    ) z, integer ldz, double precision, dimension( * ) work, integer lwork,
    integer info)
    DHGEQZ

💖 Purpose 💖

okay so DHGEQZ is out here computing the eigenvalues of a real matrix pair (H,T) — where H is an upper Hessenberg matrix and T is upper triangular — using the double-shift QZ method. 💅

and like, where do these pairs even come from? bestie they're produced by the reduction to generalized upper Hessenberg form of a real matrix pair (A,B):

   A = Q1*H*Z1**T,  B = Q1*T*Z1**T,

as computed by DGGHRD. that's the pipeline, iykyk. ✨

so if JOB='S', the Hessenberg-triangular pair (H,T) also gets reduced to generalized Schur form:

   H = Q*S*Z**T,  T = Q*P*Z**T,

where Q and Z are orthogonal matrices, P is an upper triangular matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal blocks. 🎀

the 1-by-1 blocks correspond to real eigenvalues of the matrix pair (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of eigenvalues. periodt. 🦋

additionally (yes there's more, she's thorough), the 2-by-2 upper triangular diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced to positive diagonal form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0, P(j,j) > 0, and P(j+1,j+1) > 0. 🌸

optionally, the orthogonal matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the orthogonal matrix Z may be postmultiplied into an input matrix Z1. and if Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced the matrix pair (A,B) to generalized upper Hessenberg form, then the output matrices Q1*Q and Z1*Z are the orthogonal factors from the generalized Schur factorization of (A,B):

   A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.

iconic behavior honestly. 👑

to avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of values (alpha,beta), where alpha is complex and beta real. so if beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) A*x = lambda*B*x — and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP mu*A*y = B*y. real eigenvalues can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). 🫶

Ref: C.B. Moler & G.W. Stewart, , SIAM J. Numer. Anal., 10(1973), pp. 241--256. (yes the double comma is in the original, no i'm not fixing it, it's part of the lore 🕯️)

✨ Parameters ✨

JOB — JOB is CHARACTER*1
- = 'E': Compute eigenvalues only;
- = 'S': Compute eigenvalues and the Schur form.
COMPQ — COMPQ is CHARACTER*1
- = 'N': Left Schur vectors (Q) are not computed;
- = 'I': Q is initialized to the unit matrix and the matrix Q of left Schur vectors of (H,T) is returned;
- = 'V': Q must contain an orthogonal matrix Q1 on entry and the product Q1*Q is returned.
COMPZ — COMPZ is CHARACTER*1
- = 'N': Right Schur vectors (Z) are not computed;
- = 'I': Z is initialized to the unit matrix and the matrix Z of right Schur vectors of (H,T) is returned;
- = 'V': Z must contain an orthogonal matrix Z1 on entry and the product Z1*Z is returned.
N — N is INTEGER. the order of the matrices H, T, Q, and Z. N >= 0. 💖
ILO — ILO is INTEGER
IHI — IHI is INTEGER. ILO and IHI mark the rows and columns of H which are in Hessenberg form. it is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. if N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H — H is DOUBLE PRECISION array, dimension (LDH, N). on entry, the N-by-N upper Hessenberg matrix H. on exit, if JOB = 'S', H contains the upper quasi-triangular matrix S from the generalized Schur factorization. if JOB = 'E', the diagonal blocks of H match those of S, but the rest of H is unspecified — don't go poking at it, bestie. 🙈
LDH — LDH is INTEGER. the leading dimension of the array H. LDH >= max( 1, N ).
T — T is DOUBLE PRECISION array, dimension (LDT, N). on entry, the N-by-N upper triangular matrix T. on exit, if JOB = 'S', T contains the upper triangular matrix P from the generalized Schur factorization; 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced to positive diagonal form, i.e., if H(j+1,j) is non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and T(j+1,j+1) > 0. if JOB = 'E', the diagonal blocks of T match those of P, but the rest of T is unspecified. ⭐
LDT — LDT is INTEGER. the leading dimension of the array T. LDT >= max( 1, N ).
ALPHAR — ALPHAR is DOUBLE PRECISION array, dimension (N). the real parts of each scalar alpha defining an eigenvalue of GNEP.
ALPHAI — ALPHAI is DOUBLE PRECISION array, dimension (N). the imaginary parts of each scalar alpha defining an eigenvalue of GNEP. if ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j). cute lil pair, honestly 🦋
BETA — BETA is DOUBLE PRECISION array, dimension (N). the scalars beta that define the eigenvalues of GNEP. together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. since either lambda or mu may overflow, they should not, in general, be computed. 🫶
Q — Q is DOUBLE PRECISION array, dimension (LDQ, N). on entry, if COMPQ = 'V', the orthogonal matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form. on exit, if COMPQ = 'I', the orthogonal matrix of left Schur vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix of left Schur vectors of (A,B). not referenced if COMPQ = 'N'.
LDQ — LDQ is INTEGER. the leading dimension of the array Q. LDQ >= 1. if COMPQ='V' or 'I', then LDQ >= N.
Z — Z is DOUBLE PRECISION array, dimension (LDZ, N). on entry, if COMPZ = 'V', the orthogonal matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form. on exit, if COMPZ = 'I', the orthogonal matrix of right Schur vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix of right Schur vectors of (A,B). not referenced if COMPZ = 'N'.
LDZ — LDZ is INTEGER. the leading dimension of the array Z. LDZ >= 1. if COMPZ='V' or 'I', then LDZ >= N.
WORK — WORK is DOUBLE PRECISION 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,N).
- 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. (it's giving "just asking, not committing" energy 💅)
INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- = 1,...,N: the QZ iteration did not converge. (H,T) is not in Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO+1,...,N should be correct.
- = N+1,...,2*N: the shift calculation failed. (H,T) is not in Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO-N+1,...,N should be correct.

read the INFO values carefully bestie — a nonzero return doesn't mean everything's trash, it means some of it is correct and some of it isn't, and the exact range matters. 😭

🌈 Author 🌈

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎀 Further Details 🎀

iteration counters, bestie:

JITER — counts iterations.
IITER — counts iterations run since ILAST was last changed. this is therefore reset only when a 1-by-1 or 2-by-2 block deflates off the bottom. 🍓

Definition at line 301 of file dhgeqz.f.

🦋 SHGEQZ 🦋

subroutine shgeqz (character job, character compq, character compz, integer
    n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh,
    real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) alphar,
    real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension(
    ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, real,
    dimension( * ) work, integer lwork, integer info)

✨ Purpose ✨

SHGEQZ computes the eigenvalues of a real matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the double-shift QZ method. periodt 💅

Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a real matrix pair (A,B):

A = Q1*H*Z1**T,  B = Q1*T*Z1**T,

as computed by SGGHRD.

If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form,

H = Q*S*Z**T,  T = Q*P*Z**T,

where Q and Z are orthogonal matrices, P is an upper triangular matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal blocks.

The 1-by-1 blocks correspond to real eigenvalues of the matrix pair (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of eigenvalues. the way that math lines up is honestly iconic ✨

Additionally, the 2-by-2 upper triangular diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced to positive diagonal form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0, P(j,j) > 0, and P(j+1,j+1) > 0.

Optionally, the orthogonal matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the orthogonal matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced the matrix pair (A,B) to generalized upper Hessenberg form, then the output matrices Q1*Q and Z1*Z are the orthogonal factors from the generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.

To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of values (alpha,beta), where alpha is complex and beta real. If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)

A*x = lambda*B*x

and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP

mu*A*y = B*y.

Real eigenvalues can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). iykyk 🦋

Ref: C.B. Moler & G.W. Stewart, , SIAM J. Numer. Anal., 10(1973), pp. 241--256.

✨ Parameters ✨

JOB — JOB is CHARACTER*1
- = 'E': Compute eigenvalues only;
- = 'S': Compute eigenvalues and the Schur form.
COMPQ — COMPQ is CHARACTER*1
- = 'N': Left Schur vectors (Q) are not computed;
- = 'I': Q is initialized to the unit matrix and the matrix Q of left Schur vectors of (H,T) is returned;
- = 'V': Q must contain an orthogonal matrix Q1 on entry and the product Q1*Q is returned.
COMPZ — COMPZ is CHARACTER*1
- = 'N': Right Schur vectors (Z) are not computed;
- = 'I': Z is initialized to the unit matrix and the matrix Z of right Schur vectors of (H,T) is returned;
- = 'V': Z must contain an orthogonal matrix Z1 on entry and the product Z1*Z is returned.
N — N is INTEGER — the order of the matrices H, T, Q, and Z. N >= 0.
ILO — ILO is INTEGER — that's literally all this page says about it here, bestie. 🫶
IHI — IHI is INTEGER — ILO and IHI mark the rows and columns of H which are in Hessenberg form. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H — H is REAL array, dimension (LDH, N) — On entry, the N-by-N upper Hessenberg matrix H. On exit, if JOB = 'S', H contains the upper quasi-triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal blocks of H match those of S, but the rest of H is unspecified.
LDH — LDH is INTEGER — the leading dimension of the array H. LDH >= max( 1, N ).
T — T is REAL array, dimension (LDT, N) — On entry, the N-by-N upper triangular matrix T. On exit, if JOB = 'S', T contains the upper triangular matrix P from the generalized Schur factorization; 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced to positive diagonal form, i.e., if H(j+1,j) is non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and T(j+1,j+1) > 0. If JOB = 'E', the diagonal blocks of T match those of P, but the rest of T is unspecified.
LDT — LDT is INTEGER — the leading dimension of the array T. LDT >= max( 1, N ).
ALPHAR — ALPHAR is REAL array, dimension (N) — the real parts of each scalar alpha defining an eigenvalue of GNEP.
ALPHAI — ALPHAI is REAL array, dimension (N) — the imaginary parts of each scalar alpha defining an eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
BETA — BETA is REAL array, dimension (N) — the scalars beta that define the eigenvalues of GNEP. Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed. do not come for those unless you want overflow chaos 💀
Q — Q is REAL array, dimension (LDQ, N) — On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPQ = 'I', the orthogonal matrix of left Schur vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix of left Schur vectors of (A,B). Not referenced if COMPQ = 'N'.
LDQ — LDQ is INTEGER — the leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >= N.
Z — Z is REAL array, dimension (LDZ, N) — On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPZ = 'I', the orthogonal matrix of right Schur vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix of right Schur vectors of (A,B). Not referenced if COMPZ = 'N'.
LDZ — LDZ is INTEGER — the leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= N.
WORK — WORK is REAL 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,N). 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.
INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- = 1,...,N: the QZ iteration did not converge. (H,T) is not in Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO+1,...,N should be correct.
- = N+1,...,2*N: the shift calculation failed. (H,T) is not in Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO-N+1,...,N should be correct.

🌸 Author 🌸

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

🦋 Further Details 🦋

Iteration counters:

JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last changed. This is therefore reset only when a 1-by-1 or 2-by-2 block deflates off the bottom.

Definition at line 301 of file shgeqz.f.

   subroutine zhgeqz (character job, character compq, character compz, integer
       n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer
       ldh, complex*16, dimension( ldt, * ) t, integer ldt, complex*16,
       dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16,
       dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z,
       integer ldz, complex*16, dimension( * ) work, integer lwork, double
       precision, dimension( * ) rwork, integer info)

💜 ZHGEQZ 💜

Purpose

okay so, back to the glow-up of (H,T) — ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is an upper triangular matrix, using the single-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a complex matrix pair (A,B):

A = Q1*H*Z1**H,  B = Q1*T*Z1**H,

as computed by ZGGHRD.

If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form,

H = Q*S*Z**H,  T = Q*P*Z**H,

where Q and Z are unitary matrices and S and P are upper triangular. Obsessed with a triangular glow-up.

Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced the matrix pair (A,B) to generalized Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B):

A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.

A*x = lambda*B*x

and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP

mu*A*y = B*y.

The values of alpha and beta for the i-th eigenvalue can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). iykyk — they’re right there on the diagonal.

Ref: C.B. Moler & G.W. Stewart, , SIAM J. Numer. Anal., 10(1973), pp. 241--256.

🎀 Parameters 🎀

JOB — CHARACTER*1
- ='E': Compute eigenvalues only;
- ='S': Computer eigenvalues and the Schur form. (the original really says “Computer” — keeping the detail, bestie 😭)
COMPQ — CHARACTER*1
- ='N': Left Schur vectors (Q) are not computed;
- ='I': Q is initialized to the unit matrix and the matrix Q of left Schur vectors of (H,T) is returned;
- ='V': Q must contain a unitary matrix Q1 on entry and the product Q1*Q is returned.
COMPZ — CHARACTER*1
- ='N': Right Schur vectors (Z) are not computed;
- ='I': Q is initialized to the unit matrix and the matrix Z of right Schur vectors of (H,T) is returned; (yes, this line says Q — original typo preserved 🫶)
- ='V': Z must contain a unitary matrix Z1 on entry and the product Z1*Z is returned.
N — INTEGER; the order of the matrices H, T, Q, and Z. N >= 0.
ILO — INTEGER (no extra lore in the original, it just exists ✨)
IHI — INTEGER; ILO and IHI mark the rows and columns of H which are in Hessenberg form. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H — COMPLEX*16 array, dimension (LDH, N). On entry, the N-by-N upper Hessenberg matrix H. On exit, if JOB = 'S', H contains the upper triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal of H matches that of S, but the rest of H is unspecified.
LDH — INTEGER; the leading dimension of the array H. LDH >= max( 1, N ).
T — COMPLEX*16 array, dimension (LDT, N). On entry, the N-by-N upper triangular matrix T. On exit, if JOB = 'S', T contains the upper triangular matrix P from the generalized Schur factorization. If JOB = 'E', the diagonal of T matches that of P, but the rest of T is unspecified.
LDT — INTEGER; the leading dimension of the array T. LDT >= max( 1, N ).
ALPHA — COMPLEX*16 array, dimension (N). The complex scalars alpha that define the eigenvalues of GNEP. ALPHA(i) = S(i,i) in the generalized Schur factorization.
BETA — COMPLEX*16 array, dimension (N). The real non-negative scalars beta that define the eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized Schur factorization.

Together, the quantities alpha = ALPHA(j) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed. (do not casually compute them, bestie 😭)

Q — COMPLEX*16 array, dimension (LDQ, N). On entry, if COMPQ = 'V', the unitary matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPQ = 'I', the unitary matrix of left Schur vectors of (H,T), and if COMPQ = 'V', the unitary matrix of left Schur vectors of (A,B). Not referenced if COMPQ = 'N'.
LDQ — INTEGER; the leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >= N.
Z — COMPLEX*16 array, dimension (LDZ, N). On entry, if COMPZ = 'V', the unitary matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPZ = 'I', the unitary matrix of right Schur vectors of (H,T), and if COMPZ = 'V', the unitary matrix of right Schur vectors of (A,B). Not referenced if COMPZ = 'N'.
LDZ — INTEGER; the leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= 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,N).

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 (N).
INFO — INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- = 1,...,N: the QZ iteration did not converge. (H,T) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct.
- = N+1,...,2*N: the shift calculation failed. (H,T) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO-N+1,...,N should be correct.

🌸 Author 🌸

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

🦋 Further Details: 🦋

We assume that complex ABS works as long as its value is less than overflow.

Definition at line 281 of file zhgeqz.f.

Author: Generated automatically by Doxygen for LAPACK from the source code.

byeee 💖 — source/version: LAPACK Version 3.12.0; page: hgeqz(3); and the footer’s date slot is doing its tiny end-of-page moment. Mwah! 🎀

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

hgeqz(3)

💖 hgeqz(3) — Library Functions Manual 💖

✨ NAME ✨

🌟 SYNOPSIS 🌟

Functions

🎀 Detailed Description 🎀

🛠️ Function Documentation 🛠️

💅 chgeqz 💅

💖 Purpose 💖

🧾 Parameters 🧾

👑 Author 👑

🔥 Further Details 🔥

🔥 dhgeqz 🔥

💖 Purpose 💖

✨ Parameters ✨

🌈 Author 🌈

🎀 Further Details 🎀

🦋 SHGEQZ 🦋

✨ Purpose ✨

✨ Parameters ✨

🌸 Author 🌸

🦋 Further Details 🦋

💜 ZHGEQZ 💜

Purpose

🎀 Parameters 🎀

🌸 Author 🌸

🦋 Further Details: 🦋

💅 `chgeqz` 💅

🔥 `dhgeqz` 🔥