laqz0(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

🌈 `laqz0(3)` 🌈 — your fave step in `ggev3` / `gges3` 💅

okay so welcome to the Library Functions Manual entry for laqz0(3), bestie. This is where we meet the recursive subroutine crew: claqz0, dlaqz0, slaqz0, and zlaqz0. They all handle the laqz0 step inside ggev3 and gges3, and they are all very much doing the work. ✨

🎀 NAME 🎀

laqz0 — laqz0: step in ggev3, gges3.

✨ SYNOPSIS ✨

🫶 Functions 🫶

claqz0 — CLAQZ0

recursive subroutine claqz0 (wants, wantq, wantz, n, ilo, ihi, a, lda,
    b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)

dlaqz0 — DLAQZ0

recursive subroutine dlaqz0 (wants, wantq, wantz, n, ilo, ihi, a, lda,
    b, ldb, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, rec,
    info)

slaqz0 — SLAQZ0

recursive subroutine slaqz0 (wants, wantq, wantz, n, ilo, ihi, a, lda,
    b, ldb, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, rec,
    info)

zlaqz0 — ZLAQZ0

recursive subroutine zlaqz0 (wants, wantq, wantz, n, ilo, ihi, a, lda,
    b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)

🦋 Detailed Description 🦋

No extra prose here, bestie—the real tea is in Function Documentation below. 🍵

💖 Function Documentation 💖

🌸 recursive subroutine claqz0 🌸

recursive subroutine claqz0 (character, intent(in) wants, character,
    intent(in) wantq, character, intent(in) wantz, integer, intent(in) n,
    integer, intent(in) ilo, integer, intent(in) ihi, complex, dimension(
    lda, * ), intent(inout) a, integer, intent(in) lda, complex, dimension(
    ldb, * ), intent(inout) b, integer, intent(in) ldb, complex, dimension(
    * ), intent(inout) alpha, complex, dimension( * ), intent(inout) beta,
    complex, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq,
    complex, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz,
    complex, dimension( * ), intent(inout) work, integer, intent(in) lwork,
    real, dimension( * ), intent(out) rwork, integer, intent(in) rec,
    integer, intent(out) info)

CLAQZ0

🎯 Purpose 🎯

CLAQZ0 computes the eigenvalues of a matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the double-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a 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, P and S are an upper triangular matrices.

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 upper 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 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.

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.

Ref: B. Kagstrom, D. Kressner, , SIAM J. Numer. Anal., 29(2006), pp. 199--227.

Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril

🧚 Parameters 🧚

WANTS — WANTS is CHARACTER*1.
- = 'E' — Compute eigenvalues only;
- = 'S' — Compute eigenvalues and the Schur form.
WANTQ — WANTQ 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 (A,B) is returned;
- = 'V' — Q must contain an unitary matrix Q1 on entry and the product Q1*Q is returned.
WANTZ — WANTZ 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 (A,B) is returned;
- = 'V' — Z must contain an unitary matrix Z1 on entry and the product Z1*Z is returned.
N — N is INTEGER. The order of the matrices A, B, Q, and Z. N >= 0.
ILO — ILO is INTEGER.
IHI — IHI is INTEGER. ILO and IHI mark the rows and columns of A 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.
A — A is COMPLEX array, dimension (LDA, N). On entry, the N-by-N upper Hessenberg matrix A. On exit, if JOB = 'S', A contains the upper triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal of A matches that of S, but the rest of A is unspecified.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max( 1, N ).
B — B is COMPLEX array, dimension (LDB, N). On entry, the N-by-N upper triangular matrix B. On exit, if JOB = 'S', B contains the upper triangular matrix P from the generalized Schur factorization. If JOB = 'E', the diagonal of B matches that of P, but the rest of B is unspecified.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max( 1, N ).
ALPHA — ALPHA is COMPLEX array, dimension (N). Each scalar alpha defining an eigenvalue of GNEP.
BETA — BETA is COMPLEX array, dimension (N). The scalars beta that define the eigenvalues of GNEP. 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.

🚨 No cap: 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 (A,B), 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).
REC — REC is INTEGER. REC indicates the current recursion level. Should be set to 0 on first call.
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. (A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct.

👑 Author 👑

Thijs Steel, KU Leuven

🌸 Date 🌸

May 2020

Definition at line 280 of file claqz0.f.

🔥 recursive subroutine dlaqz0 🔥

recursive subroutine dlaqz0 (character, intent(in) wants, character,
    intent(in) wantq, character, intent(in) wantz, integer, intent(in) n,
    integer, intent(in) ilo, integer, intent(in) ihi, double precision,
    dimension( lda, * ), intent(inout) a, integer, intent(in) lda, double
    precision, dimension( ldb, * ), intent(inout) b, integer, intent(in)
    ldb, double precision, dimension( * ), intent(inout) alphar, double
    precision, dimension( * ), intent(inout) alphai, double precision,
    dimension( * ), intent(inout) beta, double precision, dimension( ldq, *
    ), intent(inout) q, integer, intent(in) ldq, double precision,
    dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, double
    precision, dimension( * ), intent(inout) work, integer, intent(in)
    lwork, integer, intent(in) rec, integer, intent(out) info)

DLAQZ0

🎯 Purpose 🎯

so like, continuation time — we're picking up right where the math left off, no recap, let's go 💅

DLAQZ0 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. 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 DGGHRD.

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. quasi-triangular bestie, it's a whole vibe ✨

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. conjugates stay together, that's the rule 🫶

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. off-diagonals GONE, diagonals POSITIVE, we love a clean glow-up 💖

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 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.

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.

ok this part is lowkey the most important thing on the page: 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. neither lambda nor mu should be computed in general — either one may overflow. that's not me being dramatic, that's the actual warning 💀

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.

Ref: B. Kagstrom, D. Kressner, , SIAM J. Numer. Anal., 29(2006), pp. 199--227.

Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril

🌸 Parameters 🌸

bestie. there are SO many parameters. buckle up, we're going through every single one, no skipping 💖

✨ `WANTS` ✨

WANTS is CHARACTER*1

= 'E': Compute eigenvalues only;
= 'S': Compute eigenvalues and the Schur form.

✨ `WANTQ` ✨

WANTQ 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 (A,B) is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry and the product Q1*Q is returned.

✨ `WANTZ` ✨

WANTZ 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 (A,B) 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 A, B, Q, and Z. N >= 0. (zero is legal, zero is fine, zero is a valid lifestyle 💅)

✨ `ILO` ✨

ILO is INTEGER

no extra notes here, it's just ILO — its whole personality is defined by IHI right below 👇

✨ `IHI` ✨

IHI is INTEGER

ILO and IHI mark the rows and columns of A 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.

fr fr check those bounds before you call this, the library is not gonna hold your hand 🫶

✨ `A` ✨

A is DOUBLE PRECISION array, dimension (LDA, N)

On entry, the N-by-N upper Hessenberg matrix A.

On exit, if JOB = 'S', A contains the upper quasi-triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal blocks of A match those of S, but the rest of A is unspecified. (unspecified = do not go looking, it's giving chaos in there 🙈)

✨ `LDA` ✨

LDA is INTEGER

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

✨ `B` ✨

B is DOUBLE PRECISION array, dimension (LDB, N)

On entry, the N-by-N upper triangular matrix B.

On exit, if JOB = 'S', B 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 A(j+1,j) is non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and B(j+1,j+1) > 0. If JOB = 'E', the diagonal blocks of B match those of P, but the rest of B is unspecified.

✨ `LDB` ✨

LDB is INTEGER

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

✨ `ALPHAR` ✨

ALPHAR is DOUBLE PRECISION array, dimension (N)

The real parts of each scalar alpha defining an eigenvalue of GNEP. the R stands for Real, we stan a descriptive suffix 💖

✨ `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).

notice there's no negative case listed bestie — positive or zero, and the conjugate is stored right after with the flipped sign. that's the whole system 🦋

✨ `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.

yes I'm saying it again because it's that serious: do not just divide. overflow is real and it will humble you 😭

✨ `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 (A,B), and if COMPQ = 'V', the orthogonal matrix of left Schur vectors of (A,B).

Not referenced if COMPQ = 'N'. (and yes, that's COMPQ with an M even though the argument is named WANTQ — I'm just reading the page to you, bestie, I don't make the rules 💀)

✨ `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. it's the little info slot that tells you how much room you actually needed 🎀

✨ `LWORK` ✨

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,N).

-1 is the "just tell me the size, don't compute anything" mode. iconic honestly 🧚

✨ `REC` ✨

REC is INTEGER

REC indicates the current recursion level. Should be set to 0 on first call.

literally set it to 0 on your first call. that's not optional, that's the contract 🫶

✨ `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. (A,B) is not in Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO+1,...,N should be correct.

read that third case carefully bestie — it did not converge, the pair is not in Schur form, but the entries from INFO+1 to N are still correct. partial credit exists in LAPACK 💅

✨ Author ✨

Thijs Steel, KU Leuven we love a credited author 👑

🎀 Date 🎀

May 2020

🌈 Definition 🌈

Definition at line 302 of file dlaqz0.f.

🦋 the single precision twin has entered the chat 🦋

okay and then the page pivots to the SLAQZ0 variant, same energy, single precision:

recursive subroutine slaqz0 (character, intent(in) wants, character,
    intent(in) wantq, character, intent(in) wantz, integer, intent(in) n,
    integer, intent(in) ilo, integer, intent(in) ihi, real, dimension( lda,
    * ), intent(inout) a, integer, intent(in) lda, real, dimension( ldb, *
    ), intent(inout) b, integer, intent(in) ldb, real, dimension( * ),
    intent(inout) alphar, real, dimension( * ), intent(inout) alphai, real,
    dimension( * ), intent(inout) beta, real, dimension( ldq, * ),
    intent(inout) q, integer, intent(in) ldq, real, dimension( ldz, * ),
    intent(inout) z, integer, intent(in) ldz, real, dimension( * ),
    intent(inout) work, integer, intent(in) lwork, integer, intent(in) rec,
    integer, intent(out) info)

`SLAQZ0`

Purpose:

…and that's exactly where we're gonna pick it up next, don't go anywhere 💖

okay so SLAQZ0 is literally the routine that 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 💅.

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 😭 iconic.

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) ⭐.

Ref: C.B. Moler & G.W. Stewart, , SIAM J. Numer. Anal., 10(1973), pp. 241--256.
Ref: B. Kagstrom, D. Kressner, , SIAM J. Numer. Anal., 29(2006), pp. 199--227.
Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril

✨ Parameters ✨

WANTS — WANTS is CHARACTER*1
- 'E': Compute eigenvalues only;
- 'S': Compute eigenvalues and the Schur form.
WANTQ — WANTQ 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 (A,B) is returned;
- 'V': Q must contain an orthogonal matrix Q1 on entry and the product Q1*Q is returned.
WANTZ — WANTZ 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 (A,B) 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 A, B, Q, and Z. N >= 0.
ILO — ILO is INTEGER 🎀
IHI — IHI is INTEGER
- ILO and IHI mark the rows and columns of A 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.
A — A is REAL array, dimension (LDA, N)
- On entry, the N-by-N upper Hessenberg matrix A.
- On exit, if JOB = 'S', A contains the upper quasi-triangular matrix S from the generalized Schur factorization.
- If JOB = 'E', the diagonal blocks of A match those of S, but the rest of A is unspecified.
LDA — LDA is INTEGER
- The leading dimension of the array A. LDA >= max( 1, N ).
B — B is REAL array, dimension (LDB, N)
- On entry, the N-by-N upper triangular matrix B.
- On exit, if JOB = 'S', B 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 A(j+1,j) is non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and B(j+1,j+1) > 0.
- If JOB = 'E', the diagonal blocks of B match those of P, but the rest of B is unspecified.
LDB — LDB is INTEGER
- The leading dimension of the array B. LDB >= 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.
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 (A,B), 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.
REC — REC is INTEGER
- REC indicates the current recursion level. Should be set to 0 on first call.
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. (A,B) is not in Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO+1,...,N should be correct.

✨ Author ✨

Thijs Steel, KU Leuven

✨ Date ✨

May 2020

✨ Definition ✨

at line 300 of file slaqz0.f.

   recursive subroutine zlaqz0 (character, intent(in) wants, character,
       intent(in) wantq, character, intent(in) wantz, integer, intent(in) n,
       integer, intent(in) ilo, integer, intent(in) ihi, complex*16, dimension(
       lda, * ), intent(inout) a, integer, intent(in) lda, complex*16,
       dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb,
       complex*16, dimension( * ), intent(inout) alpha, complex*16, dimension(
       * ), intent(inout) beta, complex*16, dimension( ldq,         * ),
       intent(inout) q, integer, intent(in) ldq, complex*16, dimension( ldz, *
       ), intent(inout) z, integer, intent(in) ldz, complex*16, dimension( * ),
       intent(inout) work, integer, intent(in) lwork, double precision,
       dimension( * ), intent(out) rwork, integer, intent(in) rec, integer,
       intent(out) info)

ZLAQZ0

Purpose:

✨ Parameters ✨

Okay bestie, here's the tea on every single argument this routine takes. Don't skip any, fr fr. 💅

`WANTS`

WANTS is CHARACTER*1. It's literally you telling the routine what you want:

'E' — Compute eigenvalues only.
'S' — Compute eigenvalues and the Schur form.

Slay. ✨

`WANTQ`

WANTQ is CHARACTER*1. This one controls the left Schur vectors (Q):

'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 (A,B) is returned.
'V' — Q must contain an unitary matrix Q1 on entry and the product Q1*Q is returned.

`WANTZ`

WANTZ is CHARACTER*1. Same energy but for the right Schur vectors (Z):

'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 (A,B) is returned.
'V' — Z must contain an unitary matrix Z1 on entry and the product Z1*Z is returned.

`N`

N is INTEGER. The order of the matrices A, B, Q, and Z. N >= 0.

`ILO`

ILO is INTEGER.

`IHI`

IHI is INTEGER. ILO and IHI mark the rows and columns of A 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.

`A`

A is COMPLEX*16 array, dimension (LDA, N). On entry, the N-by-N upper Hessenberg matrix A. On exit, if JOB = 'S', A contains the upper triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal blocks of A match those of S, but the rest of A is unspecified.

`LDA`

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

`B`

B is COMPLEX*16 array, dimension (LDB, N). On entry, the N-by-N upper triangular matrix B. On exit, if JOB = 'S', B contains the upper triangular matrix P from the generalized Schur factorization; If JOB = 'E', the diagonal blocks of B match those of P, but the rest of B is unspecified.

`LDB`

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

`ALPHA`

ALPHA is COMPLEX*16 array, dimension (N). Each scalar alpha defining an eigenvalue of GNEP.

`BETA`

BETA is COMPLEX*16 array, dimension (N). The scalars beta that define the eigenvalues of GNEP. 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*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 (A,B), 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*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`

LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= N.

`WORK`

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

`RWORK`

RWORK is DOUBLE PRECISION array, dimension (N).

`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.

`REC`

REC is INTEGER. REC indicates the current recursion level. Should be set to 0 on first call.

`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. (A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct.

✨ Author ✨

Thijs Steel, KU Leuven

✨ Date ✨

May 2020

Definition at line 280 of file zlaqz0.f.

✨ Author ✨

Generated automatically by Doxygen for LAPACK from the source code.

And that's the tea! This iconic page is from LAPACK version 3.12.0, dated May 2020. Periodt. 💅✨🦋

📦 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

laqz0(3)

🌈 laqz0(3) 🌈 — your fave step in ggev3 / gges3 💅

🎀 NAME 🎀

✨ SYNOPSIS ✨

🫶 Functions 🫶

🦋 Detailed Description 🦋

💖 Function Documentation 💖

🌸 recursive subroutine claqz0 🌸

🎯 Purpose 🎯

🧚 Parameters 🧚

👑 Author 👑

🌸 Date 🌸

🔥 recursive subroutine dlaqz0 🔥

🎯 Purpose 🎯

🌸 Parameters 🌸

✨ WANTS ✨

✨ WANTQ ✨

✨ WANTZ ✨

✨ N ✨

✨ ILO ✨

✨ IHI ✨

✨ A ✨

✨ LDA ✨

✨ B ✨

✨ LDB ✨

✨ ALPHAR ✨

✨ ALPHAI ✨

✨ BETA ✨

✨ Q ✨

✨ LDQ ✨

✨ Z ✨

✨ LDZ ✨

✨ WORK ✨

✨ LWORK ✨

✨ REC ✨

✨ INFO ✨

✨ Author ✨

🎀 Date 🎀

🌈 Definition 🌈

🦋 the single precision twin has entered the chat 🦋

SLAQZ0

✨ Parameters ✨

✨ Author ✨

✨ Date ✨

✨ Definition ✨

ZLAQZ0

✨ Parameters ✨

WANTS

WANTQ

WANTZ

N

ILO

IHI

A

LDA

B

LDB

ALPHA

BETA

Q

LDQ

Z

LDZ

WORK

RWORK

LWORK

REC

INFO

✨ Author ✨

✨ Date ✨

✨ Author ✨

🌈 `laqz0(3)` 🌈 — your fave step in `ggev3` / `gges3` 💅

✨ `WANTS` ✨

✨ `WANTQ` ✨

✨ `WANTZ` ✨

✨ `N` ✨

✨ `ILO` ✨

✨ `IHI` ✨

✨ `A` ✨

✨ `LDA` ✨

✨ `B` ✨

✨ `LDB` ✨

✨ `ALPHAR` ✨

✨ `ALPHAI` ✨

✨ `BETA` ✨

✨ `Q` ✨

✨ `LDQ` ✨

✨ `Z` ✨

✨ `LDZ` ✨

✨ `WORK` ✨

✨ `LWORK` ✨

✨ `REC` ✨

✨ `INFO` ✨

`SLAQZ0`

`WANTS`

`WANTQ`

`WANTZ`

`N`

`ILO`

`IHI`

`A`

`LDA`

`B`

`LDB`

`ALPHA`

`BETA`

`Q`

`LDQ`

`Z`

`LDZ`

`WORK`

`RWORK`

`LWORK`

`REC`

`INFO`