SRC/claqz0.f(3)

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💅 SRC/claqz0.f(3) 💅

hey bestie, welcome to the Library Functions Manual page for SRC/claqz0.f — aka the CLAQZ0 page. It's giving Fortran, it's giving QZ, and it's lowkey iconic. 🦋

🦋 NAME 🦋

SRC/claqz0.f

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

okay so the whole synopsis is one recursive subroutine, and she's serving:

recursive subroutine claqz0 (wants, wantq, wantz, n, ilo, ihi, a, lda,
    b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)
    CLAQZ0

💖 Function/Subroutine Documentation 💖

here's the full signature with all the intents and dimensions, exactly as the page lays it out:

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 💡

okay so 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. No cap. 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 (yes, the page says that — love that for us 💅).

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).

The refs are giving academic baddie energy:

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

No extra lore on this one, bestie — that's literally the entry. 💀

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. 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 🧵

Definition at line 280 of file claqz0.f.

🤖 Author 🤖

Generated automatically by Doxygen for LAPACK from the source code.

okay bestie, that's the whole manual page. This one's straight from LAPACK Version 3.12.0, dated May 2020 — go off and QZ responsibly. 💖✨

📦 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