---
name: "laqz0"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/laqz0"
---

# laqz0(3)

laqz0: step in ggev3, gges3

## NAME

laqz0 - laqz0: step in ggev3, gges3

## SYNOPSIS

### Functions

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**  recursive subroutine **dlaqz0** (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, rec, info)\
 **DLAQZ0**  recursive subroutine **slaqz0** (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, rec, info)\
 **SLAQZ0**  recursive subroutine **zlaqz0** (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)\
 **ZLAQZ0**

## Detailed Description

## Function Documentation

### 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.  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 (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:**

```
!>
!> 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.
!>
!> 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.
!>
!> 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 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.
!> 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 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.
!>
```

*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.
!>
```

*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).
!>
```

*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 (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 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.
!>
```

*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 **302** of file **dlaqz0.f**.

### 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:**

```
!>
!> SLAQZ0 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.
!>
!> 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).
!>
!> 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:**

```
!>
!> ZLAQZ0 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**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, 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 ZGGHRD 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*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

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