laqz0(3)

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

  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

  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

  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

Author

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