SRC/slaqz0.f(3)

NAME
SRC/slaqz0.f

SYNOPSIS

  Functions/Subroutines
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

Function/Subroutine Documentation

  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, 'An Algorithm for Generalized Matrix
          Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
          pp. 241--256.
     Ref: B. Kagstrom, D. Kressner, 'Multishift Variants of the QZ
          Algorithm with Aggressive Early Deflation', SIAM J. Numer.
          Anal., 29(2006), pp. 199--227.
     Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril 'A multishift,
          multipole rational QZ method with aggressive early deflation'

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.

Author

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