zlaqz0.f - Man Page

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

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, '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 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.