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## dlaqz0()

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

Download DLAQZ0 + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  WANTS

      !>          WANTS is CHARACTER*1
      !>          = 'E': Compute eigenvalues only;
      !>          = 'S': Compute eigenvalues and the Schur form.
      !>

  [in]

  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.
      !>

  [in]

  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.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrices A, B, Q, and Z.  N >= 0.
      !>

  [in]

  ILO

      !>          ILO is INTEGER
      !>

  [in]

  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.
      !>

  [in,out]

  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.
      !>

  [in]

  LDA

      !>          LDA is INTEGER
      !>          The leading dimension of the array A.  LDA >= max( 1, N ).
      !>

  [in,out]

  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.
      !>

  [in]

  LDB

      !>          LDB is INTEGER
      !>          The leading dimension of the array B.  LDB >= max( 1, N ).
      !>

  [out]

  ALPHAR

      !>          ALPHAR is DOUBLE PRECISION array, dimension (N)
      !>          The real parts of each scalar alpha defining an eigenvalue
      !>          of GNEP.
      !>

  [out]

  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).
      !>

  [out]

  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.
      !>

  [in,out]

  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'.
      !>

  [in]

  LDQ

      !>          LDQ is INTEGER
      !>          The leading dimension of the array Q.  LDQ >= 1.
      !>          If COMPQ='V' or 'I', then LDQ >= N.
      !>

  [in,out]

  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'.
      !>

  [in]

  LDZ

      !>          LDZ is INTEGER
      !>          The leading dimension of the array Z.  LDZ >= 1.
      !>          If COMPZ='V' or 'I', then LDZ >= N.
      !>

  [out]

  WORK

      !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
      !>          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
      !>

  [in]

  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.
      !>

  [in]

  REC

      !>          REC is INTEGER
      !>             REC indicates the current recursion level. Should be set
      !>             to 0 on first call.
      !>

  [out]

  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