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, , 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.

Author

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