REAL routines for upper Hessenberg matrix, generalized problem (i.e a Hessenberg and a triangular matrix) matrix
shgeqz
USAGE:
alphar, alphai, beta, work, info, h, t, q, z = NumRu::Lapack.shgeqz( job, compq, compz, ilo, ihi, h, t, q, z, [:lwork => lwork, :usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO )
* Purpose
* =======
*
* SHGEQZ 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.
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* = 'E': Compute eigenvalues only;
* = 'S': Compute eigenvalues and the Schur form.
*
* COMPQ (input) 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 (H,T) is returned;
* = 'V': Q must contain an orthogonal matrix Q1 on entry and
* the product Q1*Q is returned.
*
* COMPZ (input) 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 (H,T) is returned;
* = 'V': Z must contain an orthogonal matrix Z1 on entry and
* the product Z1*Z is returned.
*
* N (input) INTEGER
* The order of the matrices H, T, Q, and Z. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* ILO and IHI mark the rows and columns of H 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.
*
* H (input/output) REAL array, dimension (LDH, N)
* On entry, the N-by-N upper Hessenberg matrix H.
* On exit, if JOB = 'S', H contains the upper quasi-triangular
* matrix S from the generalized Schur factorization;
* 2-by-2 diagonal blocks (corresponding to complex conjugate
* pairs of eigenvalues) are returned in standard form, with
* H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
* If JOB = 'E', the diagonal blocks of H match those of S, but
* the rest of H is unspecified.
*
* LDH (input) INTEGER
* The leading dimension of the array H. LDH >= max( 1, N ).
*
* T (input/output) REAL array, dimension (LDT, N)
* On entry, the N-by-N upper triangular matrix T.
* On exit, if JOB = 'S', T 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 H(j+1,j) is
* non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
* T(j+1,j+1) > 0.
* If JOB = 'E', the diagonal blocks of T match those of P, but
* the rest of T is unspecified.
*
* LDT (input) INTEGER
* The leading dimension of the array T. LDT >= max( 1, N ).
*
* ALPHAR (output) REAL array, dimension (N)
* The real parts of each scalar alpha defining an eigenvalue
* of GNEP.
*
* ALPHAI (output) 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 (output) 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 (input/output) REAL array, dimension (LDQ, N)
* On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
* the reduction of (A,B) to generalized Hessenberg form.
* On exit, if COMPZ = 'I', the orthogonal matrix of left Schur
* vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix
* of left Schur vectors of (A,B).
* Not referenced if COMPZ = 'N'.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= 1.
* If COMPQ='V' or 'I', then LDQ >= N.
*
* Z (input/output) 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 (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1.
* If COMPZ='V' or 'I', then LDZ >= N.
*
* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
* On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) 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.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* = 1,...,N: the QZ iteration did not converge. (H,T) is not
* in Schur form, but ALPHAR(i), ALPHAI(i), and
* BETA(i), i=INFO+1,...,N should be correct.
* = N+1,...,2*N: the shift calculation failed. (H,T) is not
* in Schur form, but ALPHAR(i), ALPHAI(i), and
* BETA(i), i=INFO-N+1,...,N should be correct.
*
* Further Details
* ===============
*
* Iteration counters:
*
* JITER -- counts iterations.
* IITER -- counts iterations run since ILAST was last
* changed. This is therefore reset only when a 1-by-1 or
* 2-by-2 block deflates off the bottom.
*
* =====================================================================
*
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