SRC/dhgeqz.f(3)

💖 SRC/dhgeqz.f 💖

okay bestie, welcome to the SRC/dhgeqz.f(3) page from the Library Functions Manual. The header literally repeats SRC/dhgeqz.f(3) like it’s matching its own outfit, and honestly? iconic behavior 💅.

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🌸 NAME 🌸

SRC/dhgeqz.f — that’s the name, bestie. This whole page is about the dhgeqz subroutine doing its Fortran thing.

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✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

DHGEQZ is the main-character subroutine here. The synopsis is literally:

subroutine dhgeqz (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt,
    alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)

DHGEQZ

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🎀 Function/Subroutine Documentation 🎀

Here’s the full signature, copied exactly so nobody comes for me:

subroutine dhgeqz (character job, character compq, character compz, integer
    n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h,
    integer ldh, double precision, dimension( ldt, * ) t, integer ldt,
    double precision, dimension( * ) alphar, double precision, dimension( *
    ) alphai, double precision, dimension( * ) beta, double precision,
    dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, *
    ) z, integer ldz, double precision, dimension( * ) work, integer lwork,
    integer info)

DHGEQZ

💅 Purpose 💅

okay so DHGEQZ 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 like this are produced by reducing a real matrix pair (A,B) to generalized upper Hessenberg form:

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, then lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)

A*x = lambda*B*x

and if alpha is nonzero, then 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).

Reference, exactly: C.B. Moler & G.W. Stewart, , SIAM J. Numer. Anal., 10(1973), pp. 241--256.

🌈 Parameters 🌈

• JOB — JOB is CHARACTER*1.
  • = 'E': compute eigenvalues only;
  • = 'S': compute eigenvalues and the Schur form.
• COMPQ — COMPQ 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 (H,T) is returned;
  • = 'V': Q must contain an orthogonal matrix Q1 on entry and the product Q1*Q is returned.
• COMPZ — COMPZ 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 (H,T) 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 H, T, Q, and Z. N >= 0.
• ILO — ILO is INTEGER.
• IHI — IHI is 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 — H is a DOUBLE PRECISION 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.
  • If JOB = 'E', the diagonal blocks of H match those of S, but the rest of H is unspecified.
• LDH — LDH is INTEGER. The leading dimension of the array H. LDH >= max( 1, N ).
• T — T is a DOUBLE PRECISION 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 — LDT is INTEGER. The leading dimension of the array T. LDT >= max( 1, N ).
• ALPHAR — ALPHAR is a DOUBLE PRECISION array, dimension (N). The real parts of each scalar alpha defining an eigenvalue of GNEP.
• ALPHAI — ALPHAI is a 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).
• BETA — BETA is a 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.
• Q — Q is a 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 (H,T), 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 a 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'.
• LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= N.
• WORK — WORK is a DOUBLE PRECISION 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.
• 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. (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.

👑 Author 👑

The authors are:

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

🦋 Further Details 🦋

Iteration counters, bestie:

• 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.

Definition at line 301 of file dhgeqz.f.

🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code.

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