SHGEQZ(3)

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🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/shgeqz.f

✨ SYNOPSIS ✨

🧩 Functions/Subroutines 🧩

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

🛠️ Function/Subroutine Documentation 🛠️

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

🎯 Purpose 🎯

okay so 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. iconic.

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.

📋 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 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. If JOB = 'E', the diagonal blocks of H match those of S, but the rest of H is unspecified. fair warning, bestie.

LDH

LDH is INTEGER

The leading dimension of the array H. LDH >= max( 1, N ).

T

T is 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

LDT is INTEGER

The leading dimension of the array T. LDT >= 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. like, do not casually divide these, fr fr.

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 (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 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.

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 👑

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

🔍 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.

Definition at line 301 of file shgeqz.f.

👑 Author 👑

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