# man shgeqz (Fonctions bibliothèques) - implement a single-/double-shift version of the QZ method for finding the generalized eigenvalues w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation det( A - w(i) B ) = 0 In addition, the pair A,B may be reduced to generalized Schur form

## NAME

SHGEQZ - implement a single-/double-shift version of the QZ method for finding the generalized eigenvalues w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation det( A - w(i) B ) = 0 In addition, the pair A,B may be reduced to generalized Schur form

## SYNOPSIS

- SUBROUTINE SHGEQZ(
- JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO )
- CHARACTER COMPQ, COMPZ, JOB
- INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
- REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )

## PURPOSE

SHGEQZ implements a single-/double-shift version of the QZ method for finding the generalized eigenvalues w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation det( A - w(i) B ) = 0 In addition, the pair A,B may be reduced to generalized Schur form: B is upper triangular, and A is block upper triangular, where the
diagonal blocks are either 1-by-1 or 2-by-2, the 2-by-2 blocks having
complex generalized eigenvalues (see the description of the argument
JOB.)

If JOB='S', then the pair (A,B) is simultaneously reduced to Schur
form by applying one orthogonal tranformation (usually called Q) on
the left and another (usually called Z) on the right. The 2-by-2
upper-triangular diagonal blocks of B corresponding to 2-by-2 blocks
of A will be reduced to positive diagonal matrices. (I.e.,
if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and
B(j+1,j+1) will be positive.)

If JOB='E', then at each iteration, the same transformations
are computed, but they are only applied to those parts of A and B
which are needed to compute ALPHAR, ALPHAI, and BETAR.

If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal
transformations used to reduce (A,B) are accumulated into the arrays
Q and Z s.t.:

Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*

Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*

Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1),
pp. 241--256.

## ARGUMENTS

- JOB (input) CHARACTER*1
- = 'E': compute only ALPHAR, ALPHAI, and BETA. A and B will not necessarily be put into generalized Schur form. = 'S': put A and B into generalized Schur form, as well as computing ALPHAR, ALPHAI, and BETA.
- COMPQ (input) CHARACTER*1
- = 'N': do not modify Q.

= 'V': multiply the array Q on the right by the transpose of the orthogonal tranformation that is applied to the left side of A and B to reduce them to Schur form. = 'I': like COMPQ='V', except that Q will be initialized to the identity first. - COMPZ (input) CHARACTER*1
- = 'N': do not modify Z.

= 'V': multiply the array Z on the right by the orthogonal tranformation that is applied to the right side of A and B to reduce them to Schur form. = 'I': like COMPZ='V', except that Z will be initialized to the identity first. - N (input) INTEGER
- The order of the matrices A, B, Q, and Z. N >= 0.
- ILO (input) INTEGER
- IHI (input) INTEGER It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
- A (input/output) REAL array, dimension (LDA, N)
- On entry, the N-by-N upper Hessenberg matrix A. Elements below the subdiagonal must be zero. If JOB='S', then on exit A and B will have been simultaneously reduced to generalized Schur form. If JOB='E', then on exit A will have been destroyed. The diagonal blocks will be correct, but the off-diagonal portion will be meaningless.
- LDA (input) INTEGER
- The leading dimension of the array A. LDA >= max( 1, N ).
- B (input/output) REAL array, dimension (LDB, N)
- On entry, the N-by-N upper triangular matrix B. Elements below the diagonal must be zero. 2-by-2 blocks in B corresponding to 2-by-2 blocks in A will be 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 and B(j,j) and B(j+1,j+1) will be positive.) If JOB='S', then on exit A and B will have been simultaneously reduced to Schur form. If JOB='E', then on exit B will have been destroyed. Elements corresponding to diagonal blocks of A will be correct, but the off-diagonal portion will be meaningless.
- LDB (input) INTEGER
- The leading dimension of the array B. LDB >= max( 1, N ).
- ALPHAR (output) REAL array, dimension (N)
- ALPHAR(1:N) will be set to real parts of the diagonal elements of A that would result from reducing A and B to Schur form and then further reducing them both to triangular form using unitary transformations s.t. the diagonal of B was non-negative real. Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=A(j,j). Note that the (real or complex) values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized eigenvalues of the matrix pencil A - wB.
- ALPHAI (output) REAL array, dimension (N)
- ALPHAI(1:N) will be set to imaginary parts of the diagonal elements of A that would result from reducing A and B to Schur form and then further reducing them both to triangular form using unitary transformations s.t. the diagonal of B was non-negative real. Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0. Note that the (real or complex) values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized eigenvalues of the matrix pencil A - wB.
- BETA (output) REAL array, dimension (N)
- BETA(1:N) will be set to the (real) diagonal elements of B that would result from reducing A and B to Schur form and then further reducing them both to triangular form using unitary transformations s.t. the diagonal of B was non-negative real. Thus, if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0), then BETA(j)=B(j,j). Note that the (real or complex) values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized eigenvalues of the matrix pencil A - wB. (Note that BETA(1:N) will always be non-negative, and no BETAI is necessary.)
- Q (input/output) REAL array, dimension (LDQ, N)
- If COMPQ='N', then Q will not be referenced. If COMPQ='V' or 'I', then the transpose of the orthogonal transformations which are applied to A and B on the left will be applied to the array Q on the right.
- 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)
- If COMPZ='N', then Z will not be referenced. If COMPZ='V' or 'I', then the orthogonal transformations which are applied to A and B on the right will be applied to the array Z on the right.
- 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 (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. (A,B) 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. (A,B) is not in Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO-N+1,...,N should be correct. > 2*N: various "impossible" errors.

## 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.