SUBROUTINE DHGEQZ( 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 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
DHGEQZ implements a single-/double-shift version of the QZ method for finding the generalized eigenvalues 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(1973), pp. 241--256.
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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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). 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.
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.