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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 B is upper triangular, and A is block upper triangular, where the diag- onal blocks are either 1x1 or 2x2, the 2x2 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 using one orthogonal transformation (usually called Q) on the left and another (usually called Z) on the right. The 2x2 upper-triangular diagonal blocks of B corresponding to 2x2 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 posi- tive.) 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 accumu- lated 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 General- ized Matrix Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), 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 transformation 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 transformation 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 ini- tialized to the identity first. N (input) INTEGER The number of rows and columns in the matrices A, B, Q, and Z. N must be at least 0. ILO (input) INTEGER Columns 1 through ILO-1 of A and B are assumed to be in upper triangular form already, and will not be modified. ILO must be at least 1. IHI (input) INTEGER Rows IHI+1 through N of A and B are assumed to be in upper triangular form already, and will not be touched. IHI may not be greater than N. A (input/output) REAL array, dimension (LDA, N) On entry, the N x N upper Hessenberg matrix A. Entries below the subdiagonal must be zero. If JOB='S', then on exit A and B will have been simul- taneously 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 x N upper triangular matrix B. Entries below the diagonal must be zero. 2x2 blocks in B corresponding to 2x2 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. Entries 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 diago- nal 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 transforma- tions s.t. the diagonal of B was non-negative real. Thus, if A(j,j) is in a 1x1 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 reduc- ing A and B to Schur form and then further reducing them both to triangular form using unitary transfor- mations s.t. the diagonal of B was non-negative real. Thus, if A(j,j) is in a 1x1 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 ele- ments 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 1x1 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 orthogo- nal 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 must be at least 1. If COMPQ='V' or 'I', then LDQ must also be at least 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 transforma- tions 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 must be at least 1. If COMPZ='V' or 'I', then LDZ must also be at least N. WORK (workspace) REAL array, dimension (LWORK) On exit, if INFO is not negative, WORK(1) will be set to the optimal size of the array WORK. LWORK (input) INTEGER The number of elements in WORK. It must be at least max( 1, N ). INFO (output) INTEGER < 0: if INFO = -i, the i-th argument had an illegal value = 0: successful exit. = 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 1x1 or 2x2 block deflates off the bottom.