SRC/claqz0.f(3) Library Functions Manual SRC/claqz0.f(3) NAME SRC/claqz0.f SYNOPSIS Functions/Subroutines recursive subroutine claqz0 (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info) CLAQZ0 Function/Subroutine Documentation recursive subroutine claqz0 (character, intent(in) wants, character, intent(in) wantq, character, intent(in) wantz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, complex, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex, dimension( * ), intent(inout) alpha, complex, dimension( * ), intent(inout) beta, complex, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, complex, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, complex, dimension( * ), intent(inout) work, integer, intent(in) lwork, real, dimension( * ), intent(out) rwork, integer, intent(in) rec, integer, intent(out) info) CLAQZ0 Purpose: CLAQZ0 computes the eigenvalues of a 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 matrix pair (A,B): A = Q1*H*Z1**H, B = Q1*T*Z1**H, as computed by CGGHRD. If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form, H = Q*S*Z**H, T = Q*P*Z**H, where Q and Z are unitary matrices, P and S are an upper triangular matrices. Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized upper Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B): A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. 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. 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, 'An Algorithm for Generalized Matrix Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973), pp. 241--256. Ref: B. Kagstrom, D. Kressner, 'Multishift Variants of the QZ Algorithm with Aggressive Early Deflation', SIAM J. Numer. Anal., 29(2006), pp. 199--227. Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril 'A multishift, multipole rational QZ method with aggressive early deflation' Parameters WANTS WANTS is CHARACTER*1 = 'E': Compute eigenvalues only; = 'S': Compute eigenvalues and the Schur form. WANTQ WANTQ 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 (A,B) is returned; = 'V': Q must contain an unitary matrix Q1 on entry and the product Q1*Q is returned. WANTZ WANTZ 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 (A,B) is returned; = 'V': Z must contain an unitary matrix Z1 on entry and the product Z1*Z is returned. N N is INTEGER The order of the matrices A, B, Q, and Z. N >= 0. ILO ILO is INTEGER IHI IHI is INTEGER ILO and IHI mark the rows and columns of A 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. A A is COMPLEX array, dimension (LDA, N) On entry, the N-by-N upper Hessenberg matrix A. On exit, if JOB = 'S', A contains the upper triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal of A matches that of S, but the rest of A is unspecified. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max( 1, N ). B B is COMPLEX array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, if JOB = 'S', B contains the upper triangular matrix P from the generalized Schur factorization. If JOB = 'E', the diagonal of B matches that of P, but the rest of B is unspecified. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max( 1, N ). ALPHA ALPHA is COMPLEX array, dimension (N) Each scalar alpha defining an eigenvalue of GNEP. BETA BETA is COMPLEX array, dimension (N) The scalars beta that define the eigenvalues of GNEP. Together, the quantities alpha = ALPHA(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 COMPLEX array, dimension (LDQ, N) On entry, if COMPQ = 'V', the unitary matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPQ = 'I', the unitary matrix of left Schur vectors of (A,B), and if COMPQ = 'V', the unitary 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 COMPLEX array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPZ = 'I', the unitary matrix of right Schur vectors of (H,T), and if COMPZ = 'V', the unitary 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 COMPLEX 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. RWORK RWORK is REAL array, dimension (N) REC REC is INTEGER REC indicates the current recursion level. Should be set to 0 on first call. 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. (A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct. Author Thijs Steel, KU Leuven Date May 2020 Definition at line 280 of file claqz0.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 SRC/claqz0.f(3)