USAGE: alphar, alphai, beta, work, info, h, t, q, z = NumRu::Lapack.dhgeqz( job, compq, compz, ilo, ihi, h, t, q, z, [:lwork => lwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO ) * Purpose * ======= * * DHGEQZ 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 DGGHRD. * * 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. * * 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 DGGHRD 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, "An Algorithm for Generalized Matrix * Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), * pp. 241--256. * * Arguments * ========= * * JOB (input) CHARACTER*1 * = 'E': Compute eigenvalues only; * = 'S': Compute eigenvalues and the Schur form. * * COMPQ (input) 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 (input) 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 (input) INTEGER * The order of the matrices H, T, Q, and Z. N >= 0. * * ILO (input) INTEGER * IHI (input) 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 (input/output) DOUBLE PRECISION 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; * 2-by-2 diagonal blocks (corresponding to complex conjugate * pairs of eigenvalues) are returned in standard form, with * H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. * If JOB = 'E', the diagonal blocks of H match those of S, but * the rest of H is unspecified. * * LDH (input) INTEGER * The leading dimension of the array H. LDH >= max( 1, N ). * * T (input/output) DOUBLE PRECISION 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 (input) INTEGER * The leading dimension of the array T. LDT >= max( 1, N ). * * ALPHAR (output) DOUBLE PRECISION array, dimension (N) * The real parts of each scalar alpha defining an eigenvalue * of GNEP. * * ALPHAI (output) DOUBLE PRECISION 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 (output) DOUBLE PRECISION 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. * * Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) * On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in * the reduction of (A,B) to generalized Hessenberg form. * On exit, if COMPZ = 'I', the orthogonal matrix of left Schur * vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix * of left Schur vectors of (A,B). * Not referenced if COMPZ = 'N'. * * 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) * 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 (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 (MAX(1,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. (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. * * 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. * * ===================================================================== *go to the page top

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