zgebak zgebal zgehrd zhsein zhseqr ztrevc3 ztrexc ztrsen ztrsna ztrsyl zunghr zunmhr

ztrsen() void ztrsen ( char  job, char  compq, int  select[], int  n, int  ldt, doublecomplex  t[], int  ldq, doublecomplex  q[], doublecomplex  w[], int *  m, double *  s, double *  sep, doublecomplex  work[], int  lwork, int *  info  ) Reordering of Schur factorization of complex matrix and condition numbers of cluster of eigenvalues and/or invariant subspace Purpose This routine reorders the Schur factorization of a complex matrix A = Q*T*Q^H, so that a selected cluster of eigenvalues appears in the leading positions on the diagonal of the upper triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace. Optionally the routine computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace. Parameters [in] job Specifies whether condition numbers are required for the cluster of eigenvalues (s) or the invariant subspace (sep). = 'N': none. = 'E': For eigenvalues only (s). = 'V': For invariant subspace only (sep). = 'B': For both eigenvalues and invariant subspace (s and sep). [in] compq = 'V': Update the matrix Q of Schur vectors. = 'N': Do not update Q. [in] select[] Array select[lselect] (lselect >= n) select[] specifies the eigenvalues in the selected cluster. To select the j-th eigenvalue, select[j] must be set to true. [in] n Order of the matrix T. (n >= 0) (If n = 0, returns without computation) [in] ldt Leading dimension of the two dimensional array t[][]. (ldt >= max(1, n)) [in,out] t[][] Array t[lt][ldt] (lt >= n) [in] The upper triangular matrix T. [out] T is overwritten by the reordered matrix T, with the selected eigenvalues as the leading diagonal elements. [in] ldq Leading dimension of the two dimensional array q[][]. (ldq >= 1 if compq = 'N', ld1 >= n if compq = 'V') [in] q[][] Array q[lq][ldq] (lq >= n) If compq = 'V', [in] The matrix Q of Schur vectors. [out] Q has been postmultiplied by the unitary transformation matrix which reorders T. The leading m columns of Q form an orthonormal basis for the specified invariant subspace. If compq = 'N', q[][] is not referenced. [out] w[] Array w[lw] (lw >= n) The reordered eigenvalues of T, in the same order as they appear on the diagonal of T. [out] m The dimension of the specified invariant subspace. (0 < = m <= n) [out] s If job = 'E' or 'B', s is a lower bound on the reciprocal condition number for the selected cluster of eigenvalues. s cannot underestimate the true reciprocal condition number by more than a factor of sqrt(n). If m = 0 or n, s = 1. If job = 'N' or 'V', s is not referenced. [out] sep If job = 'V' or 'B', sep is the estimated reciprocal condition number of the specified invariant subspace. If m = 0 or n, sep = norm(T). If job = 'N' or 'E', sep is not referenced. [out] work[] Array work[lwork] Complex work array. On exit, if info = 0, work[0] returns the optimal lwork. [in] lwork The dimension of the array work[]. (lwork >= 1 if job = 'N', lwork >= max(1, m*(n-m)) if job = 'E', lwork >= max(1, 2*m*(n-m)) if job = 'V' or 'B') If lwork = -1, then a workspace query is assumed. The routine only calculates the optimal size of the work[] array, and returns the value in work[0]. [out] info = 0: Successful exit = -1: The argument job had an illegal value (job != 'N', 'E', 'V' nor 'B') = -2: The argument compq had an illegal value (compq != 'V' nor 'N') = -4: The argument n had an illegal value (n < 0) = -5: The argument ldt had an illegal value (ldt < max(1, n)) = -7: The argument ldq had an illegal value (ldq too small) = -14: The argument lwork had an illegal value (lwork too small) Further Details This routine first collects the selected eigenvalues by computing an unitary transformation Z to move them to the top left corner of T. In other words, the selected eigenvalues are the eigenvalues of T11 in: Z^H * T * Z = ( T11 T12 ) n1 ( 0 T22 ) n2 n1 n2 where n = n1 + n2. The first n1 columns of Z span the specified invariant subspace of T. If T has been obtained from the Schur factorization of a matrix A = Q*T*Q^H, then the reordered Schur factorization of A is given by A = (Q*Z)*(Z^H*T*Z)*(Q*Z)^H, and the first n1 columns of Q*Z span the corresponding invariant subspace of A. The reciprocal condition number of the average of the eigenvalues of T11 may be returned in s. s lies between 0 (very badly conditioned) and 1 (very well conditioned). It is computed as follows. First we compute R so that P = ( I R ) n1 ( 0 0 ) n2 n1 n2 is the projector on the invariant subspace associated with T11. R is the solution of the Sylvester equation: T11*R - R*T22 = T12. Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote the two-norm of M. Then s is computed as the lower bound (1 + F-norm(R)^2)^(-1/2) on the reciprocal of 2-norm(P), the true reciprocal condition number. s cannot underestimate 1 / 2-norm(P) by more than a factor of sqrt(n). An approximate error bound for the computed average of the eigenvalues of T11 is eps * norm(T) / s where eps is the machine precision. The reciprocal condition number of the right invariant subspace spanned by the first n1 columns of Z (or of Q*Z) is returned in sep. sep is defined as the separation of T11 and T22: sep( T11, T22 ) = sigma-min( C ) where sigma-min(C) is the smallest singular value of the n1*n2 x n1*n2 matrix C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) I(m) is an m x m identity matrix, and kprod denotes the Kronecker product. We estimate sigma-min(C) by the reciprocal of an estimate of the 1-norm of C^(-1). The true reciprocal 1-norm of C^(-1) cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). When sep is small, small changes in T can cause large changes in the invariant subspace. An approximate bound on the maximum angular error in the computed right invariant subspace is eps * norm(T) / sep(i) Reference LAPACK