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

zgehrd() void zgehrd ( int  n, int  ilo, int  ihi, int  lda, doublecomplex  a[], doublecomplex  tau[], doublecomplex  work[], int  lwork, int *  info  ) Reduces a complex general matrix to upper Hessenberg form Purpose This routine reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation: Q^H * A * Q = H. Parameters [in] n Order of the matrix A. (n >= 0) (If n = 0, returns without computation) [in] ilo [in] ihi It is assumed that A is already upper triangular in rows and columns 1〜ilo-1 and ihi+1〜n. ilo and ihi are normally set by a previous call to zgebal. Otherwise they should be set to 1 and n respectively. See Further Details. (1 <= ilo <= ihi <= n, if n > 0. ilo = 1 and ihi = 0, if n = 0) [in] lda Leading dimension of the two dimensional array a[][]. (lda >= max(1, n)) [in,out] a[][] Array a[la][lda] (la >= n) [in] n x n general matrix to be reduced. [out] The upper triangle and the first subdiagonal of a[][] are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array tau, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. [out] tau[] Array tau[ltau] (ltau >= n - 1) The scalar factors of the elementary reflectors (see Further Details). Elements 1〜ilo-1 and ihi〜n-1 of tau are set to zero. [out] work[] Array work[lwork] Work array. On exit, if info = 0, work[0] returns the optimal lwork. [in] lwork The dimension of the array work[]. (lwork >= max(1, n)) For good performance, lwork must generally be larger. 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 n had an illegal value (n < 0) = -2: The argument ilo had an illegal value (ilo < 1 or ilo > n) = -3: The argument ihi had an illegal value (ihi < min(ilo, n) or ihi > n) = -4: The argument lda had an illegal value (lda < max(1, n)) = -8: The argument lwork had an illegal value (lwork too small) Further Details The matrix Q is represented as a product of (ihi-ilo) elementary reflectors Q = H(ilo) H(ilo+1) . . . H(ihi-1). Each H(i) has the form H(i) = I - tau * v * v^H where tau is a complex scalar, and v is a complex vector with v(1〜i) = 0, v(i+1) = 1 and v(ihi+1〜n) = 0, v(i+2〜ihi) is stored on exit in A(i+2〜ihi, i), and tau in tau[i - 1]. The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6: on entry, on exit, ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). This routine is a slight modification of LAPACK-3.0's ZGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2006). (See ZLAHR2.) Reference LAPACK