zgehrd (l) - Linux Man Pages
zgehrd: reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation
NAMEZGEHRD - reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation
- SUBROUTINE ZGEHRD(
- N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
- INTEGER IHI, ILO, INFO, LDA, LWORK, N
- COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
PURPOSEZGEHRD reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation: Qaq * A * Q = H .
- N (input) INTEGER
- The order of the matrix A. N >= 0.
- ILO (input) INTEGER
- IHI (input) INTEGER 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.
- A (input/output) COMPLEX*16 array, dimension (LDA,N)
- On entry, the N-by-N general matrix to be reduced. On exit, 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. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N).
- TAU (output) COMPLEX*16 array, dimension (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.
- WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
- On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
- LWORK (input) INTEGER
- The length of the array WORK. LWORK >= max(1,N). For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. 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.
FURTHER DETAILSThe matrix Q is represented as a product of (ihi-ilo) elementary reflectors
Each H(i) has the form
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).
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 file is a slight modification of LAPACK-3.0aqs ZGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2005).