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ZGEHRD - reduce a complex general matrix A to upper Hessenberg form H by a 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( * )
ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q’ * 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 overwrit‐ ten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array TAU, represent the uni‐ tary matrix Q as a product of elementary reflectors. See Fur‐ ther 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) 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. 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’ 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 modi‐ fied element of the upper Hessenberg matrix H, and vi denotes an ele‐ ment of the vector defining H(i).