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### NAME

ZGEHD2 - reduce a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation

### SYNOPSIS

SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO ) INTEGER IHI, ILO, INFO, LDA, N COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )

### PURPOSE

ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q’ * A * Q = H .

### ARGUMENTS

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). WORK (workspace) COMPLEX*16 array, dimension (N) 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).