NAME

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

SYNOPSIS

  SUBROUTINE CGEHRD( N, ILO, IHI, A, LDA, TAU, WORKIN, LWORKIN, INFO)
  COMPLEX A(LDA,*), TAU(*), WORKIN(*)
  INTEGER N, ILO, IHI, LDA, LWORKIN, INFO

  SUBROUTINE CGEHRD_64( N, ILO, IHI, A, LDA, TAU, WORKIN, LWORKIN, 
 *      INFO)
  COMPLEX A(LDA,*), TAU(*), WORKIN(*)
  INTEGER*8 N, ILO, IHI, LDA, LWORKIN, INFO

F95 INTERFACE

  SUBROUTINE GEHRD( [N], ILO, IHI, A, [LDA], TAU, [WORKIN], [LWORKIN], 
 *       [INFO])
  COMPLEX, DIMENSION(:) :: TAU, WORKIN
  COMPLEX, DIMENSION(:,:) :: A
  INTEGER :: N, ILO, IHI, LDA, LWORKIN, INFO

  SUBROUTINE GEHRD_64( [N], ILO, IHI, A, [LDA], TAU, [WORKIN], 
 *       [LWORKIN], [INFO])
  COMPLEX, DIMENSION(:) :: TAU, WORKIN
  COMPLEX, DIMENSION(:,:) :: A
  INTEGER(8) :: N, ILO, IHI, LDA, LWORKIN, INFO

C INTERFACE

#include <sunperf.h>

void cgehrd(int n, int ilo, int ihi, complex *a, int lda, complex *tau, int *info);

void cgehrd_64(long n, long ilo, long ihi, complex *a, long lda, complex *tau, long *info);

PURPOSE

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

ARGUMENTS

N (input)
The order of the matrix A. N > = 0.
ILO (input)
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 CGEBAL; otherwise they should be set to 1 and N respectively. See Further Details.
IHI (input)
See the description of ILO.
A (input/output)
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)
The leading dimension of the array A. LDA > = max(1,N).
TAU (output)
The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
WORKIN (workspace)
On exit, if INFO = 0, WORKIN(1) returns the optimal LWORKIN.
LWORKIN (input)
The length of the array WORKIN. LWORKIN > = max(1,N). For optimum performance LWORKIN > = N*NB, where NB is the optimal blocksize.
If LWORKIN = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORKIN array, returns this value as the first entry of the WORKIN array, and no error message related to LWORKIN is issued by XERBLA.

INFO (output)

 = 0:  successful exit

 < 0:  if INFO  = -i, the i-th argument had an illegal value.

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'

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).