sgehrd(3S)

NAME
     SGEHRD - reduce a real general matrix A to upper Hessenberg form H by an
     orthogonal similarity transformation

SYNOPSIS
     SUBROUTINE SGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )

         INTEGER        IHI, ILO, INFO, LDA, LWORK, N

         REAL           A( LDA, * ), TAU( * ), WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     SGEHRD reduces a real general matrix A to upper Hessenberg form H by an
     orthogonal 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 SGEBAL; otherwise they
             should be set to 1 and N respectively. See Further Details.

     A       (input/output) REAL 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 orthogonal
             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) REAL 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) REAL 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 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 real scalar, and v is a real 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).


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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