IRIX Base Documentation 2002 November/SGI IRIX Base Documentation 2002 November.iso/usr/share/catman/p_man/cat3/SCSL/zlahrd.z/zlahrd

Text File | 2002-10-03 | 5.5 KB | 199 lines

ZZZZLLLLAAAAHHHHRRRRDDDD((((3333SSSS))))                                                          ZZZZLLLLAAAAHHHHRRRRDDDD((((3333SSSS))))



NNNNAAAAMMMMEEEE
     ZLAHRD - reduce the first NB columns of a complex general n-by-(n-k+1)
     matrix A so that elements below the k-th subdiagonal are zero

SSSSYYYYNNNNOOOOPPPPSSSSIIIISSSS
     SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

         INTEGER        K, LDA, LDT, LDY, N, NB

         COMPLEX*16     A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

IIIIMMMMPPPPLLLLEEEEMMMMEEEENNNNTTTTAAAATTTTIIIIOOOONNNN
     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.

PPPPUUUURRRRPPPPOOOOSSSSEEEE
     ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1)
     matrix A so that elements below the k-th subdiagonal are zero. The
     reduction is performed by a unitary similarity transformation Q' * A * Q.
     The routine returns the matrices V and T which determine Q as a block
     reflector I - V*T*V', and also the matrix Y = A * V * T.

     This is an auxiliary routine called by ZGEHRD.


AAAARRRRGGGGUUUUMMMMEEEENNNNTTTTSSSS
     N       (input) INTEGER
             The order of the matrix A.

     K       (input) INTEGER
             The offset for the reduction. Elements below the k-th subdiagonal
             in the first NB columns are reduced to zero.

     NB      (input) INTEGER
             The number of columns to be reduced.

     A       (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)
             On entry, the n-by-(n-k+1) general matrix A.  On exit, the
             elements on and above the k-th subdiagonal in the first NB
             columns are overwritten with the corresponding elements of the
             reduced matrix; the elements below the k-th subdiagonal, with the
             array TAU, represent the matrix Q as a product of elementary
             reflectors. The other columns of A are unchanged. See Further



                                                                        PPPPaaaaggggeeee 1111






ZZZZLLLLAAAAHHHHRRRRDDDD((((3333SSSS))))                                                          ZZZZLLLLAAAAHHHHRRRRDDDD((((3333SSSS))))



             Details.  LDA     (input) INTEGER The leading dimension of the
             array A.  LDA >= max(1,N).

     TAU     (output) COMPLEX*16 array, dimension (NB)
             The scalar factors of the elementary reflectors. See Further
             Details.

     T       (output) COMPLEX*16 array, dimension (LDT,NB)
             The upper triangular matrix T.

     LDT     (input) INTEGER
             The leading dimension of the array T.  LDT >= NB.

     Y       (output) COMPLEX*16 array, dimension (LDY,NB)
             The n-by-nb matrix Y.

     LDY     (input) INTEGER
             The leading dimension of the array Y. LDY >= max(1,N).

FFFFUUUURRRRTTTTHHHHEEEERRRR DDDDEEEETTTTAAAAIIIILLLLSSSS
     The matrix Q is represented as a product of nb elementary reflectors

        Q = H(1) H(2) . . . H(nb).

     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+k-1)
     = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in
     TAU(i).

     The elements of the vectors v together form the (n-k+1)-by-nb matrix V
     which is needed, with T and Y, to apply the transformation to the
     unreduced part of the matrix, using an update of the form:  A := (I -
     V*T*V') * (A - Y*V').

     The contents of A on exit are illustrated by the following example with n
     = 7, k = 3 and nb = 2:

        ( a   h   a   a   a )
        ( a   h   a   a   a )
        ( a   h   a   a   a )
        ( h   h   a   a   a )
        ( v1  h   a   a   a )
        ( v1  v2  a   a   a )
        ( v1  v2  a   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).




                                                                        PPPPaaaaggggeeee 2222






ZZZZLLLLAAAAHHHHRRRRDDDD((((3333SSSS))))                                                          ZZZZLLLLAAAAHHHHRRRRDDDD((((3333SSSS))))



SSSSEEEEEEEE AAAALLLLSSSSOOOO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.



















































                                                                        PPPPaaaaggggeeee 3333