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dlatrd (3)
  • >> dlatrd (3) ( Solaris man: Библиотечные вызовы )
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    NAME
         dlatrd - reduce NB rows and  columns  of  a  real  symmetric
         matrix  A  to  symmetric  tridiagonal  form by an orthogonal
         similarity transformation Q'  *  A  *  Q,  and  returns  the
         matrices  V  and W which are needed to apply the transforma-
         tion to the unreduced part of A
    
    SYNOPSIS
         SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
    
         CHARACTER UPLO
    
         INTEGER LDA, LDW, N, NB
    
         DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
    
    
    
         #include <sunperf.h>
    
         void dlatrd(char uplo, int n, int nb, double *da,  int  lda,
                   double *e, double *tau, double *w, int ldw) ;
    
    PURPOSE
         DLATRD reduces NB rows  and  columns  of  a  real  symmetric
         matrix  A  to  symmetric  tridiagonal  form by an orthogonal
         similarity transformation Q'  *  A  *  Q,  and  returns  the
         matrices  V  and W which are needed to apply the transforma-
         tion to the unreduced part of A.
    
         If UPLO = 'U', DLATRD reduces the last NB rows  and  columns
         of a matrix, of which the upper triangle is supplied;
         if UPLO = 'L', DLATRD reduces the first NB rows and  columns
         of a matrix, of which the lower triangle is supplied.
    
         This is an auxiliary routine called by DSYTRD.
    
    
    ARGUMENTS
         UPLO      (input) CHARACTER
                   Specifies whether the upper  or  lower  triangular
                   part of the symmetric matrix A is stored:
                   = 'U': Upper triangular
                   = 'L': Lower triangular
    
         N         (input) INTEGER
                   The order of the matrix A.
    
         NB        (input) INTEGER
                   The number of rows and columns to be reduced.
    
         A         (input/output) DOUBLE PRECISION  array,  dimension
                   (LDA,N)
                   On entry, the symmetric matrix A.  If UPLO =  'U',
                   the leading n-by-n upper triangular part of A con-
                   tains the upper triangular part of the  matrix  A,
                   and the strictly lower triangular part of A is not
                   referenced.  If UPLO =  'L',  the  leading  n-by-n
                   lower triangular part of A contains the lower tri-
                   angular part of the matrix  A,  and  the  strictly
                   upper  triangular part of A is not referenced.  On
                   exit:  if UPLO = 'U', the  last  NB  columns  have
                   been  reduced to tridiagonal form, with the diago-
                   nal elements overwriting the diagonal elements  of
                   A;  the elements above the diagonal with the array
                   TAU, represent the orthogonal matrix Q as  a  pro-
                   duct  of elementary reflectors; if UPLO = 'L', the
                   first NB columns have been reduced to  tridiagonal
                   form,  with  the diagonal elements overwriting the
                   diagonal elements of A;  the  elements  below  the
                   diagonal   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
                   >= (1,N).
    
         E         (output) DOUBLE PRECISION array, dimension (N-1)
                   If UPLO = 'U', E(n-nb:n-1) contains the superdiag-
                   onal  elements  of  the  last  NB  columns  of the
                   reduced matrix; if UPLO =  'L',  E(1:nb)  contains
                   the  subdiagonal  elements of the first NB columns
                   of the reduced matrix.
    
         TAU       (output) DOUBLE PRECISION array, dimension (N-1)
                   The scalar factors of the  elementary  reflectors,
                   stored  in  TAU(n-nb:n-1)  if  UPLO  = 'U', and in
                   TAU(1:nb) if UPLO = 'L'.  See Further Details.   W
                   (output)   DOUBLE   PRECISION   array,   dimension
                   (LDW,NB) The n-by-nb matrix W required  to  update
                   the unreduced part of A.
    
         LDW       (input) INTEGER
                   The leading dimension  of  the  array  W.  LDW  >=
                   max(1,N).
    
    FURTHER DETAILS
         If UPLO = 'U', the matrix Q is represented as a  product  of
         elementary reflectors
    
            Q = H(n) H(n-1) . . . H(n-nb+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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is  stored  on  exit  in
         A(1:i-1,i), and tau in TAU(i-1).
    
         If UPLO = 'L', the matrix Q is represented as a  product  of
         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 real scalar, and v is a real vector with
         v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is  stored  on  exit  in
         A(i+1:n,i), and tau in TAU(i).
    
         The elements of the vectors  v  together  form  the  n-by-nb
         matrix  V  which is needed, with W, to apply the transforma-
         tion to the unreduced part of the matrix, using a  symmetric
         rank-2k update of the form:  A := A - V*W' - W*V'.
    
         The contents of A on exit are illustrated by  the  following
         examples with n = 5 and nb = 2:
    
         if UPLO = 'U':                       if UPLO = 'L':
    
           (  a   a   a   v4  v5 )              (  d                  )
           (      a   a   v4  v5 )              (  1   d              )
           (          a   1   v5 )              (  v1  1   a          )
           (              d   1  )              (  v1  v2  a   a      )
           (                  d  )              (  v1  v2  a   a   a  )
    
         where d denotes a diagonal element of  the  reduced  matrix,  a
         denotes  an  element  of the original matrix that is unchanged,
         and vi denotes an element of the vector defining H(i).
    
    
    
    


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