LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ dlatrd()

subroutine dlatrd ( character  UPLO,
integer  N,
integer  NB,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( * )  E,
double precision, dimension( * )  TAU,
double precision, dimension( ldw, * )  W,
integer  LDW 
)

DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Download DLATRD + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DLATRD reduces NB rows and columns of a real symmetric matrix A to
 symmetric tridiagonal form by an orthogonal similarity
 transformation Q**T * A * Q, and returns the matrices V and W which are
 needed to apply the transformation 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.
Parameters
[in]UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U': Upper triangular
          = 'L': Lower triangular
[in]N
          N is INTEGER
          The order of the matrix A.
[in]NB
          NB is INTEGER
          The number of rows and columns to be reduced.
[in,out]A
          A is 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 contains 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
          triangular 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 diagonal elements overwriting
            the diagonal elements of A; the elements above the diagonal
            with the array TAU, represent the orthogonal matrix Q as a
            product 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.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= (1,N).
[out]E
          E is DOUBLE PRECISION array, dimension (N-1)
          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
          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.
[out]TAU
          TAU is 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.
[out]W
          W is DOUBLE PRECISION array, dimension (LDW,NB)
          The n-by-nb matrix W required to update the unreduced part
          of A.
[in]LDW
          LDW is INTEGER
          The leading dimension of the array W. LDW >= max(1,N).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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**T

  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**T

  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 transformation to the unreduced
  part of the matrix, using a symmetric rank-2k update of the form:
  A := A - V*W**T - W*V**T.

  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).

Definition at line 197 of file dlatrd.f.

198*
199* -- LAPACK auxiliary routine --
200* -- LAPACK is a software package provided by Univ. of Tennessee, --
201* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
202*
203* .. Scalar Arguments ..
204 CHARACTER UPLO
205 INTEGER LDA, LDW, N, NB
206* ..
207* .. Array Arguments ..
208 DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
209* ..
210*
211* =====================================================================
212*
213* .. Parameters ..
214 DOUBLE PRECISION ZERO, ONE, HALF
215 parameter( zero = 0.0d+0, one = 1.0d+0, half = 0.5d+0 )
216* ..
217* .. Local Scalars ..
218 INTEGER I, IW
219 DOUBLE PRECISION ALPHA
220* ..
221* .. External Subroutines ..
222 EXTERNAL daxpy, dgemv, dlarfg, dscal, dsymv
223* ..
224* .. External Functions ..
225 LOGICAL LSAME
226 DOUBLE PRECISION DDOT
227 EXTERNAL lsame, ddot
228* ..
229* .. Intrinsic Functions ..
230 INTRINSIC min
231* ..
232* .. Executable Statements ..
233*
234* Quick return if possible
235*
236 IF( n.LE.0 )
237 $ RETURN
238*
239 IF( lsame( uplo, 'U' ) ) THEN
240*
241* Reduce last NB columns of upper triangle
242*
243 DO 10 i = n, n - nb + 1, -1
244 iw = i - n + nb
245 IF( i.LT.n ) THEN
246*
247* Update A(1:i,i)
248*
249 CALL dgemv( 'No transpose', i, n-i, -one, a( 1, i+1 ),
250 $ lda, w( i, iw+1 ), ldw, one, a( 1, i ), 1 )
251 CALL dgemv( 'No transpose', i, n-i, -one, w( 1, iw+1 ),
252 $ ldw, a( i, i+1 ), lda, one, a( 1, i ), 1 )
253 END IF
254 IF( i.GT.1 ) THEN
255*
256* Generate elementary reflector H(i) to annihilate
257* A(1:i-2,i)
258*
259 CALL dlarfg( i-1, a( i-1, i ), a( 1, i ), 1, tau( i-1 ) )
260 e( i-1 ) = a( i-1, i )
261 a( i-1, i ) = one
262*
263* Compute W(1:i-1,i)
264*
265 CALL dsymv( 'Upper', i-1, one, a, lda, a( 1, i ), 1,
266 $ zero, w( 1, iw ), 1 )
267 IF( i.LT.n ) THEN
268 CALL dgemv( 'Transpose', i-1, n-i, one, w( 1, iw+1 ),
269 $ ldw, a( 1, i ), 1, zero, w( i+1, iw ), 1 )
270 CALL dgemv( 'No transpose', i-1, n-i, -one,
271 $ a( 1, i+1 ), lda, w( i+1, iw ), 1, one,
272 $ w( 1, iw ), 1 )
273 CALL dgemv( 'Transpose', i-1, n-i, one, a( 1, i+1 ),
274 $ lda, a( 1, i ), 1, zero, w( i+1, iw ), 1 )
275 CALL dgemv( 'No transpose', i-1, n-i, -one,
276 $ w( 1, iw+1 ), ldw, w( i+1, iw ), 1, one,
277 $ w( 1, iw ), 1 )
278 END IF
279 CALL dscal( i-1, tau( i-1 ), w( 1, iw ), 1 )
280 alpha = -half*tau( i-1 )*ddot( i-1, w( 1, iw ), 1,
281 $ a( 1, i ), 1 )
282 CALL daxpy( i-1, alpha, a( 1, i ), 1, w( 1, iw ), 1 )
283 END IF
284*
285 10 CONTINUE
286 ELSE
287*
288* Reduce first NB columns of lower triangle
289*
290 DO 20 i = 1, nb
291*
292* Update A(i:n,i)
293*
294 CALL dgemv( 'No transpose', n-i+1, i-1, -one, a( i, 1 ),
295 $ lda, w( i, 1 ), ldw, one, a( i, i ), 1 )
296 CALL dgemv( 'No transpose', n-i+1, i-1, -one, w( i, 1 ),
297 $ ldw, a( i, 1 ), lda, one, a( i, i ), 1 )
298 IF( i.LT.n ) THEN
299*
300* Generate elementary reflector H(i) to annihilate
301* A(i+2:n,i)
302*
303 CALL dlarfg( n-i, a( i+1, i ), a( min( i+2, n ), i ), 1,
304 $ tau( i ) )
305 e( i ) = a( i+1, i )
306 a( i+1, i ) = one
307*
308* Compute W(i+1:n,i)
309*
310 CALL dsymv( 'Lower', n-i, one, a( i+1, i+1 ), lda,
311 $ a( i+1, i ), 1, zero, w( i+1, i ), 1 )
312 CALL dgemv( 'Transpose', n-i, i-1, one, w( i+1, 1 ), ldw,
313 $ a( i+1, i ), 1, zero, w( 1, i ), 1 )
314 CALL dgemv( 'No transpose', n-i, i-1, -one, a( i+1, 1 ),
315 $ lda, w( 1, i ), 1, one, w( i+1, i ), 1 )
316 CALL dgemv( 'Transpose', n-i, i-1, one, a( i+1, 1 ), lda,
317 $ a( i+1, i ), 1, zero, w( 1, i ), 1 )
318 CALL dgemv( 'No transpose', n-i, i-1, -one, w( i+1, 1 ),
319 $ ldw, w( 1, i ), 1, one, w( i+1, i ), 1 )
320 CALL dscal( n-i, tau( i ), w( i+1, i ), 1 )
321 alpha = -half*tau( i )*ddot( n-i, w( i+1, i ), 1,
322 $ a( i+1, i ), 1 )
323 CALL daxpy( n-i, alpha, a( i+1, i ), 1, w( i+1, i ), 1 )
324 END IF
325*
326 20 CONTINUE
327 END IF
328*
329 RETURN
330*
331* End of DLATRD
332*
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function ddot(N, DX, INCX, DY, INCY)
DDOT
Definition: ddot.f:82
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DSYMV
Definition: dsymv.f:152
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:156
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
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