LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ slatrd()

subroutine slatrd ( character  UPLO,
integer  N,
integer  NB,
real, dimension( lda, * )  A,
integer  LDA,
real, dimension( * )  E,
real, dimension( * )  TAU,
real, dimension( ldw, * )  W,
integer  LDW 
)

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

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

Purpose:
 SLATRD 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', SLATRD reduces the last NB rows and columns of a
 matrix, of which the upper triangle is supplied;
 if UPLO = 'L', SLATRD reduces the first NB rows and columns of a
 matrix, of which the lower triangle is supplied.

 This is an auxiliary routine called by SSYTRD.
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 REAL 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 REAL 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 REAL 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 REAL 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 slatrd.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  REAL A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
209 * ..
210 *
211 * =====================================================================
212 *
213 * .. Parameters ..
214  REAL ZERO, ONE, HALF
215  parameter( zero = 0.0e+0, one = 1.0e+0, half = 0.5e+0 )
216 * ..
217 * .. Local Scalars ..
218  INTEGER I, IW
219  REAL ALPHA
220 * ..
221 * .. External Subroutines ..
222  EXTERNAL saxpy, sgemv, slarfg, sscal, ssymv
223 * ..
224 * .. External Functions ..
225  LOGICAL LSAME
226  REAL SDOT
227  EXTERNAL lsame, sdot
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 sgemv( 'No transpose', i, n-i, -one, a( 1, i+1 ),
250  $ lda, w( i, iw+1 ), ldw, one, a( 1, i ), 1 )
251  CALL sgemv( '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 slarfg( 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 ssymv( 'Upper', i-1, one, a, lda, a( 1, i ), 1,
266  $ zero, w( 1, iw ), 1 )
267  IF( i.LT.n ) THEN
268  CALL sgemv( 'Transpose', i-1, n-i, one, w( 1, iw+1 ),
269  $ ldw, a( 1, i ), 1, zero, w( i+1, iw ), 1 )
270  CALL sgemv( '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 sgemv( 'Transpose', i-1, n-i, one, a( 1, i+1 ),
274  $ lda, a( 1, i ), 1, zero, w( i+1, iw ), 1 )
275  CALL sgemv( '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 sscal( i-1, tau( i-1 ), w( 1, iw ), 1 )
280  alpha = -half*tau( i-1 )*sdot( i-1, w( 1, iw ), 1,
281  $ a( 1, i ), 1 )
282  CALL saxpy( 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 sgemv( 'No transpose', n-i+1, i-1, -one, a( i, 1 ),
295  $ lda, w( i, 1 ), ldw, one, a( i, i ), 1 )
296  CALL sgemv( '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 slarfg( 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 ssymv( 'Lower', n-i, one, a( i+1, i+1 ), lda,
311  $ a( i+1, i ), 1, zero, w( i+1, i ), 1 )
312  CALL sgemv( 'Transpose', n-i, i-1, one, w( i+1, 1 ), ldw,
313  $ a( i+1, i ), 1, zero, w( 1, i ), 1 )
314  CALL sgemv( '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 sgemv( 'Transpose', n-i, i-1, one, a( i+1, 1 ), lda,
317  $ a( i+1, i ), 1, zero, w( 1, i ), 1 )
318  CALL sgemv( '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 sscal( n-i, tau( i ), w( i+1, i ), 1 )
321  alpha = -half*tau( i )*sdot( n-i, w( i+1, i ), 1,
322  $ a( i+1, i ), 1 )
323  CALL saxpy( 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 SLATRD
332 *
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine ssymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SSYMV
Definition: ssymv.f:152
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
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