LAPACK 3.12.0 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).```
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*
subroutine daxpy(n, da, dx, incx, dy, incy)
DAXPY
Definition daxpy.f:89
double precision function ddot(n, dx, incx, dy, incy)
DDOT
Definition ddot.f:82
subroutine dgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
DGEMV
Definition dgemv.f:158
subroutine dsymv(uplo, n, alpha, a, lda, x, incx, beta, y, incy)
DSYMV
Definition dsymv.f:152
subroutine dlarfg(n, alpha, x, incx, tau)
DLARFG generates an elementary reflector (Householder matrix).
Definition dlarfg.f:106
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
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