LAPACK  3.10.1 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.

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).```
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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