LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ clatrd()

subroutine clatrd ( character  uplo,
integer  n,
integer  nb,
complex, dimension( lda, * )  a,
integer  lda,
real, dimension( * )  e,
complex, dimension( * )  tau,
complex, dimension( ldw, * )  w,
integer  ldw 
)

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

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

Purpose:
 CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
 Hermitian tridiagonal form by a unitary similarity
 transformation Q**H * 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', CLATRD reduces the last NB rows and columns of a
 matrix, of which the upper triangle is supplied;
 if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
 matrix, of which the lower triangle is supplied.

 This is an auxiliary routine called by CHETRD.
Parameters
[in]UPLO
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian 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 COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian 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 unitary 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  unitary matrix Q as a
            product of elementary reflectors.
          See Further Details.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(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 COMPLEX 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 COMPLEX 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**H

  where tau is a complex scalar, and v is a complex 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**H

  where tau is a complex scalar, and v is a complex 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 Hermitian rank-2k update of the form:
  A := A - V*W**H - W*V**H.

  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 198 of file clatrd.f.

199*
200* -- LAPACK auxiliary routine --
201* -- LAPACK is a software package provided by Univ. of Tennessee, --
202* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
203*
204* .. Scalar Arguments ..
205 CHARACTER UPLO
206 INTEGER LDA, LDW, N, NB
207* ..
208* .. Array Arguments ..
209 REAL E( * )
210 COMPLEX A( LDA, * ), TAU( * ), W( LDW, * )
211* ..
212*
213* =====================================================================
214*
215* .. Parameters ..
216 COMPLEX ZERO, ONE, HALF
217 parameter( zero = ( 0.0e+0, 0.0e+0 ),
218 $ one = ( 1.0e+0, 0.0e+0 ),
219 $ half = ( 0.5e+0, 0.0e+0 ) )
220* ..
221* .. Local Scalars ..
222 INTEGER I, IW
223 COMPLEX ALPHA
224* ..
225* .. External Subroutines ..
226 EXTERNAL caxpy, cgemv, chemv, clacgv, clarfg, cscal
227* ..
228* .. External Functions ..
229 LOGICAL LSAME
230 COMPLEX CDOTC
231 EXTERNAL lsame, cdotc
232* ..
233* .. Intrinsic Functions ..
234 INTRINSIC min, real
235* ..
236* .. Executable Statements ..
237*
238* Quick return if possible
239*
240 IF( n.LE.0 )
241 $ RETURN
242*
243 IF( lsame( uplo, 'U' ) ) THEN
244*
245* Reduce last NB columns of upper triangle
246*
247 DO 10 i = n, n - nb + 1, -1
248 iw = i - n + nb
249 IF( i.LT.n ) THEN
250*
251* Update A(1:i,i)
252*
253 a( i, i ) = real( a( i, i ) )
254 CALL clacgv( n-i, w( i, iw+1 ), ldw )
255 CALL cgemv( 'No transpose', i, n-i, -one, a( 1, i+1 ),
256 $ lda, w( i, iw+1 ), ldw, one, a( 1, i ), 1 )
257 CALL clacgv( n-i, w( i, iw+1 ), ldw )
258 CALL clacgv( n-i, a( i, i+1 ), lda )
259 CALL cgemv( 'No transpose', i, n-i, -one, w( 1, iw+1 ),
260 $ ldw, a( i, i+1 ), lda, one, a( 1, i ), 1 )
261 CALL clacgv( n-i, a( i, i+1 ), lda )
262 a( i, i ) = real( a( i, i ) )
263 END IF
264 IF( i.GT.1 ) THEN
265*
266* Generate elementary reflector H(i) to annihilate
267* A(1:i-2,i)
268*
269 alpha = a( i-1, i )
270 CALL clarfg( i-1, alpha, a( 1, i ), 1, tau( i-1 ) )
271 e( i-1 ) = real( alpha )
272 a( i-1, i ) = one
273*
274* Compute W(1:i-1,i)
275*
276 CALL chemv( 'Upper', i-1, one, a, lda, a( 1, i ), 1,
277 $ zero, w( 1, iw ), 1 )
278 IF( i.LT.n ) THEN
279 CALL cgemv( 'Conjugate transpose', i-1, n-i, one,
280 $ w( 1, iw+1 ), ldw, a( 1, i ), 1, zero,
281 $ w( i+1, iw ), 1 )
282 CALL cgemv( 'No transpose', i-1, n-i, -one,
283 $ a( 1, i+1 ), lda, w( i+1, iw ), 1, one,
284 $ w( 1, iw ), 1 )
285 CALL cgemv( 'Conjugate transpose', i-1, n-i, one,
286 $ a( 1, i+1 ), lda, a( 1, i ), 1, zero,
287 $ w( i+1, iw ), 1 )
288 CALL cgemv( 'No transpose', i-1, n-i, -one,
289 $ w( 1, iw+1 ), ldw, w( i+1, iw ), 1, one,
290 $ w( 1, iw ), 1 )
291 END IF
292 CALL cscal( i-1, tau( i-1 ), w( 1, iw ), 1 )
293 alpha = -half*tau( i-1 )*cdotc( i-1, w( 1, iw ), 1,
294 $ a( 1, i ), 1 )
295 CALL caxpy( i-1, alpha, a( 1, i ), 1, w( 1, iw ), 1 )
296 END IF
297*
298 10 CONTINUE
299 ELSE
300*
301* Reduce first NB columns of lower triangle
302*
303 DO 20 i = 1, nb
304*
305* Update A(i:n,i)
306*
307 a( i, i ) = real( a( i, i ) )
308 CALL clacgv( i-1, w( i, 1 ), ldw )
309 CALL cgemv( 'No transpose', n-i+1, i-1, -one, a( i, 1 ),
310 $ lda, w( i, 1 ), ldw, one, a( i, i ), 1 )
311 CALL clacgv( i-1, w( i, 1 ), ldw )
312 CALL clacgv( i-1, a( i, 1 ), lda )
313 CALL cgemv( 'No transpose', n-i+1, i-1, -one, w( i, 1 ),
314 $ ldw, a( i, 1 ), lda, one, a( i, i ), 1 )
315 CALL clacgv( i-1, a( i, 1 ), lda )
316 a( i, i ) = real( a( i, i ) )
317 IF( i.LT.n ) THEN
318*
319* Generate elementary reflector H(i) to annihilate
320* A(i+2:n,i)
321*
322 alpha = a( i+1, i )
323 CALL clarfg( n-i, alpha, a( min( i+2, n ), i ), 1,
324 $ tau( i ) )
325 e( i ) = real( alpha )
326 a( i+1, i ) = one
327*
328* Compute W(i+1:n,i)
329*
330 CALL chemv( 'Lower', n-i, one, a( i+1, i+1 ), lda,
331 $ a( i+1, i ), 1, zero, w( i+1, i ), 1 )
332 CALL cgemv( 'Conjugate transpose', n-i, i-1, one,
333 $ w( i+1, 1 ), ldw, a( i+1, i ), 1, zero,
334 $ w( 1, i ), 1 )
335 CALL cgemv( 'No transpose', n-i, i-1, -one, a( i+1, 1 ),
336 $ lda, w( 1, i ), 1, one, w( i+1, i ), 1 )
337 CALL cgemv( 'Conjugate transpose', n-i, i-1, one,
338 $ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
339 $ w( 1, i ), 1 )
340 CALL cgemv( 'No transpose', n-i, i-1, -one, w( i+1, 1 ),
341 $ ldw, w( 1, i ), 1, one, w( i+1, i ), 1 )
342 CALL cscal( n-i, tau( i ), w( i+1, i ), 1 )
343 alpha = -half*tau( i )*cdotc( n-i, w( i+1, i ), 1,
344 $ a( i+1, i ), 1 )
345 CALL caxpy( n-i, alpha, a( i+1, i ), 1, w( i+1, i ), 1 )
346 END IF
347*
348 20 CONTINUE
349 END IF
350*
351 RETURN
352*
353* End of CLATRD
354*
subroutine caxpy(n, ca, cx, incx, cy, incy)
CAXPY
Definition caxpy.f:88
complex function cdotc(n, cx, incx, cy, incy)
CDOTC
Definition cdotc.f:83
subroutine cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
CGEMV
Definition cgemv.f:160
subroutine chemv(uplo, n, alpha, a, lda, x, incx, beta, y, incy)
CHEMV
Definition chemv.f:154
subroutine clacgv(n, x, incx)
CLACGV conjugates a complex vector.
Definition clacgv.f:74
subroutine clarfg(n, alpha, x, incx, tau)
CLARFG generates an elementary reflector (Householder matrix).
Definition clarfg.f:106
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine cscal(n, ca, cx, incx)
CSCAL
Definition cscal.f:78
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