LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ slahrd()

subroutine slahrd ( integer  n,
integer  k,
integer  nb,
real, dimension( lda, * )  a,
integer  lda,
real, dimension( nb )  tau,
real, dimension( ldt, nb )  t,
integer  ldt,
real, dimension( ldy, nb )  y,
integer  ldy 
)

SLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

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

Purpose:
 This routine is deprecated and has been replaced by routine SLAHR2.

 SLAHRD reduces the first NB columns of a real general n-by-(n-k+1)
 matrix A so that elements below the k-th subdiagonal are zero. The
 reduction is performed by an orthogonal similarity transformation
 Q**T * A * Q. The routine returns the matrices V and T which determine
 Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
Parameters
[in]N
          N is INTEGER
          The order of the matrix A.
[in]K
          K is INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.
[in]NB
          NB is INTEGER
          The number of columns to be reduced.
[in,out]A
          A is REAL array, dimension (LDA,N-K+1)
          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a
          product of elementary reflectors. The other columns of A are
          unchanged. See Further Details.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
[out]TAU
          TAU is REAL array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further
          Details.
[out]T
          T is REAL array, dimension (LDT,NB)
          The upper triangular matrix T.
[in]LDT
          LDT is INTEGER
          The leading dimension of the array T.  LDT >= NB.
[out]Y
          Y is REAL array, dimension (LDY,NB)
          The n-by-nb matrix Y.
[in]LDY
          LDY is INTEGER
          The leading dimension of the array Y. LDY >= N.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  The matrix Q is represented as a product of nb 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+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).

  The elements of the vectors v together form the (n-k+1)-by-nb matrix
  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V**T) * (A - Y*V**T).

  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:

     ( a   h   a   a   a )
     ( a   h   a   a   a )
     ( a   h   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an
  element of the vector defining H(i).

Definition at line 166 of file slahrd.f.

167*
168* -- LAPACK auxiliary routine --
169* -- LAPACK is a software package provided by Univ. of Tennessee, --
170* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
171*
172* .. Scalar Arguments ..
173 INTEGER K, LDA, LDT, LDY, N, NB
174* ..
175* .. Array Arguments ..
176 REAL A( LDA, * ), T( LDT, NB ), TAU( NB ),
177 $ Y( LDY, NB )
178* ..
179*
180* =====================================================================
181*
182* .. Parameters ..
183 REAL ZERO, ONE
184 parameter( zero = 0.0e+0, one = 1.0e+0 )
185* ..
186* .. Local Scalars ..
187 INTEGER I
188 REAL EI
189* ..
190* .. External Subroutines ..
191 EXTERNAL saxpy, scopy, sgemv, slarfg, sscal, strmv
192* ..
193* .. Intrinsic Functions ..
194 INTRINSIC min
195* ..
196* .. Executable Statements ..
197*
198* Quick return if possible
199*
200 IF( n.LE.1 )
201 $ RETURN
202*
203 DO 10 i = 1, nb
204 IF( i.GT.1 ) THEN
205*
206* Update A(1:n,i)
207*
208* Compute i-th column of A - Y * V**T
209*
210 CALL sgemv( 'No transpose', n, i-1, -one, y, ldy,
211 $ a( k+i-1, 1 ), lda, one, a( 1, i ), 1 )
212*
213* Apply I - V * T**T * V**T to this column (call it b) from the
214* left, using the last column of T as workspace
215*
216* Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
217* ( V2 ) ( b2 )
218*
219* where V1 is unit lower triangular
220*
221* w := V1**T * b1
222*
223 CALL scopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
224 CALL strmv( 'Lower', 'Transpose', 'Unit', i-1, a( k+1, 1 ),
225 $ lda, t( 1, nb ), 1 )
226*
227* w := w + V2**T *b2
228*
229 CALL sgemv( 'Transpose', n-k-i+1, i-1, one, a( k+i, 1 ),
230 $ lda, a( k+i, i ), 1, one, t( 1, nb ), 1 )
231*
232* w := T**T *w
233*
234 CALL strmv( 'Upper', 'Transpose', 'Non-unit', i-1, t, ldt,
235 $ t( 1, nb ), 1 )
236*
237* b2 := b2 - V2*w
238*
239 CALL sgemv( 'No transpose', n-k-i+1, i-1, -one, a( k+i, 1 ),
240 $ lda, t( 1, nb ), 1, one, a( k+i, i ), 1 )
241*
242* b1 := b1 - V1*w
243*
244 CALL strmv( 'Lower', 'No transpose', 'Unit', i-1,
245 $ a( k+1, 1 ), lda, t( 1, nb ), 1 )
246 CALL saxpy( i-1, -one, t( 1, nb ), 1, a( k+1, i ), 1 )
247*
248 a( k+i-1, i-1 ) = ei
249 END IF
250*
251* Generate the elementary reflector H(i) to annihilate
252* A(k+i+1:n,i)
253*
254 CALL slarfg( n-k-i+1, a( k+i, i ), a( min( k+i+1, n ), i ), 1,
255 $ tau( i ) )
256 ei = a( k+i, i )
257 a( k+i, i ) = one
258*
259* Compute Y(1:n,i)
260*
261 CALL sgemv( 'No transpose', n, n-k-i+1, one, a( 1, i+1 ), lda,
262 $ a( k+i, i ), 1, zero, y( 1, i ), 1 )
263 CALL sgemv( 'Transpose', n-k-i+1, i-1, one, a( k+i, 1 ), lda,
264 $ a( k+i, i ), 1, zero, t( 1, i ), 1 )
265 CALL sgemv( 'No transpose', n, i-1, -one, y, ldy, t( 1, i ), 1,
266 $ one, y( 1, i ), 1 )
267 CALL sscal( n, tau( i ), y( 1, i ), 1 )
268*
269* Compute T(1:i,i)
270*
271 CALL sscal( i-1, -tau( i ), t( 1, i ), 1 )
272 CALL strmv( 'Upper', 'No transpose', 'Non-unit', i-1, t, ldt,
273 $ t( 1, i ), 1 )
274 t( i, i ) = tau( i )
275*
276 10 CONTINUE
277 a( k+nb, nb ) = ei
278*
279 RETURN
280*
281* End of SLAHRD
282*
subroutine saxpy(n, sa, sx, incx, sy, incy)
SAXPY
Definition saxpy.f:89
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine sgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
SGEMV
Definition sgemv.f:158
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
subroutine strmv(uplo, trans, diag, n, a, lda, x, incx)
STRMV
Definition strmv.f:147
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