 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dlabrd()

 subroutine dlabrd ( integer M, integer N, integer NB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( ldy, * ) Y, integer LDY )

DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

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Purpose:
``` DLABRD reduces the first NB rows and columns of a real general
m by n matrix A to upper or lower bidiagonal form by an orthogonal
transformation Q**T * A * P, and returns the matrices X and Y which
are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
bidiagonal form.

This is an auxiliary routine called by DGEBRD```
Parameters
 [in] M ``` M is INTEGER The number of rows in the matrix A.``` [in] N ``` N is INTEGER The number of columns in the matrix A.``` [in] NB ``` NB is INTEGER The number of leading rows and columns of A to be reduced.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] D ``` D is DOUBLE PRECISION array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).``` [out] E ``` E is DOUBLE PRECISION array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.``` [out] TAUQ ``` TAUQ is DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.``` [out] TAUP ``` TAUP is DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.``` [out] X ``` X is DOUBLE PRECISION array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).``` [out] Y ``` Y is DOUBLE PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.``` [in] LDY ``` LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).```
Further Details:
```  The matrices Q and P are represented as products of elementary
reflectors:

Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix
V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
the transformation to the unreduced part of the matrix, using a block
update of the form:  A := A - V*Y**T - X*U**T.

The contents of A on exit are illustrated by the following examples
with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )

where a denotes an element of the original matrix which is unchanged,
vi denotes an element of the vector defining H(i), and ui an element
of the vector defining G(i).```

Definition at line 208 of file dlabrd.f.

210 *
211 * -- LAPACK auxiliary routine --
212 * -- LAPACK is a software package provided by Univ. of Tennessee, --
213 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
214 *
215 * .. Scalar Arguments ..
216  INTEGER LDA, LDX, LDY, M, N, NB
217 * ..
218 * .. Array Arguments ..
219  DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
220  \$ TAUQ( * ), X( LDX, * ), Y( LDY, * )
221 * ..
222 *
223 * =====================================================================
224 *
225 * .. Parameters ..
226  DOUBLE PRECISION ZERO, ONE
227  parameter( zero = 0.0d0, one = 1.0d0 )
228 * ..
229 * .. Local Scalars ..
230  INTEGER I
231 * ..
232 * .. External Subroutines ..
233  EXTERNAL dgemv, dlarfg, dscal
234 * ..
235 * .. Intrinsic Functions ..
236  INTRINSIC min
237 * ..
238 * .. Executable Statements ..
239 *
240 * Quick return if possible
241 *
242  IF( m.LE.0 .OR. n.LE.0 )
243  \$ RETURN
244 *
245  IF( m.GE.n ) THEN
246 *
247 * Reduce to upper bidiagonal form
248 *
249  DO 10 i = 1, nb
250 *
251 * Update A(i:m,i)
252 *
253  CALL dgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
254  \$ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
255  CALL dgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
256  \$ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
257 *
258 * Generate reflection Q(i) to annihilate A(i+1:m,i)
259 *
260  CALL dlarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
261  \$ tauq( i ) )
262  d( i ) = a( i, i )
263  IF( i.LT.n ) THEN
264  a( i, i ) = one
265 *
266 * Compute Y(i+1:n,i)
267 *
268  CALL dgemv( 'Transpose', m-i+1, n-i, one, a( i, i+1 ),
269  \$ lda, a( i, i ), 1, zero, y( i+1, i ), 1 )
270  CALL dgemv( 'Transpose', m-i+1, i-1, one, a( i, 1 ), lda,
271  \$ a( i, i ), 1, zero, y( 1, i ), 1 )
272  CALL dgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
273  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
274  CALL dgemv( 'Transpose', m-i+1, i-1, one, x( i, 1 ), ldx,
275  \$ a( i, i ), 1, zero, y( 1, i ), 1 )
276  CALL dgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
277  \$ lda, y( 1, i ), 1, one, y( i+1, i ), 1 )
278  CALL dscal( n-i, tauq( i ), y( i+1, i ), 1 )
279 *
280 * Update A(i,i+1:n)
281 *
282  CALL dgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
283  \$ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
284  CALL dgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
285  \$ lda, x( i, 1 ), ldx, one, a( i, i+1 ), lda )
286 *
287 * Generate reflection P(i) to annihilate A(i,i+2:n)
288 *
289  CALL dlarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),
290  \$ lda, taup( i ) )
291  e( i ) = a( i, i+1 )
292  a( i, i+1 ) = one
293 *
294 * Compute X(i+1:m,i)
295 *
296  CALL dgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
297  \$ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
298  CALL dgemv( 'Transpose', n-i, i, one, y( i+1, 1 ), ldy,
299  \$ a( i, i+1 ), lda, zero, x( 1, i ), 1 )
300  CALL dgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
301  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
302  CALL dgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
303  \$ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
304  CALL dgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
305  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
306  CALL dscal( m-i, taup( i ), x( i+1, i ), 1 )
307  END IF
308  10 CONTINUE
309  ELSE
310 *
311 * Reduce to lower bidiagonal form
312 *
313  DO 20 i = 1, nb
314 *
315 * Update A(i,i:n)
316 *
317  CALL dgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
318  \$ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
319  CALL dgemv( 'Transpose', i-1, n-i+1, -one, a( 1, i ), lda,
320  \$ x( i, 1 ), ldx, one, a( i, i ), lda )
321 *
322 * Generate reflection P(i) to annihilate A(i,i+1:n)
323 *
324  CALL dlarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
325  \$ taup( i ) )
326  d( i ) = a( i, i )
327  IF( i.LT.m ) THEN
328  a( i, i ) = one
329 *
330 * Compute X(i+1:m,i)
331 *
332  CALL dgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
333  \$ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
334  CALL dgemv( 'Transpose', n-i+1, i-1, one, y( i, 1 ), ldy,
335  \$ a( i, i ), lda, zero, x( 1, i ), 1 )
336  CALL dgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
337  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
338  CALL dgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
339  \$ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
340  CALL dgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
341  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
342  CALL dscal( m-i, taup( i ), x( i+1, i ), 1 )
343 *
344 * Update A(i+1:m,i)
345 *
346  CALL dgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
347  \$ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
348  CALL dgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
349  \$ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
350 *
351 * Generate reflection Q(i) to annihilate A(i+2:m,i)
352 *
353  CALL dlarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1,
354  \$ tauq( i ) )
355  e( i ) = a( i+1, i )
356  a( i+1, i ) = one
357 *
358 * Compute Y(i+1:n,i)
359 *
360  CALL dgemv( 'Transpose', m-i, n-i, one, a( i+1, i+1 ),
361  \$ lda, a( i+1, i ), 1, zero, y( i+1, i ), 1 )
362  CALL dgemv( 'Transpose', m-i, i-1, one, a( i+1, 1 ), lda,
363  \$ a( i+1, i ), 1, zero, y( 1, i ), 1 )
364  CALL dgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
365  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
366  CALL dgemv( 'Transpose', m-i, i, one, x( i+1, 1 ), ldx,
367  \$ a( i+1, i ), 1, zero, y( 1, i ), 1 )
368  CALL dgemv( 'Transpose', i, n-i, -one, a( 1, i+1 ), lda,
369  \$ y( 1, i ), 1, one, y( i+1, i ), 1 )
370  CALL dscal( n-i, tauq( i ), y( i+1, i ), 1 )
371  END IF
372  20 CONTINUE
373  END IF
374  RETURN
375 *
376 * End of DLABRD
377 *
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:156
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
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