 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zlabrd()

 subroutine zlabrd ( integer M, integer N, integer NB, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( * ) TAUQ, complex*16, dimension( * ) TAUP, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldy, * ) Y, integer LDY )

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

Purpose:
``` ZLABRD reduces the first NB rows and columns of a complex general
m by n matrix A to upper or lower real bidiagonal form by a unitary
transformation Q**H * 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 ZGEBRD```
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 COMPLEX*16 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 unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary 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 unitary 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 unitary 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 COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.``` [out] TAUP ``` TAUP is COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.``` [out] X ``` X is COMPLEX*16 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 COMPLEX*16 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**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex
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**H 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**H - X*U**H.

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 210 of file zlabrd.f.

212 *
213 * -- LAPACK auxiliary routine --
214 * -- LAPACK is a software package provided by Univ. of Tennessee, --
215 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216 *
217 * .. Scalar Arguments ..
218  INTEGER LDA, LDX, LDY, M, N, NB
219 * ..
220 * .. Array Arguments ..
221  DOUBLE PRECISION D( * ), E( * )
222  COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
223  \$ Y( LDY, * )
224 * ..
225 *
226 * =====================================================================
227 *
228 * .. Parameters ..
229  COMPLEX*16 ZERO, ONE
230  parameter( zero = ( 0.0d+0, 0.0d+0 ),
231  \$ one = ( 1.0d+0, 0.0d+0 ) )
232 * ..
233 * .. Local Scalars ..
234  INTEGER I
235  COMPLEX*16 ALPHA
236 * ..
237 * .. External Subroutines ..
238  EXTERNAL zgemv, zlacgv, zlarfg, zscal
239 * ..
240 * .. Intrinsic Functions ..
241  INTRINSIC min
242 * ..
243 * .. Executable Statements ..
244 *
245 * Quick return if possible
246 *
247  IF( m.LE.0 .OR. n.LE.0 )
248  \$ RETURN
249 *
250  IF( m.GE.n ) THEN
251 *
252 * Reduce to upper bidiagonal form
253 *
254  DO 10 i = 1, nb
255 *
256 * Update A(i:m,i)
257 *
258  CALL zlacgv( i-1, y( i, 1 ), ldy )
259  CALL zgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
260  \$ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
261  CALL zlacgv( i-1, y( i, 1 ), ldy )
262  CALL zgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
263  \$ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
264 *
265 * Generate reflection Q(i) to annihilate A(i+1:m,i)
266 *
267  alpha = a( i, i )
268  CALL zlarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
269  \$ tauq( i ) )
270  d( i ) = dble( alpha )
271  IF( i.LT.n ) THEN
272  a( i, i ) = one
273 *
274 * Compute Y(i+1:n,i)
275 *
276  CALL zgemv( 'Conjugate transpose', m-i+1, n-i, one,
277  \$ a( i, i+1 ), lda, a( i, i ), 1, zero,
278  \$ y( i+1, i ), 1 )
279  CALL zgemv( 'Conjugate transpose', m-i+1, i-1, one,
280  \$ a( i, 1 ), lda, a( i, i ), 1, zero,
281  \$ y( 1, i ), 1 )
282  CALL zgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
283  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
284  CALL zgemv( 'Conjugate transpose', m-i+1, i-1, one,
285  \$ x( i, 1 ), ldx, a( i, i ), 1, zero,
286  \$ y( 1, i ), 1 )
287  CALL zgemv( 'Conjugate transpose', i-1, n-i, -one,
288  \$ a( 1, i+1 ), lda, y( 1, i ), 1, one,
289  \$ y( i+1, i ), 1 )
290  CALL zscal( n-i, tauq( i ), y( i+1, i ), 1 )
291 *
292 * Update A(i,i+1:n)
293 *
294  CALL zlacgv( n-i, a( i, i+1 ), lda )
295  CALL zlacgv( i, a( i, 1 ), lda )
296  CALL zgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
297  \$ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
298  CALL zlacgv( i, a( i, 1 ), lda )
299  CALL zlacgv( i-1, x( i, 1 ), ldx )
300  CALL zgemv( 'Conjugate transpose', i-1, n-i, -one,
301  \$ a( 1, i+1 ), lda, x( i, 1 ), ldx, one,
302  \$ a( i, i+1 ), lda )
303  CALL zlacgv( i-1, x( i, 1 ), ldx )
304 *
305 * Generate reflection P(i) to annihilate A(i,i+2:n)
306 *
307  alpha = a( i, i+1 )
308  CALL zlarfg( n-i, alpha, a( i, min( i+2, n ) ), lda,
309  \$ taup( i ) )
310  e( i ) = dble( alpha )
311  a( i, i+1 ) = one
312 *
313 * Compute X(i+1:m,i)
314 *
315  CALL zgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
316  \$ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
317  CALL zgemv( 'Conjugate transpose', n-i, i, one,
318  \$ y( i+1, 1 ), ldy, a( i, i+1 ), lda, zero,
319  \$ x( 1, i ), 1 )
320  CALL zgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
321  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
322  CALL zgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
323  \$ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
324  CALL zgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
325  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
326  CALL zscal( m-i, taup( i ), x( i+1, i ), 1 )
327  CALL zlacgv( n-i, a( i, i+1 ), lda )
328  END IF
329  10 CONTINUE
330  ELSE
331 *
332 * Reduce to lower bidiagonal form
333 *
334  DO 20 i = 1, nb
335 *
336 * Update A(i,i:n)
337 *
338  CALL zlacgv( n-i+1, a( i, i ), lda )
339  CALL zlacgv( i-1, a( i, 1 ), lda )
340  CALL zgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
341  \$ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
342  CALL zlacgv( i-1, a( i, 1 ), lda )
343  CALL zlacgv( i-1, x( i, 1 ), ldx )
344  CALL zgemv( 'Conjugate transpose', i-1, n-i+1, -one,
345  \$ a( 1, i ), lda, x( i, 1 ), ldx, one, a( i, i ),
346  \$ lda )
347  CALL zlacgv( i-1, x( i, 1 ), ldx )
348 *
349 * Generate reflection P(i) to annihilate A(i,i+1:n)
350 *
351  alpha = a( i, i )
352  CALL zlarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
353  \$ taup( i ) )
354  d( i ) = dble( alpha )
355  IF( i.LT.m ) THEN
356  a( i, i ) = one
357 *
358 * Compute X(i+1:m,i)
359 *
360  CALL zgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
361  \$ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
362  CALL zgemv( 'Conjugate transpose', n-i+1, i-1, one,
363  \$ y( i, 1 ), ldy, a( i, i ), lda, zero,
364  \$ x( 1, i ), 1 )
365  CALL zgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
366  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
367  CALL zgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
368  \$ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
369  CALL zgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
370  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
371  CALL zscal( m-i, taup( i ), x( i+1, i ), 1 )
372  CALL zlacgv( n-i+1, a( i, i ), lda )
373 *
374 * Update A(i+1:m,i)
375 *
376  CALL zlacgv( i-1, y( i, 1 ), ldy )
377  CALL zgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
378  \$ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
379  CALL zlacgv( i-1, y( i, 1 ), ldy )
380  CALL zgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
381  \$ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
382 *
383 * Generate reflection Q(i) to annihilate A(i+2:m,i)
384 *
385  alpha = a( i+1, i )
386  CALL zlarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
387  \$ tauq( i ) )
388  e( i ) = dble( alpha )
389  a( i+1, i ) = one
390 *
391 * Compute Y(i+1:n,i)
392 *
393  CALL zgemv( 'Conjugate transpose', m-i, n-i, one,
394  \$ a( i+1, i+1 ), lda, a( i+1, i ), 1, zero,
395  \$ y( i+1, i ), 1 )
396  CALL zgemv( 'Conjugate transpose', m-i, i-1, one,
397  \$ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
398  \$ y( 1, i ), 1 )
399  CALL zgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
400  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
401  CALL zgemv( 'Conjugate transpose', m-i, i, one,
402  \$ x( i+1, 1 ), ldx, a( i+1, i ), 1, zero,
403  \$ y( 1, i ), 1 )
404  CALL zgemv( 'Conjugate transpose', i, n-i, -one,
405  \$ a( 1, i+1 ), lda, y( 1, i ), 1, one,
406  \$ y( i+1, i ), 1 )
407  CALL zscal( n-i, tauq( i ), y( i+1, i ), 1 )
408  ELSE
409  CALL zlacgv( n-i+1, a( i, i ), lda )
410  END IF
411  20 CONTINUE
412  END IF
413  RETURN
414 *
415 * End of ZLABRD
416 *
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106
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