 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zgebd2()

 subroutine zgebd2 ( integer M, integer N, 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( * ) WORK, integer INFO )

ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Download ZGEBD2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` ZGEBD2 reduces a complex general m by n matrix A to upper or lower
real bidiagonal form B by a unitary transformation: Q**H * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.```
Parameters
 [in] M ``` M is INTEGER The number of rows in the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns in the matrix A. N >= 0.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors; if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, 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 (min(M,N)) The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).``` [out] E ``` E is DOUBLE PRECISION array, dimension (min(M,N)-1) The off-diagonal elements of the bidiagonal matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.``` [out] TAUQ ``` TAUQ is COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.``` [out] TAUP ``` TAUP is COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (max(M,N))` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value.```
Further Details:
```  The matrices Q and P are represented as products of elementary
reflectors:

If m >= n,

Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

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

If m < n,

Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

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, v and u are complex vectors;
v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 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).

The contents of A on exit are illustrated by the following examples:

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

(  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
(  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
(  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
(  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
(  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
(  v1  v2  v3  v4  v5 )

where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).```

Definition at line 188 of file zgebd2.f.

189 *
190 * -- LAPACK computational routine --
191 * -- LAPACK is a software package provided by Univ. of Tennessee, --
192 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
193 *
194 * .. Scalar Arguments ..
195  INTEGER INFO, LDA, M, N
196 * ..
197 * .. Array Arguments ..
198  DOUBLE PRECISION D( * ), E( * )
199  COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
200 * ..
201 *
202 * =====================================================================
203 *
204 * .. Parameters ..
205  COMPLEX*16 ZERO, ONE
206  parameter( zero = ( 0.0d+0, 0.0d+0 ),
207  \$ one = ( 1.0d+0, 0.0d+0 ) )
208 * ..
209 * .. Local Scalars ..
210  INTEGER I
211  COMPLEX*16 ALPHA
212 * ..
213 * .. External Subroutines ..
214  EXTERNAL xerbla, zlacgv, zlarf, zlarfg
215 * ..
216 * .. Intrinsic Functions ..
217  INTRINSIC dconjg, max, min
218 * ..
219 * .. Executable Statements ..
220 *
221 * Test the input parameters
222 *
223  info = 0
224  IF( m.LT.0 ) THEN
225  info = -1
226  ELSE IF( n.LT.0 ) THEN
227  info = -2
228  ELSE IF( lda.LT.max( 1, m ) ) THEN
229  info = -4
230  END IF
231  IF( info.LT.0 ) THEN
232  CALL xerbla( 'ZGEBD2', -info )
233  RETURN
234  END IF
235 *
236  IF( m.GE.n ) THEN
237 *
238 * Reduce to upper bidiagonal form
239 *
240  DO 10 i = 1, n
241 *
242 * Generate elementary reflector H(i) to annihilate A(i+1:m,i)
243 *
244  alpha = a( i, i )
245  CALL zlarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
246  \$ tauq( i ) )
247  d( i ) = dble( alpha )
248  a( i, i ) = one
249 *
250 * Apply H(i)**H to A(i:m,i+1:n) from the left
251 *
252  IF( i.LT.n )
253  \$ CALL zlarf( 'Left', m-i+1, n-i, a( i, i ), 1,
254  \$ dconjg( tauq( i ) ), a( i, i+1 ), lda, work )
255  a( i, i ) = d( i )
256 *
257  IF( i.LT.n ) THEN
258 *
259 * Generate elementary reflector G(i) to annihilate
260 * A(i,i+2:n)
261 *
262  CALL zlacgv( n-i, a( i, i+1 ), lda )
263  alpha = a( i, i+1 )
264  CALL zlarfg( n-i, alpha, a( i, min( i+2, n ) ), lda,
265  \$ taup( i ) )
266  e( i ) = dble( alpha )
267  a( i, i+1 ) = one
268 *
269 * Apply G(i) to A(i+1:m,i+1:n) from the right
270 *
271  CALL zlarf( 'Right', m-i, n-i, a( i, i+1 ), lda,
272  \$ taup( i ), a( i+1, i+1 ), lda, work )
273  CALL zlacgv( n-i, a( i, i+1 ), lda )
274  a( i, i+1 ) = e( i )
275  ELSE
276  taup( i ) = zero
277  END IF
278  10 CONTINUE
279  ELSE
280 *
281 * Reduce to lower bidiagonal form
282 *
283  DO 20 i = 1, m
284 *
285 * Generate elementary reflector G(i) to annihilate A(i,i+1:n)
286 *
287  CALL zlacgv( n-i+1, a( i, i ), lda )
288  alpha = a( i, i )
289  CALL zlarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
290  \$ taup( i ) )
291  d( i ) = dble( alpha )
292  a( i, i ) = one
293 *
294 * Apply G(i) to A(i+1:m,i:n) from the right
295 *
296  IF( i.LT.m )
297  \$ CALL zlarf( 'Right', m-i, n-i+1, a( i, i ), lda,
298  \$ taup( i ), a( i+1, i ), lda, work )
299  CALL zlacgv( n-i+1, a( i, i ), lda )
300  a( i, i ) = d( i )
301 *
302  IF( i.LT.m ) THEN
303 *
304 * Generate elementary reflector H(i) to annihilate
305 * A(i+2:m,i)
306 *
307  alpha = a( i+1, i )
308  CALL zlarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
309  \$ tauq( i ) )
310  e( i ) = dble( alpha )
311  a( i+1, i ) = one
312 *
313 * Apply H(i)**H to A(i+1:m,i+1:n) from the left
314 *
315  CALL zlarf( 'Left', m-i, n-i, a( i+1, i ), 1,
316  \$ dconjg( tauq( i ) ), a( i+1, i+1 ), lda,
317  \$ work )
318  a( i+1, i ) = e( i )
319  ELSE
320  tauq( i ) = zero
321  END IF
322  20 CONTINUE
323  END IF
324  RETURN
325 *
326 * End of ZGEBD2
327 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
subroutine zlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
ZLARF applies an elementary reflector to a general rectangular matrix.
Definition: zlarf.f:128
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106
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