LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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.

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.```
Date
September 2012
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 191 of file zgebd2.f.

191 *
192 * -- LAPACK computational routine (version 3.4.2) --
193 * -- LAPACK is a software package provided by Univ. of Tennessee, --
194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195 * September 2012
196 *
197 * .. Scalar Arguments ..
198  INTEGER info, lda, m, n
199 * ..
200 * .. Array Arguments ..
201  DOUBLE PRECISION d( * ), e( * )
202  COMPLEX*16 a( lda, * ), taup( * ), tauq( * ), work( * )
203 * ..
204 *
205 * =====================================================================
206 *
207 * .. Parameters ..
208  COMPLEX*16 zero, one
209  parameter ( zero = ( 0.0d+0, 0.0d+0 ),
210  \$ one = ( 1.0d+0, 0.0d+0 ) )
211 * ..
212 * .. Local Scalars ..
213  INTEGER i
214  COMPLEX*16 alpha
215 * ..
216 * .. External Subroutines ..
217  EXTERNAL xerbla, zlacgv, zlarf, zlarfg
218 * ..
219 * .. Intrinsic Functions ..
220  INTRINSIC dconjg, max, min
221 * ..
222 * .. Executable Statements ..
223 *
224 * Test the input parameters
225 *
226  info = 0
227  IF( m.LT.0 ) THEN
228  info = -1
229  ELSE IF( n.LT.0 ) THEN
230  info = -2
231  ELSE IF( lda.LT.max( 1, m ) ) THEN
232  info = -4
233  END IF
234  IF( info.LT.0 ) THEN
235  CALL xerbla( 'ZGEBD2', -info )
236  RETURN
237  END IF
238 *
239  IF( m.GE.n ) THEN
240 *
241 * Reduce to upper bidiagonal form
242 *
243  DO 10 i = 1, n
244 *
245 * Generate elementary reflector H(i) to annihilate A(i+1:m,i)
246 *
247  alpha = a( i, i )
248  CALL zlarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
249  \$ tauq( i ) )
250  d( i ) = alpha
251  a( i, i ) = one
252 *
253 * Apply H(i)**H to A(i:m,i+1:n) from the left
254 *
255  IF( i.LT.n )
256  \$ CALL zlarf( 'Left', m-i+1, n-i, a( i, i ), 1,
257  \$ dconjg( tauq( i ) ), a( i, i+1 ), lda, work )
258  a( i, i ) = d( i )
259 *
260  IF( i.LT.n ) THEN
261 *
262 * Generate elementary reflector G(i) to annihilate
263 * A(i,i+2:n)
264 *
265  CALL zlacgv( n-i, a( i, i+1 ), lda )
266  alpha = a( i, i+1 )
267  CALL zlarfg( n-i, alpha, a( i, min( i+2, n ) ), lda,
268  \$ taup( i ) )
269  e( i ) = alpha
270  a( i, i+1 ) = one
271 *
272 * Apply G(i) to A(i+1:m,i+1:n) from the right
273 *
274  CALL zlarf( 'Right', m-i, n-i, a( i, i+1 ), lda,
275  \$ taup( i ), a( i+1, i+1 ), lda, work )
276  CALL zlacgv( n-i, a( i, i+1 ), lda )
277  a( i, i+1 ) = e( i )
278  ELSE
279  taup( i ) = zero
280  END IF
281  10 CONTINUE
282  ELSE
283 *
284 * Reduce to lower bidiagonal form
285 *
286  DO 20 i = 1, m
287 *
288 * Generate elementary reflector G(i) to annihilate A(i,i+1:n)
289 *
290  CALL zlacgv( n-i+1, a( i, i ), lda )
291  alpha = a( i, i )
292  CALL zlarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
293  \$ taup( i ) )
294  d( i ) = alpha
295  a( i, i ) = one
296 *
297 * Apply G(i) to A(i+1:m,i:n) from the right
298 *
299  IF( i.LT.m )
300  \$ CALL zlarf( 'Right', m-i, n-i+1, a( i, i ), lda,
301  \$ taup( i ), a( i+1, i ), lda, work )
302  CALL zlacgv( n-i+1, a( i, i ), lda )
303  a( i, i ) = d( i )
304 *
305  IF( i.LT.m ) THEN
306 *
307 * Generate elementary reflector H(i) to annihilate
308 * A(i+2:m,i)
309 *
310  alpha = a( i+1, i )
311  CALL zlarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
312  \$ tauq( i ) )
313  e( i ) = alpha
314  a( i+1, i ) = one
315 *
316 * Apply H(i)**H to A(i+1:m,i+1:n) from the left
317 *
318  CALL zlarf( 'Left', m-i, n-i, a( i+1, i ), 1,
319  \$ dconjg( tauq( i ) ), a( i+1, i+1 ), lda,
320  \$ work )
321  a( i+1, i ) = e( i )
322  ELSE
323  tauq( i ) = zero
324  END IF
325  20 CONTINUE
326  END IF
327  RETURN
328 *
329 * End of ZGEBD2
330 *
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:108
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
ZLARF applies an elementary reflector to a general rectangular matrix.
Definition: zlarf.f:130
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76

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