 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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.

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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