 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ cgebd2()

 subroutine cgebd2 ( integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( * ) WORK, integer INFO )

CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:
``` CGEBD2 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 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 REAL array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).``` [out] E ``` E is REAL 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 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 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 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 189 of file cgebd2.f.

190 *
191 * -- LAPACK computational routine --
192 * -- LAPACK is a software package provided by Univ. of Tennessee, --
193 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
194 *
195 * .. Scalar Arguments ..
196  INTEGER INFO, LDA, M, N
197 * ..
198 * .. Array Arguments ..
199  REAL D( * ), E( * )
200  COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
201 * ..
202 *
203 * =====================================================================
204 *
205 * .. Parameters ..
206  COMPLEX ZERO, ONE
207  parameter( zero = ( 0.0e+0, 0.0e+0 ),
208  \$ one = ( 1.0e+0, 0.0e+0 ) )
209 * ..
210 * .. Local Scalars ..
211  INTEGER I
212  COMPLEX ALPHA
213 * ..
214 * .. External Subroutines ..
215  EXTERNAL clacgv, clarf, clarfg, xerbla
216 * ..
217 * .. Intrinsic Functions ..
218  INTRINSIC conjg, max, min
219 * ..
220 * .. Executable Statements ..
221 *
222 * Test the input parameters
223 *
224  info = 0
225  IF( m.LT.0 ) THEN
226  info = -1
227  ELSE IF( n.LT.0 ) THEN
228  info = -2
229  ELSE IF( lda.LT.max( 1, m ) ) THEN
230  info = -4
231  END IF
232  IF( info.LT.0 ) THEN
233  CALL xerbla( 'CGEBD2', -info )
234  RETURN
235  END IF
236 *
237  IF( m.GE.n ) THEN
238 *
239 * Reduce to upper bidiagonal form
240 *
241  DO 10 i = 1, n
242 *
243 * Generate elementary reflector H(i) to annihilate A(i+1:m,i)
244 *
245  alpha = a( i, i )
246  CALL clarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
247  \$ tauq( i ) )
248  d( i ) = real( alpha )
249  a( i, i ) = one
250 *
251 * Apply H(i)**H to A(i:m,i+1:n) from the left
252 *
253  IF( i.LT.n )
254  \$ CALL clarf( 'Left', m-i+1, n-i, a( i, i ), 1,
255  \$ conjg( tauq( i ) ), a( i, i+1 ), lda, work )
256  a( i, i ) = d( i )
257 *
258  IF( i.LT.n ) THEN
259 *
260 * Generate elementary reflector G(i) to annihilate
261 * A(i,i+2:n)
262 *
263  CALL clacgv( n-i, a( i, i+1 ), lda )
264  alpha = a( i, i+1 )
265  CALL clarfg( n-i, alpha, a( i, min( i+2, n ) ),
266  \$ lda, taup( i ) )
267  e( i ) = real( alpha )
268  a( i, i+1 ) = one
269 *
270 * Apply G(i) to A(i+1:m,i+1:n) from the right
271 *
272  CALL clarf( 'Right', m-i, n-i, a( i, i+1 ), lda,
273  \$ taup( i ), a( i+1, i+1 ), lda, work )
274  CALL clacgv( n-i, a( i, i+1 ), lda )
275  a( i, i+1 ) = e( i )
276  ELSE
277  taup( i ) = zero
278  END IF
279  10 CONTINUE
280  ELSE
281 *
282 * Reduce to lower bidiagonal form
283 *
284  DO 20 i = 1, m
285 *
286 * Generate elementary reflector G(i) to annihilate A(i,i+1:n)
287 *
288  CALL clacgv( n-i+1, a( i, i ), lda )
289  alpha = a( i, i )
290  CALL clarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
291  \$ taup( i ) )
292  d( i ) = real( alpha )
293  a( i, i ) = one
294 *
295 * Apply G(i) to A(i+1:m,i:n) from the right
296 *
297  IF( i.LT.m )
298  \$ CALL clarf( 'Right', m-i, n-i+1, a( i, i ), lda,
299  \$ taup( i ), a( i+1, i ), lda, work )
300  CALL clacgv( n-i+1, a( i, i ), lda )
301  a( i, i ) = d( i )
302 *
303  IF( i.LT.m ) THEN
304 *
305 * Generate elementary reflector H(i) to annihilate
306 * A(i+2:m,i)
307 *
308  alpha = a( i+1, i )
309  CALL clarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
310  \$ tauq( i ) )
311  e( i ) = real( alpha )
312  a( i+1, i ) = one
313 *
314 * Apply H(i)**H to A(i+1:m,i+1:n) from the left
315 *
316  CALL clarf( 'Left', m-i, n-i, a( i+1, i ), 1,
317  \$ conjg( tauq( i ) ), a( i+1, i+1 ), lda,
318  \$ work )
319  a( i+1, i ) = e( i )
320  ELSE
321  tauq( i ) = zero
322  END IF
323  20 CONTINUE
324  END IF
325  RETURN
326 *
327 * End of CGEBD2
328 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:74
subroutine clarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix.
Definition: clarf.f:128
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
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