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.

Download CGEBD2 + dependencies [TGZ] [ZIP] [TXT]

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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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
Here is the call graph for this function:
Here is the caller graph for this function: