LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

◆ sgebd2()

 subroutine sgebd2 ( integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( * ) WORK, integer INFO )

SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:
``` SGEBD2 reduces a real general m by n matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q**T * 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 REAL 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal 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 REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.``` [out] TAUP ``` TAUP is REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.``` [out] WORK ` WORK is REAL 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
June 2017
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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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 sgebd2.f.

191 *
192 * -- LAPACK computational routine (version 3.7.1) --
193 * -- LAPACK is a software package provided by Univ. of Tennessee, --
194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195 * June 2017
196 *
197 * .. Scalar Arguments ..
198  INTEGER info, lda, m, n
199 * ..
200 * .. Array Arguments ..
201  REAL a( lda, * ), d( * ), e( * ), taup( * ),
202  \$ tauq( * ), work( * )
203 * ..
204 *
205 * =====================================================================
206 *
207 * .. Parameters ..
208  REAL zero, one
209  parameter( zero = 0.0e+0, one = 1.0e+0 )
210 * ..
211 * .. Local Scalars ..
212  INTEGER i
213 * ..
214 * .. External Subroutines ..
215  EXTERNAL slarf, slarfg, xerbla
216 * ..
217 * .. Intrinsic Functions ..
218  INTRINSIC 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( 'SGEBD2', -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  CALL slarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
246  \$ tauq( i ) )
247  d( i ) = a( i, i )
248  a( i, i ) = one
249 *
250 * Apply H(i) to A(i:m,i+1:n) from the left
251 *
252  IF( i.LT.n )
253  \$ CALL slarf( 'Left', m-i+1, n-i, a( i, i ), 1, tauq( i ),
254  \$ 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 slarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),
263  \$ lda, taup( i ) )
264  e( i ) = a( i, i+1 )
265  a( i, i+1 ) = one
266 *
267 * Apply G(i) to A(i+1:m,i+1:n) from the right
268 *
269  CALL slarf( 'Right', m-i, n-i, a( i, i+1 ), lda,
270  \$ taup( i ), a( i+1, i+1 ), lda, work )
271  a( i, i+1 ) = e( i )
272  ELSE
273  taup( i ) = zero
274  END IF
275  10 CONTINUE
276  ELSE
277 *
278 * Reduce to lower bidiagonal form
279 *
280  DO 20 i = 1, m
281 *
282 * Generate elementary reflector G(i) to annihilate A(i,i+1:n)
283 *
284  CALL slarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
285  \$ taup( i ) )
286  d( i ) = a( i, i )
287  a( i, i ) = one
288 *
289 * Apply G(i) to A(i+1:m,i:n) from the right
290 *
291  IF( i.LT.m )
292  \$ CALL slarf( 'Right', m-i, n-i+1, a( i, i ), lda,
293  \$ taup( i ), a( i+1, i ), lda, work )
294  a( i, i ) = d( i )
295 *
296  IF( i.LT.m ) THEN
297 *
298 * Generate elementary reflector H(i) to annihilate
299 * A(i+2:m,i)
300 *
301  CALL slarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1,
302  \$ tauq( i ) )
303  e( i ) = a( i+1, i )
304  a( i+1, i ) = one
305 *
306 * Apply H(i) to A(i+1:m,i+1:n) from the left
307 *
308  CALL slarf( 'Left', m-i, n-i, a( i+1, i ), 1, tauq( i ),
309  \$ a( i+1, i+1 ), lda, work )
310  a( i+1, i ) = e( i )
311  ELSE
312  tauq( i ) = zero
313  END IF
314  20 CONTINUE
315  END IF
316  RETURN
317 *
318 * End of SGEBD2
319 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
SLARF applies an elementary reflector to a general rectangular matrix.
Definition: slarf.f:126
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:108
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