LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dgebd2()

 subroutine dgebd2 ( integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double precision, dimension( * ) WORK, integer INFO )

DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:
``` DGEBD2 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 DOUBLE PRECISION 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 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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**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 188 of file dgebd2.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 A( LDA, * ), D( * ), E( * ), TAUP( * ),
199 \$ TAUQ( * ), WORK( * )
200* ..
201*
202* =====================================================================
203*
204* .. Parameters ..
205 DOUBLE PRECISION ZERO, ONE
206 parameter( zero = 0.0d+0, one = 1.0d+0 )
207* ..
208* .. Local Scalars ..
209 INTEGER I
210* ..
211* .. External Subroutines ..
212 EXTERNAL dlarf, dlarfg, xerbla
213* ..
214* .. Intrinsic Functions ..
215 INTRINSIC max, min
216* ..
217* .. Executable Statements ..
218*
219* Test the input parameters
220*
221 info = 0
222 IF( m.LT.0 ) THEN
223 info = -1
224 ELSE IF( n.LT.0 ) THEN
225 info = -2
226 ELSE IF( lda.LT.max( 1, m ) ) THEN
227 info = -4
228 END IF
229 IF( info.LT.0 ) THEN
230 CALL xerbla( 'DGEBD2', -info )
231 RETURN
232 END IF
233*
234 IF( m.GE.n ) THEN
235*
236* Reduce to upper bidiagonal form
237*
238 DO 10 i = 1, n
239*
240* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
241*
242 CALL dlarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
243 \$ tauq( i ) )
244 d( i ) = a( i, i )
245 a( i, i ) = one
246*
247* Apply H(i) to A(i:m,i+1:n) from the left
248*
249 IF( i.LT.n )
250 \$ CALL dlarf( 'Left', m-i+1, n-i, a( i, i ), 1, tauq( i ),
251 \$ a( i, i+1 ), lda, work )
252 a( i, i ) = d( i )
253*
254 IF( i.LT.n ) THEN
255*
256* Generate elementary reflector G(i) to annihilate
257* A(i,i+2:n)
258*
259 CALL dlarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),
260 \$ lda, taup( i ) )
261 e( i ) = a( i, i+1 )
262 a( i, i+1 ) = one
263*
264* Apply G(i) to A(i+1:m,i+1:n) from the right
265*
266 CALL dlarf( 'Right', m-i, n-i, a( i, i+1 ), lda,
267 \$ taup( i ), a( i+1, i+1 ), lda, work )
268 a( i, i+1 ) = e( i )
269 ELSE
270 taup( i ) = zero
271 END IF
272 10 CONTINUE
273 ELSE
274*
275* Reduce to lower bidiagonal form
276*
277 DO 20 i = 1, m
278*
279* Generate elementary reflector G(i) to annihilate A(i,i+1:n)
280*
281 CALL dlarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
282 \$ taup( i ) )
283 d( i ) = a( i, i )
284 a( i, i ) = one
285*
286* Apply G(i) to A(i+1:m,i:n) from the right
287*
288 IF( i.LT.m )
289 \$ CALL dlarf( 'Right', m-i, n-i+1, a( i, i ), lda,
290 \$ taup( i ), a( i+1, i ), lda, work )
291 a( i, i ) = d( i )
292*
293 IF( i.LT.m ) THEN
294*
295* Generate elementary reflector H(i) to annihilate
296* A(i+2:m,i)
297*
298 CALL dlarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1,
299 \$ tauq( i ) )
300 e( i ) = a( i+1, i )
301 a( i+1, i ) = one
302*
303* Apply H(i) to A(i+1:m,i+1:n) from the left
304*
305 CALL dlarf( 'Left', m-i, n-i, a( i+1, i ), 1, tauq( i ),
306 \$ a( i+1, i+1 ), lda, work )
307 a( i+1, i ) = e( i )
308 ELSE
309 tauq( i ) = zero
310 END IF
311 20 CONTINUE
312 END IF
313 RETURN
314*
315* End of DGEBD2
316*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:124
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
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