LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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dgebd2.f
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1*> \brief \b DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
22*
23* .. Scalar Arguments ..
24* INTEGER INFO, LDA, M, N
25* ..
26* .. Array Arguments ..
27* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
28* \$ TAUQ( * ), WORK( * )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> DGEBD2 reduces a real general m by n matrix A to upper or lower
38*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
39*>
40*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
41*> \endverbatim
42*
43* Arguments:
44* ==========
45*
46*> \param[in] M
47*> \verbatim
48*> M is INTEGER
49*> The number of rows in the matrix A. M >= 0.
50*> \endverbatim
51*>
52*> \param[in] N
53*> \verbatim
54*> N is INTEGER
55*> The number of columns in the matrix A. N >= 0.
56*> \endverbatim
57*>
58*> \param[in,out] A
59*> \verbatim
60*> A is DOUBLE PRECISION array, dimension (LDA,N)
61*> On entry, the m by n general matrix to be reduced.
62*> On exit,
63*> if m >= n, the diagonal and the first superdiagonal are
64*> overwritten with the upper bidiagonal matrix B; the
65*> elements below the diagonal, with the array TAUQ, represent
66*> the orthogonal matrix Q as a product of elementary
67*> reflectors, and the elements above the first superdiagonal,
68*> with the array TAUP, represent the orthogonal matrix P as
69*> a product of elementary reflectors;
70*> if m < n, the diagonal and the first subdiagonal are
71*> overwritten with the lower bidiagonal matrix B; the
72*> elements below the first subdiagonal, with the array TAUQ,
73*> represent the orthogonal matrix Q as a product of
74*> elementary reflectors, and the elements above the diagonal,
75*> with the array TAUP, represent the orthogonal matrix P as
76*> a product of elementary reflectors.
77*> See Further Details.
78*> \endverbatim
79*>
80*> \param[in] LDA
81*> \verbatim
82*> LDA is INTEGER
83*> The leading dimension of the array A. LDA >= max(1,M).
84*> \endverbatim
85*>
86*> \param[out] D
87*> \verbatim
88*> D is DOUBLE PRECISION array, dimension (min(M,N))
89*> The diagonal elements of the bidiagonal matrix B:
90*> D(i) = A(i,i).
91*> \endverbatim
92*>
93*> \param[out] E
94*> \verbatim
95*> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
96*> The off-diagonal elements of the bidiagonal matrix B:
97*> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
98*> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
99*> \endverbatim
100*>
101*> \param[out] TAUQ
102*> \verbatim
103*> TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
104*> The scalar factors of the elementary reflectors which
105*> represent the orthogonal matrix Q. See Further Details.
106*> \endverbatim
107*>
108*> \param[out] TAUP
109*> \verbatim
110*> TAUP is DOUBLE PRECISION array, dimension (min(M,N))
111*> The scalar factors of the elementary reflectors which
112*> represent the orthogonal matrix P. See Further Details.
113*> \endverbatim
114*>
115*> \param[out] WORK
116*> \verbatim
117*> WORK is DOUBLE PRECISION array, dimension (max(M,N))
118*> \endverbatim
119*>
120*> \param[out] INFO
121*> \verbatim
122*> INFO is INTEGER
123*> = 0: successful exit.
124*> < 0: if INFO = -i, the i-th argument had an illegal value.
125*> \endverbatim
126*
127* Authors:
128* ========
129*
130*> \author Univ. of Tennessee
131*> \author Univ. of California Berkeley
132*> \author Univ. of Colorado Denver
133*> \author NAG Ltd.
134*
135*> \ingroup doubleGEcomputational
136*
137*> \par Further Details:
138* =====================
139*>
140*> \verbatim
141*>
142*> The matrices Q and P are represented as products of elementary
143*> reflectors:
144*>
145*> If m >= n,
146*>
147*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
148*>
149*> Each H(i) and G(i) has the form:
150*>
151*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
152*>
153*> where tauq and taup are real scalars, and v and u are real vectors;
154*> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
155*> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
156*> tauq is stored in TAUQ(i) and taup in TAUP(i).
157*>
158*> If m < n,
159*>
160*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
161*>
162*> Each H(i) and G(i) has the form:
163*>
164*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
165*>
166*> where tauq and taup are real scalars, and v and u are real vectors;
167*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
168*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
169*> tauq is stored in TAUQ(i) and taup in TAUP(i).
170*>
171*> The contents of A on exit are illustrated by the following examples:
172*>
173*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
174*>
175*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
176*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
177*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
178*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
179*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
180*> ( v1 v2 v3 v4 v5 )
181*>
182*> where d and e denote diagonal and off-diagonal elements of B, vi
183*> denotes an element of the vector defining H(i), and ui an element of
184*> the vector defining G(i).
185*> \endverbatim
186*>
187* =====================================================================
188 SUBROUTINE dgebd2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
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*
317 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dgebd2(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
Definition: dgebd2.f:189
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