1*> \brief \b SGEBD2 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*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE SGEBD2( 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*       REAL               A( LDA, * ), D( * ), E( * ), TAUP( * ),
28*      $                   TAUQ( * ), WORK( * )
29*       ..
30*
31*
32*> \par Purpose:
33*  =============
34*>
35*> \verbatim
36*>
37*> SGEBD2 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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*> \date June 2017
136*
137*> \ingroup realGEcomputational
138*
139*> \par Further Details:
140*  =====================
141*>
142*> \verbatim
143*>
144*>  The matrices Q and P are represented as products of elementary
145*>  reflectors:
146*>
147*>  If m >= n,
148*>
149*>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
150*>
151*>  Each H(i) and G(i) has the form:
152*>
153*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
154*>
155*>  where tauq and taup are real scalars, and v and u are real vectors;
156*>  v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
157*>  u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
158*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
159*>
160*>  If m < n,
161*>
162*>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
163*>
164*>  Each H(i) and G(i) has the form:
165*>
166*>     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
167*>
168*>  where tauq and taup are real scalars, and v and u are real vectors;
169*>  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
170*>  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
171*>  tauq is stored in TAUQ(i) and taup in TAUP(i).
172*>
173*>  The contents of A on exit are illustrated by the following examples:
174*>
175*>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
176*>
177*>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
178*>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
179*>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
180*>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
181*>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
182*>    (  v1  v2  v3  v4  v5 )
183*>
184*>  where d and e denote diagonal and off-diagonal elements of B, vi
185*>  denotes an element of the vector defining H(i), and ui an element of
186*>  the vector defining G(i).
187*> \endverbatim
188*>
189*  =====================================================================
190      SUBROUTINE SGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
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*
320      END
321