1*> \brief \b SLAGSY
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*  Definition:
9*  ===========
10*
11*       SUBROUTINE SLAGSY( N, K, D, A, LDA, ISEED, WORK, INFO )
12*
13*       .. Scalar Arguments ..
14*       INTEGER            INFO, K, LDA, N
15*       ..
16*       .. Array Arguments ..
17*       INTEGER            ISEED( 4 )
18*       REAL               A( LDA, * ), D( * ), WORK( * )
19*       ..
20*
21*
22*> \par Purpose:
23*  =============
24*>
25*> \verbatim
26*>
27*> SLAGSY generates a real symmetric matrix A, by pre- and post-
28*> multiplying a real diagonal matrix D with a random orthogonal matrix:
29*> A = U*D*U'. The semi-bandwidth may then be reduced to k by additional
30*> orthogonal transformations.
31*> \endverbatim
32*
33*  Arguments:
34*  ==========
35*
36*> \param[in] N
37*> \verbatim
38*>          N is INTEGER
39*>          The order of the matrix A.  N >= 0.
40*> \endverbatim
41*>
42*> \param[in] K
43*> \verbatim
44*>          K is INTEGER
45*>          The number of nonzero subdiagonals within the band of A.
46*>          0 <= K <= N-1.
47*> \endverbatim
48*>
49*> \param[in] D
50*> \verbatim
51*>          D is REAL array, dimension (N)
52*>          The diagonal elements of the diagonal matrix D.
53*> \endverbatim
54*>
55*> \param[out] A
56*> \verbatim
57*>          A is REAL array, dimension (LDA,N)
58*>          The generated n by n symmetric matrix A (the full matrix is
59*>          stored).
60*> \endverbatim
61*>
62*> \param[in] LDA
63*> \verbatim
64*>          LDA is INTEGER
65*>          The leading dimension of the array A.  LDA >= N.
66*> \endverbatim
67*>
68*> \param[in,out] ISEED
69*> \verbatim
70*>          ISEED is INTEGER array, dimension (4)
71*>          On entry, the seed of the random number generator; the array
72*>          elements must be between 0 and 4095, and ISEED(4) must be
73*>          odd.
74*>          On exit, the seed is updated.
75*> \endverbatim
76*>
77*> \param[out] WORK
78*> \verbatim
79*>          WORK is REAL array, dimension (2*N)
80*> \endverbatim
81*>
82*> \param[out] INFO
83*> \verbatim
84*>          INFO is INTEGER
85*>          = 0: successful exit
86*>          < 0: if INFO = -i, the i-th argument had an illegal value
87*> \endverbatim
88*
89*  Authors:
90*  ========
91*
92*> \author Univ. of Tennessee
93*> \author Univ. of California Berkeley
94*> \author Univ. of Colorado Denver
95*> \author NAG Ltd.
96*
97*> \ingroup real_matgen
98*
99*  =====================================================================
100      SUBROUTINE SLAGSY( N, K, D, A, LDA, ISEED, WORK, INFO )
101*
102*  -- LAPACK auxiliary routine --
103*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
104*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
105*
106*     .. Scalar Arguments ..
107      INTEGER            INFO, K, LDA, N
108*     ..
109*     .. Array Arguments ..
110      INTEGER            ISEED( 4 )
111      REAL               A( LDA, * ), D( * ), WORK( * )
112*     ..
113*
114*  =====================================================================
115*
116*     .. Parameters ..
117      REAL               ZERO, ONE, HALF
118      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, HALF = 0.5E+0 )
119*     ..
120*     .. Local Scalars ..
121      INTEGER            I, J
122      REAL               ALPHA, TAU, WA, WB, WN
123*     ..
124*     .. External Subroutines ..
125      EXTERNAL           SAXPY, SGEMV, SGER, SLARNV, SSCAL, SSYMV,
126     $                   SSYR2, XERBLA
127*     ..
128*     .. External Functions ..
129      REAL               SDOT, SNRM2
130      EXTERNAL           SDOT, SNRM2
131*     ..
132*     .. Intrinsic Functions ..
133      INTRINSIC          MAX, SIGN
134*     ..
135*     .. Executable Statements ..
136*
137*     Test the input arguments
138*
139      INFO = 0
140      IF( N.LT.0 ) THEN
141         INFO = -1
142      ELSE IF( K.LT.0 .OR. K.GT.N-1 ) THEN
143         INFO = -2
144      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
145         INFO = -5
146      END IF
147      IF( INFO.LT.0 ) THEN
148         CALL XERBLA( 'SLAGSY', -INFO )
149         RETURN
150      END IF
151*
152*     initialize lower triangle of A to diagonal matrix
153*
154      DO 20 J = 1, N
155         DO 10 I = J + 1, N
156            A( I, J ) = ZERO
157   10    CONTINUE
158   20 CONTINUE
159      DO 30 I = 1, N
160         A( I, I ) = D( I )
161   30 CONTINUE
162*
163*     Generate lower triangle of symmetric matrix
164*
165      DO 40 I = N - 1, 1, -1
166*
167*        generate random reflection
168*
169         CALL SLARNV( 3, ISEED, N-I+1, WORK )
170         WN = SNRM2( N-I+1, WORK, 1 )
171         WA = SIGN( WN, WORK( 1 ) )
172         IF( WN.EQ.ZERO ) THEN
173            TAU = ZERO
174         ELSE
175            WB = WORK( 1 ) + WA
176            CALL SSCAL( N-I, ONE / WB, WORK( 2 ), 1 )
177            WORK( 1 ) = ONE
178            TAU = WB / WA
179         END IF
180*
181*        apply random reflection to A(i:n,i:n) from the left
182*        and the right
183*
184*        compute  y := tau * A * u
185*
186         CALL SSYMV( 'Lower', N-I+1, TAU, A( I, I ), LDA, WORK, 1, ZERO,
187     $               WORK( N+1 ), 1 )
188*
189*        compute  v := y - 1/2 * tau * ( y, u ) * u
190*
191         ALPHA = -HALF*TAU*SDOT( N-I+1, WORK( N+1 ), 1, WORK, 1 )
192         CALL SAXPY( N-I+1, ALPHA, WORK, 1, WORK( N+1 ), 1 )
193*
194*        apply the transformation as a rank-2 update to A(i:n,i:n)
195*
196         CALL SSYR2( 'Lower', N-I+1, -ONE, WORK, 1, WORK( N+1 ), 1,
197     $               A( I, I ), LDA )
198   40 CONTINUE
199*
200*     Reduce number of subdiagonals to K
201*
202      DO 60 I = 1, N - 1 - K
203*
204*        generate reflection to annihilate A(k+i+1:n,i)
205*
206         WN = SNRM2( N-K-I+1, A( K+I, I ), 1 )
207         WA = SIGN( WN, A( K+I, I ) )
208         IF( WN.EQ.ZERO ) THEN
209            TAU = ZERO
210         ELSE
211            WB = A( K+I, I ) + WA
212            CALL SSCAL( N-K-I, ONE / WB, A( K+I+1, I ), 1 )
213            A( K+I, I ) = ONE
214            TAU = WB / WA
215         END IF
216*
217*        apply reflection to A(k+i:n,i+1:k+i-1) from the left
218*
219         CALL SGEMV( 'Transpose', N-K-I+1, K-1, ONE, A( K+I, I+1 ), LDA,
220     $               A( K+I, I ), 1, ZERO, WORK, 1 )
221         CALL SGER( N-K-I+1, K-1, -TAU, A( K+I, I ), 1, WORK, 1,
222     $              A( K+I, I+1 ), LDA )
223*
224*        apply reflection to A(k+i:n,k+i:n) from the left and the right
225*
226*        compute  y := tau * A * u
227*
228         CALL SSYMV( 'Lower', N-K-I+1, TAU, A( K+I, K+I ), LDA,
229     $               A( K+I, I ), 1, ZERO, WORK, 1 )
230*
231*        compute  v := y - 1/2 * tau * ( y, u ) * u
232*
233         ALPHA = -HALF*TAU*SDOT( N-K-I+1, WORK, 1, A( K+I, I ), 1 )
234         CALL SAXPY( N-K-I+1, ALPHA, A( K+I, I ), 1, WORK, 1 )
235*
236*        apply symmetric rank-2 update to A(k+i:n,k+i:n)
237*
238         CALL SSYR2( 'Lower', N-K-I+1, -ONE, A( K+I, I ), 1, WORK, 1,
239     $               A( K+I, K+I ), LDA )
240*
241         A( K+I, I ) = -WA
242         DO 50 J = K + I + 1, N
243            A( J, I ) = ZERO
244   50    CONTINUE
245   60 CONTINUE
246*
247*     Store full symmetric matrix
248*
249      DO 80 J = 1, N
250         DO 70 I = J + 1, N
251            A( J, I ) = A( I, J )
252   70    CONTINUE
253   80 CONTINUE
254      RETURN
255*
256*     End of SLAGSY
257*
258      END
259