1*> \brief \b DSBT21
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 DSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
12*                          RESULT )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          UPLO
16*       INTEGER            KA, KS, LDA, LDU, N
17*       ..
18*       .. Array Arguments ..
19*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
20*      $                   U( LDU, * ), WORK( * )
21*       ..
22*
23*
24*> \par Purpose:
25*  =============
26*>
27*> \verbatim
28*>
29*> DSBT21  generally checks a decomposition of the form
30*>
31*>         A = U S U**T
32*>
33*> where **T means transpose, A is symmetric banded, U is
34*> orthogonal, and S is diagonal (if KS=0) or symmetric
35*> tridiagonal (if KS=1).
36*>
37*> Specifically:
38*>
39*>         RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and
40*>         RESULT(2) = | I - U U**T | / ( n ulp )
41*> \endverbatim
42*
43*  Arguments:
44*  ==========
45*
46*> \param[in] UPLO
47*> \verbatim
48*>          UPLO is CHARACTER
49*>          If UPLO='U', the upper triangle of A and V will be used and
50*>          the (strictly) lower triangle will not be referenced.
51*>          If UPLO='L', the lower triangle of A and V will be used and
52*>          the (strictly) upper triangle will not be referenced.
53*> \endverbatim
54*>
55*> \param[in] N
56*> \verbatim
57*>          N is INTEGER
58*>          The size of the matrix.  If it is zero, DSBT21 does nothing.
59*>          It must be at least zero.
60*> \endverbatim
61*>
62*> \param[in] KA
63*> \verbatim
64*>          KA is INTEGER
65*>          The bandwidth of the matrix A.  It must be at least zero.  If
66*>          it is larger than N-1, then max( 0, N-1 ) will be used.
67*> \endverbatim
68*>
69*> \param[in] KS
70*> \verbatim
71*>          KS is INTEGER
72*>          The bandwidth of the matrix S.  It may only be zero or one.
73*>          If zero, then S is diagonal, and E is not referenced.  If
74*>          one, then S is symmetric tri-diagonal.
75*> \endverbatim
76*>
77*> \param[in] A
78*> \verbatim
79*>          A is DOUBLE PRECISION array, dimension (LDA, N)
80*>          The original (unfactored) matrix.  It is assumed to be
81*>          symmetric, and only the upper (UPLO='U') or only the lower
82*>          (UPLO='L') will be referenced.
83*> \endverbatim
84*>
85*> \param[in] LDA
86*> \verbatim
87*>          LDA is INTEGER
88*>          The leading dimension of A.  It must be at least 1
89*>          and at least min( KA, N-1 ).
90*> \endverbatim
91*>
92*> \param[in] D
93*> \verbatim
94*>          D is DOUBLE PRECISION array, dimension (N)
95*>          The diagonal of the (symmetric tri-) diagonal matrix S.
96*> \endverbatim
97*>
98*> \param[in] E
99*> \verbatim
100*>          E is DOUBLE PRECISION array, dimension (N-1)
101*>          The off-diagonal of the (symmetric tri-) diagonal matrix S.
102*>          E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
103*>          (3,2) element, etc.
104*>          Not referenced if KS=0.
105*> \endverbatim
106*>
107*> \param[in] U
108*> \verbatim
109*>          U is DOUBLE PRECISION array, dimension (LDU, N)
110*>          The orthogonal matrix in the decomposition, expressed as a
111*>          dense matrix (i.e., not as a product of Householder
112*>          transformations, Givens transformations, etc.)
113*> \endverbatim
114*>
115*> \param[in] LDU
116*> \verbatim
117*>          LDU is INTEGER
118*>          The leading dimension of U.  LDU must be at least N and
119*>          at least 1.
120*> \endverbatim
121*>
122*> \param[out] WORK
123*> \verbatim
124*>          WORK is DOUBLE PRECISION array, dimension (N**2+N)
125*> \endverbatim
126*>
127*> \param[out] RESULT
128*> \verbatim
129*>          RESULT is DOUBLE PRECISION array, dimension (2)
130*>          The values computed by the two tests described above.  The
131*>          values are currently limited to 1/ulp, to avoid overflow.
132*> \endverbatim
133*
134*  Authors:
135*  ========
136*
137*> \author Univ. of Tennessee
138*> \author Univ. of California Berkeley
139*> \author Univ. of Colorado Denver
140*> \author NAG Ltd.
141*
142*> \ingroup double_eig
143*
144*  =====================================================================
145      SUBROUTINE DSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
146     $                   RESULT )
147*
148*  -- LAPACK test routine --
149*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
150*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
151*
152*     .. Scalar Arguments ..
153      CHARACTER          UPLO
154      INTEGER            KA, KS, LDA, LDU, N
155*     ..
156*     .. Array Arguments ..
157      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
158     $                   U( LDU, * ), WORK( * )
159*     ..
160*
161*  =====================================================================
162*
163*     .. Parameters ..
164      DOUBLE PRECISION   ZERO, ONE
165      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
166*     ..
167*     .. Local Scalars ..
168      LOGICAL            LOWER
169      CHARACTER          CUPLO
170      INTEGER            IKA, J, JC, JR, LW
171      DOUBLE PRECISION   ANORM, ULP, UNFL, WNORM
172*     ..
173*     .. External Functions ..
174      LOGICAL            LSAME
175      DOUBLE PRECISION   DLAMCH, DLANGE, DLANSB, DLANSP
176      EXTERNAL           LSAME, DLAMCH, DLANGE, DLANSB, DLANSP
177*     ..
178*     .. External Subroutines ..
179      EXTERNAL           DGEMM, DSPR, DSPR2
180*     ..
181*     .. Intrinsic Functions ..
182      INTRINSIC          DBLE, MAX, MIN
183*     ..
184*     .. Executable Statements ..
185*
186*     Constants
187*
188      RESULT( 1 ) = ZERO
189      RESULT( 2 ) = ZERO
190      IF( N.LE.0 )
191     $   RETURN
192*
193      IKA = MAX( 0, MIN( N-1, KA ) )
194      LW = ( N*( N+1 ) ) / 2
195*
196      IF( LSAME( UPLO, 'U' ) ) THEN
197         LOWER = .FALSE.
198         CUPLO = 'U'
199      ELSE
200         LOWER = .TRUE.
201         CUPLO = 'L'
202      END IF
203*
204      UNFL = DLAMCH( 'Safe minimum' )
205      ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
206*
207*     Some Error Checks
208*
209*     Do Test 1
210*
211*     Norm of A:
212*
213      ANORM = MAX( DLANSB( '1', CUPLO, N, IKA, A, LDA, WORK ), UNFL )
214*
215*     Compute error matrix:    Error = A - U S U**T
216*
217*     Copy A from SB to SP storage format.
218*
219      J = 0
220      DO 50 JC = 1, N
221         IF( LOWER ) THEN
222            DO 10 JR = 1, MIN( IKA+1, N+1-JC )
223               J = J + 1
224               WORK( J ) = A( JR, JC )
225   10       CONTINUE
226            DO 20 JR = IKA + 2, N + 1 - JC
227               J = J + 1
228               WORK( J ) = ZERO
229   20       CONTINUE
230         ELSE
231            DO 30 JR = IKA + 2, JC
232               J = J + 1
233               WORK( J ) = ZERO
234   30       CONTINUE
235            DO 40 JR = MIN( IKA, JC-1 ), 0, -1
236               J = J + 1
237               WORK( J ) = A( IKA+1-JR, JC )
238   40       CONTINUE
239         END IF
240   50 CONTINUE
241*
242      DO 60 J = 1, N
243         CALL DSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
244   60 CONTINUE
245*
246      IF( N.GT.1 .AND. KS.EQ.1 ) THEN
247         DO 70 J = 1, N - 1
248            CALL DSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), 1,
249     $                  WORK )
250   70    CONTINUE
251      END IF
252      WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( LW+1 ) )
253*
254      IF( ANORM.GT.WNORM ) THEN
255         RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
256      ELSE
257         IF( ANORM.LT.ONE ) THEN
258            RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
259         ELSE
260            RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
261         END IF
262      END IF
263*
264*     Do Test 2
265*
266*     Compute  U U**T - I
267*
268      CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
269     $            N )
270*
271      DO 80 J = 1, N
272         WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
273   80 CONTINUE
274*
275      RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ),
276     $              DBLE( N ) ) / ( N*ULP )
277*
278      RETURN
279*
280*     End of DSBT21
281*
282      END
283