1*> \brief \b ZHBT21
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 ZHBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
12*                          RWORK, RESULT )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          UPLO
16*       INTEGER            KA, KS, LDA, LDU, N
17*       ..
18*       .. Array Arguments ..
19*       DOUBLE PRECISION   D( * ), E( * ), RESULT( 2 ), RWORK( * )
20*       COMPLEX*16         A( LDA, * ), U( LDU, * ), WORK( * )
21*       ..
22*
23*
24*> \par Purpose:
25*  =============
26*>
27*> \verbatim
28*>
29*> ZHBT21  generally checks a decomposition of the form
30*>
31*>         A = U S U**H
32*>
33*> where **H means conjugate transpose, A is hermitian banded, U is
34*> unitary, 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**H | / ( |A| n ulp ) and
40*>         RESULT(2) = | I - U U**H | / ( 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, ZHBT21 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 COMPLEX*16 array, dimension (LDA, N)
80*>          The original (unfactored) matrix.  It is assumed to be
81*>          hermitian, 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 COMPLEX*16 array, dimension (LDU, N)
110*>          The unitary 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 COMPLEX*16 array, dimension (N**2)
125*> \endverbatim
126*>
127*> \param[out] RWORK
128*> \verbatim
129*>          RWORK is DOUBLE PRECISION array, dimension (N)
130*> \endverbatim
131*>
132*> \param[out] RESULT
133*> \verbatim
134*>          RESULT is DOUBLE PRECISION array, dimension (2)
135*>          The values computed by the two tests described above.  The
136*>          values are currently limited to 1/ulp, to avoid overflow.
137*> \endverbatim
138*
139*  Authors:
140*  ========
141*
142*> \author Univ. of Tennessee
143*> \author Univ. of California Berkeley
144*> \author Univ. of Colorado Denver
145*> \author NAG Ltd.
146*
147*> \ingroup complex16_eig
148*
149*  =====================================================================
150      SUBROUTINE ZHBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
151     $                   RWORK, RESULT )
152*
153*  -- LAPACK test routine --
154*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
155*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
156*
157*     .. Scalar Arguments ..
158      CHARACTER          UPLO
159      INTEGER            KA, KS, LDA, LDU, N
160*     ..
161*     .. Array Arguments ..
162      DOUBLE PRECISION   D( * ), E( * ), RESULT( 2 ), RWORK( * )
163      COMPLEX*16         A( LDA, * ), U( LDU, * ), WORK( * )
164*     ..
165*
166*  =====================================================================
167*
168*     .. Parameters ..
169      COMPLEX*16         CZERO, CONE
170      PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
171     $                   CONE = ( 1.0D+0, 0.0D+0 ) )
172      DOUBLE PRECISION   ZERO, ONE
173      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
174*     ..
175*     .. Local Scalars ..
176      LOGICAL            LOWER
177      CHARACTER          CUPLO
178      INTEGER            IKA, J, JC, JR
179      DOUBLE PRECISION   ANORM, ULP, UNFL, WNORM
180*     ..
181*     .. External Functions ..
182      LOGICAL            LSAME
183      DOUBLE PRECISION   DLAMCH, ZLANGE, ZLANHB, ZLANHP
184      EXTERNAL           LSAME, DLAMCH, ZLANGE, ZLANHB, ZLANHP
185*     ..
186*     .. External Subroutines ..
187      EXTERNAL           ZGEMM, ZHPR, ZHPR2
188*     ..
189*     .. Intrinsic Functions ..
190      INTRINSIC          DBLE, DCMPLX, MAX, MIN
191*     ..
192*     .. Executable Statements ..
193*
194*     Constants
195*
196      RESULT( 1 ) = ZERO
197      RESULT( 2 ) = ZERO
198      IF( N.LE.0 )
199     $   RETURN
200*
201      IKA = MAX( 0, MIN( N-1, KA ) )
202*
203      IF( LSAME( UPLO, 'U' ) ) THEN
204         LOWER = .FALSE.
205         CUPLO = 'U'
206      ELSE
207         LOWER = .TRUE.
208         CUPLO = 'L'
209      END IF
210*
211      UNFL = DLAMCH( 'Safe minimum' )
212      ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
213*
214*     Some Error Checks
215*
216*     Do Test 1
217*
218*     Norm of A:
219*
220      ANORM = MAX( ZLANHB( '1', CUPLO, N, IKA, A, LDA, RWORK ), UNFL )
221*
222*     Compute error matrix:    Error = A - U S U**H
223*
224*     Copy A from SB to SP storage format.
225*
226      J = 0
227      DO 50 JC = 1, N
228         IF( LOWER ) THEN
229            DO 10 JR = 1, MIN( IKA+1, N+1-JC )
230               J = J + 1
231               WORK( J ) = A( JR, JC )
232   10       CONTINUE
233            DO 20 JR = IKA + 2, N + 1 - JC
234               J = J + 1
235               WORK( J ) = ZERO
236   20       CONTINUE
237         ELSE
238            DO 30 JR = IKA + 2, JC
239               J = J + 1
240               WORK( J ) = ZERO
241   30       CONTINUE
242            DO 40 JR = MIN( IKA, JC-1 ), 0, -1
243               J = J + 1
244               WORK( J ) = A( IKA+1-JR, JC )
245   40       CONTINUE
246         END IF
247   50 CONTINUE
248*
249      DO 60 J = 1, N
250         CALL ZHPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
251   60 CONTINUE
252*
253      IF( N.GT.1 .AND. KS.EQ.1 ) THEN
254         DO 70 J = 1, N - 1
255            CALL ZHPR2( CUPLO, N, -DCMPLX( E( J ) ), U( 1, J ), 1,
256     $                  U( 1, J+1 ), 1, WORK )
257   70    CONTINUE
258      END IF
259      WNORM = ZLANHP( '1', CUPLO, N, WORK, RWORK )
260*
261      IF( ANORM.GT.WNORM ) THEN
262         RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
263      ELSE
264         IF( ANORM.LT.ONE ) THEN
265            RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
266         ELSE
267            RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
268         END IF
269      END IF
270*
271*     Do Test 2
272*
273*     Compute  U U**H - I
274*
275      CALL ZGEMM( 'N', 'C', N, N, N, CONE, U, LDU, U, LDU, CZERO, WORK,
276     $            N )
277*
278      DO 80 J = 1, N
279         WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - CONE
280   80 CONTINUE
281*
282      RESULT( 2 ) = MIN( ZLANGE( '1', N, N, WORK, N, RWORK ),
283     $              DBLE( N ) ) / ( N*ULP )
284*
285      RETURN
286*
287*     End of ZHBT21
288*
289      END
290