LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
ssbt21.f
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1 *> \brief \b SSBT21
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 SSBT21( 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 * REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
20 * $ U( LDU, * ), WORK( * )
21 * ..
22 *
23 *
24 *> \par Purpose:
25 * =============
26 *>
27 *> \verbatim
28 *>
29 *> SSBT21 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, SSBT21 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (N**2+N)
125 *> \endverbatim
126 *>
127 *> \param[out] RESULT
128 *> \verbatim
129 *> RESULT is REAL 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 single_eig
143 *
144 * =====================================================================
145  SUBROUTINE ssbt21( 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  REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
158  $ u( ldu, * ), work( * )
159 * ..
160 *
161 * =====================================================================
162 *
163 * .. Parameters ..
164  REAL ZERO, ONE
165  parameter( zero = 0.0e0, one = 1.0e0 )
166 * ..
167 * .. Local Scalars ..
168  LOGICAL LOWER
169  CHARACTER CUPLO
170  INTEGER IKA, J, JC, JR, LW
171  REAL ANORM, ULP, UNFL, WNORM
172 * ..
173 * .. External Functions ..
174  LOGICAL LSAME
175  REAL SLAMCH, SLANGE, SLANSB, SLANSP
176  EXTERNAL lsame, slamch, slange, slansb, slansp
177 * ..
178 * .. External Subroutines ..
179  EXTERNAL sgemm, sspr, sspr2
180 * ..
181 * .. Intrinsic Functions ..
182  INTRINSIC max, min, real
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 = slamch( 'Safe minimum' )
205  ulp = slamch( 'Epsilon' )*slamch( 'Base' )
206 *
207 * Some Error Checks
208 *
209 * Do Test 1
210 *
211 * Norm of A:
212 *
213  anorm = max( slansb( '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 sspr( 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 sspr2( cuplo, n, -e( j ), u( 1, j ), 1, u( 1, j+1 ), 1,
249  $ work )
250  70 CONTINUE
251  END IF
252  wnorm = slansp( '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, real( n ) ) / ( n*ulp )
261  END IF
262  END IF
263 *
264 * Do Test 2
265 *
266 * Compute U U**T - I
267 *
268  CALL sgemm( '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( slange( '1', n, n, work, n, work( n**2+1 ) ),
276  $ real( n ) ) / ( n*ulp )
277 *
278  RETURN
279 *
280 * End of SSBT21
281 *
282  END
subroutine sspr(UPLO, N, ALPHA, X, INCX, AP)
SSPR
Definition: sspr.f:127
subroutine sspr2(UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
SSPR2
Definition: sspr2.f:142
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
subroutine ssbt21(UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK, RESULT)
SSBT21
Definition: ssbt21.f:147