LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
dpbt01.f
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1 *> \brief \b DPBT01
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 DPBT01( UPLO, N, KD, A, LDA, AFAC, LDAFAC, RWORK,
12 * RESID )
13 *
14 * .. Scalar Arguments ..
15 * CHARACTER UPLO
16 * INTEGER KD, LDA, LDAFAC, N
17 * DOUBLE PRECISION RESID
18 * ..
19 * .. Array Arguments ..
20 * DOUBLE PRECISION A( LDA, * ), AFAC( LDAFAC, * ), RWORK( * )
21 * ..
22 *
23 *
24 *> \par Purpose:
25 * =============
26 *>
27 *> \verbatim
28 *>
29 *> DPBT01 reconstructs a symmetric positive definite band matrix A from
30 *> its L*L' or U'*U factorization and computes the residual
31 *> norm( L*L' - A ) / ( N * norm(A) * EPS ) or
32 *> norm( U'*U - A ) / ( N * norm(A) * EPS ),
33 *> where EPS is the machine epsilon, L' is the conjugate transpose of
34 *> L, and U' is the conjugate transpose of U.
35 *> \endverbatim
36 *
37 * Arguments:
38 * ==========
39 *
40 *> \param[in] UPLO
41 *> \verbatim
42 *> UPLO is CHARACTER*1
43 *> Specifies whether the upper or lower triangular part of the
44 *> symmetric matrix A is stored:
45 *> = 'U': Upper triangular
46 *> = 'L': Lower triangular
47 *> \endverbatim
48 *>
49 *> \param[in] N
50 *> \verbatim
51 *> N is INTEGER
52 *> The number of rows and columns of the matrix A. N >= 0.
53 *> \endverbatim
54 *>
55 *> \param[in] KD
56 *> \verbatim
57 *> KD is INTEGER
58 *> The number of super-diagonals of the matrix A if UPLO = 'U',
59 *> or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
60 *> \endverbatim
61 *>
62 *> \param[in] A
63 *> \verbatim
64 *> A is DOUBLE PRECISION array, dimension (LDA,N)
65 *> The original symmetric band matrix A. If UPLO = 'U', the
66 *> upper triangular part of A is stored as a band matrix; if
67 *> UPLO = 'L', the lower triangular part of A is stored. The
68 *> columns of the appropriate triangle are stored in the columns
69 *> of A and the diagonals of the triangle are stored in the rows
70 *> of A. See DPBTRF for further details.
71 *> \endverbatim
72 *>
73 *> \param[in] LDA
74 *> \verbatim
75 *> LDA is INTEGER.
76 *> The leading dimension of the array A. LDA >= max(1,KD+1).
77 *> \endverbatim
78 *>
79 *> \param[in] AFAC
80 *> \verbatim
81 *> AFAC is DOUBLE PRECISION array, dimension (LDAFAC,N)
82 *> The factored form of the matrix A. AFAC contains the factor
83 *> L or U from the L*L' or U'*U factorization in band storage
84 *> format, as computed by DPBTRF.
85 *> \endverbatim
86 *>
87 *> \param[in] LDAFAC
88 *> \verbatim
89 *> LDAFAC is INTEGER
90 *> The leading dimension of the array AFAC.
91 *> LDAFAC >= max(1,KD+1).
92 *> \endverbatim
93 *>
94 *> \param[out] RWORK
95 *> \verbatim
96 *> RWORK is DOUBLE PRECISION array, dimension (N)
97 *> \endverbatim
98 *>
99 *> \param[out] RESID
100 *> \verbatim
101 *> RESID is DOUBLE PRECISION
102 *> If UPLO = 'L', norm(L*L' - A) / ( N * norm(A) * EPS )
103 *> If UPLO = 'U', norm(U'*U - A) / ( N * norm(A) * EPS )
104 *> \endverbatim
105 *
106 * Authors:
107 * ========
108 *
109 *> \author Univ. of Tennessee
110 *> \author Univ. of California Berkeley
111 *> \author Univ. of Colorado Denver
112 *> \author NAG Ltd.
113 *
114 *> \date December 2016
115 *
116 *> \ingroup double_lin
117 *
118 * =====================================================================
119  SUBROUTINE dpbt01( UPLO, N, KD, A, LDA, AFAC, LDAFAC, RWORK,
120  $ RESID )
121 *
122 * -- LAPACK test routine (version 3.7.0) --
123 * -- LAPACK is a software package provided by Univ. of Tennessee, --
124 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
125 * December 2016
126 *
127 * .. Scalar Arguments ..
128  CHARACTER UPLO
129  INTEGER KD, LDA, LDAFAC, N
130  DOUBLE PRECISION RESID
131 * ..
132 * .. Array Arguments ..
133  DOUBLE PRECISION A( lda, * ), AFAC( ldafac, * ), RWORK( * )
134 * ..
135 *
136 * =====================================================================
137 *
138 *
139 * .. Parameters ..
140  DOUBLE PRECISION ZERO, ONE
141  parameter( zero = 0.0d+0, one = 1.0d+0 )
142 * ..
143 * .. Local Scalars ..
144  INTEGER I, J, K, KC, KLEN, ML, MU
145  DOUBLE PRECISION ANORM, EPS, T
146 * ..
147 * .. External Functions ..
148  LOGICAL LSAME
149  DOUBLE PRECISION DDOT, DLAMCH, DLANSB
150  EXTERNAL lsame, ddot, dlamch, dlansb
151 * ..
152 * .. External Subroutines ..
153  EXTERNAL dscal, dsyr, dtrmv
154 * ..
155 * .. Intrinsic Functions ..
156  INTRINSIC dble, max, min
157 * ..
158 * .. Executable Statements ..
159 *
160 * Quick exit if N = 0.
161 *
162  IF( n.LE.0 ) THEN
163  resid = zero
164  RETURN
165  END IF
166 *
167 * Exit with RESID = 1/EPS if ANORM = 0.
168 *
169  eps = dlamch( 'Epsilon' )
170  anorm = dlansb( '1', uplo, n, kd, a, lda, rwork )
171  IF( anorm.LE.zero ) THEN
172  resid = one / eps
173  RETURN
174  END IF
175 *
176 * Compute the product U'*U, overwriting U.
177 *
178  IF( lsame( uplo, 'U' ) ) THEN
179  DO 10 k = n, 1, -1
180  kc = max( 1, kd+2-k )
181  klen = kd + 1 - kc
182 *
183 * Compute the (K,K) element of the result.
184 *
185  t = ddot( klen+1, afac( kc, k ), 1, afac( kc, k ), 1 )
186  afac( kd+1, k ) = t
187 *
188 * Compute the rest of column K.
189 *
190  IF( klen.GT.0 )
191  $ CALL dtrmv( 'Upper', 'Transpose', 'Non-unit', klen,
192  $ afac( kd+1, k-klen ), ldafac-1,
193  $ afac( kc, k ), 1 )
194 *
195  10 CONTINUE
196 *
197 * UPLO = 'L': Compute the product L*L', overwriting L.
198 *
199  ELSE
200  DO 20 k = n, 1, -1
201  klen = min( kd, n-k )
202 *
203 * Add a multiple of column K of the factor L to each of
204 * columns K+1 through N.
205 *
206  IF( klen.GT.0 )
207  $ CALL dsyr( 'Lower', klen, one, afac( 2, k ), 1,
208  $ afac( 1, k+1 ), ldafac-1 )
209 *
210 * Scale column K by the diagonal element.
211 *
212  t = afac( 1, k )
213  CALL dscal( klen+1, t, afac( 1, k ), 1 )
214 *
215  20 CONTINUE
216  END IF
217 *
218 * Compute the difference L*L' - A or U'*U - A.
219 *
220  IF( lsame( uplo, 'U' ) ) THEN
221  DO 40 j = 1, n
222  mu = max( 1, kd+2-j )
223  DO 30 i = mu, kd + 1
224  afac( i, j ) = afac( i, j ) - a( i, j )
225  30 CONTINUE
226  40 CONTINUE
227  ELSE
228  DO 60 j = 1, n
229  ml = min( kd+1, n-j+1 )
230  DO 50 i = 1, ml
231  afac( i, j ) = afac( i, j ) - a( i, j )
232  50 CONTINUE
233  60 CONTINUE
234  END IF
235 *
236 * Compute norm( L*L' - A ) / ( N * norm(A) * EPS )
237 *
238  resid = dlansb( 'I', uplo, n, kd, afac, ldafac, rwork )
239 *
240  resid = ( ( resid / dble( n ) ) / anorm ) / eps
241 *
242  RETURN
243 *
244 * End of DPBT01
245 *
246  END
subroutine dsyr(UPLO, N, ALPHA, X, INCX, A, LDA)
DSYR
Definition: dsyr.f:134
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:81
subroutine dpbt01(UPLO, N, KD, A, LDA, AFAC, LDAFAC, RWORK, RESID)
DPBT01
Definition: dpbt01.f:121
subroutine dtrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
DTRMV
Definition: dtrmv.f:149