LAPACK  3.7.0 LAPACK: Linear Algebra PACKage
dsyt22.f
Go to the documentation of this file.
1 *> \brief \b DSYT22
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 DSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
12 * V, LDV, TAU, WORK, RESULT )
13 *
14 * .. Scalar Arguments ..
15 * CHARACTER UPLO
16 * INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
17 * ..
18 * .. Array Arguments ..
19 * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
20 * \$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
21 * ..
22 *
23 *
24 *> \par Purpose:
25 * =============
26 *>
27 *> \verbatim
28 *>
29 *> DSYT22 generally checks a decomposition of the form
30 *>
31 *> A U = U S
32 *>
33 *> where A is symmetric, the columns of U are orthonormal, and S
34 *> is diagonal (if KBAND=0) or symmetric tridiagonal (if
35 *> KBAND=1). If ITYPE=1, then U is represented as a dense matrix,
36 *> otherwise the U is expressed as a product of Householder
37 *> transformations, whose vectors are stored in the array "V" and
38 *> whose scaling constants are in "TAU"; we shall use the letter
39 *> "V" to refer to the product of Householder transformations
40 *> (which should be equal to U).
41 *>
42 *> Specifically, if ITYPE=1, then:
43 *>
44 *> RESULT(1) = | U' A U - S | / ( |A| m ulp ) *andC> RESULT(2) = | I - U'U | / ( m ulp )
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \verbatim
51 *> ITYPE INTEGER
52 *> Specifies the type of tests to be performed.
53 *> 1: U expressed as a dense orthogonal matrix:
54 *> RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( n ulp )
55 *>
56 *> UPLO CHARACTER
57 *> If UPLO='U', the upper triangle of A will be used and the
58 *> (strictly) lower triangle will not be referenced. If
59 *> UPLO='L', the lower triangle of A will be used and the
60 *> (strictly) upper triangle will not be referenced.
61 *> Not modified.
62 *>
63 *> N INTEGER
64 *> The size of the matrix. If it is zero, DSYT22 does nothing.
65 *> It must be at least zero.
66 *> Not modified.
67 *>
68 *> M INTEGER
69 *> The number of columns of U. If it is zero, DSYT22 does
70 *> nothing. It must be at least zero.
71 *> Not modified.
72 *>
73 *> KBAND INTEGER
74 *> The bandwidth of the matrix. It may only be zero or one.
75 *> If zero, then S is diagonal, and E is not referenced. If
76 *> one, then S is symmetric tri-diagonal.
77 *> Not modified.
78 *>
79 *> A 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 *> Not modified.
84 *>
85 *> LDA INTEGER
86 *> The leading dimension of A. It must be at least 1
87 *> and at least N.
88 *> Not modified.
89 *>
90 *> D DOUBLE PRECISION array, dimension (N)
91 *> The diagonal of the (symmetric tri-) diagonal matrix.
92 *> Not modified.
93 *>
94 *> E DOUBLE PRECISION array, dimension (N)
95 *> The off-diagonal of the (symmetric tri-) diagonal matrix.
96 *> E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
97 *> Not referenced if KBAND=0.
98 *> Not modified.
99 *>
100 *> U DOUBLE PRECISION array, dimension (LDU, N)
101 *> If ITYPE=1 or 3, this contains the orthogonal matrix in
102 *> the decomposition, expressed as a dense matrix. If ITYPE=2,
103 *> then it is not referenced.
104 *> Not modified.
105 *>
106 *> LDU INTEGER
107 *> The leading dimension of U. LDU must be at least N and
108 *> at least 1.
109 *> Not modified.
110 *>
111 *> V DOUBLE PRECISION array, dimension (LDV, N)
112 *> If ITYPE=2 or 3, the lower triangle of this array contains
113 *> the Householder vectors used to describe the orthogonal
114 *> matrix in the decomposition. If ITYPE=1, then it is not
115 *> referenced.
116 *> Not modified.
117 *>
118 *> LDV INTEGER
119 *> The leading dimension of V. LDV must be at least N and
120 *> at least 1.
121 *> Not modified.
122 *>
123 *> TAU DOUBLE PRECISION array, dimension (N)
124 *> If ITYPE >= 2, then TAU(j) is the scalar factor of
125 *> v(j) v(j)' in the Householder transformation H(j) of
126 *> the product U = H(1)...H(n-2)
127 *> If ITYPE < 2, then TAU is not referenced.
128 *> Not modified.
129 *>
130 *> WORK DOUBLE PRECISION array, dimension (2*N**2)
131 *> Workspace.
132 *> Modified.
133 *>
134 *> RESULT 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 *> RESULT(1) is always modified. RESULT(2) is modified only
138 *> if LDU is at least N.
139 *> Modified.
140 *> \endverbatim
141 *
142 * Authors:
143 * ========
144 *
145 *> \author Univ. of Tennessee
146 *> \author Univ. of California Berkeley
147 *> \author Univ. of Colorado Denver
148 *> \author NAG Ltd.
149 *
150 *> \date December 2016
151 *
152 *> \ingroup double_eig
153 *
154 * =====================================================================
155  SUBROUTINE dsyt22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
156  \$ v, ldv, tau, work, result )
157 *
158 * -- LAPACK test routine (version 3.7.0) --
159 * -- LAPACK is a software package provided by Univ. of Tennessee, --
160 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161 * December 2016
162 *
163 * .. Scalar Arguments ..
164  CHARACTER UPLO
165  INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
166 * ..
167 * .. Array Arguments ..
168  DOUBLE PRECISION A( lda, * ), D( * ), E( * ), RESULT( 2 ),
169  \$ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
170 * ..
171 *
172 * =====================================================================
173 *
174 * .. Parameters ..
175  DOUBLE PRECISION ZERO, ONE
176  parameter ( zero = 0.0d0, one = 1.0d0 )
177 * ..
178 * .. Local Scalars ..
179  INTEGER J, JJ, JJ1, JJ2, NN, NNP1
180  DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
181 * ..
182 * .. External Functions ..
183  DOUBLE PRECISION DLAMCH, DLANSY
184  EXTERNAL dlamch, dlansy
185 * ..
186 * .. External Subroutines ..
187  EXTERNAL dgemm, dort01, dsymm
188 * ..
189 * .. Intrinsic Functions ..
190  INTRINSIC dble, max, min
191 * ..
192 * .. Executable Statements ..
193 *
194  result( 1 ) = zero
195  result( 2 ) = zero
196  IF( n.LE.0 .OR. m.LE.0 )
197  \$ RETURN
198 *
199  unfl = dlamch( 'Safe minimum' )
200  ulp = dlamch( 'Precision' )
201 *
202 * Do Test 1
203 *
204 * Norm of A:
205 *
206  anorm = max( dlansy( '1', uplo, n, a, lda, work ), unfl )
207 *
208 * Compute error matrix:
209 *
210 * ITYPE=1: error = U' A U - S
211 *
212  CALL dsymm( 'L', uplo, n, m, one, a, lda, u, ldu, zero, work, n )
213  nn = n*n
214  nnp1 = nn + 1
215  CALL dgemm( 'T', 'N', m, m, n, one, u, ldu, work, n, zero,
216  \$ work( nnp1 ), n )
217  DO 10 j = 1, m
218  jj = nn + ( j-1 )*n + j
219  work( jj ) = work( jj ) - d( j )
220  10 CONTINUE
221  IF( kband.EQ.1 .AND. n.GT.1 ) THEN
222  DO 20 j = 2, m
223  jj1 = nn + ( j-1 )*n + j - 1
224  jj2 = nn + ( j-2 )*n + j
225  work( jj1 ) = work( jj1 ) - e( j-1 )
226  work( jj2 ) = work( jj2 ) - e( j-1 )
227  20 CONTINUE
228  END IF
229  wnorm = dlansy( '1', uplo, m, work( nnp1 ), n, work( 1 ) )
230 *
231  IF( anorm.GT.wnorm ) THEN
232  result( 1 ) = ( wnorm / anorm ) / ( m*ulp )
233  ELSE
234  IF( anorm.LT.one ) THEN
235  result( 1 ) = ( min( wnorm, m*anorm ) / anorm ) / ( m*ulp )
236  ELSE
237  result( 1 ) = min( wnorm / anorm, dble( m ) ) / ( m*ulp )
238  END IF
239  END IF
240 *
241 * Do Test 2
242 *
243 * Compute U'U - I
244 *
245  IF( itype.EQ.1 )
246  \$ CALL dort01( 'Columns', n, m, u, ldu, work, 2*n*n,
247  \$ result( 2 ) )
248 *
249  RETURN
250 *
251 * End of DSYT22
252 *
253  END
subroutine dsymm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DSYMM
Definition: dsymm.f:191
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:189
subroutine dsyt22(ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK, RESULT)
DSYT22
Definition: dsyt22.f:157
subroutine dort01(ROWCOL, M, N, U, LDU, WORK, LWORK, RESID)
DORT01
Definition: dort01.f:118