LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
chet22.f
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1 *> \brief \b CHET22
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 CHET22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
12 * V, LDV, TAU, WORK, RWORK, RESULT )
13 *
14 * .. Scalar Arguments ..
15 * CHARACTER UPLO
16 * INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
17 * ..
18 * .. Array Arguments ..
19 * REAL D( * ), E( * ), RESULT( 2 ), RWORK( * )
20 * COMPLEX A( LDA, * ), TAU( * ), U( LDU, * ),
21 * $ V( LDV, * ), WORK( * )
22 * ..
23 *
24 *
25 *> \par Purpose:
26 * =============
27 *>
28 *> \verbatim
29 *>
30 *> CHET22 generally checks a decomposition of the form
31 *>
32 *> A U = U S
33 *>
34 *> where A is complex Hermitian, the columns of U are orthonormal,
35 *> and S is diagonal (if KBAND=0) or symmetric tridiagonal (if
36 *> KBAND=1). If ITYPE=1, then U is represented as a dense matrix,
37 *> otherwise the U is expressed as a product of Householder
38 *> transformations, whose vectors are stored in the array "V" and
39 *> whose scaling constants are in "TAU"; we shall use the letter
40 *> "V" to refer to the product of Householder transformations
41 *> (which should be equal to U).
42 *>
43 *> Specifically, if ITYPE=1, then:
44 *>
45 *> RESULT(1) = | U**H A U - S | / ( |A| m ulp ) and
46 *> RESULT(2) = | I - U**H U | / ( m ulp )
47 *> \endverbatim
48 *
49 * Arguments:
50 * ==========
51 *
52 *> \verbatim
53 *> ITYPE INTEGER
54 *> Specifies the type of tests to be performed.
55 *> 1: U expressed as a dense orthogonal matrix:
56 *> RESULT(1) = | A - U S U**H | / ( |A| n ulp ) and
57 *> RESULT(2) = | I - U U**H | / ( n ulp )
58 *>
59 *> UPLO CHARACTER
60 *> If UPLO='U', the upper triangle of A will be used and the
61 *> (strictly) lower triangle will not be referenced. If
62 *> UPLO='L', the lower triangle of A will be used and the
63 *> (strictly) upper triangle will not be referenced.
64 *> Not modified.
65 *>
66 *> N INTEGER
67 *> The size of the matrix. If it is zero, CHET22 does nothing.
68 *> It must be at least zero.
69 *> Not modified.
70 *>
71 *> M INTEGER
72 *> The number of columns of U. If it is zero, CHET22 does
73 *> nothing. It must be at least zero.
74 *> Not modified.
75 *>
76 *> KBAND INTEGER
77 *> The bandwidth of the matrix. It may only be zero or one.
78 *> If zero, then S is diagonal, and E is not referenced. If
79 *> one, then S is symmetric tri-diagonal.
80 *> Not modified.
81 *>
82 *> A COMPLEX array, dimension (LDA , N)
83 *> The original (unfactored) matrix. It is assumed to be
84 *> symmetric, and only the upper (UPLO='U') or only the lower
85 *> (UPLO='L') will be referenced.
86 *> Not modified.
87 *>
88 *> LDA INTEGER
89 *> The leading dimension of A. It must be at least 1
90 *> and at least N.
91 *> Not modified.
92 *>
93 *> D REAL array, dimension (N)
94 *> The diagonal of the (symmetric tri-) diagonal matrix.
95 *> Not modified.
96 *>
97 *> E REAL array, dimension (N)
98 *> The off-diagonal of the (symmetric tri-) diagonal matrix.
99 *> E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
100 *> Not referenced if KBAND=0.
101 *> Not modified.
102 *>
103 *> U COMPLEX array, dimension (LDU, N)
104 *> If ITYPE=1, this contains the orthogonal matrix in
105 *> the decomposition, expressed as a dense matrix.
106 *> Not modified.
107 *>
108 *> LDU INTEGER
109 *> The leading dimension of U. LDU must be at least N and
110 *> at least 1.
111 *> Not modified.
112 *>
113 *> V COMPLEX array, dimension (LDV, N)
114 *> If ITYPE=2 or 3, the lower triangle of this array contains
115 *> the Householder vectors used to describe the orthogonal
116 *> matrix in the decomposition. If ITYPE=1, then it is not
117 *> referenced.
118 *> Not modified.
119 *>
120 *> LDV INTEGER
121 *> The leading dimension of V. LDV must be at least N and
122 *> at least 1.
123 *> Not modified.
124 *>
125 *> TAU COMPLEX array, dimension (N)
126 *> If ITYPE >= 2, then TAU(j) is the scalar factor of
127 *> v(j) v(j)**H in the Householder transformation H(j) of
128 *> the product U = H(1)...H(n-2)
129 *> If ITYPE < 2, then TAU is not referenced.
130 *> Not modified.
131 *>
132 *> WORK COMPLEX array, dimension (2*N**2)
133 *> Workspace.
134 *> Modified.
135 *>
136 *> RWORK REAL array, dimension (N)
137 *> Workspace.
138 *> Modified.
139 *>
140 *> RESULT REAL array, dimension (2)
141 *> The values computed by the two tests described above. The
142 *> values are currently limited to 1/ulp, to avoid overflow.
143 *> RESULT(1) is always modified. RESULT(2) is modified only
144 *> if LDU is at least N.
145 *> Modified.
146 *> \endverbatim
147 *
148 * Authors:
149 * ========
150 *
151 *> \author Univ. of Tennessee
152 *> \author Univ. of California Berkeley
153 *> \author Univ. of Colorado Denver
154 *> \author NAG Ltd.
155 *
156 *> \ingroup complex_eig
157 *
158 * =====================================================================
159  SUBROUTINE chet22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
160  $ V, LDV, TAU, WORK, RWORK, RESULT )
161 *
162 * -- LAPACK test routine --
163 * -- LAPACK is a software package provided by Univ. of Tennessee, --
164 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
165 *
166 * .. Scalar Arguments ..
167  CHARACTER UPLO
168  INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
169 * ..
170 * .. Array Arguments ..
171  REAL D( * ), E( * ), RESULT( 2 ), RWORK( * )
172  COMPLEX A( LDA, * ), TAU( * ), U( LDU, * ),
173  $ v( ldv, * ), work( * )
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  REAL ZERO, ONE
180  parameter( zero = 0.0e0, one = 1.0e0 )
181  COMPLEX CZERO, CONE
182  parameter( czero = ( 0.0e0, 0.0e0 ),
183  $ cone = ( 1.0e0, 0.0e0 ) )
184 * ..
185 * .. Local Scalars ..
186  INTEGER J, JJ, JJ1, JJ2, NN, NNP1
187  REAL ANORM, ULP, UNFL, WNORM
188 * ..
189 * .. External Functions ..
190  REAL CLANHE, SLAMCH
191  EXTERNAL clanhe, slamch
192 * ..
193 * .. External Subroutines ..
194  EXTERNAL cgemm, chemm
195 * ..
196 * .. Intrinsic Functions ..
197  INTRINSIC max, min, real
198 * ..
199 * .. Executable Statements ..
200 *
201  result( 1 ) = zero
202  result( 2 ) = zero
203  IF( n.LE.0 .OR. m.LE.0 )
204  $ RETURN
205 *
206  unfl = slamch( 'Safe minimum' )
207  ulp = slamch( 'Precision' )
208 *
209 * Do Test 1
210 *
211 * Norm of A:
212 *
213  anorm = max( clanhe( '1', uplo, n, a, lda, rwork ), unfl )
214 *
215 * Compute error matrix:
216 *
217 * ITYPE=1: error = U**H A U - S
218 *
219  CALL chemm( 'L', uplo, n, m, cone, a, lda, u, ldu, czero, work,
220  $ n )
221  nn = n*n
222  nnp1 = nn + 1
223  CALL cgemm( 'C', 'N', m, m, n, cone, u, ldu, work, n, czero,
224  $ work( nnp1 ), n )
225  DO 10 j = 1, m
226  jj = nn + ( j-1 )*n + j
227  work( jj ) = work( jj ) - d( j )
228  10 CONTINUE
229  IF( kband.EQ.1 .AND. n.GT.1 ) THEN
230  DO 20 j = 2, m
231  jj1 = nn + ( j-1 )*n + j - 1
232  jj2 = nn + ( j-2 )*n + j
233  work( jj1 ) = work( jj1 ) - e( j-1 )
234  work( jj2 ) = work( jj2 ) - e( j-1 )
235  20 CONTINUE
236  END IF
237  wnorm = clanhe( '1', uplo, m, work( nnp1 ), n, rwork )
238 *
239  IF( anorm.GT.wnorm ) THEN
240  result( 1 ) = ( wnorm / anorm ) / ( m*ulp )
241  ELSE
242  IF( anorm.LT.one ) THEN
243  result( 1 ) = ( min( wnorm, m*anorm ) / anorm ) / ( m*ulp )
244  ELSE
245  result( 1 ) = min( wnorm / anorm, real( m ) ) / ( m*ulp )
246  END IF
247  END IF
248 *
249 * Do Test 2
250 *
251 * Compute U**H U - I
252 *
253  IF( itype.EQ.1 )
254  $ CALL cunt01( 'Columns', n, m, u, ldu, work, 2*n*n, rwork,
255  $ result( 2 ) )
256 *
257  RETURN
258 *
259 * End of CHET22
260 *
261  END
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
subroutine chemm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CHEMM
Definition: chemm.f:191
subroutine chet22(ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK, RWORK, RESULT)
CHET22
Definition: chet22.f:161
subroutine cunt01(ROWCOL, M, N, U, LDU, WORK, LWORK, RWORK, RESID)
CUNT01
Definition: cunt01.f:126