LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
ssyt22.f
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1 *> \brief \b SSYT22
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 SSYT22( 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 * REAL 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 *> SSYT22 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**T A U - S | / ( |A| m ulp ) and
45 *> RESULT(2) = | I - U**T U | / ( m ulp )
46 *> \endverbatim
47 *
48 * Arguments:
49 * ==========
50 *
51 *> \verbatim
52 *> ITYPE INTEGER
53 *> Specifies the type of tests to be performed.
54 *> 1: U expressed as a dense orthogonal matrix:
55 *> RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and
56 *> RESULT(2) = | I - U U**T | / ( n ulp )
57 *>
58 *> UPLO CHARACTER
59 *> If UPLO='U', the upper triangle of A will be used and the
60 *> (strictly) lower triangle will not be referenced. If
61 *> UPLO='L', the lower triangle of A will be used and the
62 *> (strictly) upper triangle will not be referenced.
63 *> Not modified.
64 *>
65 *> N INTEGER
66 *> The size of the matrix. If it is zero, SSYT22 does nothing.
67 *> It must be at least zero.
68 *> Not modified.
69 *>
70 *> M INTEGER
71 *> The number of columns of U. If it is zero, SSYT22 does
72 *> nothing. It must be at least zero.
73 *> Not modified.
74 *>
75 *> KBAND INTEGER
76 *> The bandwidth of the matrix. It may only be zero or one.
77 *> If zero, then S is diagonal, and E is not referenced. If
78 *> one, then S is symmetric tri-diagonal.
79 *> Not modified.
80 *>
81 *> A REAL array, dimension (LDA , N)
82 *> The original (unfactored) matrix. It is assumed to be
83 *> symmetric, and only the upper (UPLO='U') or only the lower
84 *> (UPLO='L') will be referenced.
85 *> Not modified.
86 *>
87 *> LDA INTEGER
88 *> The leading dimension of A. It must be at least 1
89 *> and at least N.
90 *> Not modified.
91 *>
92 *> D REAL array, dimension (N)
93 *> The diagonal of the (symmetric tri-) diagonal matrix.
94 *> Not modified.
95 *>
96 *> E REAL array, dimension (N)
97 *> The off-diagonal of the (symmetric tri-) diagonal matrix.
98 *> E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
99 *> Not referenced if KBAND=0.
100 *> Not modified.
101 *>
102 *> U REAL array, dimension (LDU, N)
103 *> If ITYPE=1 or 3, this contains the orthogonal matrix in
104 *> the decomposition, expressed as a dense matrix. If ITYPE=2,
105 *> then it is not referenced.
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 REAL 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 REAL array, dimension (N)
126 *> If ITYPE >= 2, then TAU(j) is the scalar factor of
127 *> v(j) v(j)**T 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 REAL array, dimension (2*N**2)
133 *> Workspace.
134 *> Modified.
135 *>
136 *> RESULT REAL array, dimension (2)
137 *> The values computed by the two tests described above. The
138 *> values are currently limited to 1/ulp, to avoid overflow.
139 *> RESULT(1) is always modified. RESULT(2) is modified only
140 *> if LDU is at least N.
141 *> Modified.
142 *> \endverbatim
143 *
144 * Authors:
145 * ========
146 *
147 *> \author Univ. of Tennessee
148 *> \author Univ. of California Berkeley
149 *> \author Univ. of Colorado Denver
150 *> \author NAG Ltd.
151 *
152 *> \ingroup single_eig
153 *
154 * =====================================================================
155  SUBROUTINE ssyt22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
156  $ V, LDV, TAU, WORK, RESULT )
157 *
158 * -- LAPACK test routine --
159 * -- LAPACK is a software package provided by Univ. of Tennessee, --
160 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161 *
162 * .. Scalar Arguments ..
163  CHARACTER UPLO
164  INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
165 * ..
166 * .. Array Arguments ..
167  REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
168  $ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
169 * ..
170 *
171 * =====================================================================
172 *
173 * .. Parameters ..
174  REAL ZERO, ONE
175  parameter( zero = 0.0e0, one = 1.0e0 )
176 * ..
177 * .. Local Scalars ..
178  INTEGER J, JJ, JJ1, JJ2, NN, NNP1
179  REAL ANORM, ULP, UNFL, WNORM
180 * ..
181 * .. External Functions ..
182  REAL SLAMCH, SLANSY
183  EXTERNAL slamch, slansy
184 * ..
185 * .. External Subroutines ..
186  EXTERNAL sgemm, ssymm
187 * ..
188 * .. Intrinsic Functions ..
189  INTRINSIC max, min, real
190 * ..
191 * .. Executable Statements ..
192 *
193  result( 1 ) = zero
194  result( 2 ) = zero
195  IF( n.LE.0 .OR. m.LE.0 )
196  $ RETURN
197 *
198  unfl = slamch( 'Safe minimum' )
199  ulp = slamch( 'Precision' )
200 *
201 * Do Test 1
202 *
203 * Norm of A:
204 *
205  anorm = max( slansy( '1', uplo, n, a, lda, work ), unfl )
206 *
207 * Compute error matrix:
208 *
209 * ITYPE=1: error = U**T A U - S
210 *
211  CALL ssymm( 'L', uplo, n, m, one, a, lda, u, ldu, zero, work, n )
212  nn = n*n
213  nnp1 = nn + 1
214  CALL sgemm( 'T', 'N', m, m, n, one, u, ldu, work, n, zero,
215  $ work( nnp1 ), n )
216  DO 10 j = 1, m
217  jj = nn + ( j-1 )*n + j
218  work( jj ) = work( jj ) - d( j )
219  10 CONTINUE
220  IF( kband.EQ.1 .AND. n.GT.1 ) THEN
221  DO 20 j = 2, m
222  jj1 = nn + ( j-1 )*n + j - 1
223  jj2 = nn + ( j-2 )*n + j
224  work( jj1 ) = work( jj1 ) - e( j-1 )
225  work( jj2 ) = work( jj2 ) - e( j-1 )
226  20 CONTINUE
227  END IF
228  wnorm = slansy( '1', uplo, m, work( nnp1 ), n, work( 1 ) )
229 *
230  IF( anorm.GT.wnorm ) THEN
231  result( 1 ) = ( wnorm / anorm ) / ( m*ulp )
232  ELSE
233  IF( anorm.LT.one ) THEN
234  result( 1 ) = ( min( wnorm, m*anorm ) / anorm ) / ( m*ulp )
235  ELSE
236  result( 1 ) = min( wnorm / anorm, real( m ) ) / ( m*ulp )
237  END IF
238  END IF
239 *
240 * Do Test 2
241 *
242 * Compute U**T U - I
243 *
244  IF( itype.EQ.1 )
245  $ CALL sort01( 'Columns', n, m, u, ldu, work, 2*n*n,
246  $ result( 2 ) )
247 *
248  RETURN
249 *
250 * End of SSYT22
251 *
252  END
subroutine ssymm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SSYMM
Definition: ssymm.f:189
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
subroutine sort01(ROWCOL, M, N, U, LDU, WORK, LWORK, RESID)
SORT01
Definition: sort01.f:116
subroutine ssyt22(ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK, RESULT)
SSYT22
Definition: ssyt22.f:157