1 SUBROUTINE DSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
2 $ RESULT )
3 *
4 * -- LAPACK test routine (version 3.1) --
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
6 * November 2006
7 *
8 * .. Scalar Arguments ..
9 CHARACTER UPLO
10 INTEGER KA, KS, LDA, LDU, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
14 $ U( LDU, * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DSBT21 generally checks a decomposition of the form
21 *
22 * A = U S U'
23 *
24 * where ' means transpose, A is symmetric banded, U is
25 * orthogonal, and S is diagonal (if KS=0) or symmetric
26 * tridiagonal (if KS=1).
27 *
28 * Specifically:
29 *
30 * RESULT(1) = | A - U S U' | / ( |A| n ulp ) *and*
31 * RESULT(2) = | I - UU' | / ( n ulp )
32 *
33 * Arguments
34 * =========
35 *
36 * UPLO (input) CHARACTER
37 * If UPLO='U', the upper triangle of A and V will be used and
38 * the (strictly) lower triangle will not be referenced.
39 * If UPLO='L', the lower triangle of A and V will be used and
40 * the (strictly) upper triangle will not be referenced.
41 *
42 * N (input) INTEGER
43 * The size of the matrix. If it is zero, DSBT21 does nothing.
44 * It must be at least zero.
45 *
46 * KA (input) INTEGER
47 * The bandwidth of the matrix A. It must be at least zero. If
48 * it is larger than N-1, then max( 0, N-1 ) will be used.
49 *
50 * KS (input) INTEGER
51 * The bandwidth of the matrix S. It may only be zero or one.
52 * If zero, then S is diagonal, and E is not referenced. If
53 * one, then S is symmetric tri-diagonal.
54 *
55 * A (input) DOUBLE PRECISION array, dimension (LDA, N)
56 * The original (unfactored) matrix. It is assumed to be
57 * symmetric, and only the upper (UPLO='U') or only the lower
58 * (UPLO='L') will be referenced.
59 *
60 * LDA (input) INTEGER
61 * The leading dimension of A. It must be at least 1
62 * and at least min( KA, N-1 ).
63 *
64 * D (input) DOUBLE PRECISION array, dimension (N)
65 * The diagonal of the (symmetric tri-) diagonal matrix S.
66 *
67 * E (input) DOUBLE PRECISION array, dimension (N-1)
68 * The off-diagonal of the (symmetric tri-) diagonal matrix S.
69 * E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
70 * (3,2) element, etc.
71 * Not referenced if KS=0.
72 *
73 * U (input) DOUBLE PRECISION array, dimension (LDU, N)
74 * The orthogonal matrix in the decomposition, expressed as a
75 * dense matrix (i.e., not as a product of Householder
76 * transformations, Givens transformations, etc.)
77 *
78 * LDU (input) INTEGER
79 * The leading dimension of U. LDU must be at least N and
80 * at least 1.
81 *
82 * WORK (workspace) DOUBLE PRECISION array, dimension (N**2+N)
83 *
84 * RESULT (output) DOUBLE PRECISION array, dimension (2)
85 * The values computed by the two tests described above. The
86 * values are currently limited to 1/ulp, to avoid overflow.
87 *
88 * =====================================================================
89 *
90 * .. Parameters ..
91 DOUBLE PRECISION ZERO, ONE
92 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
93 * ..
94 * .. Local Scalars ..
95 LOGICAL LOWER
96 CHARACTER CUPLO
97 INTEGER IKA, J, JC, JR, LW
98 DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
99 * ..
100 * .. External Functions ..
101 LOGICAL LSAME
102 DOUBLE PRECISION DLAMCH, DLANGE, DLANSB, DLANSP
103 EXTERNAL LSAME, DLAMCH, DLANGE, DLANSB, DLANSP
104 * ..
105 * .. External Subroutines ..
106 EXTERNAL DGEMM, DSPR, DSPR2
107 * ..
108 * .. Intrinsic Functions ..
109 INTRINSIC DBLE, MAX, MIN
110 * ..
111 * .. Executable Statements ..
112 *
113 * Constants
114 *
115 RESULT( 1 ) = ZERO
116 RESULT( 2 ) = ZERO
117 IF( N.LE.0 )
118 $ RETURN
119 *
120 IKA = MAX( 0, MIN( N-1, KA ) )
121 LW = ( N*( N+1 ) ) / 2
122 *
123 IF( LSAME( UPLO, 'U' ) ) THEN
124 LOWER = .FALSE.
125 CUPLO = 'U'
126 ELSE
127 LOWER = .TRUE.
128 CUPLO = 'L'
129 END IF
130 *
131 UNFL = DLAMCH( 'Safe minimum' )
132 ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
133 *
134 * Some Error Checks
135 *
136 * Do Test 1
137 *
138 * Norm of A:
139 *
140 ANORM = MAX( DLANSB( '1', CUPLO, N, IKA, A, LDA, WORK ), UNFL )
141 *
142 * Compute error matrix: Error = A - U S U'
143 *
144 * Copy A from SB to SP storage format.
145 *
146 J = 0
147 DO 50 JC = 1, N
148 IF( LOWER ) THEN
149 DO 10 JR = 1, MIN( IKA+1, N+1-JC )
150 J = J + 1
151 WORK( J ) = A( JR, JC )
152 10 CONTINUE
153 DO 20 JR = IKA + 2, N + 1 - JC
154 J = J + 1
155 WORK( J ) = ZERO
156 20 CONTINUE
157 ELSE
158 DO 30 JR = IKA + 2, JC
159 J = J + 1
160 WORK( J ) = ZERO
161 30 CONTINUE
162 DO 40 JR = MIN( IKA, JC-1 ), 0, -1
163 J = J + 1
164 WORK( J ) = A( IKA+1-JR, JC )
165 40 CONTINUE
166 END IF
167 50 CONTINUE
168 *
169 DO 60 J = 1, N
170 CALL DSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
171 60 CONTINUE
172 *
173 IF( N.GT.1 .AND. KS.EQ.1 ) THEN
174 DO 70 J = 1, N - 1
175 CALL DSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), 1,
176 $ WORK )
177 70 CONTINUE
178 END IF
179 WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( LW+1 ) )
180 *
181 IF( ANORM.GT.WNORM ) THEN
182 RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
183 ELSE
184 IF( ANORM.LT.ONE ) THEN
185 RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
186 ELSE
187 RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
188 END IF
189 END IF
190 *
191 * Do Test 2
192 *
193 * Compute UU' - I
194 *
195 CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
196 $ N )
197 *
198 DO 80 J = 1, N
199 WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
200 80 CONTINUE
201 *
202 RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ),
203 $ DBLE( N ) ) / ( N*ULP )
204 *
205 RETURN
206 *
207 * End of DSBT21
208 *
209 END
2 $ RESULT )
3 *
4 * -- LAPACK test routine (version 3.1) --
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
6 * November 2006
7 *
8 * .. Scalar Arguments ..
9 CHARACTER UPLO
10 INTEGER KA, KS, LDA, LDU, N
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
14 $ U( LDU, * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DSBT21 generally checks a decomposition of the form
21 *
22 * A = U S U'
23 *
24 * where ' means transpose, A is symmetric banded, U is
25 * orthogonal, and S is diagonal (if KS=0) or symmetric
26 * tridiagonal (if KS=1).
27 *
28 * Specifically:
29 *
30 * RESULT(1) = | A - U S U' | / ( |A| n ulp ) *and*
31 * RESULT(2) = | I - UU' | / ( n ulp )
32 *
33 * Arguments
34 * =========
35 *
36 * UPLO (input) CHARACTER
37 * If UPLO='U', the upper triangle of A and V will be used and
38 * the (strictly) lower triangle will not be referenced.
39 * If UPLO='L', the lower triangle of A and V will be used and
40 * the (strictly) upper triangle will not be referenced.
41 *
42 * N (input) INTEGER
43 * The size of the matrix. If it is zero, DSBT21 does nothing.
44 * It must be at least zero.
45 *
46 * KA (input) INTEGER
47 * The bandwidth of the matrix A. It must be at least zero. If
48 * it is larger than N-1, then max( 0, N-1 ) will be used.
49 *
50 * KS (input) INTEGER
51 * The bandwidth of the matrix S. It may only be zero or one.
52 * If zero, then S is diagonal, and E is not referenced. If
53 * one, then S is symmetric tri-diagonal.
54 *
55 * A (input) DOUBLE PRECISION array, dimension (LDA, N)
56 * The original (unfactored) matrix. It is assumed to be
57 * symmetric, and only the upper (UPLO='U') or only the lower
58 * (UPLO='L') will be referenced.
59 *
60 * LDA (input) INTEGER
61 * The leading dimension of A. It must be at least 1
62 * and at least min( KA, N-1 ).
63 *
64 * D (input) DOUBLE PRECISION array, dimension (N)
65 * The diagonal of the (symmetric tri-) diagonal matrix S.
66 *
67 * E (input) DOUBLE PRECISION array, dimension (N-1)
68 * The off-diagonal of the (symmetric tri-) diagonal matrix S.
69 * E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
70 * (3,2) element, etc.
71 * Not referenced if KS=0.
72 *
73 * U (input) DOUBLE PRECISION array, dimension (LDU, N)
74 * The orthogonal matrix in the decomposition, expressed as a
75 * dense matrix (i.e., not as a product of Householder
76 * transformations, Givens transformations, etc.)
77 *
78 * LDU (input) INTEGER
79 * The leading dimension of U. LDU must be at least N and
80 * at least 1.
81 *
82 * WORK (workspace) DOUBLE PRECISION array, dimension (N**2+N)
83 *
84 * RESULT (output) DOUBLE PRECISION array, dimension (2)
85 * The values computed by the two tests described above. The
86 * values are currently limited to 1/ulp, to avoid overflow.
87 *
88 * =====================================================================
89 *
90 * .. Parameters ..
91 DOUBLE PRECISION ZERO, ONE
92 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
93 * ..
94 * .. Local Scalars ..
95 LOGICAL LOWER
96 CHARACTER CUPLO
97 INTEGER IKA, J, JC, JR, LW
98 DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
99 * ..
100 * .. External Functions ..
101 LOGICAL LSAME
102 DOUBLE PRECISION DLAMCH, DLANGE, DLANSB, DLANSP
103 EXTERNAL LSAME, DLAMCH, DLANGE, DLANSB, DLANSP
104 * ..
105 * .. External Subroutines ..
106 EXTERNAL DGEMM, DSPR, DSPR2
107 * ..
108 * .. Intrinsic Functions ..
109 INTRINSIC DBLE, MAX, MIN
110 * ..
111 * .. Executable Statements ..
112 *
113 * Constants
114 *
115 RESULT( 1 ) = ZERO
116 RESULT( 2 ) = ZERO
117 IF( N.LE.0 )
118 $ RETURN
119 *
120 IKA = MAX( 0, MIN( N-1, KA ) )
121 LW = ( N*( N+1 ) ) / 2
122 *
123 IF( LSAME( UPLO, 'U' ) ) THEN
124 LOWER = .FALSE.
125 CUPLO = 'U'
126 ELSE
127 LOWER = .TRUE.
128 CUPLO = 'L'
129 END IF
130 *
131 UNFL = DLAMCH( 'Safe minimum' )
132 ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
133 *
134 * Some Error Checks
135 *
136 * Do Test 1
137 *
138 * Norm of A:
139 *
140 ANORM = MAX( DLANSB( '1', CUPLO, N, IKA, A, LDA, WORK ), UNFL )
141 *
142 * Compute error matrix: Error = A - U S U'
143 *
144 * Copy A from SB to SP storage format.
145 *
146 J = 0
147 DO 50 JC = 1, N
148 IF( LOWER ) THEN
149 DO 10 JR = 1, MIN( IKA+1, N+1-JC )
150 J = J + 1
151 WORK( J ) = A( JR, JC )
152 10 CONTINUE
153 DO 20 JR = IKA + 2, N + 1 - JC
154 J = J + 1
155 WORK( J ) = ZERO
156 20 CONTINUE
157 ELSE
158 DO 30 JR = IKA + 2, JC
159 J = J + 1
160 WORK( J ) = ZERO
161 30 CONTINUE
162 DO 40 JR = MIN( IKA, JC-1 ), 0, -1
163 J = J + 1
164 WORK( J ) = A( IKA+1-JR, JC )
165 40 CONTINUE
166 END IF
167 50 CONTINUE
168 *
169 DO 60 J = 1, N
170 CALL DSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
171 60 CONTINUE
172 *
173 IF( N.GT.1 .AND. KS.EQ.1 ) THEN
174 DO 70 J = 1, N - 1
175 CALL DSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), 1,
176 $ WORK )
177 70 CONTINUE
178 END IF
179 WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( LW+1 ) )
180 *
181 IF( ANORM.GT.WNORM ) THEN
182 RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
183 ELSE
184 IF( ANORM.LT.ONE ) THEN
185 RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
186 ELSE
187 RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
188 END IF
189 END IF
190 *
191 * Do Test 2
192 *
193 * Compute UU' - I
194 *
195 CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
196 $ N )
197 *
198 DO 80 J = 1, N
199 WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
200 80 CONTINUE
201 *
202 RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ),
203 $ DBLE( N ) ) / ( N*ULP )
204 *
205 RETURN
206 *
207 * End of DSBT21
208 *
209 END