1 SUBROUTINE SSTT21( N, KBAND, AD, AE, SD, SE, 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 INTEGER KBAND, LDU, N
10 * ..
11 * .. Array Arguments ..
12 REAL AD( * ), AE( * ), RESULT( 2 ), SD( * ),
13 $ SE( * ), U( LDU, * ), WORK( * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * SSTT21 checks a decomposition of the form
20 *
21 * A = U S U'
22 *
23 * where ' means transpose, A is symmetric tridiagonal, U is orthogonal,
24 * and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1).
25 * Two tests are performed:
26 *
27 * RESULT(1) = | A - U S U' | / ( |A| n ulp )
28 *
29 * RESULT(2) = | I - UU' | / ( n ulp )
30 *
31 * Arguments
32 * =========
33 *
34 * N (input) INTEGER
35 * The size of the matrix. If it is zero, SSTT21 does nothing.
36 * It must be at least zero.
37 *
38 * KBAND (input) INTEGER
39 * The bandwidth of the matrix S. It may only be zero or one.
40 * If zero, then S is diagonal, and SE is not referenced. If
41 * one, then S is symmetric tri-diagonal.
42 *
43 * AD (input) REAL array, dimension (N)
44 * The diagonal of the original (unfactored) matrix A. A is
45 * assumed to be symmetric tridiagonal.
46 *
47 * AE (input) REAL array, dimension (N-1)
48 * The off-diagonal of the original (unfactored) matrix A. A
49 * is assumed to be symmetric tridiagonal. AE(1) is the (1,2)
50 * and (2,1) element, AE(2) is the (2,3) and (3,2) element, etc.
51 *
52 * SD (input) REAL array, dimension (N)
53 * The diagonal of the (symmetric tri-) diagonal matrix S.
54 *
55 * SE (input) REAL array, dimension (N-1)
56 * The off-diagonal of the (symmetric tri-) diagonal matrix S.
57 * Not referenced if KBSND=0. If KBAND=1, then AE(1) is the
58 * (1,2) and (2,1) element, SE(2) is the (2,3) and (3,2)
59 * element, etc.
60 *
61 * U (input) REAL array, dimension (LDU, N)
62 * The orthogonal matrix in the decomposition.
63 *
64 * LDU (input) INTEGER
65 * The leading dimension of U. LDU must be at least N.
66 *
67 * WORK (workspace) REAL array, dimension (N*(N+1))
68 *
69 * RESULT (output) REAL array, dimension (2)
70 * The values computed by the two tests described above. The
71 * values are currently limited to 1/ulp, to avoid overflow.
72 * RESULT(1) is always modified.
73 *
74 * =====================================================================
75 *
76 * .. Parameters ..
77 REAL ZERO, ONE
78 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
79 * ..
80 * .. Local Scalars ..
81 INTEGER J
82 REAL ANORM, TEMP1, TEMP2, ULP, UNFL, WNORM
83 * ..
84 * .. External Functions ..
85 REAL SLAMCH, SLANGE, SLANSY
86 EXTERNAL SLAMCH, SLANGE, SLANSY
87 * ..
88 * .. External Subroutines ..
89 EXTERNAL SGEMM, SLASET, SSYR, SSYR2
90 * ..
91 * .. Intrinsic Functions ..
92 INTRINSIC ABS, MAX, MIN, REAL
93 * ..
94 * .. Executable Statements ..
95 *
96 * 1) Constants
97 *
98 RESULT( 1 ) = ZERO
99 RESULT( 2 ) = ZERO
100 IF( N.LE.0 )
101 $ RETURN
102 *
103 UNFL = SLAMCH( 'Safe minimum' )
104 ULP = SLAMCH( 'Precision' )
105 *
106 * Do Test 1
107 *
108 * Copy A & Compute its 1-Norm:
109 *
110 CALL SLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
111 *
112 ANORM = ZERO
113 TEMP1 = ZERO
114 *
115 DO 10 J = 1, N - 1
116 WORK( ( N+1 )*( J-1 )+1 ) = AD( J )
117 WORK( ( N+1 )*( J-1 )+2 ) = AE( J )
118 TEMP2 = ABS( AE( J ) )
119 ANORM = MAX( ANORM, ABS( AD( J ) )+TEMP1+TEMP2 )
120 TEMP1 = TEMP2
121 10 CONTINUE
122 *
123 WORK( N**2 ) = AD( N )
124 ANORM = MAX( ANORM, ABS( AD( N ) )+TEMP1, UNFL )
125 *
126 * Norm of A - USU'
127 *
128 DO 20 J = 1, N
129 CALL SSYR( 'L', N, -SD( J ), U( 1, J ), 1, WORK, N )
130 20 CONTINUE
131 *
132 IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
133 DO 30 J = 1, N - 1
134 CALL SSYR2( 'L', N, -SE( J ), U( 1, J ), 1, U( 1, J+1 ), 1,
135 $ WORK, N )
136 30 CONTINUE
137 END IF
138 *
139 WNORM = SLANSY( '1', 'L', N, WORK, N, WORK( N**2+1 ) )
140 *
141 IF( ANORM.GT.WNORM ) THEN
142 RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
143 ELSE
144 IF( ANORM.LT.ONE ) THEN
145 RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
146 ELSE
147 RESULT( 1 ) = MIN( WNORM / ANORM, REAL( N ) ) / ( N*ULP )
148 END IF
149 END IF
150 *
151 * Do Test 2
152 *
153 * Compute UU' - I
154 *
155 CALL SGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
156 $ N )
157 *
158 DO 40 J = 1, N
159 WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
160 40 CONTINUE
161 *
162 RESULT( 2 ) = MIN( REAL( N ), SLANGE( '1', N, N, WORK, N,
163 $ WORK( N**2+1 ) ) ) / ( N*ULP )
164 *
165 RETURN
166 *
167 * End of SSTT21
168 *
169 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 INTEGER KBAND, LDU, N
10 * ..
11 * .. Array Arguments ..
12 REAL AD( * ), AE( * ), RESULT( 2 ), SD( * ),
13 $ SE( * ), U( LDU, * ), WORK( * )
14 * ..
15 *
16 * Purpose
17 * =======
18 *
19 * SSTT21 checks a decomposition of the form
20 *
21 * A = U S U'
22 *
23 * where ' means transpose, A is symmetric tridiagonal, U is orthogonal,
24 * and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1).
25 * Two tests are performed:
26 *
27 * RESULT(1) = | A - U S U' | / ( |A| n ulp )
28 *
29 * RESULT(2) = | I - UU' | / ( n ulp )
30 *
31 * Arguments
32 * =========
33 *
34 * N (input) INTEGER
35 * The size of the matrix. If it is zero, SSTT21 does nothing.
36 * It must be at least zero.
37 *
38 * KBAND (input) INTEGER
39 * The bandwidth of the matrix S. It may only be zero or one.
40 * If zero, then S is diagonal, and SE is not referenced. If
41 * one, then S is symmetric tri-diagonal.
42 *
43 * AD (input) REAL array, dimension (N)
44 * The diagonal of the original (unfactored) matrix A. A is
45 * assumed to be symmetric tridiagonal.
46 *
47 * AE (input) REAL array, dimension (N-1)
48 * The off-diagonal of the original (unfactored) matrix A. A
49 * is assumed to be symmetric tridiagonal. AE(1) is the (1,2)
50 * and (2,1) element, AE(2) is the (2,3) and (3,2) element, etc.
51 *
52 * SD (input) REAL array, dimension (N)
53 * The diagonal of the (symmetric tri-) diagonal matrix S.
54 *
55 * SE (input) REAL array, dimension (N-1)
56 * The off-diagonal of the (symmetric tri-) diagonal matrix S.
57 * Not referenced if KBSND=0. If KBAND=1, then AE(1) is the
58 * (1,2) and (2,1) element, SE(2) is the (2,3) and (3,2)
59 * element, etc.
60 *
61 * U (input) REAL array, dimension (LDU, N)
62 * The orthogonal matrix in the decomposition.
63 *
64 * LDU (input) INTEGER
65 * The leading dimension of U. LDU must be at least N.
66 *
67 * WORK (workspace) REAL array, dimension (N*(N+1))
68 *
69 * RESULT (output) REAL array, dimension (2)
70 * The values computed by the two tests described above. The
71 * values are currently limited to 1/ulp, to avoid overflow.
72 * RESULT(1) is always modified.
73 *
74 * =====================================================================
75 *
76 * .. Parameters ..
77 REAL ZERO, ONE
78 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
79 * ..
80 * .. Local Scalars ..
81 INTEGER J
82 REAL ANORM, TEMP1, TEMP2, ULP, UNFL, WNORM
83 * ..
84 * .. External Functions ..
85 REAL SLAMCH, SLANGE, SLANSY
86 EXTERNAL SLAMCH, SLANGE, SLANSY
87 * ..
88 * .. External Subroutines ..
89 EXTERNAL SGEMM, SLASET, SSYR, SSYR2
90 * ..
91 * .. Intrinsic Functions ..
92 INTRINSIC ABS, MAX, MIN, REAL
93 * ..
94 * .. Executable Statements ..
95 *
96 * 1) Constants
97 *
98 RESULT( 1 ) = ZERO
99 RESULT( 2 ) = ZERO
100 IF( N.LE.0 )
101 $ RETURN
102 *
103 UNFL = SLAMCH( 'Safe minimum' )
104 ULP = SLAMCH( 'Precision' )
105 *
106 * Do Test 1
107 *
108 * Copy A & Compute its 1-Norm:
109 *
110 CALL SLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
111 *
112 ANORM = ZERO
113 TEMP1 = ZERO
114 *
115 DO 10 J = 1, N - 1
116 WORK( ( N+1 )*( J-1 )+1 ) = AD( J )
117 WORK( ( N+1 )*( J-1 )+2 ) = AE( J )
118 TEMP2 = ABS( AE( J ) )
119 ANORM = MAX( ANORM, ABS( AD( J ) )+TEMP1+TEMP2 )
120 TEMP1 = TEMP2
121 10 CONTINUE
122 *
123 WORK( N**2 ) = AD( N )
124 ANORM = MAX( ANORM, ABS( AD( N ) )+TEMP1, UNFL )
125 *
126 * Norm of A - USU'
127 *
128 DO 20 J = 1, N
129 CALL SSYR( 'L', N, -SD( J ), U( 1, J ), 1, WORK, N )
130 20 CONTINUE
131 *
132 IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
133 DO 30 J = 1, N - 1
134 CALL SSYR2( 'L', N, -SE( J ), U( 1, J ), 1, U( 1, J+1 ), 1,
135 $ WORK, N )
136 30 CONTINUE
137 END IF
138 *
139 WNORM = SLANSY( '1', 'L', N, WORK, N, WORK( N**2+1 ) )
140 *
141 IF( ANORM.GT.WNORM ) THEN
142 RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
143 ELSE
144 IF( ANORM.LT.ONE ) THEN
145 RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
146 ELSE
147 RESULT( 1 ) = MIN( WNORM / ANORM, REAL( N ) ) / ( N*ULP )
148 END IF
149 END IF
150 *
151 * Do Test 2
152 *
153 * Compute UU' - I
154 *
155 CALL SGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
156 $ N )
157 *
158 DO 40 J = 1, N
159 WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
160 40 CONTINUE
161 *
162 RESULT( 2 ) = MIN( REAL( N ), SLANGE( '1', N, N, WORK, N,
163 $ WORK( N**2+1 ) ) ) / ( N*ULP )
164 *
165 RETURN
166 *
167 * End of SSTT21
168 *
169 END