1 SUBROUTINE STBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
2 $ SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
3 $ RESID )
4 *
5 * -- LAPACK test routine (version 3.1) --
6 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER DIAG, TRANS, UPLO
11 INTEGER KD, LDAB, LDB, LDX, N, NRHS
12 REAL RESID, SCALE, TSCAL
13 * ..
14 * .. Array Arguments ..
15 REAL AB( LDAB, * ), B( LDB, * ), CNORM( * ),
16 $ WORK( * ), X( LDX, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * STBT03 computes the residual for the solution to a scaled triangular
23 * system of equations A*x = s*b or A'*x = s*b when A is a
24 * triangular band matrix. Here A' is the transpose of A, s is a scalar,
25 * and x and b are N by NRHS matrices. The test ratio is the maximum
26 * over the number of right hand sides of
27 * norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
28 * where op(A) denotes A or A' and EPS is the machine epsilon.
29 *
30 * Arguments
31 * =========
32 *
33 * UPLO (input) CHARACTER*1
34 * Specifies whether the matrix A is upper or lower triangular.
35 * = 'U': Upper triangular
36 * = 'L': Lower triangular
37 *
38 * TRANS (input) CHARACTER*1
39 * Specifies the operation applied to A.
40 * = 'N': A *x = b (No transpose)
41 * = 'T': A'*x = b (Transpose)
42 * = 'C': A'*x = b (Conjugate transpose = Transpose)
43 *
44 * DIAG (input) CHARACTER*1
45 * Specifies whether or not the matrix A is unit triangular.
46 * = 'N': Non-unit triangular
47 * = 'U': Unit triangular
48 *
49 * N (input) INTEGER
50 * The order of the matrix A. N >= 0.
51 *
52 * KD (input) INTEGER
53 * The number of superdiagonals or subdiagonals of the
54 * triangular band matrix A. KD >= 0.
55 *
56 * NRHS (input) INTEGER
57 * The number of right hand sides, i.e., the number of columns
58 * of the matrices X and B. NRHS >= 0.
59 *
60 * AB (input) REAL array, dimension (LDAB,N)
61 * The upper or lower triangular band matrix A, stored in the
62 * first kd+1 rows of the array. The j-th column of A is stored
63 * in the j-th column of the array AB as follows:
64 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
65 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
66 *
67 * LDAB (input) INTEGER
68 * The leading dimension of the array AB. LDAB >= KD+1.
69 *
70 * SCALE (input) REAL
71 * The scaling factor s used in solving the triangular system.
72 *
73 * CNORM (input) REAL array, dimension (N)
74 * The 1-norms of the columns of A, not counting the diagonal.
75 *
76 * TSCAL (input) REAL
77 * The scaling factor used in computing the 1-norms in CNORM.
78 * CNORM actually contains the column norms of TSCAL*A.
79 *
80 * X (input) REAL array, dimension (LDX,NRHS)
81 * The computed solution vectors for the system of linear
82 * equations.
83 *
84 * LDX (input) INTEGER
85 * The leading dimension of the array X. LDX >= max(1,N).
86 *
87 * B (input) REAL array, dimension (LDB,NRHS)
88 * The right hand side vectors for the system of linear
89 * equations.
90 *
91 * LDB (input) INTEGER
92 * The leading dimension of the array B. LDB >= max(1,N).
93 *
94 * WORK (workspace) REAL array, dimension (N)
95 *
96 * RESID (output) REAL
97 * The maximum over the number of right hand sides of
98 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
99 *
100 * =====================================================================
101 *
102 * .. Parameters ..
103 REAL ONE, ZERO
104 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
105 * ..
106 * .. Local Scalars ..
107 INTEGER IX, J
108 REAL BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
109 * ..
110 * .. External Functions ..
111 LOGICAL LSAME
112 INTEGER ISAMAX
113 REAL SLAMCH
114 EXTERNAL LSAME, ISAMAX, SLAMCH
115 * ..
116 * .. External Subroutines ..
117 EXTERNAL SAXPY, SCOPY, SLABAD, SSCAL, STBMV
118 * ..
119 * .. Intrinsic Functions ..
120 INTRINSIC ABS, MAX, REAL
121 * ..
122 * .. Executable Statements ..
123 *
124 * Quick exit if N = 0
125 *
126 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
127 RESID = ZERO
128 RETURN
129 END IF
130 EPS = SLAMCH( 'Epsilon' )
131 SMLNUM = SLAMCH( 'Safe minimum' )
132 BIGNUM = ONE / SMLNUM
133 CALL SLABAD( SMLNUM, BIGNUM )
134 *
135 * Compute the norm of the triangular matrix A using the column
136 * norms already computed by SLATBS.
137 *
138 TNORM = ZERO
139 IF( LSAME( DIAG, 'N' ) ) THEN
140 IF( LSAME( UPLO, 'U' ) ) THEN
141 DO 10 J = 1, N
142 TNORM = MAX( TNORM, TSCAL*ABS( AB( KD+1, J ) )+
143 $ CNORM( J ) )
144 10 CONTINUE
145 ELSE
146 DO 20 J = 1, N
147 TNORM = MAX( TNORM, TSCAL*ABS( AB( 1, J ) )+CNORM( J ) )
148 20 CONTINUE
149 END IF
150 ELSE
151 DO 30 J = 1, N
152 TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
153 30 CONTINUE
154 END IF
155 *
156 * Compute the maximum over the number of right hand sides of
157 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
158 *
159 RESID = ZERO
160 DO 40 J = 1, NRHS
161 CALL SCOPY( N, X( 1, J ), 1, WORK, 1 )
162 IX = ISAMAX( N, WORK, 1 )
163 XNORM = MAX( ONE, ABS( X( IX, J ) ) )
164 XSCAL = ( ONE / XNORM ) / REAL( KD+1 )
165 CALL SSCAL( N, XSCAL, WORK, 1 )
166 CALL STBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
167 CALL SAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 )
168 IX = ISAMAX( N, WORK, 1 )
169 ERR = TSCAL*ABS( WORK( IX ) )
170 IX = ISAMAX( N, X( 1, J ), 1 )
171 XNORM = ABS( X( IX, J ) )
172 IF( ERR*SMLNUM.LE.XNORM ) THEN
173 IF( XNORM.GT.ZERO )
174 $ ERR = ERR / XNORM
175 ELSE
176 IF( ERR.GT.ZERO )
177 $ ERR = ONE / EPS
178 END IF
179 IF( ERR*SMLNUM.LE.TNORM ) THEN
180 IF( TNORM.GT.ZERO )
181 $ ERR = ERR / TNORM
182 ELSE
183 IF( ERR.GT.ZERO )
184 $ ERR = ONE / EPS
185 END IF
186 RESID = MAX( RESID, ERR )
187 40 CONTINUE
188 *
189 RETURN
190 *
191 * End of STBT03
192 *
193 END
2 $ SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
3 $ RESID )
4 *
5 * -- LAPACK test routine (version 3.1) --
6 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER DIAG, TRANS, UPLO
11 INTEGER KD, LDAB, LDB, LDX, N, NRHS
12 REAL RESID, SCALE, TSCAL
13 * ..
14 * .. Array Arguments ..
15 REAL AB( LDAB, * ), B( LDB, * ), CNORM( * ),
16 $ WORK( * ), X( LDX, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * STBT03 computes the residual for the solution to a scaled triangular
23 * system of equations A*x = s*b or A'*x = s*b when A is a
24 * triangular band matrix. Here A' is the transpose of A, s is a scalar,
25 * and x and b are N by NRHS matrices. The test ratio is the maximum
26 * over the number of right hand sides of
27 * norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
28 * where op(A) denotes A or A' and EPS is the machine epsilon.
29 *
30 * Arguments
31 * =========
32 *
33 * UPLO (input) CHARACTER*1
34 * Specifies whether the matrix A is upper or lower triangular.
35 * = 'U': Upper triangular
36 * = 'L': Lower triangular
37 *
38 * TRANS (input) CHARACTER*1
39 * Specifies the operation applied to A.
40 * = 'N': A *x = b (No transpose)
41 * = 'T': A'*x = b (Transpose)
42 * = 'C': A'*x = b (Conjugate transpose = Transpose)
43 *
44 * DIAG (input) CHARACTER*1
45 * Specifies whether or not the matrix A is unit triangular.
46 * = 'N': Non-unit triangular
47 * = 'U': Unit triangular
48 *
49 * N (input) INTEGER
50 * The order of the matrix A. N >= 0.
51 *
52 * KD (input) INTEGER
53 * The number of superdiagonals or subdiagonals of the
54 * triangular band matrix A. KD >= 0.
55 *
56 * NRHS (input) INTEGER
57 * The number of right hand sides, i.e., the number of columns
58 * of the matrices X and B. NRHS >= 0.
59 *
60 * AB (input) REAL array, dimension (LDAB,N)
61 * The upper or lower triangular band matrix A, stored in the
62 * first kd+1 rows of the array. The j-th column of A is stored
63 * in the j-th column of the array AB as follows:
64 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
65 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
66 *
67 * LDAB (input) INTEGER
68 * The leading dimension of the array AB. LDAB >= KD+1.
69 *
70 * SCALE (input) REAL
71 * The scaling factor s used in solving the triangular system.
72 *
73 * CNORM (input) REAL array, dimension (N)
74 * The 1-norms of the columns of A, not counting the diagonal.
75 *
76 * TSCAL (input) REAL
77 * The scaling factor used in computing the 1-norms in CNORM.
78 * CNORM actually contains the column norms of TSCAL*A.
79 *
80 * X (input) REAL array, dimension (LDX,NRHS)
81 * The computed solution vectors for the system of linear
82 * equations.
83 *
84 * LDX (input) INTEGER
85 * The leading dimension of the array X. LDX >= max(1,N).
86 *
87 * B (input) REAL array, dimension (LDB,NRHS)
88 * The right hand side vectors for the system of linear
89 * equations.
90 *
91 * LDB (input) INTEGER
92 * The leading dimension of the array B. LDB >= max(1,N).
93 *
94 * WORK (workspace) REAL array, dimension (N)
95 *
96 * RESID (output) REAL
97 * The maximum over the number of right hand sides of
98 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
99 *
100 * =====================================================================
101 *
102 * .. Parameters ..
103 REAL ONE, ZERO
104 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
105 * ..
106 * .. Local Scalars ..
107 INTEGER IX, J
108 REAL BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
109 * ..
110 * .. External Functions ..
111 LOGICAL LSAME
112 INTEGER ISAMAX
113 REAL SLAMCH
114 EXTERNAL LSAME, ISAMAX, SLAMCH
115 * ..
116 * .. External Subroutines ..
117 EXTERNAL SAXPY, SCOPY, SLABAD, SSCAL, STBMV
118 * ..
119 * .. Intrinsic Functions ..
120 INTRINSIC ABS, MAX, REAL
121 * ..
122 * .. Executable Statements ..
123 *
124 * Quick exit if N = 0
125 *
126 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
127 RESID = ZERO
128 RETURN
129 END IF
130 EPS = SLAMCH( 'Epsilon' )
131 SMLNUM = SLAMCH( 'Safe minimum' )
132 BIGNUM = ONE / SMLNUM
133 CALL SLABAD( SMLNUM, BIGNUM )
134 *
135 * Compute the norm of the triangular matrix A using the column
136 * norms already computed by SLATBS.
137 *
138 TNORM = ZERO
139 IF( LSAME( DIAG, 'N' ) ) THEN
140 IF( LSAME( UPLO, 'U' ) ) THEN
141 DO 10 J = 1, N
142 TNORM = MAX( TNORM, TSCAL*ABS( AB( KD+1, J ) )+
143 $ CNORM( J ) )
144 10 CONTINUE
145 ELSE
146 DO 20 J = 1, N
147 TNORM = MAX( TNORM, TSCAL*ABS( AB( 1, J ) )+CNORM( J ) )
148 20 CONTINUE
149 END IF
150 ELSE
151 DO 30 J = 1, N
152 TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
153 30 CONTINUE
154 END IF
155 *
156 * Compute the maximum over the number of right hand sides of
157 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
158 *
159 RESID = ZERO
160 DO 40 J = 1, NRHS
161 CALL SCOPY( N, X( 1, J ), 1, WORK, 1 )
162 IX = ISAMAX( N, WORK, 1 )
163 XNORM = MAX( ONE, ABS( X( IX, J ) ) )
164 XSCAL = ( ONE / XNORM ) / REAL( KD+1 )
165 CALL SSCAL( N, XSCAL, WORK, 1 )
166 CALL STBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
167 CALL SAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 )
168 IX = ISAMAX( N, WORK, 1 )
169 ERR = TSCAL*ABS( WORK( IX ) )
170 IX = ISAMAX( N, X( 1, J ), 1 )
171 XNORM = ABS( X( IX, J ) )
172 IF( ERR*SMLNUM.LE.XNORM ) THEN
173 IF( XNORM.GT.ZERO )
174 $ ERR = ERR / XNORM
175 ELSE
176 IF( ERR.GT.ZERO )
177 $ ERR = ONE / EPS
178 END IF
179 IF( ERR*SMLNUM.LE.TNORM ) THEN
180 IF( TNORM.GT.ZERO )
181 $ ERR = ERR / TNORM
182 ELSE
183 IF( ERR.GT.ZERO )
184 $ ERR = ONE / EPS
185 END IF
186 RESID = MAX( RESID, ERR )
187 40 CONTINUE
188 *
189 RETURN
190 *
191 * End of STBT03
192 *
193 END