1 SUBROUTINE STPT03( UPLO, TRANS, DIAG, N, NRHS, AP, SCALE, CNORM,
2 $ TSCAL, X, LDX, B, LDB, WORK, RESID )
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 DIAG, TRANS, UPLO
10 INTEGER LDB, LDX, N, NRHS
11 REAL RESID, SCALE, TSCAL
12 * ..
13 * .. Array Arguments ..
14 REAL AP( * ), B( LDB, * ), CNORM( * ), WORK( * ),
15 $ X( LDX, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * STPT03 computes the residual for the solution to a scaled triangular
22 * system of equations A*x = s*b or A'*x = s*b when the triangular
23 * matrix A is stored in packed format. Here A' is the transpose of A,
24 * s is a scalar, and x and b are N by NRHS matrices. The test ratio is
25 * the maximum over the number of right hand sides of
26 * norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
27 * where op(A) denotes A or A' and EPS is the machine epsilon.
28 *
29 * Arguments
30 * =========
31 *
32 * UPLO (input) CHARACTER*1
33 * Specifies whether the matrix A is upper or lower triangular.
34 * = 'U': Upper triangular
35 * = 'L': Lower triangular
36 *
37 * TRANS (input) CHARACTER*1
38 * Specifies the operation applied to A.
39 * = 'N': A *x = s*b (No transpose)
40 * = 'T': A'*x = s*b (Transpose)
41 * = 'C': A'*x = s*b (Conjugate transpose = Transpose)
42 *
43 * DIAG (input) CHARACTER*1
44 * Specifies whether or not the matrix A is unit triangular.
45 * = 'N': Non-unit triangular
46 * = 'U': Unit triangular
47 *
48 * N (input) INTEGER
49 * The order of the matrix A. N >= 0.
50 *
51 * NRHS (input) INTEGER
52 * The number of right hand sides, i.e., the number of columns
53 * of the matrices X and B. NRHS >= 0.
54 *
55 * AP (input) REAL array, dimension (N*(N+1)/2)
56 * The upper or lower triangular matrix A, packed columnwise in
57 * a linear array. The j-th column of A is stored in the array
58 * AP as follows:
59 * if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j;
60 * if UPLO = 'L',
61 * AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=n.
62 *
63 * SCALE (input) REAL
64 * The scaling factor s used in solving the triangular system.
65 *
66 * CNORM (input) REAL array, dimension (N)
67 * The 1-norms of the columns of A, not counting the diagonal.
68 *
69 * TSCAL (input) REAL
70 * The scaling factor used in computing the 1-norms in CNORM.
71 * CNORM actually contains the column norms of TSCAL*A.
72 *
73 * X (input) REAL array, dimension (LDX,NRHS)
74 * The computed solution vectors for the system of linear
75 * equations.
76 *
77 * LDX (input) INTEGER
78 * The leading dimension of the array X. LDX >= max(1,N).
79 *
80 * B (input) REAL array, dimension (LDB,NRHS)
81 * The right hand side vectors for the system of linear
82 * equations.
83 *
84 * LDB (input) INTEGER
85 * The leading dimension of the array B. LDB >= max(1,N).
86 *
87 * WORK (workspace) REAL array, dimension (N)
88 *
89 * RESID (output) REAL
90 * The maximum over the number of right hand sides of
91 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
92 *
93 * =====================================================================
94 *
95 * .. Parameters ..
96 REAL ONE, ZERO
97 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
98 * ..
99 * .. Local Scalars ..
100 INTEGER IX, J, JJ
101 REAL BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
102 * ..
103 * .. External Functions ..
104 LOGICAL LSAME
105 INTEGER ISAMAX
106 REAL SLAMCH
107 EXTERNAL LSAME, ISAMAX, SLAMCH
108 * ..
109 * .. External Subroutines ..
110 EXTERNAL SAXPY, SCOPY, SLABAD, SSCAL, STPMV
111 * ..
112 * .. Intrinsic Functions ..
113 INTRINSIC ABS, MAX, REAL
114 * ..
115 * .. Executable Statements ..
116 *
117 * Quick exit if N = 0.
118 *
119 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
120 RESID = ZERO
121 RETURN
122 END IF
123 EPS = SLAMCH( 'Epsilon' )
124 SMLNUM = SLAMCH( 'Safe minimum' )
125 BIGNUM = ONE / SMLNUM
126 CALL SLABAD( SMLNUM, BIGNUM )
127 *
128 * Compute the norm of the triangular matrix A using the column
129 * norms already computed by SLATPS.
130 *
131 TNORM = ZERO
132 IF( LSAME( DIAG, 'N' ) ) THEN
133 IF( LSAME( UPLO, 'U' ) ) THEN
134 JJ = 1
135 DO 10 J = 1, N
136 TNORM = MAX( TNORM, TSCAL*ABS( AP( JJ ) )+CNORM( J ) )
137 JJ = JJ + J + 1
138 10 CONTINUE
139 ELSE
140 JJ = 1
141 DO 20 J = 1, N
142 TNORM = MAX( TNORM, TSCAL*ABS( AP( JJ ) )+CNORM( J ) )
143 JJ = JJ + N - J + 1
144 20 CONTINUE
145 END IF
146 ELSE
147 DO 30 J = 1, N
148 TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
149 30 CONTINUE
150 END IF
151 *
152 * Compute the maximum over the number of right hand sides of
153 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
154 *
155 RESID = ZERO
156 DO 40 J = 1, NRHS
157 CALL SCOPY( N, X( 1, J ), 1, WORK, 1 )
158 IX = ISAMAX( N, WORK, 1 )
159 XNORM = MAX( ONE, ABS( X( IX, J ) ) )
160 XSCAL = ( ONE / XNORM ) / REAL( N )
161 CALL SSCAL( N, XSCAL, WORK, 1 )
162 CALL STPMV( UPLO, TRANS, DIAG, N, AP, WORK, 1 )
163 CALL SAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 )
164 IX = ISAMAX( N, WORK, 1 )
165 ERR = TSCAL*ABS( WORK( IX ) )
166 IX = ISAMAX( N, X( 1, J ), 1 )
167 XNORM = ABS( X( IX, J ) )
168 IF( ERR*SMLNUM.LE.XNORM ) THEN
169 IF( XNORM.GT.ZERO )
170 $ ERR = ERR / XNORM
171 ELSE
172 IF( ERR.GT.ZERO )
173 $ ERR = ONE / EPS
174 END IF
175 IF( ERR*SMLNUM.LE.TNORM ) THEN
176 IF( TNORM.GT.ZERO )
177 $ ERR = ERR / TNORM
178 ELSE
179 IF( ERR.GT.ZERO )
180 $ ERR = ONE / EPS
181 END IF
182 RESID = MAX( RESID, ERR )
183 40 CONTINUE
184 *
185 RETURN
186 *
187 * End of STPT03
188 *
189 END
2 $ TSCAL, X, LDX, B, LDB, WORK, RESID )
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 DIAG, TRANS, UPLO
10 INTEGER LDB, LDX, N, NRHS
11 REAL RESID, SCALE, TSCAL
12 * ..
13 * .. Array Arguments ..
14 REAL AP( * ), B( LDB, * ), CNORM( * ), WORK( * ),
15 $ X( LDX, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * STPT03 computes the residual for the solution to a scaled triangular
22 * system of equations A*x = s*b or A'*x = s*b when the triangular
23 * matrix A is stored in packed format. Here A' is the transpose of A,
24 * s is a scalar, and x and b are N by NRHS matrices. The test ratio is
25 * the maximum over the number of right hand sides of
26 * norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
27 * where op(A) denotes A or A' and EPS is the machine epsilon.
28 *
29 * Arguments
30 * =========
31 *
32 * UPLO (input) CHARACTER*1
33 * Specifies whether the matrix A is upper or lower triangular.
34 * = 'U': Upper triangular
35 * = 'L': Lower triangular
36 *
37 * TRANS (input) CHARACTER*1
38 * Specifies the operation applied to A.
39 * = 'N': A *x = s*b (No transpose)
40 * = 'T': A'*x = s*b (Transpose)
41 * = 'C': A'*x = s*b (Conjugate transpose = Transpose)
42 *
43 * DIAG (input) CHARACTER*1
44 * Specifies whether or not the matrix A is unit triangular.
45 * = 'N': Non-unit triangular
46 * = 'U': Unit triangular
47 *
48 * N (input) INTEGER
49 * The order of the matrix A. N >= 0.
50 *
51 * NRHS (input) INTEGER
52 * The number of right hand sides, i.e., the number of columns
53 * of the matrices X and B. NRHS >= 0.
54 *
55 * AP (input) REAL array, dimension (N*(N+1)/2)
56 * The upper or lower triangular matrix A, packed columnwise in
57 * a linear array. The j-th column of A is stored in the array
58 * AP as follows:
59 * if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j;
60 * if UPLO = 'L',
61 * AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=n.
62 *
63 * SCALE (input) REAL
64 * The scaling factor s used in solving the triangular system.
65 *
66 * CNORM (input) REAL array, dimension (N)
67 * The 1-norms of the columns of A, not counting the diagonal.
68 *
69 * TSCAL (input) REAL
70 * The scaling factor used in computing the 1-norms in CNORM.
71 * CNORM actually contains the column norms of TSCAL*A.
72 *
73 * X (input) REAL array, dimension (LDX,NRHS)
74 * The computed solution vectors for the system of linear
75 * equations.
76 *
77 * LDX (input) INTEGER
78 * The leading dimension of the array X. LDX >= max(1,N).
79 *
80 * B (input) REAL array, dimension (LDB,NRHS)
81 * The right hand side vectors for the system of linear
82 * equations.
83 *
84 * LDB (input) INTEGER
85 * The leading dimension of the array B. LDB >= max(1,N).
86 *
87 * WORK (workspace) REAL array, dimension (N)
88 *
89 * RESID (output) REAL
90 * The maximum over the number of right hand sides of
91 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
92 *
93 * =====================================================================
94 *
95 * .. Parameters ..
96 REAL ONE, ZERO
97 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
98 * ..
99 * .. Local Scalars ..
100 INTEGER IX, J, JJ
101 REAL BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
102 * ..
103 * .. External Functions ..
104 LOGICAL LSAME
105 INTEGER ISAMAX
106 REAL SLAMCH
107 EXTERNAL LSAME, ISAMAX, SLAMCH
108 * ..
109 * .. External Subroutines ..
110 EXTERNAL SAXPY, SCOPY, SLABAD, SSCAL, STPMV
111 * ..
112 * .. Intrinsic Functions ..
113 INTRINSIC ABS, MAX, REAL
114 * ..
115 * .. Executable Statements ..
116 *
117 * Quick exit if N = 0.
118 *
119 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
120 RESID = ZERO
121 RETURN
122 END IF
123 EPS = SLAMCH( 'Epsilon' )
124 SMLNUM = SLAMCH( 'Safe minimum' )
125 BIGNUM = ONE / SMLNUM
126 CALL SLABAD( SMLNUM, BIGNUM )
127 *
128 * Compute the norm of the triangular matrix A using the column
129 * norms already computed by SLATPS.
130 *
131 TNORM = ZERO
132 IF( LSAME( DIAG, 'N' ) ) THEN
133 IF( LSAME( UPLO, 'U' ) ) THEN
134 JJ = 1
135 DO 10 J = 1, N
136 TNORM = MAX( TNORM, TSCAL*ABS( AP( JJ ) )+CNORM( J ) )
137 JJ = JJ + J + 1
138 10 CONTINUE
139 ELSE
140 JJ = 1
141 DO 20 J = 1, N
142 TNORM = MAX( TNORM, TSCAL*ABS( AP( JJ ) )+CNORM( J ) )
143 JJ = JJ + N - J + 1
144 20 CONTINUE
145 END IF
146 ELSE
147 DO 30 J = 1, N
148 TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
149 30 CONTINUE
150 END IF
151 *
152 * Compute the maximum over the number of right hand sides of
153 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
154 *
155 RESID = ZERO
156 DO 40 J = 1, NRHS
157 CALL SCOPY( N, X( 1, J ), 1, WORK, 1 )
158 IX = ISAMAX( N, WORK, 1 )
159 XNORM = MAX( ONE, ABS( X( IX, J ) ) )
160 XSCAL = ( ONE / XNORM ) / REAL( N )
161 CALL SSCAL( N, XSCAL, WORK, 1 )
162 CALL STPMV( UPLO, TRANS, DIAG, N, AP, WORK, 1 )
163 CALL SAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 )
164 IX = ISAMAX( N, WORK, 1 )
165 ERR = TSCAL*ABS( WORK( IX ) )
166 IX = ISAMAX( N, X( 1, J ), 1 )
167 XNORM = ABS( X( IX, J ) )
168 IF( ERR*SMLNUM.LE.XNORM ) THEN
169 IF( XNORM.GT.ZERO )
170 $ ERR = ERR / XNORM
171 ELSE
172 IF( ERR.GT.ZERO )
173 $ ERR = ONE / EPS
174 END IF
175 IF( ERR*SMLNUM.LE.TNORM ) THEN
176 IF( TNORM.GT.ZERO )
177 $ ERR = ERR / TNORM
178 ELSE
179 IF( ERR.GT.ZERO )
180 $ ERR = ONE / EPS
181 END IF
182 RESID = MAX( RESID, ERR )
183 40 CONTINUE
184 *
185 RETURN
186 *
187 * End of STPT03
188 *
189 END