1 SUBROUTINE CTPT03( 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 CNORM( * )
15 COMPLEX AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * CTPT03 computes the residual for the solution to a scaled triangular
22 * system of equations A*x = s*b, A**T *x = s*b, or A**H *x = s*b,
23 * when the triangular matrix A is stored in packed format. Here A**T
24 * denotes the transpose of A, A**H denotes the conjugate transpose of
25 * A, s is a scalar, and x and b are N by NRHS matrices. The test ratio
26 * is the maximum 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, A**T, or A**H, 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 = s*b (No transpose)
41 * = 'T': A**T *x = s*b (Transpose)
42 * = 'C': A**H *x = s*b (Conjugate 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 * NRHS (input) INTEGER
53 * The number of right hand sides, i.e., the number of columns
54 * of the matrices X and B. NRHS >= 0.
55 *
56 * AP (input) COMPLEX array, dimension (N*(N+1)/2)
57 * The upper or lower triangular matrix A, packed columnwise in
58 * a linear array. The j-th column of A is stored in the array
59 * AP as follows:
60 * if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j;
61 * if UPLO = 'L',
62 * AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=n.
63 *
64 * SCALE (input) REAL
65 * The scaling factor s used in solving the triangular system.
66 *
67 * CNORM (input) REAL array, dimension (N)
68 * The 1-norms of the columns of A, not counting the diagonal.
69 *
70 * TSCAL (input) REAL
71 * The scaling factor used in computing the 1-norms in CNORM.
72 * CNORM actually contains the column norms of TSCAL*A.
73 *
74 * X (input) COMPLEX array, dimension (LDX,NRHS)
75 * The computed solution vectors for the system of linear
76 * equations.
77 *
78 * LDX (input) INTEGER
79 * The leading dimension of the array X. LDX >= max(1,N).
80 *
81 * B (input) COMPLEX array, dimension (LDB,NRHS)
82 * The right hand side vectors for the system of linear
83 * equations.
84 *
85 * LDB (input) INTEGER
86 * The leading dimension of the array B. LDB >= max(1,N).
87 *
88 * WORK (workspace) COMPLEX array, dimension (N)
89 *
90 * RESID (output) REAL
91 * The maximum over the number of right hand sides of
92 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
93 *
94 * =====================================================================
95 *
96 * .. Parameters ..
97 REAL ONE, ZERO
98 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
99 * ..
100 * .. Local Scalars ..
101 INTEGER IX, J, JJ
102 REAL EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
103 * ..
104 * .. External Functions ..
105 LOGICAL LSAME
106 INTEGER ICAMAX
107 REAL SLAMCH
108 EXTERNAL LSAME, ICAMAX, SLAMCH
109 * ..
110 * .. External Subroutines ..
111 EXTERNAL CAXPY, CCOPY, CSSCAL, CTPMV
112 * ..
113 * .. Intrinsic Functions ..
114 INTRINSIC ABS, CMPLX, MAX, REAL
115 * ..
116 * .. Executable Statements ..
117 *
118 * Quick exit if N = 0.
119 *
120 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
121 RESID = ZERO
122 RETURN
123 END IF
124 EPS = SLAMCH( 'Epsilon' )
125 SMLNUM = SLAMCH( 'Safe minimum' )
126 *
127 * Compute the norm of the triangular matrix A using the column
128 * norms already computed by CLATPS.
129 *
130 TNORM = 0.
131 IF( LSAME( DIAG, 'N' ) ) THEN
132 IF( LSAME( UPLO, 'U' ) ) THEN
133 JJ = 1
134 DO 10 J = 1, N
135 TNORM = MAX( TNORM, TSCAL*ABS( AP( JJ ) )+CNORM( J ) )
136 JJ = JJ + J + 1
137 10 CONTINUE
138 ELSE
139 JJ = 1
140 DO 20 J = 1, N
141 TNORM = MAX( TNORM, TSCAL*ABS( AP( JJ ) )+CNORM( J ) )
142 JJ = JJ + N - J + 1
143 20 CONTINUE
144 END IF
145 ELSE
146 DO 30 J = 1, N
147 TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
148 30 CONTINUE
149 END IF
150 *
151 * Compute the maximum over the number of right hand sides of
152 * norm(op(A)*x - s*b) / ( norm(A) * norm(x) * EPS ).
153 *
154 RESID = ZERO
155 DO 40 J = 1, NRHS
156 CALL CCOPY( N, X( 1, J ), 1, WORK, 1 )
157 IX = ICAMAX( N, WORK, 1 )
158 XNORM = MAX( ONE, ABS( X( IX, J ) ) )
159 XSCAL = ( ONE / XNORM ) / REAL( N )
160 CALL CSSCAL( N, XSCAL, WORK, 1 )
161 CALL CTPMV( UPLO, TRANS, DIAG, N, AP, WORK, 1 )
162 CALL CAXPY( N, CMPLX( -SCALE*XSCAL ), B( 1, J ), 1, WORK, 1 )
163 IX = ICAMAX( N, WORK, 1 )
164 ERR = TSCAL*ABS( WORK( IX ) )
165 IX = ICAMAX( N, X( 1, J ), 1 )
166 XNORM = ABS( X( IX, J ) )
167 IF( ERR*SMLNUM.LE.XNORM ) THEN
168 IF( XNORM.GT.ZERO )
169 $ ERR = ERR / XNORM
170 ELSE
171 IF( ERR.GT.ZERO )
172 $ ERR = ONE / EPS
173 END IF
174 IF( ERR*SMLNUM.LE.TNORM ) THEN
175 IF( TNORM.GT.ZERO )
176 $ ERR = ERR / TNORM
177 ELSE
178 IF( ERR.GT.ZERO )
179 $ ERR = ONE / EPS
180 END IF
181 RESID = MAX( RESID, ERR )
182 40 CONTINUE
183 *
184 RETURN
185 *
186 * End of CTPT03
187 *
188 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 CNORM( * )
15 COMPLEX AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * CTPT03 computes the residual for the solution to a scaled triangular
22 * system of equations A*x = s*b, A**T *x = s*b, or A**H *x = s*b,
23 * when the triangular matrix A is stored in packed format. Here A**T
24 * denotes the transpose of A, A**H denotes the conjugate transpose of
25 * A, s is a scalar, and x and b are N by NRHS matrices. The test ratio
26 * is the maximum 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, A**T, or A**H, 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 = s*b (No transpose)
41 * = 'T': A**T *x = s*b (Transpose)
42 * = 'C': A**H *x = s*b (Conjugate 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 * NRHS (input) INTEGER
53 * The number of right hand sides, i.e., the number of columns
54 * of the matrices X and B. NRHS >= 0.
55 *
56 * AP (input) COMPLEX array, dimension (N*(N+1)/2)
57 * The upper or lower triangular matrix A, packed columnwise in
58 * a linear array. The j-th column of A is stored in the array
59 * AP as follows:
60 * if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j;
61 * if UPLO = 'L',
62 * AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=n.
63 *
64 * SCALE (input) REAL
65 * The scaling factor s used in solving the triangular system.
66 *
67 * CNORM (input) REAL array, dimension (N)
68 * The 1-norms of the columns of A, not counting the diagonal.
69 *
70 * TSCAL (input) REAL
71 * The scaling factor used in computing the 1-norms in CNORM.
72 * CNORM actually contains the column norms of TSCAL*A.
73 *
74 * X (input) COMPLEX array, dimension (LDX,NRHS)
75 * The computed solution vectors for the system of linear
76 * equations.
77 *
78 * LDX (input) INTEGER
79 * The leading dimension of the array X. LDX >= max(1,N).
80 *
81 * B (input) COMPLEX array, dimension (LDB,NRHS)
82 * The right hand side vectors for the system of linear
83 * equations.
84 *
85 * LDB (input) INTEGER
86 * The leading dimension of the array B. LDB >= max(1,N).
87 *
88 * WORK (workspace) COMPLEX array, dimension (N)
89 *
90 * RESID (output) REAL
91 * The maximum over the number of right hand sides of
92 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
93 *
94 * =====================================================================
95 *
96 * .. Parameters ..
97 REAL ONE, ZERO
98 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
99 * ..
100 * .. Local Scalars ..
101 INTEGER IX, J, JJ
102 REAL EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
103 * ..
104 * .. External Functions ..
105 LOGICAL LSAME
106 INTEGER ICAMAX
107 REAL SLAMCH
108 EXTERNAL LSAME, ICAMAX, SLAMCH
109 * ..
110 * .. External Subroutines ..
111 EXTERNAL CAXPY, CCOPY, CSSCAL, CTPMV
112 * ..
113 * .. Intrinsic Functions ..
114 INTRINSIC ABS, CMPLX, MAX, REAL
115 * ..
116 * .. Executable Statements ..
117 *
118 * Quick exit if N = 0.
119 *
120 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
121 RESID = ZERO
122 RETURN
123 END IF
124 EPS = SLAMCH( 'Epsilon' )
125 SMLNUM = SLAMCH( 'Safe minimum' )
126 *
127 * Compute the norm of the triangular matrix A using the column
128 * norms already computed by CLATPS.
129 *
130 TNORM = 0.
131 IF( LSAME( DIAG, 'N' ) ) THEN
132 IF( LSAME( UPLO, 'U' ) ) THEN
133 JJ = 1
134 DO 10 J = 1, N
135 TNORM = MAX( TNORM, TSCAL*ABS( AP( JJ ) )+CNORM( J ) )
136 JJ = JJ + J + 1
137 10 CONTINUE
138 ELSE
139 JJ = 1
140 DO 20 J = 1, N
141 TNORM = MAX( TNORM, TSCAL*ABS( AP( JJ ) )+CNORM( J ) )
142 JJ = JJ + N - J + 1
143 20 CONTINUE
144 END IF
145 ELSE
146 DO 30 J = 1, N
147 TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
148 30 CONTINUE
149 END IF
150 *
151 * Compute the maximum over the number of right hand sides of
152 * norm(op(A)*x - s*b) / ( norm(A) * norm(x) * EPS ).
153 *
154 RESID = ZERO
155 DO 40 J = 1, NRHS
156 CALL CCOPY( N, X( 1, J ), 1, WORK, 1 )
157 IX = ICAMAX( N, WORK, 1 )
158 XNORM = MAX( ONE, ABS( X( IX, J ) ) )
159 XSCAL = ( ONE / XNORM ) / REAL( N )
160 CALL CSSCAL( N, XSCAL, WORK, 1 )
161 CALL CTPMV( UPLO, TRANS, DIAG, N, AP, WORK, 1 )
162 CALL CAXPY( N, CMPLX( -SCALE*XSCAL ), B( 1, J ), 1, WORK, 1 )
163 IX = ICAMAX( N, WORK, 1 )
164 ERR = TSCAL*ABS( WORK( IX ) )
165 IX = ICAMAX( N, X( 1, J ), 1 )
166 XNORM = ABS( X( IX, J ) )
167 IF( ERR*SMLNUM.LE.XNORM ) THEN
168 IF( XNORM.GT.ZERO )
169 $ ERR = ERR / XNORM
170 ELSE
171 IF( ERR.GT.ZERO )
172 $ ERR = ONE / EPS
173 END IF
174 IF( ERR*SMLNUM.LE.TNORM ) THEN
175 IF( TNORM.GT.ZERO )
176 $ ERR = ERR / TNORM
177 ELSE
178 IF( ERR.GT.ZERO )
179 $ ERR = ONE / EPS
180 END IF
181 RESID = MAX( RESID, ERR )
182 40 CONTINUE
183 *
184 RETURN
185 *
186 * End of CTPT03
187 *
188 END