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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DTRT03 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 * Definition: 00009 * =========== 00010 * 00011 * SUBROUTINE DTRT03( UPLO, TRANS, DIAG, N, NRHS, A, LDA, SCALE, 00012 * CNORM, TSCAL, X, LDX, B, LDB, WORK, RESID ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER DIAG, TRANS, UPLO 00016 * INTEGER LDA, LDB, LDX, N, NRHS 00017 * DOUBLE PRECISION RESID, SCALE, TSCAL 00018 * .. 00019 * .. Array Arguments .. 00020 * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), CNORM( * ), 00021 * $ WORK( * ), X( LDX, * ) 00022 * .. 00023 * 00024 * 00025 *> \par Purpose: 00026 * ============= 00027 *> 00028 *> \verbatim 00029 *> 00030 *> DTRT03 computes the residual for the solution to a scaled triangular 00031 *> system of equations A*x = s*b or A'*x = s*b. 00032 *> Here A is a triangular matrix, A' is the transpose of A, s is a 00033 *> scalar, and x and b are N by NRHS matrices. The test ratio is the 00034 *> maximum over the number of right hand sides of 00035 *> norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ), 00036 *> where op(A) denotes A or A' and EPS is the machine epsilon. 00037 *> \endverbatim 00038 * 00039 * Arguments: 00040 * ========== 00041 * 00042 *> \param[in] UPLO 00043 *> \verbatim 00044 *> UPLO is CHARACTER*1 00045 *> Specifies whether the matrix A is upper or lower triangular. 00046 *> = 'U': Upper triangular 00047 *> = 'L': Lower triangular 00048 *> \endverbatim 00049 *> 00050 *> \param[in] TRANS 00051 *> \verbatim 00052 *> TRANS is CHARACTER*1 00053 *> Specifies the operation applied to A. 00054 *> = 'N': A *x = s*b (No transpose) 00055 *> = 'T': A'*x = s*b (Transpose) 00056 *> = 'C': A'*x = s*b (Conjugate transpose = Transpose) 00057 *> \endverbatim 00058 *> 00059 *> \param[in] DIAG 00060 *> \verbatim 00061 *> DIAG is CHARACTER*1 00062 *> Specifies whether or not the matrix A is unit triangular. 00063 *> = 'N': Non-unit triangular 00064 *> = 'U': Unit triangular 00065 *> \endverbatim 00066 *> 00067 *> \param[in] N 00068 *> \verbatim 00069 *> N is INTEGER 00070 *> The order of the matrix A. N >= 0. 00071 *> \endverbatim 00072 *> 00073 *> \param[in] NRHS 00074 *> \verbatim 00075 *> NRHS is INTEGER 00076 *> The number of right hand sides, i.e., the number of columns 00077 *> of the matrices X and B. NRHS >= 0. 00078 *> \endverbatim 00079 *> 00080 *> \param[in] A 00081 *> \verbatim 00082 *> A is DOUBLE PRECISION array, dimension (LDA,N) 00083 *> The triangular matrix A. If UPLO = 'U', the leading n by n 00084 *> upper triangular part of the array A contains the upper 00085 *> triangular matrix, and the strictly lower triangular part of 00086 *> A is not referenced. If UPLO = 'L', the leading n by n lower 00087 *> triangular part of the array A contains the lower triangular 00088 *> matrix, and the strictly upper triangular part of A is not 00089 *> referenced. If DIAG = 'U', the diagonal elements of A are 00090 *> also not referenced and are assumed to be 1. 00091 *> \endverbatim 00092 *> 00093 *> \param[in] LDA 00094 *> \verbatim 00095 *> LDA is INTEGER 00096 *> The leading dimension of the array A. LDA >= max(1,N). 00097 *> \endverbatim 00098 *> 00099 *> \param[in] SCALE 00100 *> \verbatim 00101 *> SCALE is DOUBLE PRECISION 00102 *> The scaling factor s used in solving the triangular system. 00103 *> \endverbatim 00104 *> 00105 *> \param[in] CNORM 00106 *> \verbatim 00107 *> CNORM is DOUBLE PRECISION array, dimension (N) 00108 *> The 1-norms of the columns of A, not counting the diagonal. 00109 *> \endverbatim 00110 *> 00111 *> \param[in] TSCAL 00112 *> \verbatim 00113 *> TSCAL is DOUBLE PRECISION 00114 *> The scaling factor used in computing the 1-norms in CNORM. 00115 *> CNORM actually contains the column norms of TSCAL*A. 00116 *> \endverbatim 00117 *> 00118 *> \param[in] X 00119 *> \verbatim 00120 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 00121 *> The computed solution vectors for the system of linear 00122 *> equations. 00123 *> \endverbatim 00124 *> 00125 *> \param[in] LDX 00126 *> \verbatim 00127 *> LDX is INTEGER 00128 *> The leading dimension of the array X. LDX >= max(1,N). 00129 *> \endverbatim 00130 *> 00131 *> \param[in] B 00132 *> \verbatim 00133 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 00134 *> The right hand side vectors for the system of linear 00135 *> equations. 00136 *> \endverbatim 00137 *> 00138 *> \param[in] LDB 00139 *> \verbatim 00140 *> LDB is INTEGER 00141 *> The leading dimension of the array B. LDB >= max(1,N). 00142 *> \endverbatim 00143 *> 00144 *> \param[out] WORK 00145 *> \verbatim 00146 *> WORK is DOUBLE PRECISION array, dimension (N) 00147 *> \endverbatim 00148 *> 00149 *> \param[out] RESID 00150 *> \verbatim 00151 *> RESID is DOUBLE PRECISION 00152 *> The maximum over the number of right hand sides of 00153 *> norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ). 00154 *> \endverbatim 00155 * 00156 * Authors: 00157 * ======== 00158 * 00159 *> \author Univ. of Tennessee 00160 *> \author Univ. of California Berkeley 00161 *> \author Univ. of Colorado Denver 00162 *> \author NAG Ltd. 00163 * 00164 *> \date November 2011 00165 * 00166 *> \ingroup double_lin 00167 * 00168 * ===================================================================== 00169 SUBROUTINE DTRT03( UPLO, TRANS, DIAG, N, NRHS, A, LDA, SCALE, 00170 $ CNORM, TSCAL, X, LDX, B, LDB, WORK, RESID ) 00171 * 00172 * -- LAPACK test routine (version 3.4.0) -- 00173 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00174 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00175 * November 2011 00176 * 00177 * .. Scalar Arguments .. 00178 CHARACTER DIAG, TRANS, UPLO 00179 INTEGER LDA, LDB, LDX, N, NRHS 00180 DOUBLE PRECISION RESID, SCALE, TSCAL 00181 * .. 00182 * .. Array Arguments .. 00183 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), CNORM( * ), 00184 $ WORK( * ), X( LDX, * ) 00185 * .. 00186 * 00187 * ===================================================================== 00188 * 00189 * .. Parameters .. 00190 DOUBLE PRECISION ONE, ZERO 00191 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) 00192 * .. 00193 * .. Local Scalars .. 00194 INTEGER IX, J 00195 DOUBLE PRECISION BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL 00196 * .. 00197 * .. External Functions .. 00198 LOGICAL LSAME 00199 INTEGER IDAMAX 00200 DOUBLE PRECISION DLAMCH 00201 EXTERNAL LSAME, IDAMAX, DLAMCH 00202 * .. 00203 * .. External Subroutines .. 00204 EXTERNAL DAXPY, DCOPY, DLABAD, DSCAL, DTRMV 00205 * .. 00206 * .. Intrinsic Functions .. 00207 INTRINSIC ABS, DBLE, MAX 00208 * .. 00209 * .. Executable Statements .. 00210 * 00211 * Quick exit if N = 0 00212 * 00213 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00214 RESID = ZERO 00215 RETURN 00216 END IF 00217 EPS = DLAMCH( 'Epsilon' ) 00218 SMLNUM = DLAMCH( 'Safe minimum' ) 00219 BIGNUM = ONE / SMLNUM 00220 CALL DLABAD( SMLNUM, BIGNUM ) 00221 * 00222 * Compute the norm of the triangular matrix A using the column 00223 * norms already computed by DLATRS. 00224 * 00225 TNORM = ZERO 00226 IF( LSAME( DIAG, 'N' ) ) THEN 00227 DO 10 J = 1, N 00228 TNORM = MAX( TNORM, TSCAL*ABS( A( J, J ) )+CNORM( J ) ) 00229 10 CONTINUE 00230 ELSE 00231 DO 20 J = 1, N 00232 TNORM = MAX( TNORM, TSCAL+CNORM( J ) ) 00233 20 CONTINUE 00234 END IF 00235 * 00236 * Compute the maximum over the number of right hand sides of 00237 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ). 00238 * 00239 RESID = ZERO 00240 DO 30 J = 1, NRHS 00241 CALL DCOPY( N, X( 1, J ), 1, WORK, 1 ) 00242 IX = IDAMAX( N, WORK, 1 ) 00243 XNORM = MAX( ONE, ABS( X( IX, J ) ) ) 00244 XSCAL = ( ONE / XNORM ) / DBLE( N ) 00245 CALL DSCAL( N, XSCAL, WORK, 1 ) 00246 CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 ) 00247 CALL DAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 ) 00248 IX = IDAMAX( N, WORK, 1 ) 00249 ERR = TSCAL*ABS( WORK( IX ) ) 00250 IX = IDAMAX( N, X( 1, J ), 1 ) 00251 XNORM = ABS( X( IX, J ) ) 00252 IF( ERR*SMLNUM.LE.XNORM ) THEN 00253 IF( XNORM.GT.ZERO ) 00254 $ ERR = ERR / XNORM 00255 ELSE 00256 IF( ERR.GT.ZERO ) 00257 $ ERR = ONE / EPS 00258 END IF 00259 IF( ERR*SMLNUM.LE.TNORM ) THEN 00260 IF( TNORM.GT.ZERO ) 00261 $ ERR = ERR / TNORM 00262 ELSE 00263 IF( ERR.GT.ZERO ) 00264 $ ERR = ONE / EPS 00265 END IF 00266 RESID = MAX( RESID, ERR ) 00267 30 CONTINUE 00268 * 00269 RETURN 00270 * 00271 * End of DTRT03 00272 * 00273 END