00001 SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, 00002 + SWORK, ITER, INFO ) 00003 * 00004 * -- LAPACK PROTOTYPE driver routine (version 3.2.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * February 2007 00008 * 00009 * .. 00010 * .. Scalar Arguments .. 00011 INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS 00012 * .. 00013 * .. Array Arguments .. 00014 INTEGER IPIV( * ) 00015 REAL SWORK( * ) 00016 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ), 00017 + X( LDX, * ) 00018 * .. 00019 * 00020 * Purpose 00021 * ======= 00022 * 00023 * DSGESV computes the solution to a real system of linear equations 00024 * A * X = B, 00025 * where A is an N-by-N matrix and X and B are N-by-NRHS matrices. 00026 * 00027 * DSGESV first attempts to factorize the matrix in SINGLE PRECISION 00028 * and use this factorization within an iterative refinement procedure 00029 * to produce a solution with DOUBLE PRECISION normwise backward error 00030 * quality (see below). If the approach fails the method switches to a 00031 * DOUBLE PRECISION factorization and solve. 00032 * 00033 * The iterative refinement is not going to be a winning strategy if 00034 * the ratio SINGLE PRECISION performance over DOUBLE PRECISION 00035 * performance is too small. A reasonable strategy should take the 00036 * number of right-hand sides and the size of the matrix into account. 00037 * This might be done with a call to ILAENV in the future. Up to now, we 00038 * always try iterative refinement. 00039 * 00040 * The iterative refinement process is stopped if 00041 * ITER > ITERMAX 00042 * or for all the RHS we have: 00043 * RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX 00044 * where 00045 * o ITER is the number of the current iteration in the iterative 00046 * refinement process 00047 * o RNRM is the infinity-norm of the residual 00048 * o XNRM is the infinity-norm of the solution 00049 * o ANRM is the infinity-operator-norm of the matrix A 00050 * o EPS is the machine epsilon returned by DLAMCH('Epsilon') 00051 * The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 00052 * respectively. 00053 * 00054 * Arguments 00055 * ========= 00056 * 00057 * N (input) INTEGER 00058 * The number of linear equations, i.e., the order of the 00059 * matrix A. N >= 0. 00060 * 00061 * NRHS (input) INTEGER 00062 * The number of right hand sides, i.e., the number of columns 00063 * of the matrix B. NRHS >= 0. 00064 * 00065 * A (input/output) DOUBLE PRECISION array, 00066 * dimension (LDA,N) 00067 * On entry, the N-by-N coefficient matrix A. 00068 * On exit, if iterative refinement has been successfully used 00069 * (INFO.EQ.0 and ITER.GE.0, see description below), then A is 00070 * unchanged, if double precision factorization has been used 00071 * (INFO.EQ.0 and ITER.LT.0, see description below), then the 00072 * array A contains the factors L and U from the factorization 00073 * A = P*L*U; the unit diagonal elements of L are not stored. 00074 * 00075 * LDA (input) INTEGER 00076 * The leading dimension of the array A. LDA >= max(1,N). 00077 * 00078 * IPIV (output) INTEGER array, dimension (N) 00079 * The pivot indices that define the permutation matrix P; 00080 * row i of the matrix was interchanged with row IPIV(i). 00081 * Corresponds either to the single precision factorization 00082 * (if INFO.EQ.0 and ITER.GE.0) or the double precision 00083 * factorization (if INFO.EQ.0 and ITER.LT.0). 00084 * 00085 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 00086 * The N-by-NRHS right hand side matrix B. 00087 * 00088 * LDB (input) INTEGER 00089 * The leading dimension of the array B. LDB >= max(1,N). 00090 * 00091 * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) 00092 * If INFO = 0, the N-by-NRHS solution matrix X. 00093 * 00094 * LDX (input) INTEGER 00095 * The leading dimension of the array X. LDX >= max(1,N). 00096 * 00097 * WORK (workspace) DOUBLE PRECISION array, dimension (N,NRHS) 00098 * This array is used to hold the residual vectors. 00099 * 00100 * SWORK (workspace) REAL array, dimension (N*(N+NRHS)) 00101 * This array is used to use the single precision matrix and the 00102 * right-hand sides or solutions in single precision. 00103 * 00104 * ITER (output) INTEGER 00105 * < 0: iterative refinement has failed, double precision 00106 * factorization has been performed 00107 * -1 : the routine fell back to full precision for 00108 * implementation- or machine-specific reasons 00109 * -2 : narrowing the precision induced an overflow, 00110 * the routine fell back to full precision 00111 * -3 : failure of SGETRF 00112 * -31: stop the iterative refinement after the 30th 00113 * iterations 00114 * > 0: iterative refinement has been sucessfully used. 00115 * Returns the number of iterations 00116 * 00117 * INFO (output) INTEGER 00118 * = 0: successful exit 00119 * < 0: if INFO = -i, the i-th argument had an illegal value 00120 * > 0: if INFO = i, U(i,i) computed in DOUBLE PRECISION is 00121 * exactly zero. The factorization has been completed, 00122 * but the factor U is exactly singular, so the solution 00123 * could not be computed. 00124 * 00125 * ========= 00126 * 00127 * .. Parameters .. 00128 LOGICAL DOITREF 00129 PARAMETER ( DOITREF = .TRUE. ) 00130 * 00131 INTEGER ITERMAX 00132 PARAMETER ( ITERMAX = 30 ) 00133 * 00134 DOUBLE PRECISION BWDMAX 00135 PARAMETER ( BWDMAX = 1.0E+00 ) 00136 * 00137 DOUBLE PRECISION NEGONE, ONE 00138 PARAMETER ( NEGONE = -1.0D+0, ONE = 1.0D+0 ) 00139 * 00140 * .. Local Scalars .. 00141 INTEGER I, IITER, PTSA, PTSX 00142 DOUBLE PRECISION ANRM, CTE, EPS, RNRM, XNRM 00143 * 00144 * .. External Subroutines .. 00145 EXTERNAL DAXPY, DGEMM, DLACPY, DLAG2S, SLAG2D, SGETRF, 00146 + SGETRS, XERBLA 00147 * .. 00148 * .. External Functions .. 00149 INTEGER IDAMAX 00150 DOUBLE PRECISION DLAMCH, DLANGE 00151 EXTERNAL IDAMAX, DLAMCH, DLANGE 00152 * .. 00153 * .. Intrinsic Functions .. 00154 INTRINSIC ABS, DBLE, MAX, SQRT 00155 * .. 00156 * .. Executable Statements .. 00157 * 00158 INFO = 0 00159 ITER = 0 00160 * 00161 * Test the input parameters. 00162 * 00163 IF( N.LT.0 ) THEN 00164 INFO = -1 00165 ELSE IF( NRHS.LT.0 ) THEN 00166 INFO = -2 00167 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00168 INFO = -4 00169 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00170 INFO = -7 00171 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00172 INFO = -9 00173 END IF 00174 IF( INFO.NE.0 ) THEN 00175 CALL XERBLA( 'DSGESV', -INFO ) 00176 RETURN 00177 END IF 00178 * 00179 * Quick return if (N.EQ.0). 00180 * 00181 IF( N.EQ.0 ) 00182 + RETURN 00183 * 00184 * Skip single precision iterative refinement if a priori slower 00185 * than double precision factorization. 00186 * 00187 IF( .NOT.DOITREF ) THEN 00188 ITER = -1 00189 GO TO 40 00190 END IF 00191 * 00192 * Compute some constants. 00193 * 00194 ANRM = DLANGE( 'I', N, N, A, LDA, WORK ) 00195 EPS = DLAMCH( 'Epsilon' ) 00196 CTE = ANRM*EPS*SQRT( DBLE( N ) )*BWDMAX 00197 * 00198 * Set the indices PTSA, PTSX for referencing SA and SX in SWORK. 00199 * 00200 PTSA = 1 00201 PTSX = PTSA + N*N 00202 * 00203 * Convert B from double precision to single precision and store the 00204 * result in SX. 00205 * 00206 CALL DLAG2S( N, NRHS, B, LDB, SWORK( PTSX ), N, INFO ) 00207 * 00208 IF( INFO.NE.0 ) THEN 00209 ITER = -2 00210 GO TO 40 00211 END IF 00212 * 00213 * Convert A from double precision to single precision and store the 00214 * result in SA. 00215 * 00216 CALL DLAG2S( N, N, A, LDA, SWORK( PTSA ), N, INFO ) 00217 * 00218 IF( INFO.NE.0 ) THEN 00219 ITER = -2 00220 GO TO 40 00221 END IF 00222 * 00223 * Compute the LU factorization of SA. 00224 * 00225 CALL SGETRF( N, N, SWORK( PTSA ), N, IPIV, INFO ) 00226 * 00227 IF( INFO.NE.0 ) THEN 00228 ITER = -3 00229 GO TO 40 00230 END IF 00231 * 00232 * Solve the system SA*SX = SB. 00233 * 00234 CALL SGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV, 00235 + SWORK( PTSX ), N, INFO ) 00236 * 00237 * Convert SX back to double precision 00238 * 00239 CALL SLAG2D( N, NRHS, SWORK( PTSX ), N, X, LDX, INFO ) 00240 * 00241 * Compute R = B - AX (R is WORK). 00242 * 00243 CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N ) 00244 * 00245 CALL DGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, A, 00246 + LDA, X, LDX, ONE, WORK, N ) 00247 * 00248 * Check whether the NRHS normwise backward errors satisfy the 00249 * stopping criterion. If yes, set ITER=0 and return. 00250 * 00251 DO I = 1, NRHS 00252 XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) ) 00253 RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) ) 00254 IF( RNRM.GT.XNRM*CTE ) 00255 + GO TO 10 00256 END DO 00257 * 00258 * If we are here, the NRHS normwise backward errors satisfy the 00259 * stopping criterion. We are good to exit. 00260 * 00261 ITER = 0 00262 RETURN 00263 * 00264 10 CONTINUE 00265 * 00266 DO 30 IITER = 1, ITERMAX 00267 * 00268 * Convert R (in WORK) from double precision to single precision 00269 * and store the result in SX. 00270 * 00271 CALL DLAG2S( N, NRHS, WORK, N, SWORK( PTSX ), N, INFO ) 00272 * 00273 IF( INFO.NE.0 ) THEN 00274 ITER = -2 00275 GO TO 40 00276 END IF 00277 * 00278 * Solve the system SA*SX = SR. 00279 * 00280 CALL SGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV, 00281 + SWORK( PTSX ), N, INFO ) 00282 * 00283 * Convert SX back to double precision and update the current 00284 * iterate. 00285 * 00286 CALL SLAG2D( N, NRHS, SWORK( PTSX ), N, WORK, N, INFO ) 00287 * 00288 DO I = 1, NRHS 00289 CALL DAXPY( N, ONE, WORK( 1, I ), 1, X( 1, I ), 1 ) 00290 END DO 00291 * 00292 * Compute R = B - AX (R is WORK). 00293 * 00294 CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N ) 00295 * 00296 CALL DGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, 00297 + A, LDA, X, LDX, ONE, WORK, N ) 00298 * 00299 * Check whether the NRHS normwise backward errors satisfy the 00300 * stopping criterion. If yes, set ITER=IITER>0 and return. 00301 * 00302 DO I = 1, NRHS 00303 XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) ) 00304 RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) ) 00305 IF( RNRM.GT.XNRM*CTE ) 00306 + GO TO 20 00307 END DO 00308 * 00309 * If we are here, the NRHS normwise backward errors satisfy the 00310 * stopping criterion, we are good to exit. 00311 * 00312 ITER = IITER 00313 * 00314 RETURN 00315 * 00316 20 CONTINUE 00317 * 00318 30 CONTINUE 00319 * 00320 * If we are at this place of the code, this is because we have 00321 * performed ITER=ITERMAX iterations and never satisified the 00322 * stopping criterion, set up the ITER flag accordingly and follow up 00323 * on double precision routine. 00324 * 00325 ITER = -ITERMAX - 1 00326 * 00327 40 CONTINUE 00328 * 00329 * Single-precision iterative refinement failed to converge to a 00330 * satisfactory solution, so we resort to double precision. 00331 * 00332 CALL DGETRF( N, N, A, LDA, IPIV, INFO ) 00333 * 00334 IF( INFO.NE.0 ) 00335 + RETURN 00336 * 00337 CALL DLACPY( 'All', N, NRHS, B, LDB, X, LDX ) 00338 CALL DGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, X, LDX, 00339 + INFO ) 00340 * 00341 RETURN 00342 * 00343 * End of DSGESV. 00344 * 00345 END
1.7.2