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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE CGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, 00002 $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, 00003 $ WORK, RWORK, INFO ) 00004 * 00005 * -- LAPACK driver routine (version 3.2) -- 00006 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00007 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00008 * November 2006 00009 * 00010 * .. Scalar Arguments .. 00011 CHARACTER EQUED, FACT, TRANS 00012 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00013 REAL RCOND 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IPIV( * ) 00017 REAL BERR( * ), C( * ), FERR( * ), R( * ), 00018 $ RWORK( * ) 00019 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00020 $ WORK( * ), X( LDX, * ) 00021 * .. 00022 * 00023 * Purpose 00024 * ======= 00025 * 00026 * CGESVX uses the LU factorization to compute the solution to a complex 00027 * system of linear equations 00028 * A * X = B, 00029 * where A is an N-by-N matrix and X and B are N-by-NRHS matrices. 00030 * 00031 * Error bounds on the solution and a condition estimate are also 00032 * provided. 00033 * 00034 * Description 00035 * =========== 00036 * 00037 * The following steps are performed: 00038 * 00039 * 1. If FACT = 'E', real scaling factors are computed to equilibrate 00040 * the system: 00041 * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B 00042 * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B 00043 * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B 00044 * Whether or not the system will be equilibrated depends on the 00045 * scaling of the matrix A, but if equilibration is used, A is 00046 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') 00047 * or diag(C)*B (if TRANS = 'T' or 'C'). 00048 * 00049 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the 00050 * matrix A (after equilibration if FACT = 'E') as 00051 * A = P * L * U, 00052 * where P is a permutation matrix, L is a unit lower triangular 00053 * matrix, and U is upper triangular. 00054 * 00055 * 3. If some U(i,i)=0, so that U is exactly singular, then the routine 00056 * returns with INFO = i. Otherwise, the factored form of A is used 00057 * to estimate the condition number of the matrix A. If the 00058 * reciprocal of the condition number is less than machine precision, 00059 * INFO = N+1 is returned as a warning, but the routine still goes on 00060 * to solve for X and compute error bounds as described below. 00061 * 00062 * 4. The system of equations is solved for X using the factored form 00063 * of A. 00064 * 00065 * 5. Iterative refinement is applied to improve the computed solution 00066 * matrix and calculate error bounds and backward error estimates 00067 * for it. 00068 * 00069 * 6. If equilibration was used, the matrix X is premultiplied by 00070 * diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so 00071 * that it solves the original system before equilibration. 00072 * 00073 * Arguments 00074 * ========= 00075 * 00076 * FACT (input) CHARACTER*1 00077 * Specifies whether or not the factored form of the matrix A is 00078 * supplied on entry, and if not, whether the matrix A should be 00079 * equilibrated before it is factored. 00080 * = 'F': On entry, AF and IPIV contain the factored form of A. 00081 * If EQUED is not 'N', the matrix A has been 00082 * equilibrated with scaling factors given by R and C. 00083 * A, AF, and IPIV are not modified. 00084 * = 'N': The matrix A will be copied to AF and factored. 00085 * = 'E': The matrix A will be equilibrated if necessary, then 00086 * copied to AF and factored. 00087 * 00088 * TRANS (input) CHARACTER*1 00089 * Specifies the form of the system of equations: 00090 * = 'N': A * X = B (No transpose) 00091 * = 'T': A**T * X = B (Transpose) 00092 * = 'C': A**H * X = B (Conjugate transpose) 00093 * 00094 * N (input) INTEGER 00095 * The number of linear equations, i.e., the order of the 00096 * matrix A. N >= 0. 00097 * 00098 * NRHS (input) INTEGER 00099 * The number of right hand sides, i.e., the number of columns 00100 * of the matrices B and X. NRHS >= 0. 00101 * 00102 * A (input/output) COMPLEX array, dimension (LDA,N) 00103 * On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is 00104 * not 'N', then A must have been equilibrated by the scaling 00105 * factors in R and/or C. A is not modified if FACT = 'F' or 00106 * 'N', or if FACT = 'E' and EQUED = 'N' on exit. 00107 * 00108 * On exit, if EQUED .ne. 'N', A is scaled as follows: 00109 * EQUED = 'R': A := diag(R) * A 00110 * EQUED = 'C': A := A * diag(C) 00111 * EQUED = 'B': A := diag(R) * A * diag(C). 00112 * 00113 * LDA (input) INTEGER 00114 * The leading dimension of the array A. LDA >= max(1,N). 00115 * 00116 * AF (input or output) COMPLEX array, dimension (LDAF,N) 00117 * If FACT = 'F', then AF is an input argument and on entry 00118 * contains the factors L and U from the factorization 00119 * A = P*L*U as computed by CGETRF. If EQUED .ne. 'N', then 00120 * AF is the factored form of the equilibrated matrix A. 00121 * 00122 * If FACT = 'N', then AF is an output argument and on exit 00123 * returns the factors L and U from the factorization A = P*L*U 00124 * of the original matrix A. 00125 * 00126 * If FACT = 'E', then AF is an output argument and on exit 00127 * returns the factors L and U from the factorization A = P*L*U 00128 * of the equilibrated matrix A (see the description of A for 00129 * the form of the equilibrated matrix). 00130 * 00131 * LDAF (input) INTEGER 00132 * The leading dimension of the array AF. LDAF >= max(1,N). 00133 * 00134 * IPIV (input or output) INTEGER array, dimension (N) 00135 * If FACT = 'F', then IPIV is an input argument and on entry 00136 * contains the pivot indices from the factorization A = P*L*U 00137 * as computed by CGETRF; row i of the matrix was interchanged 00138 * with row IPIV(i). 00139 * 00140 * If FACT = 'N', then IPIV is an output argument and on exit 00141 * contains the pivot indices from the factorization A = P*L*U 00142 * of the original matrix A. 00143 * 00144 * If FACT = 'E', then IPIV is an output argument and on exit 00145 * contains the pivot indices from the factorization A = P*L*U 00146 * of the equilibrated matrix A. 00147 * 00148 * EQUED (input or output) CHARACTER*1 00149 * Specifies the form of equilibration that was done. 00150 * = 'N': No equilibration (always true if FACT = 'N'). 00151 * = 'R': Row equilibration, i.e., A has been premultiplied by 00152 * diag(R). 00153 * = 'C': Column equilibration, i.e., A has been postmultiplied 00154 * by diag(C). 00155 * = 'B': Both row and column equilibration, i.e., A has been 00156 * replaced by diag(R) * A * diag(C). 00157 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00158 * output argument. 00159 * 00160 * R (input or output) REAL array, dimension (N) 00161 * The row scale factors for A. If EQUED = 'R' or 'B', A is 00162 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R 00163 * is not accessed. R is an input argument if FACT = 'F'; 00164 * otherwise, R is an output argument. If FACT = 'F' and 00165 * EQUED = 'R' or 'B', each element of R must be positive. 00166 * 00167 * C (input or output) REAL array, dimension (N) 00168 * The column scale factors for A. If EQUED = 'C' or 'B', A is 00169 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C 00170 * is not accessed. C is an input argument if FACT = 'F'; 00171 * otherwise, C is an output argument. If FACT = 'F' and 00172 * EQUED = 'C' or 'B', each element of C must be positive. 00173 * 00174 * B (input/output) COMPLEX array, dimension (LDB,NRHS) 00175 * On entry, the N-by-NRHS right hand side matrix B. 00176 * On exit, 00177 * if EQUED = 'N', B is not modified; 00178 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by 00179 * diag(R)*B; 00180 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is 00181 * overwritten by diag(C)*B. 00182 * 00183 * LDB (input) INTEGER 00184 * The leading dimension of the array B. LDB >= max(1,N). 00185 * 00186 * X (output) COMPLEX array, dimension (LDX,NRHS) 00187 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X 00188 * to the original system of equations. Note that A and B are 00189 * modified on exit if EQUED .ne. 'N', and the solution to the 00190 * equilibrated system is inv(diag(C))*X if TRANS = 'N' and 00191 * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' 00192 * and EQUED = 'R' or 'B'. 00193 * 00194 * LDX (input) INTEGER 00195 * The leading dimension of the array X. LDX >= max(1,N). 00196 * 00197 * RCOND (output) REAL 00198 * The estimate of the reciprocal condition number of the matrix 00199 * A after equilibration (if done). If RCOND is less than the 00200 * machine precision (in particular, if RCOND = 0), the matrix 00201 * is singular to working precision. This condition is 00202 * indicated by a return code of INFO > 0. 00203 * 00204 * FERR (output) REAL array, dimension (NRHS) 00205 * The estimated forward error bound for each solution vector 00206 * X(j) (the j-th column of the solution matrix X). 00207 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00208 * is an estimated upper bound for the magnitude of the largest 00209 * element in (X(j) - XTRUE) divided by the magnitude of the 00210 * largest element in X(j). The estimate is as reliable as 00211 * the estimate for RCOND, and is almost always a slight 00212 * overestimate of the true error. 00213 * 00214 * BERR (output) REAL array, dimension (NRHS) 00215 * The componentwise relative backward error of each solution 00216 * vector X(j) (i.e., the smallest relative change in 00217 * any element of A or B that makes X(j) an exact solution). 00218 * 00219 * WORK (workspace) COMPLEX array, dimension (2*N) 00220 * 00221 * RWORK (workspace/output) REAL array, dimension (2*N) 00222 * On exit, RWORK(1) contains the reciprocal pivot growth 00223 * factor norm(A)/norm(U). The "max absolute element" norm is 00224 * used. If RWORK(1) is much less than 1, then the stability 00225 * of the LU factorization of the (equilibrated) matrix A 00226 * could be poor. This also means that the solution X, condition 00227 * estimator RCOND, and forward error bound FERR could be 00228 * unreliable. If factorization fails with 0<INFO<=N, then 00229 * RWORK(1) contains the reciprocal pivot growth factor for the 00230 * leading INFO columns of A. 00231 * 00232 * INFO (output) INTEGER 00233 * = 0: successful exit 00234 * < 0: if INFO = -i, the i-th argument had an illegal value 00235 * > 0: if INFO = i, and i is 00236 * <= N: U(i,i) is exactly zero. The factorization has 00237 * been completed, but the factor U is exactly 00238 * singular, so the solution and error bounds 00239 * could not be computed. RCOND = 0 is returned. 00240 * = N+1: U is nonsingular, but RCOND is less than machine 00241 * precision, meaning that the matrix is singular 00242 * to working precision. Nevertheless, the 00243 * solution and error bounds are computed because 00244 * there are a number of situations where the 00245 * computed solution can be more accurate than the 00246 * value of RCOND would suggest. 00247 * 00248 * ===================================================================== 00249 * 00250 * .. Parameters .. 00251 REAL ZERO, ONE 00252 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00253 * .. 00254 * .. Local Scalars .. 00255 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU 00256 CHARACTER NORM 00257 INTEGER I, INFEQU, J 00258 REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN, 00259 $ ROWCND, RPVGRW, SMLNUM 00260 * .. 00261 * .. External Functions .. 00262 LOGICAL LSAME 00263 REAL CLANGE, CLANTR, SLAMCH 00264 EXTERNAL LSAME, CLANGE, CLANTR, SLAMCH 00265 * .. 00266 * .. External Subroutines .. 00267 EXTERNAL CGECON, CGEEQU, CGERFS, CGETRF, CGETRS, CLACPY, 00268 $ CLAQGE, XERBLA 00269 * .. 00270 * .. Intrinsic Functions .. 00271 INTRINSIC MAX, MIN 00272 * .. 00273 * .. Executable Statements .. 00274 * 00275 INFO = 0 00276 NOFACT = LSAME( FACT, 'N' ) 00277 EQUIL = LSAME( FACT, 'E' ) 00278 NOTRAN = LSAME( TRANS, 'N' ) 00279 IF( NOFACT .OR. EQUIL ) THEN 00280 EQUED = 'N' 00281 ROWEQU = .FALSE. 00282 COLEQU = .FALSE. 00283 ELSE 00284 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 00285 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 00286 SMLNUM = SLAMCH( 'Safe minimum' ) 00287 BIGNUM = ONE / SMLNUM 00288 END IF 00289 * 00290 * Test the input parameters. 00291 * 00292 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 00293 $ THEN 00294 INFO = -1 00295 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00296 $ LSAME( TRANS, 'C' ) ) THEN 00297 INFO = -2 00298 ELSE IF( N.LT.0 ) THEN 00299 INFO = -3 00300 ELSE IF( NRHS.LT.0 ) THEN 00301 INFO = -4 00302 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00303 INFO = -6 00304 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00305 INFO = -8 00306 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00307 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00308 INFO = -10 00309 ELSE 00310 IF( ROWEQU ) THEN 00311 RCMIN = BIGNUM 00312 RCMAX = ZERO 00313 DO 10 J = 1, N 00314 RCMIN = MIN( RCMIN, R( J ) ) 00315 RCMAX = MAX( RCMAX, R( J ) ) 00316 10 CONTINUE 00317 IF( RCMIN.LE.ZERO ) THEN 00318 INFO = -11 00319 ELSE IF( N.GT.0 ) THEN 00320 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 00321 ELSE 00322 ROWCND = ONE 00323 END IF 00324 END IF 00325 IF( COLEQU .AND. INFO.EQ.0 ) THEN 00326 RCMIN = BIGNUM 00327 RCMAX = ZERO 00328 DO 20 J = 1, N 00329 RCMIN = MIN( RCMIN, C( J ) ) 00330 RCMAX = MAX( RCMAX, C( J ) ) 00331 20 CONTINUE 00332 IF( RCMIN.LE.ZERO ) THEN 00333 INFO = -12 00334 ELSE IF( N.GT.0 ) THEN 00335 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 00336 ELSE 00337 COLCND = ONE 00338 END IF 00339 END IF 00340 IF( INFO.EQ.0 ) THEN 00341 IF( LDB.LT.MAX( 1, N ) ) THEN 00342 INFO = -14 00343 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00344 INFO = -16 00345 END IF 00346 END IF 00347 END IF 00348 * 00349 IF( INFO.NE.0 ) THEN 00350 CALL XERBLA( 'CGESVX', -INFO ) 00351 RETURN 00352 END IF 00353 * 00354 IF( EQUIL ) THEN 00355 * 00356 * Compute row and column scalings to equilibrate the matrix A. 00357 * 00358 CALL CGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU ) 00359 IF( INFEQU.EQ.0 ) THEN 00360 * 00361 * Equilibrate the matrix. 00362 * 00363 CALL CLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, 00364 $ EQUED ) 00365 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 00366 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 00367 END IF 00368 END IF 00369 * 00370 * Scale the right hand side. 00371 * 00372 IF( NOTRAN ) THEN 00373 IF( ROWEQU ) THEN 00374 DO 40 J = 1, NRHS 00375 DO 30 I = 1, N 00376 B( I, J ) = R( I )*B( I, J ) 00377 30 CONTINUE 00378 40 CONTINUE 00379 END IF 00380 ELSE IF( COLEQU ) THEN 00381 DO 60 J = 1, NRHS 00382 DO 50 I = 1, N 00383 B( I, J ) = C( I )*B( I, J ) 00384 50 CONTINUE 00385 60 CONTINUE 00386 END IF 00387 * 00388 IF( NOFACT .OR. EQUIL ) THEN 00389 * 00390 * Compute the LU factorization of A. 00391 * 00392 CALL CLACPY( 'Full', N, N, A, LDA, AF, LDAF ) 00393 CALL CGETRF( N, N, AF, LDAF, IPIV, INFO ) 00394 * 00395 * Return if INFO is non-zero. 00396 * 00397 IF( INFO.GT.0 ) THEN 00398 * 00399 * Compute the reciprocal pivot growth factor of the 00400 * leading rank-deficient INFO columns of A. 00401 * 00402 RPVGRW = CLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF, 00403 $ RWORK ) 00404 IF( RPVGRW.EQ.ZERO ) THEN 00405 RPVGRW = ONE 00406 ELSE 00407 RPVGRW = CLANGE( 'M', N, INFO, A, LDA, RWORK ) / 00408 $ RPVGRW 00409 END IF 00410 RWORK( 1 ) = RPVGRW 00411 RCOND = ZERO 00412 RETURN 00413 END IF 00414 END IF 00415 * 00416 * Compute the norm of the matrix A and the 00417 * reciprocal pivot growth factor RPVGRW. 00418 * 00419 IF( NOTRAN ) THEN 00420 NORM = '1' 00421 ELSE 00422 NORM = 'I' 00423 END IF 00424 ANORM = CLANGE( NORM, N, N, A, LDA, RWORK ) 00425 RPVGRW = CLANTR( 'M', 'U', 'N', N, N, AF, LDAF, RWORK ) 00426 IF( RPVGRW.EQ.ZERO ) THEN 00427 RPVGRW = ONE 00428 ELSE 00429 RPVGRW = CLANGE( 'M', N, N, A, LDA, RWORK ) / RPVGRW 00430 END IF 00431 * 00432 * Compute the reciprocal of the condition number of A. 00433 * 00434 CALL CGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO ) 00435 * 00436 * Compute the solution matrix X. 00437 * 00438 CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00439 CALL CGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 00440 * 00441 * Use iterative refinement to improve the computed solution and 00442 * compute error bounds and backward error estimates for it. 00443 * 00444 CALL CGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, 00445 $ LDX, FERR, BERR, WORK, RWORK, INFO ) 00446 * 00447 * Transform the solution matrix X to a solution of the original 00448 * system. 00449 * 00450 IF( NOTRAN ) THEN 00451 IF( COLEQU ) THEN 00452 DO 80 J = 1, NRHS 00453 DO 70 I = 1, N 00454 X( I, J ) = C( I )*X( I, J ) 00455 70 CONTINUE 00456 80 CONTINUE 00457 DO 90 J = 1, NRHS 00458 FERR( J ) = FERR( J ) / COLCND 00459 90 CONTINUE 00460 END IF 00461 ELSE IF( ROWEQU ) THEN 00462 DO 110 J = 1, NRHS 00463 DO 100 I = 1, N 00464 X( I, J ) = R( I )*X( I, J ) 00465 100 CONTINUE 00466 110 CONTINUE 00467 DO 120 J = 1, NRHS 00468 FERR( J ) = FERR( J ) / ROWCND 00469 120 CONTINUE 00470 END IF 00471 * 00472 * Set INFO = N+1 if the matrix is singular to working precision. 00473 * 00474 IF( RCOND.LT.SLAMCH( 'Epsilon' ) ) 00475 $ INFO = N + 1 00476 * 00477 RWORK( 1 ) = RPVGRW 00478 RETURN 00479 * 00480 * End of CGESVX 00481 * 00482 END
1.7.3 
