00001 SUBROUTINE SGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, 00002 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, 00003 $ RCOND, FERR, BERR, WORK, IWORK, 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, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS 00013 REAL RCOND 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IPIV( * ), IWORK( * ) 00017 REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 00018 $ BERR( * ), C( * ), FERR( * ), R( * ), 00019 $ WORK( * ), X( LDX, * ) 00020 * .. 00021 * 00022 * Purpose 00023 * ======= 00024 * 00025 * SGBSVX uses the LU factorization to compute the solution to a real 00026 * system of linear equations A * X = B, A**T * X = B, or A**H * X = B, 00027 * where A is a band matrix of order N with KL subdiagonals and KU 00028 * superdiagonals, and X and B are N-by-NRHS matrices. 00029 * 00030 * Error bounds on the solution and a condition estimate are also 00031 * provided. 00032 * 00033 * Description 00034 * =========== 00035 * 00036 * The following steps are performed by this subroutine: 00037 * 00038 * 1. If FACT = 'E', real scaling factors are computed to equilibrate 00039 * the system: 00040 * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B 00041 * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B 00042 * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B 00043 * Whether or not the system will be equilibrated depends on the 00044 * scaling of the matrix A, but if equilibration is used, A is 00045 * overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') 00046 * or diag(C)*B (if TRANS = 'T' or 'C'). 00047 * 00048 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the 00049 * matrix A (after equilibration if FACT = 'E') as 00050 * A = L * U, 00051 * where L is a product of permutation and unit lower triangular 00052 * matrices with KL subdiagonals, and U is upper triangular with 00053 * KL+KU superdiagonals. 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, AFB and IPIV contain the factored form of 00081 * A. If EQUED is not 'N', the matrix A has been 00082 * equilibrated with scaling factors given by R and C. 00083 * AB, AFB, and IPIV are not modified. 00084 * = 'N': The matrix A will be copied to AFB and factored. 00085 * = 'E': The matrix A will be equilibrated if necessary, then 00086 * copied to AFB 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 (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 * KL (input) INTEGER 00099 * The number of subdiagonals within the band of A. KL >= 0. 00100 * 00101 * KU (input) INTEGER 00102 * The number of superdiagonals within the band of A. KU >= 0. 00103 * 00104 * NRHS (input) INTEGER 00105 * The number of right hand sides, i.e., the number of columns 00106 * of the matrices B and X. NRHS >= 0. 00107 * 00108 * AB (input/output) REAL array, dimension (LDAB,N) 00109 * On entry, the matrix A in band storage, in rows 1 to KL+KU+1. 00110 * The j-th column of A is stored in the j-th column of the 00111 * array AB as follows: 00112 * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) 00113 * 00114 * If FACT = 'F' and EQUED is not 'N', then A must have been 00115 * equilibrated by the scaling factors in R and/or C. AB is not 00116 * modified if FACT = 'F' or 'N', or if FACT = 'E' and 00117 * EQUED = 'N' on exit. 00118 * 00119 * On exit, if EQUED .ne. 'N', A is scaled as follows: 00120 * EQUED = 'R': A := diag(R) * A 00121 * EQUED = 'C': A := A * diag(C) 00122 * EQUED = 'B': A := diag(R) * A * diag(C). 00123 * 00124 * LDAB (input) INTEGER 00125 * The leading dimension of the array AB. LDAB >= KL+KU+1. 00126 * 00127 * AFB (input or output) REAL array, dimension (LDAFB,N) 00128 * If FACT = 'F', then AFB is an input argument and on entry 00129 * contains details of the LU factorization of the band matrix 00130 * A, as computed by SGBTRF. U is stored as an upper triangular 00131 * band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, 00132 * and the multipliers used during the factorization are stored 00133 * in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is 00134 * the factored form of the equilibrated matrix A. 00135 * 00136 * If FACT = 'N', then AFB is an output argument and on exit 00137 * returns details of the LU factorization of A. 00138 * 00139 * If FACT = 'E', then AFB is an output argument and on exit 00140 * returns details of the LU factorization of the equilibrated 00141 * matrix A (see the description of AB for the form of the 00142 * equilibrated matrix). 00143 * 00144 * LDAFB (input) INTEGER 00145 * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. 00146 * 00147 * IPIV (input or output) INTEGER array, dimension (N) 00148 * If FACT = 'F', then IPIV is an input argument and on entry 00149 * contains the pivot indices from the factorization A = L*U 00150 * as computed by SGBTRF; row i of the matrix was interchanged 00151 * with row IPIV(i). 00152 * 00153 * If FACT = 'N', then IPIV is an output argument and on exit 00154 * contains the pivot indices from the factorization A = L*U 00155 * of the original matrix A. 00156 * 00157 * If FACT = 'E', then IPIV is an output argument and on exit 00158 * contains the pivot indices from the factorization A = L*U 00159 * of the equilibrated matrix A. 00160 * 00161 * EQUED (input or output) CHARACTER*1 00162 * Specifies the form of equilibration that was done. 00163 * = 'N': No equilibration (always true if FACT = 'N'). 00164 * = 'R': Row equilibration, i.e., A has been premultiplied by 00165 * diag(R). 00166 * = 'C': Column equilibration, i.e., A has been postmultiplied 00167 * by diag(C). 00168 * = 'B': Both row and column equilibration, i.e., A has been 00169 * replaced by diag(R) * A * diag(C). 00170 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00171 * output argument. 00172 * 00173 * R (input or output) REAL array, dimension (N) 00174 * The row scale factors for A. If EQUED = 'R' or 'B', A is 00175 * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R 00176 * is not accessed. R is an input argument if FACT = 'F'; 00177 * otherwise, R is an output argument. If FACT = 'F' and 00178 * EQUED = 'R' or 'B', each element of R must be positive. 00179 * 00180 * C (input or output) REAL array, dimension (N) 00181 * The column scale factors for A. If EQUED = 'C' or 'B', A is 00182 * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C 00183 * is not accessed. C is an input argument if FACT = 'F'; 00184 * otherwise, C is an output argument. If FACT = 'F' and 00185 * EQUED = 'C' or 'B', each element of C must be positive. 00186 * 00187 * B (input/output) REAL array, dimension (LDB,NRHS) 00188 * On entry, the right hand side matrix B. 00189 * On exit, 00190 * if EQUED = 'N', B is not modified; 00191 * if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by 00192 * diag(R)*B; 00193 * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is 00194 * overwritten by diag(C)*B. 00195 * 00196 * LDB (input) INTEGER 00197 * The leading dimension of the array B. LDB >= max(1,N). 00198 * 00199 * X (output) REAL array, dimension (LDX,NRHS) 00200 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X 00201 * to the original system of equations. Note that A and B are 00202 * modified on exit if EQUED .ne. 'N', and the solution to the 00203 * equilibrated system is inv(diag(C))*X if TRANS = 'N' and 00204 * EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' 00205 * and EQUED = 'R' or 'B'. 00206 * 00207 * LDX (input) INTEGER 00208 * The leading dimension of the array X. LDX >= max(1,N). 00209 * 00210 * RCOND (output) REAL 00211 * The estimate of the reciprocal condition number of the matrix 00212 * A after equilibration (if done). If RCOND is less than the 00213 * machine precision (in particular, if RCOND = 0), the matrix 00214 * is singular to working precision. This condition is 00215 * indicated by a return code of INFO > 0. 00216 * 00217 * FERR (output) REAL array, dimension (NRHS) 00218 * The estimated forward error bound for each solution vector 00219 * X(j) (the j-th column of the solution matrix X). 00220 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00221 * is an estimated upper bound for the magnitude of the largest 00222 * element in (X(j) - XTRUE) divided by the magnitude of the 00223 * largest element in X(j). The estimate is as reliable as 00224 * the estimate for RCOND, and is almost always a slight 00225 * overestimate of the true error. 00226 * 00227 * BERR (output) REAL array, dimension (NRHS) 00228 * The componentwise relative backward error of each solution 00229 * vector X(j) (i.e., the smallest relative change in 00230 * any element of A or B that makes X(j) an exact solution). 00231 * 00232 * WORK (workspace/output) REAL array, dimension (3*N) 00233 * On exit, WORK(1) contains the reciprocal pivot growth 00234 * factor norm(A)/norm(U). The "max absolute element" norm is 00235 * used. If WORK(1) is much less than 1, then the stability 00236 * of the LU factorization of the (equilibrated) matrix A 00237 * could be poor. This also means that the solution X, condition 00238 * estimator RCOND, and forward error bound FERR could be 00239 * unreliable. If factorization fails with 0<INFO<=N, then 00240 * WORK(1) contains the reciprocal pivot growth factor for the 00241 * leading INFO columns of A. 00242 * 00243 * IWORK (workspace) INTEGER array, dimension (N) 00244 * 00245 * INFO (output) INTEGER 00246 * = 0: successful exit 00247 * < 0: if INFO = -i, the i-th argument had an illegal value 00248 * > 0: if INFO = i, and i is 00249 * <= N: U(i,i) is exactly zero. The factorization 00250 * has been completed, but the factor U is exactly 00251 * singular, so the solution and error bounds 00252 * could not be computed. RCOND = 0 is returned. 00253 * = N+1: U is nonsingular, but RCOND is less than machine 00254 * precision, meaning that the matrix is singular 00255 * to working precision. Nevertheless, the 00256 * solution and error bounds are computed because 00257 * there are a number of situations where the 00258 * computed solution can be more accurate than the 00259 * 00260 * value of RCOND would suggest. 00261 * ===================================================================== 00262 * Moved setting of INFO = N+1 so INFO does not subsequently get 00263 * overwritten. Sven, 17 Mar 05. 00264 * ===================================================================== 00265 * 00266 * .. Parameters .. 00267 REAL ZERO, ONE 00268 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00269 * .. 00270 * .. Local Scalars .. 00271 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU 00272 CHARACTER NORM 00273 INTEGER I, INFEQU, J, J1, J2 00274 REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN, 00275 $ ROWCND, RPVGRW, SMLNUM 00276 * .. 00277 * .. External Functions .. 00278 LOGICAL LSAME 00279 REAL SLAMCH, SLANGB, SLANTB 00280 EXTERNAL LSAME, SLAMCH, SLANGB, SLANTB 00281 * .. 00282 * .. External Subroutines .. 00283 EXTERNAL SCOPY, SGBCON, SGBEQU, SGBRFS, SGBTRF, SGBTRS, 00284 $ SLACPY, SLAQGB, XERBLA 00285 * .. 00286 * .. Intrinsic Functions .. 00287 INTRINSIC ABS, MAX, MIN 00288 * .. 00289 * .. Executable Statements .. 00290 * 00291 INFO = 0 00292 NOFACT = LSAME( FACT, 'N' ) 00293 EQUIL = LSAME( FACT, 'E' ) 00294 NOTRAN = LSAME( TRANS, 'N' ) 00295 IF( NOFACT .OR. EQUIL ) THEN 00296 EQUED = 'N' 00297 ROWEQU = .FALSE. 00298 COLEQU = .FALSE. 00299 ELSE 00300 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 00301 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 00302 SMLNUM = SLAMCH( 'Safe minimum' ) 00303 BIGNUM = ONE / SMLNUM 00304 END IF 00305 * 00306 * Test the input parameters. 00307 * 00308 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 00309 $ THEN 00310 INFO = -1 00311 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00312 $ LSAME( TRANS, 'C' ) ) THEN 00313 INFO = -2 00314 ELSE IF( N.LT.0 ) THEN 00315 INFO = -3 00316 ELSE IF( KL.LT.0 ) THEN 00317 INFO = -4 00318 ELSE IF( KU.LT.0 ) THEN 00319 INFO = -5 00320 ELSE IF( NRHS.LT.0 ) THEN 00321 INFO = -6 00322 ELSE IF( LDAB.LT.KL+KU+1 ) THEN 00323 INFO = -8 00324 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN 00325 INFO = -10 00326 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00327 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00328 INFO = -12 00329 ELSE 00330 IF( ROWEQU ) THEN 00331 RCMIN = BIGNUM 00332 RCMAX = ZERO 00333 DO 10 J = 1, N 00334 RCMIN = MIN( RCMIN, R( J ) ) 00335 RCMAX = MAX( RCMAX, R( J ) ) 00336 10 CONTINUE 00337 IF( RCMIN.LE.ZERO ) THEN 00338 INFO = -13 00339 ELSE IF( N.GT.0 ) THEN 00340 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 00341 ELSE 00342 ROWCND = ONE 00343 END IF 00344 END IF 00345 IF( COLEQU .AND. INFO.EQ.0 ) THEN 00346 RCMIN = BIGNUM 00347 RCMAX = ZERO 00348 DO 20 J = 1, N 00349 RCMIN = MIN( RCMIN, C( J ) ) 00350 RCMAX = MAX( RCMAX, C( J ) ) 00351 20 CONTINUE 00352 IF( RCMIN.LE.ZERO ) THEN 00353 INFO = -14 00354 ELSE IF( N.GT.0 ) THEN 00355 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 00356 ELSE 00357 COLCND = ONE 00358 END IF 00359 END IF 00360 IF( INFO.EQ.0 ) THEN 00361 IF( LDB.LT.MAX( 1, N ) ) THEN 00362 INFO = -16 00363 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00364 INFO = -18 00365 END IF 00366 END IF 00367 END IF 00368 * 00369 IF( INFO.NE.0 ) THEN 00370 CALL XERBLA( 'SGBSVX', -INFO ) 00371 RETURN 00372 END IF 00373 * 00374 IF( EQUIL ) THEN 00375 * 00376 * Compute row and column scalings to equilibrate the matrix A. 00377 * 00378 CALL SGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, 00379 $ AMAX, INFEQU ) 00380 IF( INFEQU.EQ.0 ) THEN 00381 * 00382 * Equilibrate the matrix. 00383 * 00384 CALL SLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, 00385 $ AMAX, EQUED ) 00386 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 00387 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 00388 END IF 00389 END IF 00390 * 00391 * Scale the right hand side. 00392 * 00393 IF( NOTRAN ) THEN 00394 IF( ROWEQU ) THEN 00395 DO 40 J = 1, NRHS 00396 DO 30 I = 1, N 00397 B( I, J ) = R( I )*B( I, J ) 00398 30 CONTINUE 00399 40 CONTINUE 00400 END IF 00401 ELSE IF( COLEQU ) THEN 00402 DO 60 J = 1, NRHS 00403 DO 50 I = 1, N 00404 B( I, J ) = C( I )*B( I, J ) 00405 50 CONTINUE 00406 60 CONTINUE 00407 END IF 00408 * 00409 IF( NOFACT .OR. EQUIL ) THEN 00410 * 00411 * Compute the LU factorization of the band matrix A. 00412 * 00413 DO 70 J = 1, N 00414 J1 = MAX( J-KU, 1 ) 00415 J2 = MIN( J+KL, N ) 00416 CALL SCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1, 00417 $ AFB( KL+KU+1-J+J1, J ), 1 ) 00418 70 CONTINUE 00419 * 00420 CALL SGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO ) 00421 * 00422 * Return if INFO is non-zero. 00423 * 00424 IF( INFO.GT.0 ) THEN 00425 * 00426 * Compute the reciprocal pivot growth factor of the 00427 * leading rank-deficient INFO columns of A. 00428 * 00429 ANORM = ZERO 00430 DO 90 J = 1, INFO 00431 DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 ) 00432 ANORM = MAX( ANORM, ABS( AB( I, J ) ) ) 00433 80 CONTINUE 00434 90 CONTINUE 00435 RPVGRW = SLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ), 00436 $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB, 00437 $ WORK ) 00438 IF( RPVGRW.EQ.ZERO ) THEN 00439 RPVGRW = ONE 00440 ELSE 00441 RPVGRW = ANORM / RPVGRW 00442 END IF 00443 WORK( 1 ) = RPVGRW 00444 RCOND = ZERO 00445 RETURN 00446 END IF 00447 END IF 00448 * 00449 * Compute the norm of the matrix A and the 00450 * reciprocal pivot growth factor RPVGRW. 00451 * 00452 IF( NOTRAN ) THEN 00453 NORM = '1' 00454 ELSE 00455 NORM = 'I' 00456 END IF 00457 ANORM = SLANGB( NORM, N, KL, KU, AB, LDAB, WORK ) 00458 RPVGRW = SLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK ) 00459 IF( RPVGRW.EQ.ZERO ) THEN 00460 RPVGRW = ONE 00461 ELSE 00462 RPVGRW = SLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW 00463 END IF 00464 * 00465 * Compute the reciprocal of the condition number of A. 00466 * 00467 CALL SGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND, 00468 $ WORK, IWORK, INFO ) 00469 * 00470 * Compute the solution matrix X. 00471 * 00472 CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00473 CALL SGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX, 00474 $ INFO ) 00475 * 00476 * Use iterative refinement to improve the computed solution and 00477 * compute error bounds and backward error estimates for it. 00478 * 00479 CALL SGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, 00480 $ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) 00481 * 00482 * Transform the solution matrix X to a solution of the original 00483 * system. 00484 * 00485 IF( NOTRAN ) THEN 00486 IF( COLEQU ) THEN 00487 DO 110 J = 1, NRHS 00488 DO 100 I = 1, N 00489 X( I, J ) = C( I )*X( I, J ) 00490 100 CONTINUE 00491 110 CONTINUE 00492 DO 120 J = 1, NRHS 00493 FERR( J ) = FERR( J ) / COLCND 00494 120 CONTINUE 00495 END IF 00496 ELSE IF( ROWEQU ) THEN 00497 DO 140 J = 1, NRHS 00498 DO 130 I = 1, N 00499 X( I, J ) = R( I )*X( I, J ) 00500 130 CONTINUE 00501 140 CONTINUE 00502 DO 150 J = 1, NRHS 00503 FERR( J ) = FERR( J ) / ROWCND 00504 150 CONTINUE 00505 END IF 00506 * 00507 * Set INFO = N+1 if the matrix is singular to working precision. 00508 * 00509 IF( RCOND.LT.SLAMCH( 'Epsilon' ) ) 00510 $ INFO = N + 1 00511 * 00512 WORK( 1 ) = RPVGRW 00513 RETURN 00514 * 00515 * End of SGBSVX 00516 * 00517 END
1.7.2