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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE ZPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 00002 $ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, 00003 $ 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, UPLO 00012 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 00013 DOUBLE PRECISION RCOND 00014 * .. 00015 * .. Array Arguments .. 00016 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * ) 00017 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00018 $ WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to 00025 * compute the solution to a complex system of linear equations 00026 * A * X = B, 00027 * where A is an N-by-N Hermitian positive definite matrix and X and B 00028 * 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: 00037 * 00038 * 1. If FACT = 'E', real scaling factors are computed to equilibrate 00039 * the system: 00040 * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B 00041 * Whether or not the system will be equilibrated depends on the 00042 * scaling of the matrix A, but if equilibration is used, A is 00043 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 00044 * 00045 * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 00046 * factor the matrix A (after equilibration if FACT = 'E') as 00047 * A = U**H* U, if UPLO = 'U', or 00048 * A = L * L**H, if UPLO = 'L', 00049 * where U is an upper triangular matrix and L is a lower triangular 00050 * matrix. 00051 * 00052 * 3. If the leading i-by-i principal minor is not positive definite, 00053 * then the routine returns with INFO = i. Otherwise, the factored 00054 * form of A is used to estimate the condition number of the matrix 00055 * A. If the reciprocal of the condition number is less than machine 00056 * precision, INFO = N+1 is returned as a warning, but the routine 00057 * still goes on to solve for X and compute error bounds as 00058 * described below. 00059 * 00060 * 4. The system of equations is solved for X using the factored form 00061 * of A. 00062 * 00063 * 5. Iterative refinement is applied to improve the computed solution 00064 * matrix and calculate error bounds and backward error estimates 00065 * for it. 00066 * 00067 * 6. If equilibration was used, the matrix X is premultiplied by 00068 * diag(S) so that it solves the original system before 00069 * equilibration. 00070 * 00071 * Arguments 00072 * ========= 00073 * 00074 * FACT (input) CHARACTER*1 00075 * Specifies whether or not the factored form of the matrix A is 00076 * supplied on entry, and if not, whether the matrix A should be 00077 * equilibrated before it is factored. 00078 * = 'F': On entry, AF contains the factored form of A. 00079 * If EQUED = 'Y', the matrix A has been equilibrated 00080 * with scaling factors given by S. A and AF will not 00081 * be modified. 00082 * = 'N': The matrix A will be copied to AF and factored. 00083 * = 'E': The matrix A will be equilibrated if necessary, then 00084 * copied to AF and factored. 00085 * 00086 * UPLO (input) CHARACTER*1 00087 * = 'U': Upper triangle of A is stored; 00088 * = 'L': Lower triangle of A is stored. 00089 * 00090 * N (input) INTEGER 00091 * The number of linear equations, i.e., the order of the 00092 * matrix A. N >= 0. 00093 * 00094 * NRHS (input) INTEGER 00095 * The number of right hand sides, i.e., the number of columns 00096 * of the matrices B and X. NRHS >= 0. 00097 * 00098 * A (input/output) COMPLEX*16 array, dimension (LDA,N) 00099 * On entry, the Hermitian matrix A, except if FACT = 'F' and 00100 * EQUED = 'Y', then A must contain the equilibrated matrix 00101 * diag(S)*A*diag(S). If UPLO = 'U', the leading 00102 * N-by-N upper triangular part of A contains the upper 00103 * triangular part of the matrix A, and the strictly lower 00104 * triangular part of A is not referenced. If UPLO = 'L', the 00105 * leading N-by-N lower triangular part of A contains the lower 00106 * triangular part of the matrix A, and the strictly upper 00107 * triangular part of A is not referenced. A is not modified if 00108 * FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. 00109 * 00110 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00111 * diag(S)*A*diag(S). 00112 * 00113 * LDA (input) INTEGER 00114 * The leading dimension of the array A. LDA >= max(1,N). 00115 * 00116 * AF (input or output) COMPLEX*16 array, dimension (LDAF,N) 00117 * If FACT = 'F', then AF is an input argument and on entry 00118 * contains the triangular factor U or L from the Cholesky 00119 * factorization A = U**H*U or A = L*L**H, in the same storage 00120 * format as A. If EQUED .ne. 'N', then AF is the factored form 00121 * of the equilibrated matrix diag(S)*A*diag(S). 00122 * 00123 * If FACT = 'N', then AF is an output argument and on exit 00124 * returns the triangular factor U or L from the Cholesky 00125 * factorization A = U**H*U or A = L*L**H of the original 00126 * matrix A. 00127 * 00128 * If FACT = 'E', then AF is an output argument and on exit 00129 * returns the triangular factor U or L from the Cholesky 00130 * factorization A = U**H*U or A = L*L**H of the equilibrated 00131 * matrix A (see the description of A for the form of the 00132 * equilibrated matrix). 00133 * 00134 * LDAF (input) INTEGER 00135 * The leading dimension of the array AF. LDAF >= max(1,N). 00136 * 00137 * EQUED (input or output) CHARACTER*1 00138 * Specifies the form of equilibration that was done. 00139 * = 'N': No equilibration (always true if FACT = 'N'). 00140 * = 'Y': Equilibration was done, i.e., A has been replaced by 00141 * diag(S) * A * diag(S). 00142 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00143 * output argument. 00144 * 00145 * S (input or output) DOUBLE PRECISION array, dimension (N) 00146 * The scale factors for A; not accessed if EQUED = 'N'. S is 00147 * an input argument if FACT = 'F'; otherwise, S is an output 00148 * argument. If FACT = 'F' and EQUED = 'Y', each element of S 00149 * must be positive. 00150 * 00151 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) 00152 * On entry, the N-by-NRHS righthand side matrix B. 00153 * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', 00154 * B is overwritten by diag(S) * B. 00155 * 00156 * LDB (input) INTEGER 00157 * The leading dimension of the array B. LDB >= max(1,N). 00158 * 00159 * X (output) COMPLEX*16 array, dimension (LDX,NRHS) 00160 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to 00161 * the original system of equations. Note that if EQUED = 'Y', 00162 * A and B are modified on exit, and the solution to the 00163 * equilibrated system is inv(diag(S))*X. 00164 * 00165 * LDX (input) INTEGER 00166 * The leading dimension of the array X. LDX >= max(1,N). 00167 * 00168 * RCOND (output) DOUBLE PRECISION 00169 * The estimate of the reciprocal condition number of the matrix 00170 * A after equilibration (if done). If RCOND is less than the 00171 * machine precision (in particular, if RCOND = 0), the matrix 00172 * is singular to working precision. This condition is 00173 * indicated by a return code of INFO > 0. 00174 * 00175 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00176 * The estimated forward error bound for each solution vector 00177 * X(j) (the j-th column of the solution matrix X). 00178 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00179 * is an estimated upper bound for the magnitude of the largest 00180 * element in (X(j) - XTRUE) divided by the magnitude of the 00181 * largest element in X(j). The estimate is as reliable as 00182 * the estimate for RCOND, and is almost always a slight 00183 * overestimate of the true error. 00184 * 00185 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00186 * The componentwise relative backward error of each solution 00187 * vector X(j) (i.e., the smallest relative change in 00188 * any element of A or B that makes X(j) an exact solution). 00189 * 00190 * WORK (workspace) COMPLEX*16 array, dimension (2*N) 00191 * 00192 * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 00193 * 00194 * INFO (output) INTEGER 00195 * = 0: successful exit 00196 * < 0: if INFO = -i, the i-th argument had an illegal value 00197 * > 0: if INFO = i, and i is 00198 * <= N: the leading minor of order i of A is 00199 * not positive definite, so the factorization 00200 * could not be completed, and the solution has not 00201 * been computed. RCOND = 0 is returned. 00202 * = N+1: U is nonsingular, but RCOND is less than machine 00203 * precision, meaning that the matrix is singular 00204 * to working precision. Nevertheless, the 00205 * solution and error bounds are computed because 00206 * there are a number of situations where the 00207 * computed solution can be more accurate than the 00208 * value of RCOND would suggest. 00209 * 00210 * ===================================================================== 00211 * 00212 * .. Parameters .. 00213 DOUBLE PRECISION ZERO, ONE 00214 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00215 * .. 00216 * .. Local Scalars .. 00217 LOGICAL EQUIL, NOFACT, RCEQU 00218 INTEGER I, INFEQU, J 00219 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM 00220 * .. 00221 * .. External Functions .. 00222 LOGICAL LSAME 00223 DOUBLE PRECISION DLAMCH, ZLANHE 00224 EXTERNAL LSAME, DLAMCH, ZLANHE 00225 * .. 00226 * .. External Subroutines .. 00227 EXTERNAL XERBLA, ZLACPY, ZLAQHE, ZPOCON, ZPOEQU, ZPORFS, 00228 $ ZPOTRF, ZPOTRS 00229 * .. 00230 * .. Intrinsic Functions .. 00231 INTRINSIC MAX, MIN 00232 * .. 00233 * .. Executable Statements .. 00234 * 00235 INFO = 0 00236 NOFACT = LSAME( FACT, 'N' ) 00237 EQUIL = LSAME( FACT, 'E' ) 00238 IF( NOFACT .OR. EQUIL ) THEN 00239 EQUED = 'N' 00240 RCEQU = .FALSE. 00241 ELSE 00242 RCEQU = LSAME( EQUED, 'Y' ) 00243 SMLNUM = DLAMCH( 'Safe minimum' ) 00244 BIGNUM = ONE / SMLNUM 00245 END IF 00246 * 00247 * Test the input parameters. 00248 * 00249 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 00250 $ THEN 00251 INFO = -1 00252 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) 00253 $ THEN 00254 INFO = -2 00255 ELSE IF( N.LT.0 ) THEN 00256 INFO = -3 00257 ELSE IF( NRHS.LT.0 ) THEN 00258 INFO = -4 00259 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00260 INFO = -6 00261 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00262 INFO = -8 00263 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00264 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00265 INFO = -9 00266 ELSE 00267 IF( RCEQU ) THEN 00268 SMIN = BIGNUM 00269 SMAX = ZERO 00270 DO 10 J = 1, N 00271 SMIN = MIN( SMIN, S( J ) ) 00272 SMAX = MAX( SMAX, S( J ) ) 00273 10 CONTINUE 00274 IF( SMIN.LE.ZERO ) THEN 00275 INFO = -10 00276 ELSE IF( N.GT.0 ) THEN 00277 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00278 ELSE 00279 SCOND = ONE 00280 END IF 00281 END IF 00282 IF( INFO.EQ.0 ) THEN 00283 IF( LDB.LT.MAX( 1, N ) ) THEN 00284 INFO = -12 00285 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00286 INFO = -14 00287 END IF 00288 END IF 00289 END IF 00290 * 00291 IF( INFO.NE.0 ) THEN 00292 CALL XERBLA( 'ZPOSVX', -INFO ) 00293 RETURN 00294 END IF 00295 * 00296 IF( EQUIL ) THEN 00297 * 00298 * Compute row and column scalings to equilibrate the matrix A. 00299 * 00300 CALL ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU ) 00301 IF( INFEQU.EQ.0 ) THEN 00302 * 00303 * Equilibrate the matrix. 00304 * 00305 CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 00306 RCEQU = LSAME( EQUED, 'Y' ) 00307 END IF 00308 END IF 00309 * 00310 * Scale the right hand side. 00311 * 00312 IF( RCEQU ) THEN 00313 DO 30 J = 1, NRHS 00314 DO 20 I = 1, N 00315 B( I, J ) = S( I )*B( I, J ) 00316 20 CONTINUE 00317 30 CONTINUE 00318 END IF 00319 * 00320 IF( NOFACT .OR. EQUIL ) THEN 00321 * 00322 * Compute the Cholesky factorization A = U'*U or A = L*L'. 00323 * 00324 CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 00325 CALL ZPOTRF( UPLO, N, AF, LDAF, INFO ) 00326 * 00327 * Return if INFO is non-zero. 00328 * 00329 IF( INFO.GT.0 )THEN 00330 RCOND = ZERO 00331 RETURN 00332 END IF 00333 END IF 00334 * 00335 * Compute the norm of the matrix A. 00336 * 00337 ANORM = ZLANHE( '1', UPLO, N, A, LDA, RWORK ) 00338 * 00339 * Compute the reciprocal of the condition number of A. 00340 * 00341 CALL ZPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO ) 00342 * 00343 * Compute the solution matrix X. 00344 * 00345 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00346 CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) 00347 * 00348 * Use iterative refinement to improve the computed solution and 00349 * compute error bounds and backward error estimates for it. 00350 * 00351 CALL ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, 00352 $ FERR, BERR, WORK, RWORK, INFO ) 00353 * 00354 * Transform the solution matrix X to a solution of the original 00355 * system. 00356 * 00357 IF( RCEQU ) THEN 00358 DO 50 J = 1, NRHS 00359 DO 40 I = 1, N 00360 X( I, J ) = S( I )*X( I, J ) 00361 40 CONTINUE 00362 50 CONTINUE 00363 DO 60 J = 1, NRHS 00364 FERR( J ) = FERR( J ) / SCOND 00365 60 CONTINUE 00366 END IF 00367 * 00368 * Set INFO = N+1 if the matrix is singular to working precision. 00369 * 00370 IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 00371 $ INFO = N + 1 00372 * 00373 RETURN 00374 * 00375 * End of ZPOSVX 00376 * 00377 END
1.7.3 
