00001 SUBROUTINE DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, 00002 $ X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) 00003 * 00004 * -- LAPACK driver routine (version 3.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * November 2006 00008 * 00009 * .. Scalar Arguments .. 00010 CHARACTER EQUED, FACT, UPLO 00011 INTEGER INFO, LDB, LDX, N, NRHS 00012 DOUBLE PRECISION RCOND 00013 * .. 00014 * .. Array Arguments .. 00015 INTEGER IWORK( * ) 00016 DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), 00017 $ FERR( * ), S( * ), WORK( * ), X( LDX, * ) 00018 * .. 00019 * 00020 * Purpose 00021 * ======= 00022 * 00023 * DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to 00024 * compute the solution to a real system of linear equations 00025 * A * X = B, 00026 * where A is an N-by-N symmetric positive definite matrix stored in 00027 * packed format and X and B are N-by-NRHS matrices. 00028 * 00029 * Error bounds on the solution and a condition estimate are also 00030 * provided. 00031 * 00032 * Description 00033 * =========== 00034 * 00035 * The following steps are performed: 00036 * 00037 * 1. If FACT = 'E', real scaling factors are computed to equilibrate 00038 * the system: 00039 * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B 00040 * Whether or not the system will be equilibrated depends on the 00041 * scaling of the matrix A, but if equilibration is used, A is 00042 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 00043 * 00044 * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 00045 * factor the matrix A (after equilibration if FACT = 'E') as 00046 * A = U**T* U, if UPLO = 'U', or 00047 * A = L * L**T, if UPLO = 'L', 00048 * where U is an upper triangular matrix and L is a lower triangular 00049 * matrix. 00050 * 00051 * 3. If the leading i-by-i principal minor is not positive definite, 00052 * then the routine returns with INFO = i. Otherwise, the factored 00053 * form of A is used to estimate the condition number of the matrix 00054 * A. If the reciprocal of the condition number is less than machine 00055 * precision, INFO = N+1 is returned as a warning, but the routine 00056 * still goes on to solve for X and compute error bounds as 00057 * described below. 00058 * 00059 * 4. The system of equations is solved for X using the factored form 00060 * of A. 00061 * 00062 * 5. Iterative refinement is applied to improve the computed solution 00063 * matrix and calculate error bounds and backward error estimates 00064 * for it. 00065 * 00066 * 6. If equilibration was used, the matrix X is premultiplied by 00067 * diag(S) so that it solves the original system before 00068 * equilibration. 00069 * 00070 * Arguments 00071 * ========= 00072 * 00073 * FACT (input) CHARACTER*1 00074 * Specifies whether or not the factored form of the matrix A is 00075 * supplied on entry, and if not, whether the matrix A should be 00076 * equilibrated before it is factored. 00077 * = 'F': On entry, AFP contains the factored form of A. 00078 * If EQUED = 'Y', the matrix A has been equilibrated 00079 * with scaling factors given by S. AP and AFP will not 00080 * be modified. 00081 * = 'N': The matrix A will be copied to AFP and factored. 00082 * = 'E': The matrix A will be equilibrated if necessary, then 00083 * copied to AFP and factored. 00084 * 00085 * UPLO (input) CHARACTER*1 00086 * = 'U': Upper triangle of A is stored; 00087 * = 'L': Lower triangle of A is stored. 00088 * 00089 * N (input) INTEGER 00090 * The number of linear equations, i.e., the order of the 00091 * matrix A. N >= 0. 00092 * 00093 * NRHS (input) INTEGER 00094 * The number of right hand sides, i.e., the number of columns 00095 * of the matrices B and X. NRHS >= 0. 00096 * 00097 * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) 00098 * On entry, the upper or lower triangle of the symmetric matrix 00099 * A, packed columnwise in a linear array, except if FACT = 'F' 00100 * and EQUED = 'Y', then A must contain the equilibrated matrix 00101 * diag(S)*A*diag(S). The j-th column of A is stored in the 00102 * array AP as follows: 00103 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00104 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00105 * See below for further details. A is not modified if 00106 * FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. 00107 * 00108 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00109 * diag(S)*A*diag(S). 00110 * 00111 * AFP (input or output) DOUBLE PRECISION array, dimension 00112 * (N*(N+1)/2) 00113 * If FACT = 'F', then AFP is an input argument and on entry 00114 * contains the triangular factor U or L from the Cholesky 00115 * factorization A = U'*U or A = L*L', in the same storage 00116 * format as A. If EQUED .ne. 'N', then AFP is the factored 00117 * form of the equilibrated matrix A. 00118 * 00119 * If FACT = 'N', then AFP is an output argument and on exit 00120 * returns the triangular factor U or L from the Cholesky 00121 * factorization A = U'*U or A = L*L' of the original matrix A. 00122 * 00123 * If FACT = 'E', then AFP is an output argument and on exit 00124 * returns the triangular factor U or L from the Cholesky 00125 * factorization A = U'*U or A = L*L' of the equilibrated 00126 * matrix A (see the description of AP for the form of the 00127 * equilibrated matrix). 00128 * 00129 * EQUED (input or output) CHARACTER*1 00130 * Specifies the form of equilibration that was done. 00131 * = 'N': No equilibration (always true if FACT = 'N'). 00132 * = 'Y': Equilibration was done, i.e., A has been replaced by 00133 * diag(S) * A * diag(S). 00134 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00135 * output argument. 00136 * 00137 * S (input or output) DOUBLE PRECISION array, dimension (N) 00138 * The scale factors for A; not accessed if EQUED = 'N'. S is 00139 * an input argument if FACT = 'F'; otherwise, S is an output 00140 * argument. If FACT = 'F' and EQUED = 'Y', each element of S 00141 * must be positive. 00142 * 00143 * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) 00144 * On entry, the N-by-NRHS right hand side matrix B. 00145 * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', 00146 * B is overwritten by diag(S) * B. 00147 * 00148 * LDB (input) INTEGER 00149 * The leading dimension of the array B. LDB >= max(1,N). 00150 * 00151 * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) 00152 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to 00153 * the original system of equations. Note that if EQUED = 'Y', 00154 * A and B are modified on exit, and the solution to the 00155 * equilibrated system is inv(diag(S))*X. 00156 * 00157 * LDX (input) INTEGER 00158 * The leading dimension of the array X. LDX >= max(1,N). 00159 * 00160 * RCOND (output) DOUBLE PRECISION 00161 * The estimate of the reciprocal condition number of the matrix 00162 * A after equilibration (if done). If RCOND is less than the 00163 * machine precision (in particular, if RCOND = 0), the matrix 00164 * is singular to working precision. This condition is 00165 * indicated by a return code of INFO > 0. 00166 * 00167 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00168 * The estimated forward error bound for each solution vector 00169 * X(j) (the j-th column of the solution matrix X). 00170 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00171 * is an estimated upper bound for the magnitude of the largest 00172 * element in (X(j) - XTRUE) divided by the magnitude of the 00173 * largest element in X(j). The estimate is as reliable as 00174 * the estimate for RCOND, and is almost always a slight 00175 * overestimate of the true error. 00176 * 00177 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00178 * The componentwise relative backward error of each solution 00179 * vector X(j) (i.e., the smallest relative change in 00180 * any element of A or B that makes X(j) an exact solution). 00181 * 00182 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 00183 * 00184 * IWORK (workspace) INTEGER array, dimension (N) 00185 * 00186 * INFO (output) INTEGER 00187 * = 0: successful exit 00188 * < 0: if INFO = -i, the i-th argument had an illegal value 00189 * > 0: if INFO = i, and i is 00190 * <= N: the leading minor of order i of A is 00191 * not positive definite, so the factorization 00192 * could not be completed, and the solution has not 00193 * been computed. RCOND = 0 is returned. 00194 * = N+1: U is nonsingular, but RCOND is less than machine 00195 * precision, meaning that the matrix is singular 00196 * to working precision. Nevertheless, the 00197 * solution and error bounds are computed because 00198 * there are a number of situations where the 00199 * computed solution can be more accurate than the 00200 * value of RCOND would suggest. 00201 * 00202 * Further Details 00203 * =============== 00204 * 00205 * The packed storage scheme is illustrated by the following example 00206 * when N = 4, UPLO = 'U': 00207 * 00208 * Two-dimensional storage of the symmetric matrix A: 00209 * 00210 * a11 a12 a13 a14 00211 * a22 a23 a24 00212 * a33 a34 (aij = conjg(aji)) 00213 * a44 00214 * 00215 * Packed storage of the upper triangle of A: 00216 * 00217 * AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] 00218 * 00219 * ===================================================================== 00220 * 00221 * .. Parameters .. 00222 DOUBLE PRECISION ZERO, ONE 00223 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00224 * .. 00225 * .. Local Scalars .. 00226 LOGICAL EQUIL, NOFACT, RCEQU 00227 INTEGER I, INFEQU, J 00228 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM 00229 * .. 00230 * .. External Functions .. 00231 LOGICAL LSAME 00232 DOUBLE PRECISION DLAMCH, DLANSP 00233 EXTERNAL LSAME, DLAMCH, DLANSP 00234 * .. 00235 * .. External Subroutines .. 00236 EXTERNAL DCOPY, DLACPY, DLAQSP, DPPCON, DPPEQU, DPPRFS, 00237 $ DPPTRF, DPPTRS, XERBLA 00238 * .. 00239 * .. Intrinsic Functions .. 00240 INTRINSIC MAX, MIN 00241 * .. 00242 * .. Executable Statements .. 00243 * 00244 INFO = 0 00245 NOFACT = LSAME( FACT, 'N' ) 00246 EQUIL = LSAME( FACT, 'E' ) 00247 IF( NOFACT .OR. EQUIL ) THEN 00248 EQUED = 'N' 00249 RCEQU = .FALSE. 00250 ELSE 00251 RCEQU = LSAME( EQUED, 'Y' ) 00252 SMLNUM = DLAMCH( 'Safe minimum' ) 00253 BIGNUM = ONE / SMLNUM 00254 END IF 00255 * 00256 * Test the input parameters. 00257 * 00258 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 00259 $ THEN 00260 INFO = -1 00261 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) 00262 $ THEN 00263 INFO = -2 00264 ELSE IF( N.LT.0 ) THEN 00265 INFO = -3 00266 ELSE IF( NRHS.LT.0 ) THEN 00267 INFO = -4 00268 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00269 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00270 INFO = -7 00271 ELSE 00272 IF( RCEQU ) THEN 00273 SMIN = BIGNUM 00274 SMAX = ZERO 00275 DO 10 J = 1, N 00276 SMIN = MIN( SMIN, S( J ) ) 00277 SMAX = MAX( SMAX, S( J ) ) 00278 10 CONTINUE 00279 IF( SMIN.LE.ZERO ) THEN 00280 INFO = -8 00281 ELSE IF( N.GT.0 ) THEN 00282 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00283 ELSE 00284 SCOND = ONE 00285 END IF 00286 END IF 00287 IF( INFO.EQ.0 ) THEN 00288 IF( LDB.LT.MAX( 1, N ) ) THEN 00289 INFO = -10 00290 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00291 INFO = -12 00292 END IF 00293 END IF 00294 END IF 00295 * 00296 IF( INFO.NE.0 ) THEN 00297 CALL XERBLA( 'DPPSVX', -INFO ) 00298 RETURN 00299 END IF 00300 * 00301 IF( EQUIL ) THEN 00302 * 00303 * Compute row and column scalings to equilibrate the matrix A. 00304 * 00305 CALL DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU ) 00306 IF( INFEQU.EQ.0 ) THEN 00307 * 00308 * Equilibrate the matrix. 00309 * 00310 CALL DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED ) 00311 RCEQU = LSAME( EQUED, 'Y' ) 00312 END IF 00313 END IF 00314 * 00315 * Scale the right-hand side. 00316 * 00317 IF( RCEQU ) THEN 00318 DO 30 J = 1, NRHS 00319 DO 20 I = 1, N 00320 B( I, J ) = S( I )*B( I, J ) 00321 20 CONTINUE 00322 30 CONTINUE 00323 END IF 00324 * 00325 IF( NOFACT .OR. EQUIL ) THEN 00326 * 00327 * Compute the Cholesky factorization A = U'*U or A = L*L'. 00328 * 00329 CALL DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 ) 00330 CALL DPPTRF( UPLO, N, AFP, INFO ) 00331 * 00332 * Return if INFO is non-zero. 00333 * 00334 IF( INFO.GT.0 )THEN 00335 RCOND = ZERO 00336 RETURN 00337 END IF 00338 END IF 00339 * 00340 * Compute the norm of the matrix A. 00341 * 00342 ANORM = DLANSP( 'I', UPLO, N, AP, WORK ) 00343 * 00344 * Compute the reciprocal of the condition number of A. 00345 * 00346 CALL DPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, IWORK, INFO ) 00347 * 00348 * Compute the solution matrix X. 00349 * 00350 CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00351 CALL DPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO ) 00352 * 00353 * Use iterative refinement to improve the computed solution and 00354 * compute error bounds and backward error estimates for it. 00355 * 00356 CALL DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, 00357 $ WORK, IWORK, INFO ) 00358 * 00359 * Transform the solution matrix X to a solution of the original 00360 * system. 00361 * 00362 IF( RCEQU ) THEN 00363 DO 50 J = 1, NRHS 00364 DO 40 I = 1, N 00365 X( I, J ) = S( I )*X( I, J ) 00366 40 CONTINUE 00367 50 CONTINUE 00368 DO 60 J = 1, NRHS 00369 FERR( J ) = FERR( J ) / SCOND 00370 60 CONTINUE 00371 END IF 00372 * 00373 * Set INFO = N+1 if the matrix is singular to working precision. 00374 * 00375 IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 00376 $ INFO = N + 1 00377 * 00378 RETURN 00379 * 00380 * End of DPPSVX 00381 * 00382 END
1.7.2