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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 SPPSVX( 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 3.3.1) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * -- April 2011 -- 00008 * 00009 * .. Scalar Arguments .. 00010 CHARACTER EQUED, FACT, UPLO 00011 INTEGER INFO, LDB, LDX, N, NRHS 00012 REAL RCOND 00013 * .. 00014 * .. Array Arguments .. 00015 INTEGER IWORK( * ) 00016 REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ), 00017 $ FERR( * ), S( * ), WORK( * ), X( LDX, * ) 00018 * .. 00019 * 00020 * Purpose 00021 * ======= 00022 * 00023 * SPPSVX 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) REAL 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) REAL 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**T*U or A = L*L**T, 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**T * U or A = L * L**T of the original 00122 * matrix A. 00123 * 00124 * If FACT = 'E', then AFP is an output argument and on exit 00125 * returns the triangular factor U or L from the Cholesky 00126 * factorization A = U**T * U or A = L * L**T of the equilibrated 00127 * matrix A (see the description of AP for the form of the 00128 * equilibrated matrix). 00129 * 00130 * EQUED (input or output) CHARACTER*1 00131 * Specifies the form of equilibration that was done. 00132 * = 'N': No equilibration (always true if FACT = 'N'). 00133 * = 'Y': Equilibration was done, i.e., A has been replaced by 00134 * diag(S) * A * diag(S). 00135 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00136 * output argument. 00137 * 00138 * S (input or output) REAL array, dimension (N) 00139 * The scale factors for A; not accessed if EQUED = 'N'. S is 00140 * an input argument if FACT = 'F'; otherwise, S is an output 00141 * argument. If FACT = 'F' and EQUED = 'Y', each element of S 00142 * must be positive. 00143 * 00144 * B (input/output) REAL array, dimension (LDB,NRHS) 00145 * On entry, the N-by-NRHS right hand side matrix B. 00146 * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', 00147 * B is overwritten by diag(S) * B. 00148 * 00149 * LDB (input) INTEGER 00150 * The leading dimension of the array B. LDB >= max(1,N). 00151 * 00152 * X (output) REAL array, dimension (LDX,NRHS) 00153 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to 00154 * the original system of equations. Note that if EQUED = 'Y', 00155 * A and B are modified on exit, and the solution to the 00156 * equilibrated system is inv(diag(S))*X. 00157 * 00158 * LDX (input) INTEGER 00159 * The leading dimension of the array X. LDX >= max(1,N). 00160 * 00161 * RCOND (output) REAL 00162 * The estimate of the reciprocal condition number of the matrix 00163 * A after equilibration (if done). If RCOND is less than the 00164 * machine precision (in particular, if RCOND = 0), the matrix 00165 * is singular to working precision. This condition is 00166 * indicated by a return code of INFO > 0. 00167 * 00168 * FERR (output) REAL array, dimension (NRHS) 00169 * The estimated forward error bound for each solution vector 00170 * X(j) (the j-th column of the solution matrix X). 00171 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00172 * is an estimated upper bound for the magnitude of the largest 00173 * element in (X(j) - XTRUE) divided by the magnitude of the 00174 * largest element in X(j). The estimate is as reliable as 00175 * the estimate for RCOND, and is almost always a slight 00176 * overestimate of the true error. 00177 * 00178 * BERR (output) REAL array, dimension (NRHS) 00179 * The componentwise relative backward error of each solution 00180 * vector X(j) (i.e., the smallest relative change in 00181 * any element of A or B that makes X(j) an exact solution). 00182 * 00183 * WORK (workspace) REAL array, dimension (3*N) 00184 * 00185 * IWORK (workspace) INTEGER array, dimension (N) 00186 * 00187 * INFO (output) INTEGER 00188 * = 0: successful exit 00189 * < 0: if INFO = -i, the i-th argument had an illegal value 00190 * > 0: if INFO = i, and i is 00191 * <= N: the leading minor of order i of A is 00192 * not positive definite, so the factorization 00193 * could not be completed, and the solution has not 00194 * been computed. RCOND = 0 is returned. 00195 * = N+1: U is nonsingular, but RCOND is less than machine 00196 * precision, meaning that the matrix is singular 00197 * to working precision. Nevertheless, the 00198 * solution and error bounds are computed because 00199 * there are a number of situations where the 00200 * computed solution can be more accurate than the 00201 * value of RCOND would suggest. 00202 * 00203 * Further Details 00204 * =============== 00205 * 00206 * The packed storage scheme is illustrated by the following example 00207 * when N = 4, UPLO = 'U': 00208 * 00209 * Two-dimensional storage of the symmetric matrix A: 00210 * 00211 * a11 a12 a13 a14 00212 * a22 a23 a24 00213 * a33 a34 (aij = conjg(aji)) 00214 * a44 00215 * 00216 * Packed storage of the upper triangle of A: 00217 * 00218 * AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] 00219 * 00220 * ===================================================================== 00221 * 00222 * .. Parameters .. 00223 REAL ZERO, ONE 00224 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00225 * .. 00226 * .. Local Scalars .. 00227 LOGICAL EQUIL, NOFACT, RCEQU 00228 INTEGER I, INFEQU, J 00229 REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM 00230 * .. 00231 * .. External Functions .. 00232 LOGICAL LSAME 00233 REAL SLAMCH, SLANSP 00234 EXTERNAL LSAME, SLAMCH, SLANSP 00235 * .. 00236 * .. External Subroutines .. 00237 EXTERNAL SCOPY, SLACPY, SLAQSP, SPPCON, SPPEQU, SPPRFS, 00238 $ SPPTRF, SPPTRS, XERBLA 00239 * .. 00240 * .. Intrinsic Functions .. 00241 INTRINSIC MAX, MIN 00242 * .. 00243 * .. Executable Statements .. 00244 * 00245 INFO = 0 00246 NOFACT = LSAME( FACT, 'N' ) 00247 EQUIL = LSAME( FACT, 'E' ) 00248 IF( NOFACT .OR. EQUIL ) THEN 00249 EQUED = 'N' 00250 RCEQU = .FALSE. 00251 ELSE 00252 RCEQU = LSAME( EQUED, 'Y' ) 00253 SMLNUM = SLAMCH( 'Safe minimum' ) 00254 BIGNUM = ONE / SMLNUM 00255 END IF 00256 * 00257 * Test the input parameters. 00258 * 00259 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 00260 $ THEN 00261 INFO = -1 00262 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) 00263 $ THEN 00264 INFO = -2 00265 ELSE IF( N.LT.0 ) THEN 00266 INFO = -3 00267 ELSE IF( NRHS.LT.0 ) THEN 00268 INFO = -4 00269 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00270 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00271 INFO = -7 00272 ELSE 00273 IF( RCEQU ) THEN 00274 SMIN = BIGNUM 00275 SMAX = ZERO 00276 DO 10 J = 1, N 00277 SMIN = MIN( SMIN, S( J ) ) 00278 SMAX = MAX( SMAX, S( J ) ) 00279 10 CONTINUE 00280 IF( SMIN.LE.ZERO ) THEN 00281 INFO = -8 00282 ELSE IF( N.GT.0 ) THEN 00283 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00284 ELSE 00285 SCOND = ONE 00286 END IF 00287 END IF 00288 IF( INFO.EQ.0 ) THEN 00289 IF( LDB.LT.MAX( 1, N ) ) THEN 00290 INFO = -10 00291 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00292 INFO = -12 00293 END IF 00294 END IF 00295 END IF 00296 * 00297 IF( INFO.NE.0 ) THEN 00298 CALL XERBLA( 'SPPSVX', -INFO ) 00299 RETURN 00300 END IF 00301 * 00302 IF( EQUIL ) THEN 00303 * 00304 * Compute row and column scalings to equilibrate the matrix A. 00305 * 00306 CALL SPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU ) 00307 IF( INFEQU.EQ.0 ) THEN 00308 * 00309 * Equilibrate the matrix. 00310 * 00311 CALL SLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED ) 00312 RCEQU = LSAME( EQUED, 'Y' ) 00313 END IF 00314 END IF 00315 * 00316 * Scale the right-hand side. 00317 * 00318 IF( RCEQU ) THEN 00319 DO 30 J = 1, NRHS 00320 DO 20 I = 1, N 00321 B( I, J ) = S( I )*B( I, J ) 00322 20 CONTINUE 00323 30 CONTINUE 00324 END IF 00325 * 00326 IF( NOFACT .OR. EQUIL ) THEN 00327 * 00328 * Compute the Cholesky factorization A = U**T * U or A = L * L**T. 00329 * 00330 CALL SCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 ) 00331 CALL SPPTRF( UPLO, N, AFP, INFO ) 00332 * 00333 * Return if INFO is non-zero. 00334 * 00335 IF( INFO.GT.0 )THEN 00336 RCOND = ZERO 00337 RETURN 00338 END IF 00339 END IF 00340 * 00341 * Compute the norm of the matrix A. 00342 * 00343 ANORM = SLANSP( 'I', UPLO, N, AP, WORK ) 00344 * 00345 * Compute the reciprocal of the condition number of A. 00346 * 00347 CALL SPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, IWORK, INFO ) 00348 * 00349 * Compute the solution matrix X. 00350 * 00351 CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00352 CALL SPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO ) 00353 * 00354 * Use iterative refinement to improve the computed solution and 00355 * compute error bounds and backward error estimates for it. 00356 * 00357 CALL SPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, 00358 $ WORK, IWORK, INFO ) 00359 * 00360 * Transform the solution matrix X to a solution of the original 00361 * system. 00362 * 00363 IF( RCEQU ) THEN 00364 DO 50 J = 1, NRHS 00365 DO 40 I = 1, N 00366 X( I, J ) = S( I )*X( I, J ) 00367 40 CONTINUE 00368 50 CONTINUE 00369 DO 60 J = 1, NRHS 00370 FERR( J ) = FERR( J ) / SCOND 00371 60 CONTINUE 00372 END IF 00373 * 00374 * Set INFO = N+1 if the matrix is singular to working precision. 00375 * 00376 IF( RCOND.LT.SLAMCH( 'Epsilon' ) ) 00377 $ INFO = N + 1 00378 * 00379 RETURN 00380 * 00381 * End of SPPSVX 00382 * 00383 END
1.7.3 
