 |
LAPACK 3.3.1
Linear Algebra PACKage
|
|
LAPACK 3.3.1
Linear Algebra PACKage
|
00001 SUBROUTINE SPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, 00002 $ BERR, WORK, IWORK, INFO ) 00003 * 00004 * -- LAPACK 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 * Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH. 00010 * 00011 * .. Scalar Arguments .. 00012 CHARACTER UPLO 00013 INTEGER INFO, LDB, LDX, N, NRHS 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IWORK( * ) 00017 REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ), 00018 $ FERR( * ), WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * SPPRFS improves the computed solution to a system of linear 00025 * equations when the coefficient matrix is symmetric positive definite 00026 * and packed, and provides error bounds and backward error estimates 00027 * for the solution. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * UPLO (input) CHARACTER*1 00033 * = 'U': Upper triangle of A is stored; 00034 * = 'L': Lower triangle of A is stored. 00035 * 00036 * N (input) INTEGER 00037 * The order of the matrix A. N >= 0. 00038 * 00039 * NRHS (input) INTEGER 00040 * The number of right hand sides, i.e., the number of columns 00041 * of the matrices B and X. NRHS >= 0. 00042 * 00043 * AP (input) REAL array, dimension (N*(N+1)/2) 00044 * The upper or lower triangle of the symmetric matrix A, packed 00045 * columnwise in a linear array. The j-th column of A is stored 00046 * in the array AP as follows: 00047 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00048 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00049 * 00050 * AFP (input) REAL array, dimension (N*(N+1)/2) 00051 * The triangular factor U or L from the Cholesky factorization 00052 * A = U**T*U or A = L*L**T, as computed by SPPTRF/CPPTRF, 00053 * packed columnwise in a linear array in the same format as A 00054 * (see AP). 00055 * 00056 * B (input) REAL array, dimension (LDB,NRHS) 00057 * The right hand side matrix B. 00058 * 00059 * LDB (input) INTEGER 00060 * The leading dimension of the array B. LDB >= max(1,N). 00061 * 00062 * X (input/output) REAL array, dimension (LDX,NRHS) 00063 * On entry, the solution matrix X, as computed by SPPTRS. 00064 * On exit, the improved solution matrix X. 00065 * 00066 * LDX (input) INTEGER 00067 * The leading dimension of the array X. LDX >= max(1,N). 00068 * 00069 * FERR (output) REAL array, dimension (NRHS) 00070 * The estimated forward error bound for each solution vector 00071 * X(j) (the j-th column of the solution matrix X). 00072 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00073 * is an estimated upper bound for the magnitude of the largest 00074 * element in (X(j) - XTRUE) divided by the magnitude of the 00075 * largest element in X(j). The estimate is as reliable as 00076 * the estimate for RCOND, and is almost always a slight 00077 * overestimate of the true error. 00078 * 00079 * BERR (output) REAL array, dimension (NRHS) 00080 * The componentwise relative backward error of each solution 00081 * vector X(j) (i.e., the smallest relative change in 00082 * any element of A or B that makes X(j) an exact solution). 00083 * 00084 * WORK (workspace) REAL array, dimension (3*N) 00085 * 00086 * IWORK (workspace) INTEGER array, dimension (N) 00087 * 00088 * INFO (output) INTEGER 00089 * = 0: successful exit 00090 * < 0: if INFO = -i, the i-th argument had an illegal value 00091 * 00092 * Internal Parameters 00093 * =================== 00094 * 00095 * ITMAX is the maximum number of steps of iterative refinement. 00096 * 00097 * ===================================================================== 00098 * 00099 * .. Parameters .. 00100 INTEGER ITMAX 00101 PARAMETER ( ITMAX = 5 ) 00102 REAL ZERO 00103 PARAMETER ( ZERO = 0.0E+0 ) 00104 REAL ONE 00105 PARAMETER ( ONE = 1.0E+0 ) 00106 REAL TWO 00107 PARAMETER ( TWO = 2.0E+0 ) 00108 REAL THREE 00109 PARAMETER ( THREE = 3.0E+0 ) 00110 * .. 00111 * .. Local Scalars .. 00112 LOGICAL UPPER 00113 INTEGER COUNT, I, IK, J, K, KASE, KK, NZ 00114 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 00115 * .. 00116 * .. Local Arrays .. 00117 INTEGER ISAVE( 3 ) 00118 * .. 00119 * .. External Subroutines .. 00120 EXTERNAL SAXPY, SCOPY, SLACN2, SPPTRS, SSPMV, XERBLA 00121 * .. 00122 * .. Intrinsic Functions .. 00123 INTRINSIC ABS, MAX 00124 * .. 00125 * .. External Functions .. 00126 LOGICAL LSAME 00127 REAL SLAMCH 00128 EXTERNAL LSAME, SLAMCH 00129 * .. 00130 * .. Executable Statements .. 00131 * 00132 * Test the input parameters. 00133 * 00134 INFO = 0 00135 UPPER = LSAME( UPLO, 'U' ) 00136 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00137 INFO = -1 00138 ELSE IF( N.LT.0 ) THEN 00139 INFO = -2 00140 ELSE IF( NRHS.LT.0 ) THEN 00141 INFO = -3 00142 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00143 INFO = -7 00144 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00145 INFO = -9 00146 END IF 00147 IF( INFO.NE.0 ) THEN 00148 CALL XERBLA( 'SPPRFS', -INFO ) 00149 RETURN 00150 END IF 00151 * 00152 * Quick return if possible 00153 * 00154 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 00155 DO 10 J = 1, NRHS 00156 FERR( J ) = ZERO 00157 BERR( J ) = ZERO 00158 10 CONTINUE 00159 RETURN 00160 END IF 00161 * 00162 * NZ = maximum number of nonzero elements in each row of A, plus 1 00163 * 00164 NZ = N + 1 00165 EPS = SLAMCH( 'Epsilon' ) 00166 SAFMIN = SLAMCH( 'Safe minimum' ) 00167 SAFE1 = NZ*SAFMIN 00168 SAFE2 = SAFE1 / EPS 00169 * 00170 * Do for each right hand side 00171 * 00172 DO 140 J = 1, NRHS 00173 * 00174 COUNT = 1 00175 LSTRES = THREE 00176 20 CONTINUE 00177 * 00178 * Loop until stopping criterion is satisfied. 00179 * 00180 * Compute residual R = B - A * X 00181 * 00182 CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 00183 CALL SSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ), 00184 $ 1 ) 00185 * 00186 * Compute componentwise relative backward error from formula 00187 * 00188 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 00189 * 00190 * where abs(Z) is the componentwise absolute value of the matrix 00191 * or vector Z. If the i-th component of the denominator is less 00192 * than SAFE2, then SAFE1 is added to the i-th components of the 00193 * numerator and denominator before dividing. 00194 * 00195 DO 30 I = 1, N 00196 WORK( I ) = ABS( B( I, J ) ) 00197 30 CONTINUE 00198 * 00199 * Compute abs(A)*abs(X) + abs(B). 00200 * 00201 KK = 1 00202 IF( UPPER ) THEN 00203 DO 50 K = 1, N 00204 S = ZERO 00205 XK = ABS( X( K, J ) ) 00206 IK = KK 00207 DO 40 I = 1, K - 1 00208 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK 00209 S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) 00210 IK = IK + 1 00211 40 CONTINUE 00212 WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S 00213 KK = KK + K 00214 50 CONTINUE 00215 ELSE 00216 DO 70 K = 1, N 00217 S = ZERO 00218 XK = ABS( X( K, J ) ) 00219 WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK 00220 IK = KK + 1 00221 DO 60 I = K + 1, N 00222 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK 00223 S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) 00224 IK = IK + 1 00225 60 CONTINUE 00226 WORK( K ) = WORK( K ) + S 00227 KK = KK + ( N-K+1 ) 00228 70 CONTINUE 00229 END IF 00230 S = ZERO 00231 DO 80 I = 1, N 00232 IF( WORK( I ).GT.SAFE2 ) THEN 00233 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 00234 ELSE 00235 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 00236 $ ( WORK( I )+SAFE1 ) ) 00237 END IF 00238 80 CONTINUE 00239 BERR( J ) = S 00240 * 00241 * Test stopping criterion. Continue iterating if 00242 * 1) The residual BERR(J) is larger than machine epsilon, and 00243 * 2) BERR(J) decreased by at least a factor of 2 during the 00244 * last iteration, and 00245 * 3) At most ITMAX iterations tried. 00246 * 00247 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 00248 $ COUNT.LE.ITMAX ) THEN 00249 * 00250 * Update solution and try again. 00251 * 00252 CALL SPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) 00253 CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 00254 LSTRES = BERR( J ) 00255 COUNT = COUNT + 1 00256 GO TO 20 00257 END IF 00258 * 00259 * Bound error from formula 00260 * 00261 * norm(X - XTRUE) / norm(X) .le. FERR = 00262 * norm( abs(inv(A))* 00263 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 00264 * 00265 * where 00266 * norm(Z) is the magnitude of the largest component of Z 00267 * inv(A) is the inverse of A 00268 * abs(Z) is the componentwise absolute value of the matrix or 00269 * vector Z 00270 * NZ is the maximum number of nonzeros in any row of A, plus 1 00271 * EPS is machine epsilon 00272 * 00273 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 00274 * is incremented by SAFE1 if the i-th component of 00275 * abs(A)*abs(X) + abs(B) is less than SAFE2. 00276 * 00277 * Use SLACN2 to estimate the infinity-norm of the matrix 00278 * inv(A) * diag(W), 00279 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 00280 * 00281 DO 90 I = 1, N 00282 IF( WORK( I ).GT.SAFE2 ) THEN 00283 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 00284 ELSE 00285 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 00286 END IF 00287 90 CONTINUE 00288 * 00289 KASE = 0 00290 100 CONTINUE 00291 CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 00292 $ KASE, ISAVE ) 00293 IF( KASE.NE.0 ) THEN 00294 IF( KASE.EQ.1 ) THEN 00295 * 00296 * Multiply by diag(W)*inv(A**T). 00297 * 00298 CALL SPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) 00299 DO 110 I = 1, N 00300 WORK( N+I ) = WORK( I )*WORK( N+I ) 00301 110 CONTINUE 00302 ELSE IF( KASE.EQ.2 ) THEN 00303 * 00304 * Multiply by inv(A)*diag(W). 00305 * 00306 DO 120 I = 1, N 00307 WORK( N+I ) = WORK( I )*WORK( N+I ) 00308 120 CONTINUE 00309 CALL SPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) 00310 END IF 00311 GO TO 100 00312 END IF 00313 * 00314 * Normalize error. 00315 * 00316 LSTRES = ZERO 00317 DO 130 I = 1, N 00318 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 00319 130 CONTINUE 00320 IF( LSTRES.NE.ZERO ) 00321 $ FERR( J ) = FERR( J ) / LSTRES 00322 * 00323 140 CONTINUE 00324 * 00325 RETURN 00326 * 00327 * End of SPPRFS 00328 * 00329 END
1.7.3 
