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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, 2: $ X, LDX, FERR, BERR, WORK, IWORK, INFO ) 3: * 4: * -- LAPACK routine (version 3.2) -- 5: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 7: * November 2006 8: * 9: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. 10: * 11: * .. Scalar Arguments .. 12: CHARACTER UPLO 13: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 14: * .. 15: * .. Array Arguments .. 16: INTEGER IPIV( * ), IWORK( * ) 17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) 19: * .. 20: * 21: * Purpose 22: * ======= 23: * 24: * DSYRFS improves the computed solution to a system of linear 25: * equations when the coefficient matrix is symmetric indefinite, and 26: * provides error bounds and backward error estimates for the solution. 27: * 28: * Arguments 29: * ========= 30: * 31: * UPLO (input) CHARACTER*1 32: * = 'U': Upper triangle of A is stored; 33: * = 'L': Lower triangle of A is stored. 34: * 35: * N (input) INTEGER 36: * The order of the matrix A. N >= 0. 37: * 38: * NRHS (input) INTEGER 39: * The number of right hand sides, i.e., the number of columns 40: * of the matrices B and X. NRHS >= 0. 41: * 42: * A (input) DOUBLE PRECISION array, dimension (LDA,N) 43: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N 44: * upper triangular part of A contains the upper triangular part 45: * of the matrix A, and the strictly lower triangular part of A 46: * is not referenced. If UPLO = 'L', the leading N-by-N lower 47: * triangular part of A contains the lower triangular part of 48: * the matrix A, and the strictly upper triangular part of A is 49: * not referenced. 50: * 51: * LDA (input) INTEGER 52: * The leading dimension of the array A. LDA >= max(1,N). 53: * 54: * AF (input) DOUBLE PRECISION array, dimension (LDAF,N) 55: * The factored form of the matrix A. AF contains the block 56: * diagonal matrix D and the multipliers used to obtain the 57: * factor U or L from the factorization A = U*D*U**T or 58: * A = L*D*L**T as computed by DSYTRF. 59: * 60: * LDAF (input) INTEGER 61: * The leading dimension of the array AF. LDAF >= max(1,N). 62: * 63: * IPIV (input) INTEGER array, dimension (N) 64: * Details of the interchanges and the block structure of D 65: * as determined by DSYTRF. 66: * 67: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 68: * The right hand side matrix B. 69: * 70: * LDB (input) INTEGER 71: * The leading dimension of the array B. LDB >= max(1,N). 72: * 73: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) 74: * On entry, the solution matrix X, as computed by DSYTRS. 75: * On exit, the improved solution matrix X. 76: * 77: * LDX (input) INTEGER 78: * The leading dimension of the array X. LDX >= max(1,N). 79: * 80: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 81: * The estimated forward error bound for each solution vector 82: * X(j) (the j-th column of the solution matrix X). 83: * If XTRUE is the true solution corresponding to X(j), FERR(j) 84: * is an estimated upper bound for the magnitude of the largest 85: * element in (X(j) - XTRUE) divided by the magnitude of the 86: * largest element in X(j). The estimate is as reliable as 87: * the estimate for RCOND, and is almost always a slight 88: * overestimate of the true error. 89: * 90: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 91: * The componentwise relative backward error of each solution 92: * vector X(j) (i.e., the smallest relative change in 93: * any element of A or B that makes X(j) an exact solution). 94: * 95: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 96: * 97: * IWORK (workspace) INTEGER array, dimension (N) 98: * 99: * INFO (output) INTEGER 100: * = 0: successful exit 101: * < 0: if INFO = -i, the i-th argument had an illegal value 102: * 103: * Internal Parameters 104: * =================== 105: * 106: * ITMAX is the maximum number of steps of iterative refinement. 107: * 108: * ===================================================================== 109: * 110: * .. Parameters .. 111: INTEGER ITMAX 112: PARAMETER ( ITMAX = 5 ) 113: DOUBLE PRECISION ZERO 114: PARAMETER ( ZERO = 0.0D+0 ) 115: DOUBLE PRECISION ONE 116: PARAMETER ( ONE = 1.0D+0 ) 117: DOUBLE PRECISION TWO 118: PARAMETER ( TWO = 2.0D+0 ) 119: DOUBLE PRECISION THREE 120: PARAMETER ( THREE = 3.0D+0 ) 121: * .. 122: * .. Local Scalars .. 123: LOGICAL UPPER 124: INTEGER COUNT, I, J, K, KASE, NZ 125: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 126: * .. 127: * .. Local Arrays .. 128: INTEGER ISAVE( 3 ) 129: * .. 130: * .. External Subroutines .. 131: EXTERNAL DAXPY, DCOPY, DLACN2, DSYMV, DSYTRS, XERBLA 132: * .. 133: * .. Intrinsic Functions .. 134: INTRINSIC ABS, MAX 135: * .. 136: * .. External Functions .. 137: LOGICAL LSAME 138: DOUBLE PRECISION DLAMCH 139: EXTERNAL LSAME, DLAMCH 140: * .. 141: * .. Executable Statements .. 142: * 143: * Test the input parameters. 144: * 145: INFO = 0 146: UPPER = LSAME( UPLO, 'U' ) 147: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 148: INFO = -1 149: ELSE IF( N.LT.0 ) THEN 150: INFO = -2 151: ELSE IF( NRHS.LT.0 ) THEN 152: INFO = -3 153: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 154: INFO = -5 155: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 156: INFO = -7 157: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 158: INFO = -10 159: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 160: INFO = -12 161: END IF 162: IF( INFO.NE.0 ) THEN 163: CALL XERBLA( 'DSYRFS', -INFO ) 164: RETURN 165: END IF 166: * 167: * Quick return if possible 168: * 169: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 170: DO 10 J = 1, NRHS 171: FERR( J ) = ZERO 172: BERR( J ) = ZERO 173: 10 CONTINUE 174: RETURN 175: END IF 176: * 177: * NZ = maximum number of nonzero elements in each row of A, plus 1 178: * 179: NZ = N + 1 180: EPS = DLAMCH( 'Epsilon' ) 181: SAFMIN = DLAMCH( 'Safe minimum' ) 182: SAFE1 = NZ*SAFMIN 183: SAFE2 = SAFE1 / EPS 184: * 185: * Do for each right hand side 186: * 187: DO 140 J = 1, NRHS 188: * 189: COUNT = 1 190: LSTRES = THREE 191: 20 CONTINUE 192: * 193: * Loop until stopping criterion is satisfied. 194: * 195: * Compute residual R = B - A * X 196: * 197: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 198: CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, 199: $ WORK( N+1 ), 1 ) 200: * 201: * Compute componentwise relative backward error from formula 202: * 203: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 204: * 205: * where abs(Z) is the componentwise absolute value of the matrix 206: * or vector Z. If the i-th component of the denominator is less 207: * than SAFE2, then SAFE1 is added to the i-th components of the 208: * numerator and denominator before dividing. 209: * 210: DO 30 I = 1, N 211: WORK( I ) = ABS( B( I, J ) ) 212: 30 CONTINUE 213: * 214: * Compute abs(A)*abs(X) + abs(B). 215: * 216: IF( UPPER ) THEN 217: DO 50 K = 1, N 218: S = ZERO 219: XK = ABS( X( K, J ) ) 220: DO 40 I = 1, K - 1 221: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 222: S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 223: 40 CONTINUE 224: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S 225: 50 CONTINUE 226: ELSE 227: DO 70 K = 1, N 228: S = ZERO 229: XK = ABS( X( K, J ) ) 230: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK 231: DO 60 I = K + 1, N 232: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 233: S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 234: 60 CONTINUE 235: WORK( K ) = WORK( K ) + S 236: 70 CONTINUE 237: END IF 238: S = ZERO 239: DO 80 I = 1, N 240: IF( WORK( I ).GT.SAFE2 ) THEN 241: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 242: ELSE 243: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 244: $ ( WORK( I )+SAFE1 ) ) 245: END IF 246: 80 CONTINUE 247: BERR( J ) = S 248: * 249: * Test stopping criterion. Continue iterating if 250: * 1) The residual BERR(J) is larger than machine epsilon, and 251: * 2) BERR(J) decreased by at least a factor of 2 during the 252: * last iteration, and 253: * 3) At most ITMAX iterations tried. 254: * 255: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 256: $ COUNT.LE.ITMAX ) THEN 257: * 258: * Update solution and try again. 259: * 260: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 261: $ INFO ) 262: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 263: LSTRES = BERR( J ) 264: COUNT = COUNT + 1 265: GO TO 20 266: END IF 267: * 268: * Bound error from formula 269: * 270: * norm(X - XTRUE) / norm(X) .le. FERR = 271: * norm( abs(inv(A))* 272: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 273: * 274: * where 275: * norm(Z) is the magnitude of the largest component of Z 276: * inv(A) is the inverse of A 277: * abs(Z) is the componentwise absolute value of the matrix or 278: * vector Z 279: * NZ is the maximum number of nonzeros in any row of A, plus 1 280: * EPS is machine epsilon 281: * 282: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 283: * is incremented by SAFE1 if the i-th component of 284: * abs(A)*abs(X) + abs(B) is less than SAFE2. 285: * 286: * Use DLACN2 to estimate the infinity-norm of the matrix 287: * inv(A) * diag(W), 288: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 289: * 290: DO 90 I = 1, N 291: IF( WORK( I ).GT.SAFE2 ) THEN 292: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 293: ELSE 294: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 295: END IF 296: 90 CONTINUE 297: * 298: KASE = 0 299: 100 CONTINUE 300: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 301: $ KASE, ISAVE ) 302: IF( KASE.NE.0 ) THEN 303: IF( KASE.EQ.1 ) THEN 304: * 305: * Multiply by diag(W)*inv(A'). 306: * 307: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 308: $ INFO ) 309: DO 110 I = 1, N 310: WORK( N+I ) = WORK( I )*WORK( N+I ) 311: 110 CONTINUE 312: ELSE IF( KASE.EQ.2 ) THEN 313: * 314: * Multiply by inv(A)*diag(W). 315: * 316: DO 120 I = 1, N 317: WORK( N+I ) = WORK( I )*WORK( N+I ) 318: 120 CONTINUE 319: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, 320: $ INFO ) 321: END IF 322: GO TO 100 323: END IF 324: * 325: * Normalize error. 326: * 327: LSTRES = ZERO 328: DO 130 I = 1, N 329: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 330: 130 CONTINUE 331: IF( LSTRES.NE.ZERO ) 332: $ FERR( J ) = FERR( J ) / LSTRES 333: * 334: 140 CONTINUE 335: * 336: RETURN 337: * 338: * End of DSYRFS 339: * 340: END